url stringlengths 14 2.42k | text stringlengths 100 1.02M | date stringlengths 19 19 | metadata stringlengths 1.06k 1.1k |
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https://artofproblemsolving.com/wiki/index.php?title=File:Ceva1.PNG&diff=prev&oldid=60710 | # Difference between revisions of "File:Ceva1.PNG"
Let ABC be a triangle, and let D, E, F be points on lines BC, CA, AB, respectively. Lines AD, BE, CF are concurrent if and only if
$\frac{BD}{DC} \cdot \frac{CE}{EA}\cdot \frac{AF}{FB} = 1,$
where lengths are directed. This also works for the reciprocal or each of the ratios, as the reciprocal of 1 is 1.
## File history
Click on a date/time to view the file as it appeared at that time.
Date/TimeThumbnailDimensionsUserComment
current09:26, 18 August 2006344 × 186 (6 KB)Joml88 (talk | contribs)
• You cannot overwrite this file.
The following page links to this file: | 2020-11-28 18:21:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7721333503723145, "perplexity": 1540.625237548337}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141195687.51/warc/CC-MAIN-20201128155305-20201128185305-00627.warc.gz"} |
https://tongfamily.com/2022/08/09/running-ubuntu-on-an-asus-rog-x13-2022-convertible-notebook/ | Wow, the ASUS X13 (2022) convertible notebook is a nice nVidia GTX 3050Ti thin and light notebook. It is on sale for \$1300 at Best Buy right now and it has a Ryzen 6000 processor with 16GB of memory and 1TB hard drive. There are some things that are hard to do:
1. Turned the thing on. I'm too used to the Mac so it was impossible to find the on/off button or activate it. It is located as a dark button on the right side of the keyboard and is nearly impossible to find. Also, you need to long-press to make it start.
2. At least for me, the battery was not charged enough to start, but at least this is USB C powered. At least for me, the battery was so far down, it needed the included charger.
3. Once started, you boot into Windows 10 and you need your Microsoft.com password and then it will take about 30 minutes to load Windows 11.
4. The thing will ask you how to program the fingerprint reader. There are no documents in the box (but hilariously, there is a piece of cardboard with obscure Ikea instructions so you can make a triangular rest. I'm not sure why they did that.
Now to install Ubuntu things are even more mysterious:
1. You need to get a USB installer for Ubuntu, so first brew install balenaetcher which will install a USB installer.
3. Then crank up Baleen Etcher and you can flash a USB key.
4. Plug the USB key into the X13. One annoying thing is that it only one USB C and one USB A key, so I use the USB C for the battery and then you need a USB-A key
5. Now you long press to turn off the machine
6. Then short press the power on button and then hold the F12 key down and you will get to the Windows recovery menu. Click on boot from USB.
7. You will see Ubuntu come up and ask if it should boot. The nouveau default driver for the nVidia card does not work, so make sure to click, and boot with standard graphics.
8. Now you will be asked how to partition the drive. On a 1TB drive, the default is half for Linux and Windows, so that's not bad.
## Post installation with drivers
For the ROG X13, now you need to install an additional set of drivers:
1. Ubuntu nVidia driver. This is actually pretty easy and after you get Ubuntu running without GPU acceleration. But basically you run ubuntu autoinstall and it should work.
2. The fingerprint reader does not seem to work.
#### Related Posts
This site uses Akismet to reduce spam. Learn how your comment data is processed. | 2022-12-06 21:55:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19153985381126404, "perplexity": 1884.5387298611322}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711114.3/warc/CC-MAIN-20221206192947-20221206222947-00402.warc.gz"} |
https://library.kiwix.org/datascience.stackexchange.com_en_all_2021-04/A/question/53845.html | ## Python Script using pandas to plot histograms between the features
0
I am working around with data from Kaggle : Titanic Competition dataset.
Script
#cols : List of all column header
cols = list(df.columns.values)
#script to plot histograms of all columns with respect to column: "Survived"
for val in cols:
df.groupby('Survived')[val].hist(alpha = 0.2)
Code of importance (COI)
df.groupby('Survived')['Age'].hist(alpha = 0.3)
Output
The COI works but the Script is not working. The script will help to plot all the histograms in one go without manually inserting the column in the COI.
1. The script is running indefinetly.
2. The COI also runs indefinetly but once I execute the following code:
df.groupby('Survived').Age.hist(alpha = 0.2) #This runs flawlessly
COI runs perfectly. (Not able to figure out why).
1. The scripts are written in Kaggel Kernel.
Given the above details, please help me figure out a way to achieve the above script either by making improvements to the Script or an alternate way of doing the same.
I meant use seaborn or something like that for the plot and then add explicitly that – Aditya – 2019-06-15T10:47:12.193
cols = list(df.columns.values) | 2021-08-04 15:18:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5434271097183228, "perplexity": 3089.331475758456}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154878.27/warc/CC-MAIN-20210804142918-20210804172918-00243.warc.gz"} |
https://nbviewer.jupyter.org/github/fonnesbeck/Bios8366/blob/test/notebooks/Section0-IPython_and_Jupyter.ipynb | # IPython¶
IPython (Interactive Python) is an enhanced Python shell which provides a more robust and productive development environment for users. There are several key features that set it apart from the standard Python shell.
### History¶
In IPython, all your inputs and outputs are saved. There are two variables named In and Out which are assigned as you work with your results. All outputs are saved automatically to variables of the form _N, where N is the prompt number, and inputs to _iN. This allows you to recover quickly the result of a prior computation by referring to its number even if you forgot to store it as a variable.
In [ ]:
import numpy as np
np.sin(4)**2
In [ ]:
_1
In [ ]:
_i1
In [ ]:
_1 / 4.
### Output is asynchronous¶
All output is displayed asynchronously as it is generated in the Kernel. If you execute the next cell, you will see the output one piece at a time, not all at the end.
In [ ]:
import time, sys
for i in range(8):
print(i)
time.sleep(0.5)
### Introspection¶
If you want details regarding the properties and functionality of any Python objects currently loaded into IPython, you can use the ? to reveal any details that are available:
In [ ]:
some_dict = {}
some_dict?
If available, additional detail is provided with two question marks, including the source code of the object itself.
In [ ]:
from numpy.linalg import cholesky
cholesky??
This syntax can also be used to search namespaces with wildcards (*).
In [ ]:
%matplotlib inline
import pylab as plt
plt.*plot*?
### Tab completion¶
Because IPython allows for introspection, it is able to afford the user the ability to tab-complete commands that have been partially typed. This is done by pressing the <tab> key at any point during the process of typing a command:
In [ ]:
np.ar
This can even be used to help with specifying arguments to functions, which can sometimes be difficult to remember:
In [ ]:
plt.hist
### System commands¶
In IPython, you can type ls to see your files or cd to change directories, just like you would at a regular system prompt:
In [ ]:
ls /Users/fonnescj/Teaching/Bios8366/data
Virtually any system command can be accessed by prepending !, which passes any subsequent command directly to the OS.
In [ ]:
!locate python | grep pdf
You can even use Python variables in commands sent to the OS:
In [ ]:
file_type = 'csv'
!ls ../data/*$file_type The output of a system command using the exclamation point syntax can be assigned to a Python variable. In [ ]: data_files = !ls ../data/microbiome/ In [ ]: data_files ## Qt Console¶ If you type at the system prompt: $ ipython qtconsole
instead of opening in a terminal, IPython will start a graphical console that at first sight appears just like a terminal, but which is in fact much more capable than a text-only terminal. This is a specialized terminal designed for interactive scientific work, and it supports full multi-line editing with color highlighting and graphical calltips for functions, it can keep multiple IPython sessions open simultaneously in tabs, and when scripts run it can display the figures inline directly in the work area.
# Jupyter Notebook¶
Over time, the IPython project grew to include several components, including:
• an interactive shell
• a REPL protocol
• a notebook document fromat
• a notebook document conversion tool
• a web-based notebook authoring tool
• tools for building interactive UI (widgets)
• interactive parallel Python
As each component has evolved, several had grown to the point that they warrented projects of their own. For example, pieces like the notebook and protocol are not even specific to Python. As the result, the IPython team created Project Jupyter, which is the new home of language-agnostic projects that began as part of IPython, such as the notebook in which you are reading this text.
The HTML notebook that is part of the Jupyter project supports interactive data visualization and easy high-performance parallel computing.
In [ ]:
import matplotlib.pyplot as plt
plt.style.use('fivethirtyeight')
def f(x):
return (x-3)*(x-5)*(x-7)+85
import numpy as np
x = np.linspace(0, 10, 200)
y = f(x)
plt.plot(x,y)
The notebook lets you document your workflow using either HTML or Markdown.
The Jupyter Notebook consists of two related components:
• A JSON based Notebook document format for recording and distributing Python code and rich text.
• A web-based user interface for authoring and running notebook documents.
The Notebook can be used by starting the Notebook server with the command:
$ipython notebook This initiates an iPython engine, which is a Python instance that takes Python commands over a network connection. The IPython controller provides an interface for working with a set of engines, to which one or more iPython clients can connect. The Notebook gives you everything that a browser gives you. For example, you can embed images, videos, or entire websites. In [ ]: from IPython.display import HTML HTML("<iframe src=http://fonnesbeck.github.io/Bios8366 width=700 height=350></iframe>") In [ ]: from IPython.display import YouTubeVideo YouTubeVideo("rl5DaFbLc60") ### Remote Code¶ Use %load to add remote code In [ ]: %load http://matplotlib.org/mpl_examples/shapes_and_collections/scatter_demo.py ### Mathjax Support¶ Mathjax ia a javascript implementation$\alpha$of LaTeX that allows equations to be embedded into HTML. For example, this markup: """$$\int_{a}^{b} f(x)\, dx \approx \frac{1}{2} \sum_{k=1}^{N} \left( x_{k} - x_{k-1} \right) \left( f(x_{k}) + f(x_{k-1}) \right).$$""" becomes this: $$\int_{a}^{b} f(x)\, dx \approx \frac{1}{2} \sum_{k=1}^{N} \left( x_{k} - x_{k-1} \right) \left( f(x_{k}) + f(x_{k-1}) \right).$$ ## SymPy Support¶ SymPy is a Python library for symbolic mathematics. It supports: • polynomials • calculus • solving equations • discrete math • matrices In [ ]: from sympy import * init_printing() x, y = symbols("x y") In [ ]: eq = ((x+y)**2 * (x+1)) eq In [ ]: expand(eq) In [ ]: (1/cos(x)).series(x, 0, 6) In [ ]: limit((sin(x)-x)/x**3, x, 0) In [ ]: diff(cos(x**2)**2 / (1+x), x) ### Magic functions¶ IPython has a set of predefined ‘magic functions’ that you can call with a command line style syntax. These include: • %run • %edit • %debug • %timeit • %paste • %load_ext In [ ]: %lsmagic Timing the execution of code; the timeit magic exists both in line and cell form: In [ ]: %timeit np.linalg.eigvals(np.random.rand(100,100)) In [ ]: %%timeit a = np.random.rand(100, 100) np.linalg.eigvals(a) IPython also creates aliases for a few common interpreters, such as bash, ruby, perl, etc. These are all equivalent to %%script <name> In [ ]: %%ruby puts "Hello from Ruby #{RUBY_VERSION}" In [ ]: %%bash echo "hello from$BASH"
IPython has an rmagic extension that contains a some magic functions for working with R via rpy2. This extension can be loaded using the %load_ext magic as follows:
In [ ]:
%load_ext rpy2.ipython
If the above generates an error, it is likely that you do not have the rpy2 module installed. You can install this now via:
In [ ]:
!pip install rpy2
In [ ]:
x,y = np.arange(10), np.random.normal(size=10)
%R print(lm(rnorm(10)~rnorm(10)))
In [ ]:
%%R -i x,y -o XYcoef
lm.fit <- lm(y~x)
par(mfrow=c(2,2))
print(summary(lm.fit))
plot(lm.fit)
XYcoef <- coef(lm.fit)
In [ ]:
XYcoef
### LaTeX¶
In addition to MathJax support, you may declare a LaTeX cell using the %latex magic:
In [ ]:
%%latex
\begin{align}
\nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\
\nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\
\nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\
\nabla \cdot \vec{\mathbf{B}} & = 0
\end{align}
## Javscript¶
Jupyter also enables objects to declare a JavaScript representation. At first, this may seem odd as output is inherently visual and JavaScript is a programming language. However, this opens the door for rich output that leverages the full power of JavaScript and associated libraries such as D3 for output.
In [ ]:
%%javascript
## Exporting and Converting Notebooks¶
In Jupyter, one can convert an .ipynb notebook document file into various static formats via the nbconvert tool. Currently, nbconvert is a command line tool, run as a script using Jupyter.
In [ ]:
!jupyter nbconvert --to html Section0-IPython_and_Jupyter.ipynb
Currently, nbconvert supports HTML (default), LaTeX, Markdown, reStructuredText, Python and HTML5 slides for presentations. Some types can be post-processed, such as LaTeX to PDF (this requires Pandoc to be installed, however).
In [ ]:
!jupyter nbconvert --to pdf Section2_1-Introduction-to-Pandas.ipynb
A very useful online service is the IPython Notebook Viewer which allows you to display your notebook as a static HTML page, which is useful for sharing with others:
In [ ]:
%%html
<iframe src=http://nbviewer.ipython.org/2352771 width=700 height=300></iframe>
Also, GitHub supports the rendering of Jupyter Notebooks stored on its repositories.
## Reproducible Research¶
reproducing conclusions from a single experiment based on the measurements from that experiment
The most basic form of reproducibility is a complete description of the data and associated analyses (including code!) so the results can be exactly reproduced by others.
Reproducing calculations can be onerous, even with one's own work!
Scientific data are becoming larger and more complex, making simple descriptions inadequate for reproducibility. As a result, most modern research is irreproducible without tremendous effort.
Reproducible research is not yet part of the culture of science in general, or scientific computing in particular.
## Scientific Computing Workflow¶
There are a number of steps to scientific endeavors that involve computing:
Many of the standard tools impose barriers between one or more of these steps. This can make it difficult to iterate, reproduce work.
The Jupyter notebook eliminates or reduces these barriers to reproducibility.
## Parallel iPython¶
At a high level, there are three basic components to parallel IPython:
• Engine(s) - the remote or distributed processes where your code runs.
• Client - your interface to running code on Engines.
• Controller - the collection of processes that coordinate Engines and Clients.
These components live in the IPython.parallel package, which has been rolled out into its own model that requires installation.
To install ipyparallel:
pip install ipyparallel
or via conda:
conda install ipyparallel
To install the IPython Clusters tab in Jupyter Notebook, add this to your jupyter_notebook_config.py:
c.NotebookApp.server_extensions.append('ipyparallel.nbextension')
This file resides in your ~/.jupyter subdirectory of your home directory, and should be created if it does not already exist.
Before running the next cell, make sure you have first started your cluster, you can use the clusters tab in the dashboard to do so.
In [ ]:
from ipyparallel import Client
client = Client()
dv = client.direct_view()
In [ ]:
len(dv)
In [ ]:
def where_am_i():
import os
import socket
return "In process with pid {0} on host: '{1}'".format(
os.getpid(), socket.gethostname())
In [ ]:
where_am_i_direct_results = dv.apply(where_am_i)
where_am_i_direct_results.get()
IPython Notebook Viewer Displays static HTML versions of notebooks, and includes a gallery of notebook examples.
NotebookCloud A service that allows you to launch and control IPython Notebook servers on Amazon EC2 from your browser.
A Reference-Free Algorithm for Computational Normalization of Shotgun Sequencing Data A landmark example of reproducible research in genomics: Git repo, iPython notebook, data and scripts.
Jacques Ravel and K Eric Wommack. 2014. All Hail Reproducibility in Microbiome Research. Microbiome, 2:8.
Benjamin Ragan-Kelley et al.. 2013. Collaborative cloud-enabled tools allow rapid, reproducible biological insights. The ISME Journal, 7, 461–464; doi:10.1038/ismej.2012.123; | 2021-04-22 11:36:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.27493926882743835, "perplexity": 3799.7066798032433}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039603582.93/warc/CC-MAIN-20210422100106-20210422130106-00167.warc.gz"} |
https://codereview.stackexchange.com/questions/121041/find-two-numbers-that-add-up-to-a-given-total-from-a-formatted-string | # Find two numbers that add up to a given total, from a formatted string
The full challenge from codeeval.com can be viewed here.
Input
Your program should accept as its first argument a filename. This file will contain a comma separated list of sorted numbers and then the sum 'X', separated by semicolon. Ignore all empty lines. If no pair exists, print the string NULL.
Output
Print out the pairs of numbers that equal to the sum X. The pairs should themselves be printed in sorted order i.e the first number of each pair should be in ascending order.
I am writing a program to solve the above conditions. It currently works, however I believe it can be optimized to run better. This challenge is unique, since it requires you to work with a String for input, and to output a specifically formatted String as well. The problem with this is memory efficiency, as I have to convert this String to a String array, and then further convert it to a workable int array.
I am a student developer, and I don't have much experience, so any tips and tricks would be helpful.
Java
import java.util.Arrays;
import java.util.HashMap;
import java.util.Map;
public class SomeClass {
public static void main(String[] gras) {
String line = "1,2,3,4,5;6"; //replace string with test case
//code
String[] j = line.split(";")[0].split(",");
int[] input = new int[j.length];
for (int i = 0; i < j.length; i++) {
input[i] = Integer.parseInt(j[i]);
}
Arrays.sort(input);
int k = Integer.parseInt(line.split(";")[1]);
// Modified Algorithm
Map<Integer, Integer> paers = new HashMap<Integer, Integer>();
boolean d = false;
for (int i = input.length - 1; i >= 0; i--) {
paers.put(k - input[i], input[i]);
}
for (int i = 0; i < input.length; i++) {
if (paers.containsKey(input[i]) && paers.get(input[i]) > input[i]) {
if (d) {
System.out.print(";");
}
d = true;
System.out.print(input[i] + "," + paers.get(input[i]));
}
}
if (!d) System.out.println("NULL"); else System.out.println("");
}
}
As far as what I've tried, this is my final product. I changed my solution from using a naive $O(n^2)$ algorithm to this $O(3n)$ algorithm. Again, my terminology might be off, I'm just figuring this stuff out. Also, the code is tested and working.
# Improved algorithm
You can significantly reduce the memory requirements of your algorithm by not creating any objects, including avoiding Maps, Sets, and String concatenations (the +s you use in print statements). You only need to use a constant amount of memory, comprised completely of primitives stored on the stack (avoiding garbage collection).
You can significantly reduce the time requirements of your algorithm by using a simple trick that allows you to scan the input list only once by moving toward the middle from the left and the right.
public final class SumPairs {
public static final void printSumPairs(final String input) {
final int inputLength = input.length();
// On the left, move to the very first number.
int leftStartIndex = 0;
int leftEndIndex = input.indexOf(',', 1);
int leftNumber = parseIntFromSubstring(input, leftStartIndex, leftEndIndex);
// On the right, move to the very last number.
int rightEndIndex = input.lastIndexOf(';', inputLength - 2);
int rightStartIndex = input.lastIndexOf(',', rightEndIndex - 2) + 1;
int rightNumber = parseIntFromSubstring(input, rightStartIndex, rightEndIndex);
// Figure out the desired sum.
final int desiredSum = parseIntFromSubstring(input, rightEndIndex + 1, inputLength);
boolean noOutputYet = true;
while (leftStartIndex < rightStartIndex) {
final int currentSum = leftNumber + rightNumber;
if (currentSum > desiredSum) {
// On the right, move to the previous distinct number.
int oldRightNumber;
do {
oldRightNumber = rightNumber;
rightEndIndex = rightStartIndex - 1;
rightStartIndex = input.lastIndexOf(',', rightEndIndex - 2) + 1;
rightNumber = parseIntFromSubstring(input, rightStartIndex, rightEndIndex);
} while ((rightNumber == oldRightNumber) && (leftStartIndex < rightStartIndex));
}
else {
if (currentSum == desiredSum) {
if (noOutputYet) noOutputYet = false;
else System.out.print(';');
System.out.print(leftNumber);
System.out.print(',');
System.out.print(rightNumber);
}
// On the left, move to the next distinct number.
int oldLeftNumber;
do {
oldLeftNumber = leftNumber;
leftStartIndex = leftEndIndex + 1;
leftEndIndex = input.indexOf(',', leftStartIndex + 1);
leftNumber = parseIntFromSubstring(input, leftStartIndex, leftEndIndex);
} while ((leftNumber == oldLeftNumber) && (leftStartIndex < rightStartIndex));
}
}
if (noOutputYet) System.out.print("NULL");
System.out.println();
}
// Java 7 and later's String#substring creates a new char[] array just about every time it's used.
// Since Integer#parseInt requires a full String, we'd have to let String#substring create a lot of char[]s if we used Integer#parseInt.
// We'd like to avoid that to reduce memory requirements and to eliminate garbage collection.
// WARNING: This method doesn't actually check whether there is a valid number.
private static final int parseIntFromSubstring(final String str, int start, final int end) {
int result = 0;
final boolean negative = str.charAt(start) == '-';
if (negative) start++;
for (; start < end; start++)
result = 10*result + str.charAt(start) - '0';
if (negative) return -result;
else return result;
}
public static final void main(String[] args) {
printSumPairs("1,2,3,4,5,6;6");
}
}
# Critiques
### Problem statement
1. The part about outputting "NULL" should be in the section about output, not the section about input.
2. The characters in the input are not fully specified. Will there be spaces or other characters to ignore mixed in? Can numbers have a decimal point? Can numbers be negative? Are newlines separators for separate problems to solve?
3. The CodeEval challenge says not to output duplicate pairs. For example "3,3,3,3;6" should output "3,3", not "3,3;3,3;3,3;...".
1. The problem statement says to read lines from a file, but your algorithm doesn't do that.
2. You don't need to sort the array of ints. The problem statement says they'll already be sorted.
### Time complexity
Your code shouldn't be regarded as $O(3n)$ for a few reasons:
1. Big O notation ignores constant multiples like 3. It should be $O(n)$, not $O(3n)$.
2. You use Arrays#sort when you don't need to. While it uses a nicer version of quicksort than normal, it still has a $O(n^2)$ worst case time complexity.
3. If you're limiting your algorithm to a list of unique 32-bit integers, there's only so many of those available to put in your input list. This means that there will be an input that takes the longest amount of time to produce an answer. This means that you have a constant upper bound on the time needed, which makes your algorithm (including your 'naive' one) $O(1)$.
If you're not limiting your algorithm to a list of unique 32-bit integers, then you should realize that it takes a lot longer to add together two integers with billions of digits than it does to add 1 and 2. The time complexity of addition is $O(m)$, where $m$ is the number of digits in the largest number in the input list. This means your complexity will be $O(mn)$.
• Thank you! As for reading lines from a file, and all that other CodeEval jazz, that would be implemented when I submitted the file, but I should have mentioned that in my post. Sorry about that. Also, through testing, sorting the array made the code slightly more efficient. Thank you so much once again. kudos – michal Feb 26 '16 at 15:39
Map<Integer, Integer> paers = new HashMap<Integer, Integer>();
you can simply use a set
Set<Integer> set = new HashSet<>();
because you can always check whether you have seen k-input[i] before by checking the set. But, this would give pairs in reverse order. Hence, use stringbuilder to construct the string and reverse it in the end before printing it. This would also improve your time complexity because you only need to iterate through the array once.
Naming: k could be better named as target and d as foundOnePair.
Here is the code.
public static void printPairsThatMakeTarget(String line){
String[] j = line.split(";")[0].split(",");
int[] input = new int[j.length];
for (int i = 0; i < j.length; i++) {
input[i] = Integer.parseInt(j[i]);
}
Arrays.sort(input);
int target = Integer.parseInt(line.split(";")[1]);
// Modified Algorithm
Set<Integer> set = new HashSet<>();
StringBuilder result = new StringBuilder();
boolean foundOnePair = false;
for (int i = input.length - 1; i >= 0; i--) {
if(set.contains(target - input[i])){
if (foundOnePair) {
result.append(";");
}
foundOnePair = true;
result.append((target - input[i]) + "," + input[i]);
}
else | 2020-02-28 03:19:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18114717304706573, "perplexity": 2919.443714953965}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875146940.95/warc/CC-MAIN-20200228012313-20200228042313-00268.warc.gz"} |
https://collegephysicsanswers.com/openstax-solutions/inductor-designed-filter-high-frequency-noise-power-supplied-personal-computer | Question
(a) An inductor designed to filter high-frequency noise from power supplied to a personal computer is placed in series with the computer. What minimum inductance should it have to produce a $2.00 \textrm{ k}\Omega$ reactance for 15.0 kHz noise? (b) What is its reactance at 60.0 Hz?
1. $21.2 \textrm{ mH}$
2. $8.00 \textrm{ }\Omega$
Solution Video
# OpenStax College Physics Solution, Chapter 23, Problem 87 (Problems & Exercises) (1:38)
View sample solution
## Calculator Screenshots
Video Transcript
This is College Physics Answers with Shaun Dychko. An inductor is meant to filter out high frequency noise from a power supply. So, this is called a low pass filter, which means low frequency is passed through it and high frequencies get filtered out because it will have a high resistance to high frequencies, and by that we mean high reactants. Okay. So, the frequency that it's meant to filter out is 15 kilohertz. And so, if a reactant of two kiloohms is needed to filter out this 15 kilohertz frequency, what inductance is necessary? So, the reactants for an inductor is two pi times frequency times inductance, and we'll divide both sides by two pi F to solve for L. So, inductance is reactants divided by two pi times frequency. So, that's 2 times ten to the three ohms divided by two times pi times 15 kilohertz written as times ten to the three hertz, which gives 21.2 millihenries. And then the next question is, what would the reactants of this inductor be given a frequency of 60 hertz instead of 15 kilohertz. And, we want the reactants to be small for this low frequency because this thing is meant to let low frequencies pass through, it's a low pass filter. And so, we have two pi times 60 hertz times the inductance of 21.22 times 10 to the minus 3 henrys, which has a reactance of 8 ohms, which is very small compared to its reactants of two kiloohms for a 15 kilohertz frequency. | 2019-02-21 03:48:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8615188598632812, "perplexity": 1350.6782622323292}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247499009.48/warc/CC-MAIN-20190221031117-20190221053117-00257.warc.gz"} |
https://howlingpixel.com/i-en/%C4%B4 | # Ĵ
Ĵ or ĵ (J circumflex) is a letter in Esperanto orthography representing the sound [ʒ].
While Esperanto orthography uses a diacritic for its four postalveolar consonants, as do the Latin-based Slavic alphabets, the base letters are Romano-Germanic. Ĵ is based on the French pronunciation of the letter j to better preserve the shape of borrowings from that language (such as ĵurnalo from journal) than Slavic ž would.
Ĵ is the fourteenth letter of the Esperanto alphabet. Although it is written as jx and jh respectively in the x-system and h-system workarounds, it is normally written as J with a circumflex: ĵ.
J with circumflex in Doulos SIL
## Usage
### In mathematics
• The letter ${\displaystyle {\boldsymbol {\hat {\jmath }}}}$ is sometimes used to denote a unit vector in mathematics.
## Character mappings
Character Ĵ ĵ
Unicode name LATIN CAPITAL LETTER J WITH CIRCUMFLEX LATIN SMALL LETTER J WITH CIRCUMFLEX
Encodings decimal hex decimal hex
Unicode 308 U+0134 309 U+0135
UTF-8 196 180 C4 B4 196 181 C4 B5
Numeric character reference Ĵ Ĵ ĵ ĵ
2015 Cape Verdean Football Championships
The 2015 Cape Verdean Football Championship season was the 36th beginner level competition of the first-tier football in Cape Verde. Its started on 9 May and finished on 11 July. The tournament was organized by the Cape Verdean Football Federation. The schedule including its matches were created on Saturday January 10. CS Mindelense won the eleventh title and became the second club after Sporting Praia to win three in a row. Neither clubs participated in the CAF Champions League competition in 2016 and in the 2016 CAF Confederation Cup. This was the second ever and most recent finals competition that two clubs came from the same island as well as the same city.
CS Mindelense was the defending team of the title. A total of 12 clubs participated in the competition, one from each island league and one who won the last season's title.
The biggest win was Mindelense who scored 6-0 over Sporting Clube from Brava. In Group A, only one club scored more than ten goals while four clubs in Group B scored more than ten, the top two scored 14 each and the 3rd and 4th place clubs scored 12 each. Mindelense became the third and recent club to win all five matches in a six club group stage, Derby also done theirs for the second and most recent time.ĵ
The finals had two of its matches ended in a draw with a goal each, this was the first time that happened. The winner was decided on penalty kicks and Mindelense won 4-3 on penalty kicks, this was the first that ended in a penalty shootout in 21 years, the next occurred in the following season.
Circumflex
The circumflex is a diacritic in the Latin and Greek scripts that is used in the written forms of many languages and in various romanization and transcription schemes. It received its English name from Latin circumflexus "bent around"—a translation of the Greek περισπωμένη (perispōménē). The circumflex in the Latin script is chevron-shaped ( ˆ ), while the Greek circumflex may be displayed either like a tilde ( ˜ ) or like an inverted breve ( ̑ ).
In English the circumflex, like other diacritics, is sometimes retained on loanwords that used it in the original language (for example, crème brûlée).
The diacritic is also used in mathematics, where it is typically called a hat or roof or house.
Code page 853
Code page 853 (also known as CP 853 or IBM 00853) is a code page used under DOS to write Turkish, Maltese, and Esperanto. It includes all characters from ISO 8859-3.
Domari language
Domari is an endangered Indic language, spoken by older Dom people scattered across the Middle East and North Africa. The language is reported to be spoken as far north as Azerbaijan and as far south as central Sudan, in Turkey, Iran, Iraq, Palestine, Israel, Jordan, Egypt, Sudan, Libya, Tunisia, Algeria, Morocco, Syria and Lebanon. Based on the systematicity of sound changes, we know with a fair degree of certainty that the names Domari and Romani derive from the Indic word ḍom. The language itself actually derives from an Indo-Aryan language. It shares many similarities to Punjabi and Rajasthani, two languages that originated in India. The Arabs referred to them as nawar as they were a nomadic people that originally immigrated to the Middle East from India.Domari is also known as "Middle Eastern Romani", "Tsigene", "Luti", or "Mehtar". There is no standard written form. In the Arab world, it is occasionally written using the Arabic script and has many Arabic and Persian loanwords. Descriptive work was done by Yaron Matras, who published a comprehensive grammar of the language along with an historical and dialectological evaluation of secondary sources (Matras 2012).
Domari is an endangered language and is currently being shifted away from in younger generations, according to Yaron Matras. In certain areas such as Jerusalem, only about 20% of these Dom people, known as “Middle Eastern Gypsies”, speak the Domari language in everyday interactions. The language is mainly spoken by the elderly in the Jerusalem community. The younger generation are more influenced by Arabic, therefore most only know basic words and phrases. The modern-day community of Doms in Jerusalem was established by the nomadic people deciding to settle inside the Old City from 1940 until it came under Israeli administration in 1967 (Matras 1999).
EBCDIC 905
IBM code page 905 (CCSID 905) is an EBCDIC code page with full Latin-3-charset used in IBM mainframes.
Esperantido
An Esperantido is a constructed language derived from Esperanto. Esperantido originally referred to the language which is now known as Ido. The word Esperantido is derived from Esperanto plus the affix -id- (-ido), which means a "child (born to a parent), young (of an animal) or offspring" (ido). Hence, Esperantido literally means an "offspring or descendant of Esperanto".
A number of Esperantidos have been created to address a number of perceived flaws or weaknesses in Esperanto, or in other Esperantidos, attempting to improve their lexicon, grammar, pronunciation, and orthography. Others were created as language games or to add variety to Esperanto literature.
Esperanto Braille
Esperanto Braille is the braille alphabet of the Esperanto language. One Esperanto Braille magazine, Aŭroro, has been published since 1920, and another, Esperanta Ligilo, since 1904.
Esperanto orthography
Esperanto is written in a Latin-script alphabet of twenty-eight letters, with upper and lower case. This is supplemented by punctuation marks and by various logograms, such as the numerals 0–9, currency signs such as \$, and mathematical symbols.
Twenty-two of the letters are identical in form to letters of the English alphabet (q, w, x, and y being omitted). The remaining six have diacritic marks, ĉ, ĝ, ĥ, ĵ, ŝ, and ŭ (that is, c, g, h, j, and s circumflex, and u breve).
In handwritten Esperanto, the diacritics pose no problem. However, since they do not appear on standard alphanumeric keyboards, various alternative methods have been devised for representing them in printed and typed text. The original method was a set of digraphs now known as the "h-system", but with the rise of computer word processing, the so-called "x-system" has become equally popular. These systems are described below. However, with the advent of Unicode, the need for such work-arounds has lessened.
Esperanto phonology
Esperanto is a constructed international auxiliary language. The creator of Esperanto, L. L. Zamenhof, illustrated Esperanto pronunciation by comparing its letters with their equivalents in several major European languages and declaring a principle of "one letter, one sound".
With over a century of use, Esperanto has developed a phonological norm, including accepted details of phonetics, phonotactics, and intonation, so that it is now possible to speak of proper Esperanto pronunciation and properly formed words independently of the languages originally used to describe Esperanto. This norm diverges only minimally from the original ideal of "one letter, one sound"; that is, it accepts only minor allophonic variation.Before Esperanto phonotactics became fixed, foreign words were adopted with spellings that violated the apparent intentions of Zamenhof and the norms that would develop later, such as poŭpo ('poop deck'), ŭato ('Watt'), and matĉo ('sports match'). Many of these coinages have proven to be unstable, and have either fallen out of use or been replaced with pronunciations more in keeping with the developing norms, such as pobo for poŭpo, vato for ŭato, and maĉo for matĉo. On the other hand, the word jida ('Yiddish'), which was also sometimes criticized on phonotactical grounds but had been used by Zamenhof, is well established.
J
J is the tenth letter in the modern English alphabet and the ISO basic Latin alphabet. Its normal name in English is jay or, now uncommonly, jy . When used for the palatal approximant, it may be called yod ( or ) or yot ( or ).
JX
JX may refer to:
People:
JX (artist), an early alias of DJ Jake Williams, who is more recently known as Rex the Dog
J. X. Williams, a pseudonym used by several different authors during the 1960s for many adult novels
Jesus ChristTechnology:
Roland JX-3P, a MIDI capable synthesizer keyboard which debuted in 1983
JX (operating system), a Java operating system
IBM JX, a personal computer that based on IBM PCjr, released in Japan, in 1984Other uses
Jx or jx, a digraph in the Esperanto x-system orthography, representing the consonant sound [ʒ], normally spelled as Ĵ or ĵ in the Esperanto alphabet
Jambojet IATA designator code
Jiangxi, a province of China (Guobiao abbreviation JX)
Latin Extended-A
Latin Extended-A is a Unicode block and is the third block of the Unicode standard. It encodes Latin letters from the Latin ISO character sets other than Latin-1 (which is already encoded in the Latin-1 Supplement block) and also legacy characters from the ISO 6937 standard.
The Latin Extended-A block has been in the Unicode Standard since version 1.0, with its entire character repertoire, except for the Latin Small Letter Long S, which was added during unification with ISO 10646 in version 1.1.
Mac OS Maltese/Esperanto encoding
Mac OS Maltese/Esperanto, called MacOS Esperanto in older sources, is a character encoding for Esperanto, Maltese and Turkish created by Michael Everson on August 15 1997, based on the Mac OS Turkish encoding. It is used in his fonts, but not on official Mac OS fonts.ISO/IEC 8859-3 supports the same languages with a different layout.
Orbital elements
Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are generally considered in classical two-body systems, where a Kepler orbit is used. There are many different ways to mathematically describe the same orbit, but certain schemes, each consisting of a set of six parameters, are commonly used in astronomy and orbital mechanics.
A real orbit (and its elements) changes over time due to gravitational perturbations by other objects and the effects of relativity. A Keplerian orbit is merely an idealized, mathematical approximation at a particular time.
Proto-Esperanto
Proto-Esperanto (Esperanto: Pra-Esperanto) is the modern term for any of the stages in the evolution of L. L. Zamenhof's language project, prior to the publication of Unua Libro in 1887.
Proto-Indo-Iranian language
Proto-Indo-Iranian or Proto-Indo-Iranic is the reconstructed proto-language of the Indo-Iranian/Indo-Iranic branch of Indo-European. Its speakers, the hypothetical Proto-Indo-Iranians, are assumed to have lived in the late 3rd millennium BC, and are often connected with the Sintashta culture of the Eurasian Steppe and the early Andronovo archaeological horizon.
Proto-Indo-Iranian was a satem language, likely removed less than a millennium from the late Proto-Indo-European language, its ancestor, and in turn removed less than a millennium from the Vedic Sanskrit of the Rigveda, its descendant. It is the ancestor of the Indo-Aryan languages, the Iranian languages, and the Nuristani languages.
Tittle
A tittle or superscript dot is a small distinguishing mark, such as a diacritic or the dot on a lowercase i or j. The tittle is an integral part of the glyph of i and j, but diacritic dots can appear over other letters in various languages. In most languages, the tittle of i or j is omitted when a diacritic is placed in the tittle's usual position (as í or ĵ), but not when the diacritic appears elsewhere (as į, ɉ).
Udi language
The Udi language, spoken by the Udi people, is a member of the Lezgic branch of the Northeast Caucasian language family. It is believed an earlier form of it was the main language of Caucasian Albania, which stretched from south Dagestan to current day Azerbaijan. The Old Udi language is also called the Caucasian Albanian language and possibly corresponds to the "Gargarian" language identified by medieval Armenian historians. Modern Udi is known simply as Udi.
The language is spoken by about 4,000 people in the Azerbaijani village of Nij in Qabala rayon, in Oghuz rayon, as well as in parts of the North Caucasus in Russia. It is also spoken by ethnic Udis living in the villages of Debetavan, Bagratashen, Ptghavan, and Haghtanak in Tavush Province of northeastern Armenia and in the village of Zinobiani (former Oktomberi) in the Kvareli Municipality of the Kakheti province of Georgia.
Udi is endangered, classified as "severely endangered" by UNESCO's Atlas of the World's Languages in Danger.
Ĥ
Ĥ or ĥ is a consonant in Esperanto orthography, representing a voiceless velar fricative [x] or voiceless uvular fricative [χ]. Its name in Esperanto is ĥo (pronounced /xo/).
It is also used in the revised Demers/Blanchet/St Onge orthography for Chinook Jargon.In the case of the minuscule, some fonts place the circumflex centred above the entire base letter h, others over the riser of the letter, and others over the shoulder.
Ĥ is the eleventh letter of the Esperanto alphabet. Although it is written as hx and hh respectively in the x-system and h-system workarounds, it is normally written as H with a circumflex: ĥ.
Alphabets (list)
Letters (list)
Multigraphs
Keyboard layouts (list)
Standards
Lists
This page is based on a Wikipedia article written by authors (here).
Text is available under the CC BY-SA 3.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses. | 2019-03-22 22:08:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.46170881390571594, "perplexity": 6359.15157928998}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202698.22/warc/CC-MAIN-20190322220357-20190323002357-00151.warc.gz"} |
https://tamino.wordpress.com/2011/07/20/ice-forecast-update/ | # Ice Forecast Update
As many of you know, nine months ago I predicted the minimum Arctic sea ice extent for 2011 would be 4.63 +/- 0.9 million km^2. I’ll update that prediction using more recent data from NSIDC.
The earlier model predicted the September average using a model based on time evolution. In particular, it used a quadratic model:
$x = \beta_o + \beta_1 t + \beta_2 t^2$.
Fitting the model to observed September averages, then extrapolating one year into the future, gave the prediction 4.63 million km^2.
We now have monthly average data for both extent and area of sea ice through the first six months of 2011. Can these data improve the prediction? I first tried including an additional variable, the monthly average extent for June:
$x = \beta_o + \beta_1 t + \beta_2 t^2 + \beta_3 x_6$,
where $x_6$ is the monthly average for the preceding June. This gives a slightly lower prediction, 4.58 +/- 0.9 million km^2, but the coefficient of the June average extent is not statistically significant. Furthermore, the AIC (Akaike Information Criterion) for this model is worse than that for the original model. So, I don’t put much stock in this refinement.
Then I tried using the June average, not of extent but of sea ice area:
$x = \beta_o + \beta_1 t + \beta_2 t^2 + \beta_3 a_6$,
where $a_6$ is the monthly average area for the preceding June. This time, the coefficient for the additional term is statistically significant and the AIC of this model is better than that of the original model. The prediction based on this improved model is 4.57 +/- 0.72 million km^2.
Finally, I tried a model using both the area and extent of the preceding June:
$x = \beta_o + \beta_1 t + \beta_2 t^2 + \beta_3 a_6 + \beta_4 x_6$.
Both new coefficients are statistically significant, and the AIC for this model is the best yet (but not much better than the preceding model). Its prediction is for an upcoming September average of 4.66 +/- 0.66 million km^2, which is only slightly different from the original prediction of 4.63 million km^2.
RealClimate recently posted a discussion thread for Arctic sea ice. Several of the reader comments suggested that this year we’re likely to break the remarkably low 2007 value for sea ice extent minimum. Most of these prognostications are based on the fact that at present, extent is at an all-time low for this time of year and is still dropping fast (daily extent data from JAXA, 2011 in red):
However, the coefficient of June extent wasn’t statistically significant. Furthermore, the June area is quite low, but not an all-time record for this time of year (daily area data from Cryosphere Today, 2011 in red):
It seems to me that the final model (using both area and extent from the preceding June) gives the best prediction, so I’ll go with 4.66 million km^2.
I’ll also point out that even with this model, the 95% confidence interval is still quite large (+/- 0.66 million km^2). So according to the model, it could be as low as 4.00 million km^2 or as high as 5.32 million km^2 and still be within the 95% CI. Clearly, what happens with the weather over the next few months will dramatically affect this year’s minimum Arctic sea ice extent.
Just as clearly, over the long term Arctic sea ice will continue to disappear. The trend continues.
### 33 responses to “Ice Forecast Update”
1. Well done, Tamino. Top notch.
As Mike Serreze pointed out on the RC thread and I have been saying for a week on the Arctic Sea Ice blog, there is a shift in weather patterns taking place. This is already being reflected in the daily extent numbers as reported by IJIS. It remains to be seen if things stay this way. If they do, there could be sustained transport of ice through Fram and Nares Strait. This could have an effect in the final phase of the melting season.
But that’s mere speculation. For now extent decrease is going to slow down, and the trend line will probably creep closer to that of 2007.
• Did I say Mike Serreze? I meant Mark Serreze, of course.
2. Jim
Hi, Only thing that matters is the minimum and thickness nothing else and I do remember that during the MWP that Eskimos visited Scotland.
[Response: Never heard that.]
• Kidding, I hope? Personally, I suspect it’s ice *volume* that matters the most, and it’s dropping like a rock as far as we can tell right now.
• Jim Arndt
• Very interesting. . . though I didn’t find anything about Scotland in there. . .
• Phil
According to Lamb’s “Climate, History and the Modern World”, there were a number of reports of stray Eskimos arriving in the Orkney Islands between 1690 and 1728, and once on a river near Aberdeen. This is very much in the Little Ice Age not the MWP.
• Fascinating, Phil. Are these reports credible? How did the Eskimos get there, one wonders, and how on Earth did the locals figure out they were Eskimos? (Well, maybe some of the boys from the Isles went a-whaling betimes, and could recognize a Greenlander when they saw one?) Hell of a voyage in a skin boat. . . .
And if any of this is true, what became of those folks? What a story they could tell!
3. Kevin Stanley
Please forgive me if this question exposes my ignorance, but could volume estimates be incorporated as you have incorporated are and extent, and do you think it would be useful? If not now, maybe after a few years of cryosat data?
[Response: Probably yes, and that might be very fruitful. But only the numbers will tell. I plan to update this when the next month’s data is available from NSIDC, so maybe I’ll trying using volume numbers at that time.]
• TC
For me, as a layman, this graph says that “this year’s minimum volume is going to be about 2-3 thousand km^3″ below that of 2007” and that tells me that a few percent here or there won’t really make a difference compared to the 12-15 thousand km^3 minima of about twenty years ago.
4. Zinfan94
Well Tamino, I think you are too high. I like the analysis where people looked at the
melt rate from here to the end of the season for the last eight years. If we get the melt rate of ’04, ‘ 07, ’08, ’10, then 2011 will set a new low extent record. If we get the melt rate of ’03, ’05, ’06, ’09, then 2011 minimum extent won’t beat the 2007 low. So based on previous melt rates from here, it seems as though dropping below 4.25 million is about a 50/50 proposition.
The other thing to look at is where the ice is this year. There is a lot of ice area in the seas where most of it will melt out. The Beaufort, East Siberian, and Greenland seas, as well as the Canadian Archipelago have a lot of ice that will almost certainly melt out by the minimum. Adjusting out this expected loss of extent, then looking how much ice extent must fall in the central Arctic Basin leads again to a likely chance of a new low extent minimum this year (lower than 4.25 million sq km).
The odds of the extent minimum being less than 4.63 million sq km is probably over 90%, since the loss of extent in the lower latitudes is almost certain, and that alone brings us to about 5.20 million sq km. Then if the central Arctic Basin loses more than 0.57 million sq km from this point, the minimum ice extent falls below your estimate. None of the last eight years have seen that low of an extent loss for the central Arctic from this point in the season.
[Response: Could be. I’m the first to admit that this simple statistical model is quite limited in its ability, and it certainly doesn’t account for a lot of factors (like geographic distribution and ice volume). My prediction last year using a similar method was very close, but that was luck. And my error bars are quite large.]
• Kevin Stanley
Hey, the comment Zinfan94 linked was from me! My 15 minutes have arrived!
Seriously, though, Zinfan–you are kind, but probably giving way too much credit by calling that an “analysis.” It’s like Tamino’s model only in that it _also_ doesn’t account for geographic distribution and ice volume. It’s unlike Tamino’s model in that my thingy doesn’t account for current area or even current extent (!), and involves no statistics other than a mean (with no characterization of the error) and a range (of an arbitrary and tiny data set).
It was fun to think about, but IMO there’s really no reason to think the situation will evolve from here in a way that’s bounded by what happened in the same date range in the last 8 years. There’s a very different starting point, for one thing. And there’s nothing to say that weather conditions will be similar to any of those 8 preceding years.
• Zinfan94
Kevin S. – I really am just bootstrapping off your idea. But instead of looking at the entire ice area/extent, I break the ice down roughly by location. I have noticed that is very difficult to melt ice north of 80N latitude in the heart of the Arctic Basin. If and when (in future years) this ice goes, it will likely be due to loss of the buttressing effect, and with favorable wind conditions that push the central Arctic Basin ice into the Fram Strait.
So what does the ice location distribution look like this year, compared to 2007? The guys at the Univ. Illinois keep a nice site showing high concentration (30% plus versus 15% plus used by most sites) ice extent maps. This site shows comparable maps for 2011 vs 2007 and this year shows very different geographic distribution. The melt in 07 came from the direction of the Beaufort, Chukchi, and East Siberian seas. The melt in 2011 is coming from central Siberia in the Barents and Kara seas. The blockade of ice that usually exists around Severnaya Zemlya is already gone this year, and the ice pack in the Arctic Basin has fallen back above 80N in many intrusions from that direction. There must be a lot of sea surface warming in those areas, which could drive the 2nd half seasonal melt. (Zoom in on both sites to see better detail.)
The big ice extent in the East Siberian sea and the Arctic basin extending down toward the Beaufort, Chukchi, and East Siberian seas will almost certainly be decimated as we approach the September minimum. So if we correct out for most of the lower latitude melting that is almost certain, then we can calculate an estimate of the minimum extent (less the Arctic Basin) and this is about 5.2 million sq km.
At this point, I can apply your technique of using melt from this point forward, but only to the Arctic Basin ice extent. Arctic Basin ice extent is roughly the same as last year, and we only need a fall of about 1.0 million sq km from the current extent in the Arctic Basin. Generally, this is the kind of extent loss we have seen in the Arctic Basin in recent years. So the minimum ice extent is almost certain to fall below 4.6 million sq km, and has about a 50/50 chance of getting below 2007.
Why? Because a lot of the “stubborn ice pack” near Novaya Zemlya is already gone this year.
If the wind and weather set up an Arctic dipole in August, then we could blow out below the 2007 minimum by September 1st.
5. BKsea
One problem with your estimate is that by basing the estimate on June, it ignores the last 3 weeks of data. Visually (based on the JAXA plots), it seems that there is a period from April-June that is not very meaningful in where the ice extent will end up. This starts to change in the June – July time frame. I just did a quick comparison of the ice extent from June versus September compared to the most recent 30 days (June 20-July 19) versus September and found that the correlation coefficient moves from 0.34 up to 0.63. Your story might change if you based your model on the most recent 30 days.
6. GFW
Tamino, to follow up on the commenter who suggested volume data, you could use the daily (modeled) volume data available for download from the UW PIOMAS team here http://psc.apl.washington.edu/wordpress/research/projects/arctic-sea-ice-volume-anomaly/data/
You could then choose to work with some consistent average, say the average volume of days 181-200 from each year, and put that in to your basic quadratic-in-time function as the third element. Some playing around on my part finds a significant correlation between the PIOMAS July volume and Sept avg (or min) extent.
7. Jim Arndt
Please correct to her to here
8. Paul
Tamino,
Question for you. How do you get the confidence intervals for your predicted value? I mean you have to propagate the polynomial coefficient covariance matrix to a future time don’t you? If so, how was that accomplished?
[Response: You don’t propagate the covariance matrix into the future; the polynomial coefficients remain constant — that’s what the model is — and so does the covariance matrix. Instead you can define what I like to call the “uncertainty function,” a function of time which depends on the uncertainty in the polynomial coefficients. For a quadratic fit, it turns out to be a quartic function of time. And that’s just the uncertainty due to probable errors in the fit coefficients.
Then you add in the uncertainty due to plain old noise because the observed value jitters around the model values even if the model is, statistically, exactly correct. In fact in this case, the jitter due to noise overwhelms the uncertainty due to probable errors in the fit coefficients.]
9. john byatt
“On August 21, 2007, the Northwest Passage became open to ships without the need of an icebreaker.”
read at climate progress that opening was not far away”
anyone taking bets on this year’s date,
Tamino without butting in i wish that all your great analysis and graphs could be placed into one peer reviewed paper each year, bloody sceptics, “taminos just a blog where is you peer reviewed science”
10. Nick Barnes
How well does your model work if you do without the trend terms, both linear and quadratic, and simply try to predict from the observational terms (June extent and area)? How about if you add more observational terms (winter maximum extent? previous summer minimum extent? lagged ENSO?)
11. I’ve read about Inuit kayaks, sometimes containing dead Inuit, reaching Great Britain by following along currents.
Unlikely to have happened in the MWP – the Inuit had to knock off the Norse in Greenland first, and that took a little while
• Rattus Norvegicus
Like the end of MWP.
12. Gaz
Hi Tamino.
A paper on sea level rise was given prominence today in a certain newspaper. You know, the one that’s reached a score of 64 on Deltoid.
The paper concluded that there was a decelerating trend in sea level rise over the period 1940 to 2000.
Here’s the paper.
http://www.jcronline.org/doi/pdf/10.2112/JCOASTRES-D-10-00141.1
Maybe you could find the time to comment on the paper’s methodolody – 20 year moving averages of 4 long-running tide gauges, fitted with an order 2 polynomial.
There are obvious issues with the paper, eg it skirts around changes in the various forcings (anthro vs non-anthro) forcings in the 1st vs 2nd halves of C20, but I don’t think the author is trying to obfuscate, I’m just intirigued by the choice of statistical methodology.
Please keep up the good work.
13. IA
Tamino
Apologies for OT.
I wonder if you are able to help.
There is a little storm in the Australian denier press brewing stating that this paper shows the SLR acceleration is not happening the Southern hemisphere:
P. J. Watson (2011) Is There Evidence Yet of Acceleration in Mean Sea Level Rise around Mainland Australia?. Journal of Coastal Research: Volume 27, Issue 2: pp. 368 – 377.
http://www.jcronline.org/doi/full/10.2112/JCOASTRES-D-10-00141.1?prevSearch=%5BAllField%3A+watson%5D&searchHistoryKey=
I’ve had a look at the paper and some of the quadratic curve fitting looks a little odd.
Would you be able to have a look?
Thanks
14. IA
Honestly, there was no comment from Gaz when I posted…!
15. PJKar
Tamino,
Thank you for your response to my question above concerning the propagation of the covariance matrix. The motivation for the question came from my attempt to implement a Kalman filter for the purpose of tracking and predicting sea ice extent. I would be very interested to hear any comments you may have on this approach.
I had tried a polynomial filter over the full data set and computed the covariance matrix but was unsure how to use it with the predicted value so the question. The Kalman filter has a straightforward way of doing this because in the state space implementation the predicted state covariance matrix is propagated forward to the same time as the predicated state and that is what I used to determine the prediction uncertainty.
I liked your approach on this which was just to use the Sept sea ice extent data to predict the coming September value. It is a very effective method and only requires a one step prediction. The updates in the current blog entry have some interesting ideas too for sharpening the prediction.
The Kalman filter is very involved so a lot can go wrong. But anyway to briefly describe it the anomalies were computed using the entire 1978 through 2011 as the baseline, The measurements input to the filter included the anomaly, its velocity, and its acceleration. The latter two were formed by first and second differences in the month to month anomaly data. The measurement noise (uncertainty) matrix had to take into account the cross correlations in the measurements. The filter was linear in the sense that the filter states were the same as the measurements so was implemented as a 3 state constant acceleration estimator with a white noise jerk process noise model providing acceleration uncertainty. The update time was one month. As such it too was a quadratic polynomial estimator with some adaptation capability resulting from the process noise. The predicted extent was formed by summing the predicted anomaly and the monthly extent average. The extent prediction uncertainly was calculated from the RSS of the predicted anomaly standard deviation acquired from the predicted state covariance matrix and the standard deviation of the average monthly extent estimate. Filter performance was based on how well the filter reduced the measurement noise compared to the state noise.
To see how well the predictor worked the filter trained on data up through April 2011. A September prediction required 5 steps forward which was beyond the filters capability.. With no measurements from May to September the filter basically coasts the last filtered state (from April) and filtered covariance through 5 updates.
For one step prediction to May it predicted 12.92 million sq km with a prediction standard deviation of .359 million sq km. Its prediction for June was 11.675 million sq km with a prediction standard deviation of .62584 million sq km. May extent came in at 12.79 million sq km June at 11.01 million sq km.
So for one or two step prediction it seemed to do OK but the uncertainty will yield large confidence intervals. 5 months out to September yielded an estimate of 7.5 million sq km. with an uncertainty of 2.45 million square km.
With May data available the filter prediction for June improved to 11.34 million sq km with a prediction standard deviation of .354 million sq km. September’s prediction was 6.26 million sq km with a s.d. of 1.4 million sq km.
With May and June data available the July prediction is 8.95 million square km with a one s.d. prediction uncertainty of .447 million square km. Three months out to Sept the prediction is 5.47 million sq km with an uncertainty of 1.14 million sq km.
Just wondering. What is your prediction and confidence interval for July?
Anyway, I look forward to hearing any comments you or anyone else in the forum may have on the procedure or results discussed here.
Incidentally, this has been an extremely interesting and informative series of articles on Sea Ice Extent from the point of view of both time series analysis and climatology. The time and effort you put in on explaining and presenting these ideas is greatly appreciated. Also the commenters have provided some excellent links to sites and papers on Arctic Sea Ice.
16. Gaz
Great minds think alike!
17. Martin
Is there a physical reason why including the sea ice area + extent in June should improve your model?
18. Ernst K
I’m not really happy with using a time trend to forecast ice extent because there’s nothing physical about it. It’s obviously just a proxy for the steady warming of the arctic.
I’ve been trying to develop a simple relationship between arctic sea ice extent and polar temperature. Unfortunately, It has been limited by the fact that I can only find annual polar temperature data (GISS, 64 to 90 deg N). The most appropriate monthly data I can find is for the entire northern hemisphere.
While N hemisphere seasonal temps and polar annual temps both loosely correlate with ice extent, neither does any better (or worse) than Tamino’s simple time trend approach. However there are some tantalizing hints that anomalous years like 1996 (large increase in September extent) and 2007 (large decrease in September extent) might be explained by summer polar temperatures (1996 was a cool summer in the N hemisphere and 2007 was a warm year at the poles).
Does GISS publish their monthly/seasonal zonal data anywhere?
19. Ernst K,
The best I’ve been able to do is what was referred to in earlier in the thread:
Use the current extent and work out the falls from the same date to the minima for previous years. Using JAXA’s dataset from 23/7/11 I get
2002 4321875
2003 4662187
2004 4149844
2005 4534531
2006 5065000
2007 4190781
2008 3843750
2009 4597656
2010 4247344
2007’s minimum was 4254531.
In essence what that method is doing is using the weather and ice conditions of the years 2002 to 2010 to estimate a range of probable outcomes for this year.
Trying to use just temperature in the Arctic has one main problem, the temperature can be raised by the presence of open water, which itself is due to reduced sea-ice. Furthermore it doesn’t take into account issue like cloudiness (which are related to the AO and AD modes), and ocean heat transport.
The best option is a model like PIOMAS, which I consider to be the best available. e.g. PIPS overprojects thickness. However even PIOMAS suffers from the major impact that weather has, so they use the weather conditions for the past 7 years to make 7 ensemble members, like the (admittedly far simpler) method I outline above. You may have seen the PIOMAS projection, but in case you haven’t it’s here:
http://psc.apl.washington.edu/zhang/IDAO/seasonal_outlook.html
Their forecast is 4.3 +/- 0.5 million square kilometers, the hindcast runs to the end of June, so the forecast doesn’t account for the rapid losses due to clear skies in early July. But it is in the same ball-park as Tamino’s stats.
Sorry to be so negative, the only positive sugestion I can make is Megan stone’s Doctoral Thesis: It may be of interest to you.
http://edocs.nps.edu/npspubs/scholarly/theses/2010/Jun/10Jun_sTONE.pdf
20. crandles
I am glad someone with more skill is going down the route of a multiple non-linear regression model (following ;o) my example http://www.arcus.org/files/search/sea-ice-outlook/2011/07/pdf/pan-arctic/randles_panarctic_july.pdf )
For the August report, I have tried using various combinations of extent, volume, thickness and July decrease as well as area but didn’t find anything that reduced RMSE more than an average set of random numbers so have just stuck to using gompertz fit and area.
I am sure you have a lot more experience / ability to do this sort of thing than I have so any comments on the difference between our approaches would be appreciated.
I wouldn’t have a clue how to calculate an ‘Akaike Information Criterion’ but suspect it would be a significant improvement on my approach of comparing to several sets of random numbers to see if an extra data set provides significant improvement in prediction.
Should I leave this sort of thing to experts rather than having a go?
[Response: By all means keep at it. Your approach seems logical.
You can find out about AIC on Wikipedia. Numerous programs (including R) will calculate it for you automatically.]
21. crandles
Ernst,
UAH gives monthly t2lt temperature data for North Polar regions and also breaks this down between land and ocean. I realise you probably want SST / 2m height temperatures instead or perhaps other heights as well.
http://vortex.nsstc.uah.edu/data/msu/t2lt/uahncdc.lt
[Response: Note that the UAH TLT (and RSS TLT) data does not extend above latitude 82.5N.] | 2017-02-22 08:20:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 6, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.46315303444862366, "perplexity": 1377.376208309892}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501170925.44/warc/CC-MAIN-20170219104610-00049-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://puzzling.stackexchange.com/questions/110743/make-an-equation-from-expression-1-4-9-100/110744 | Make an equation from expression "1 4 9 100" [closed]
Fill the gaps of the expression $$1$$ $$4$$ $$9$$ $$100$$ with mathematical operations $$+,-,*,/$$, physical units and an $$=$$ sign, such that a correct equation is created.
Example: expression $$1$$ $$100$$ $$100$$ $$9$$ -> $$1 \text{m} = 100 \text{mm} + 100 \cdot 9 \text{mm}$$
• Would something like 1 feet = 2 * 2 * 3 inches for 1 2 2 3 be allowed? Jul 1, 2021 at 18:53
• yes, this is a valid solution for 1 2 2 3 Jul 1, 2021 at 18:56
• what about something like 4 ft * 6 in = 2 ft^2 * 1 for 4 6 2 1? where ft^2 is the same as sqft which is square feet? Jul 1, 2021 at 19:05
• @rhavelka, yes, this is a valid solution as well. Jul 1, 2021 at 21:11
• @IsaacRoanSison I do not think that the approximate symbol would work since the question strictly states an "=" sign. Jul 2, 2021 at 0:20
Hmm, let's see.
1 hour = 4 * 9 * 100 seconds
A few more possibilities:
With pints and fluid ounces
1x4 pt = 9 pt - 100 fl oz
With feet and inches
1 ft - 4 in = 9 ft - 100 in
With angles
1° = 4x9x100"
There are some non-standard "journalism units" like "Olympic swimming pool" for volume, "Texas" for area, or "Library of Congress" for amount of data. One of these, recognized by the units program, is "football field = 100 yards".
If we allow these, then 1 football field = 4 * 9 inches * 100
Alternatively, if you permit SI prefixes on very non-SI units, then 1 * 4 * 9 inches = 100 centiyards :)
if base number is a valid unit then 1 * 4 * 9 (decimal)= 100 (Senary, heximal, or seximal, base-6) | 2022-10-04 00:51:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 11, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.653878390789032, "perplexity": 2367.338288020275}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337446.8/warc/CC-MAIN-20221003231906-20221004021906-00077.warc.gz"} |
https://stats.stackexchange.com/questions/354355/what-is-the-relation-between-svd-and-als | # What is the relation between SVD and ALS?
I am trying to build a simple CF-recommender system using the small MovieLens data set. In order to do this, I tried to use ALS to factor my (user, item) matrix $A$ into a (user, latent-space) matrix $U$ and a (item, latent-space) matrix $V$ such that: $$A = UV$$ To do this I am using the ALS algorithm as defined in the Implicit package for python, which when I multiply $UV$, is wildly different from my initial $A$. My question is, am I confused about what ALS actually is? I thought it was something akin to SVD, or any other matrix factorization algorithm. Would It matter that i'm using explicit data instead of implicit data? Here is the code i'm using to perform the matrix decomposition:
from implicit.als import AlternatingLeastSquares
from scipy import sparse
def matrix_decomposition(matrix, k, i):
matrix = sparse.csr_matrix(matrix.T)
model = AlternatingLeastSquares(factors=k, iterations=i)
model.fit(matrix)
user_latent = model.user_factors
item_latent = model.item_factors
return user_latent, item_latent
... I confused about what ALS actually is? I thought it was something akin to SVD, or any other matrix factorization algorithm. Would It matter that i'm using explicit data instead of implicit data?
"Implicit vs explicit data
Explicit data is data where we have some sort of rating. Like the 1 to 5 ratings from the MovieLens or Netflix dataset. Here we know how much a user likes or dislikes an item which is great, but this data is hard to come by. Your users might not spend the time to rate items or your app might not work well with a rating approach in the first place.
Implicit data (the type of data we’re using here) is data we gather from the users behaviour, with no ratings or specific actions needed. It could be what items a user purchased, how many times they played a song or watched a movie, how long they’ve spent reading a specific article etc. The upside is that we have a lot more of this data, the downside is that it’s more noisy and not always apparent what it means.
For example, with star ratings we know that a 1 means the user did not like that item and a 5 that they really loved it. With song plays it might be that the user played a song and hated it, or loved it, or somewhere in-between. If they did not play a song it might be since they don’t like it or that they would love it if they just knew about.
So instead we focus on what we know the user has consumed and the confidence we have in whether or not they like any given item. We can for example measure how often they play a song and assume a higher confidence if they’ve listened to it 500 times vs. one time.
Implicit recommendations are becoming an increasingly important part of many recommendation systems as the amount of implicit data grows. For example the original Netflix challenge focused only on explicit data but they’re now relying more and more on implicit signals. The same thing goes for Hulu, Spotify, Etsy and many others.".
There are different ways to factor a matrix, like Singular Value Decomposition (SVD) or Probabilistic Latent Semantic Analysis (PLSA) if we’re dealing with explicit data.
A least squares approach in it’s basic forms means fitting some line to the data, measuring the sum of squared distances from all points to the line and trying to get an optimal fit for missing points.
With the alternating least squares approach we use the same idea but iteratively alternate between optimizing U and fixing V and vice versa. It is an iterative optimization process where we for every iteration try to arrive closer and closer to a factorized representation of our original data.
Singular-Value Decomposition is a matrix decomposition method for reducing a matrix to its constituent parts in order to make certain subsequent matrix calculations simpler.
The tutorials in the first two and last link should be very helpful.
• would If I could, but don't quite have the reputation yet. Will do when I get enough Jul 9 '18 at 17:18 | 2021-09-29 01:57:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.49352359771728516, "perplexity": 693.5980572348986}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780061350.42/warc/CC-MAIN-20210929004757-20210929034757-00338.warc.gz"} |
https://www.gradesaver.com/textbooks/math/applied-mathematics/elementary-technical-mathematics/chapter-1-section-1-14-rate-base-and-part-exercises-page-85/56 | ## Elementary Technical Mathematics
To find the part that is lost, multiply the base weight times the shrinkage rate. Move the decimal two places to the left to convert the percentage to a decimal. $P=BR=70\ lb\times17\%=70\ lb\times0.17=11.9\ lb$ Deduct the shrinkage from the original weight to find the remaining part. $70\ lb-11.9\ lb=58.1\ lb$ | 2021-03-03 09:29:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8719103336334229, "perplexity": 1061.080650359993}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178366477.52/warc/CC-MAIN-20210303073439-20210303103439-00437.warc.gz"} |
https://ai.stackexchange.com/questions/17510/why-does-pytorch-use-a-different-formula-for-the-cross-entropy | # Why does PyTorch use a different formula for the cross-entropy?
In my understanding, the formula to calculate the cross-entropy is
$$H(p,q) = - \sum p_i \log(q_i)$$
But in PyTorch nn.CrossEntropyLoss is calculated using this formula:
$$loss = -\log\left( \frac{\exp(x[class])}{\sum_j \exp(x_j)} \right)$$
that I think it only addresses the $$\log(q_i)$$ part in the first formula.
Why does PyTorch use a different formula for the cross-entropy?
• You are using the Softmax CE loss, use BCE loss or Binary CE loss for your formula. – DuttaA Jan 15 at 8:09
When you one-hot-encode your labels with $$p_i \in \{0,1\}$$ you get $$p_i = 0$$ iff $$i$$ is not correct and, equivalently, $$p_i =1$$ iff $$i$$ is correct.
Hence, $$p_i \log(q_i) = 0 \log(q_i) = 0$$ for all classes except the "truth" and $$p_i \log(q_i) = 1 \log(q_i) = \log(q_i)$$ for the correct prediction.
Therefore, your loss reduces to: $$H(p,q) = - \sum p_i \log(q_i) = - \log(q_{truth})$$
• @DuttaA have a look at the last part of the question: "that I think it only addresses the $\log(q_i)$ part in the first formula. So is that means Pytorch using different Cross entropy formula?". Reads to me that it actually is exactly about this. . – Sammy Jan 15 at 8:23
• OMG! yes, this is what I meant, I accidentally mix language modeling cross-entropy (that have continuous values of $p_i$) and PyTorch nn.CrossEntropyLoss. I forgot in classification $p_i$ will be 0 or 1 so the PyTorch function is valid, but of course, I can't use it directly for Language Modeling case. Thank You! – malioboro Jan 15 at 8:35 | 2020-01-19 15:42:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 11, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9039647579193115, "perplexity": 851.8269890870398}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250594662.6/warc/CC-MAIN-20200119151736-20200119175736-00360.warc.gz"} |
https://math.stackexchange.com/questions/764454/permutations-expressed-as-product-of-transpositions | # Permutations expressed as product of transpositions
There is a theorem that states that all permutations can be expressed as a product of transpositions. I have a couple of questions about this theorem:
1. Does the product which is equal to the permutation always start from the identity permutation?
In the proof for this theorem our professor has argued that every permutation can be transformed into the identity permutation by applying a certain number of transpositions, e.g. if $\sigma$ is a permutation not equal to the identity permutation then you can apply, say l transpositions, so that you get: $\tau_l \circ \tau_{l-1} \circ .... \circ \tau_1 \circ \sigma = id \Rightarrow \tau_1^{-1} \circ \tau_2^{-1} \circ ... \circ \tau_l^{-1}=\tau_1 \circ \tau_2 \circ .... \circ \tau_l$.
Is this product of transpositions always unique, or can you start from any arbitrary permutation and perform the required number of transpositions to get your permutation?
2 If I form the composition of two permutations, say $\sigma_1$ and $\sigma_2$ given by: $$\sigma_1 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 3 & 4 &5 &6 &1 &2 \\ \end{pmatrix}$$
$$\sigma_2 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 4 &6 &1 &5 &3 \\ \end{pmatrix}$$
I can express them in terms of the following transpositions $$\sigma_1 = (1,5)\circ (2,6) \circ (3,5) \circ (4,6)$$ $$\sigma_2 = (1,4) \circ (2,4) \circ (3,6)$$
When I form the composition $\sigma_1 \circ \sigma_2$ I get:
$$\sigma_1 \circ \sigma_2 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 4 & 6 &2 &3 &1 &5 \\ \end{pmatrix}$$
But since the product of the transpositions is equal to the permutations, I should get the same result when I use:
$$\sigma_1 \circ \sigma_2 = (1,5)\circ (2,6) \circ (3,5) \circ (4,6) \circ (1,4) \circ (2,4) \circ (3,6)$$
but I get:
$$\sigma_1 \circ \sigma_2 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 6 & 1 &5 &3 &2 &4 \\ \end{pmatrix}$$
Why doesn't this work?
• I'm not sure I understand your first question - but for the second one, you're just reading the transpositions in the wrong order. Your convention for $\sigma_1\circ\sigma_2$ is that $\sigma_2$ is applied first, so with the transpositions you need to start from $(3,6)$ and read from right to left. So $1\mapsto 4\mapsto 6\mapsto 2$ etc. – mdp Apr 22 '14 at 13:49
• Thanks for your comment, but I did start from the right side: $\begin{pmatrix} 1 & 2 & 3&4&5&6\end{pmatrix} \Rightarrow \begin{pmatrix} 1 & 2 & 6&4&5&3\end{pmatrix} \Rightarrow \begin{pmatrix} 1 & 4 & 6&2&5&3\end{pmatrix} \Rightarrow \begin{pmatrix} 2 & 4 & 6&1&5&3\end{pmatrix} \Rightarrow \begin{pmatrix} 2 & 4 & 6&3&5&1\end{pmatrix} \Rightarrow \begin{pmatrix} 2 & 4 & 5&3&6&1\end{pmatrix} \Rightarrow \begin{pmatrix} 2 & 1 & 5&3&6&4\end{pmatrix} \Rightarrow \begin{pmatrix} 6 & 1 & 5&3&2&4\end{pmatrix}$ I'm not sure what you mean by $1 \rightarrow 4 \rightarrow 6 \rightarrow 2...$ – eager2learn Apr 22 '14 at 13:58
• Ah, you made a different mistake that gives the same result as doing the transpositions backwards - on the third arrow you should be swapping the numbers $1$ and $4$ over, wherever they appear, not the first position with the fourth. (I was drawing my way of doing this calculation; put $1$ in on the right and see where it goes - it first gets mapped to $4$ (by $(1,4)$), then $4$ is mapped to $6$ by $(4,6)$, and finally $6$ is mapped to $2$ by $(2,6)$). – mdp Apr 22 '14 at 14:04
• Yes that was the mistake I made. Thanks a lot for your help. – eager2learn Apr 22 '14 at 14:33
Note that the product of transpositions that you used to express $\sigma_1$, when composed, does not again yield $\sigma_1$; I.e., you did not correctly express $\sigma_1$ as a product of transpositions.
Rather, $\sigma_1$ can be written $(1,5)\circ (1, 3)\circ (2, 6)\circ (2, 4)$, or $(1, 3)\circ (3, 5)\circ (2, 4)\circ(4, 6)$...
Similarly, $\sigma_2$ is incorrectly decomposed. Two correct decompositions include $(1, 4)\circ (1, 2)\circ (3, 6)$ and $(1, 2)\circ (2, 4) \circ (3, 6)$...
...which answers your question about uniqueness. When writing a permutation as a product of transpositions, there are many such ways to do this. What does not vary is the parity: an "odd" permutation is one that can only be decomposed to a product of an odd number of transpositions, and "even" permutations can only be decomposed into a product of an even number of transpositions. So, for example, $\sigma_1$ is even, and $\sigma_2$ is odd.
• I don't understand this. If I apply $(1,5)\circ (1,3) \circ (2,6) \circ (2,4)$ on the id permutation I get the permutation $\begin{pmatrix} 5 &6&1&2&3&4 \end{pmatrix}$ Why is this a transposition for $\sigma_1$? I think this goes back to my original question, from which permutation do I start when applying those transpositions? – eager2learn Apr 22 '14 at 14:26
• You start from the rightmost permutation. Where does it send $1$?. If it sends $1$ to $a$, then you move to the next transposition to its left to see where it sends $a$. Etc. – Namaste Apr 22 '14 at 14:30 | 2019-10-18 11:55:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8850502371788025, "perplexity": 136.33681366525107}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986682037.37/warc/CC-MAIN-20191018104351-20191018131851-00469.warc.gz"} |
http://www.thefullwiki.org/%C4%92 | # Ē: Wikis
Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.
# Encyclopedia
(Redirected to Macron article)
Ā ā Ǟ ǟ Ǡ ǡ Ǣ ǣ Ḇ ḇ C̄ c̄ Ḏ ḏ Ē ē Ḕ ḕ Ḗ ḗ Ḡ ḡ H̱ ẖ Ī ī Ḵ ḵ Ḻ ḻ Ḹ ḹ Ṉ ṉ Ō ō Ṓ ṓ Ṑ ṑ Ȫ ȫ Ǭ ǭ Ȭ ȭ Ȱ ȱ Ṟ ṟ Ṝ ṝ Ṯ ṯ Ū ū Ǖ ǖ Ṻ ṻ Ȳ ȳ Ẕ ẕ
A macron, from the Greek μακρόv (makrón), meaning "long", is a diacritic placed above a vowel (and, more rarely, under or above a consonant). It was originally used to mark a long syllable in Græco-Roman metrics, but now also indicates that the vowel is long. (This is the opposite of a breve ˘, used to indicate originally a short syllable and now also a short vowel.) Distinctions between long and short vowels are often phonemic. In the International Phonetic Alphabet the macron is used to indicate mid tone; the sign for a long vowel is a modified triangular colon.
## Syllable weight
In Græco-Roman metrics and in the description of the metrics of other literatures, the macron was introduced and is still widely used to mark a long (i.e., heavy) syllable. Even the best and relatively recent classical Greek and Latin dictionaries[1] are still only concerned with indicating the length (i.e., weight) of syllables; that is why most still do not indicate the length of vowels in syllables that are otherwise metrically determined. Though many textbooks about ancient Rome and Greece employ the macron, it was not actually used at that time.
## Vowel length
The following languages or transliteration systems use the macron to mark long vowels:
• Slavicists use the macron to indicate a non-tonic long vowel, or a non-tonic syllabic liquid, such as on l, lj, m, n, nj, and r. Languages with this feature include standard and jargon varieties of Serbian, Croatian, Macedonian[citation needed], Slovak[citation needed], Bulgarian.[2]
• Transcriptions of Arabic typically use macrons to indicate long vowels — ا (alif when pronounced as /aː/), و (waw, when pronounced as /uː/), and ي (ya', when pronounced as /iː/). Thus the Arabic word ثلاثة (three) is transliterated ṯalāṯah.
• Some modern dictionaries of classical Greek and Latin, where the macron is sometimes used in conjunction with the breve. However, many such dictionaries still have ambiguities in their treatment and distinction of long vowels or heavy syllables.
• The Hepburn romanization system of Japanese. Examples: kōtsū (交通) "traffic" as opposed to kotsu () "bone" or "knack" (fig.)
• Latvian. "Ā", "ē", "ī", "ū" are separate letters that sort in alphabetical order immediately after "a", "e", "i", "u" respectively. Ō was also used in Latvian, but it was discarded as of 1957.
• Lithuanian. "Ū" is a separate letter but given the same position in collation as the unaccented "u". It marks a long vowel; other long vowels are indicated with an ogonek (which used to indicate nasalization, but no longer does): "ą", "ę", "į", "ų", "o" being always long in Lithuanian except for some recent loanwords. For the long counterpart of "i", "y" is used.
• Transcriptions of Nahuatl (spoken in Mexico). Since Nahuatl (Nāhuatl) (Aztecs' language) did not have a writing system, when Spanish conquistadors arrived, they wrote the language in their own alphabet without distinguishing long vowels. Over a century later, in 1645, Horacio Carochi defined macrons to mark long vowels ā, ē, ī and ō, and short vowels with grave () accents. This is rare nowadays since many people write Nahuatl without any orthographic sign and with the letters /k/, /s/ and /w/, not present in the original alphabet. Some projects prefer macron-based writing, as in Nahuatl Wikipedia.
• Modern transcriptions of Old English.
• Latin transliteration of Pali and Sanskrit.
• Polynesian languages:
• Hawaiian. The macron is called kahakō, and it indicates vowel length, which changes meaning and the placement of stress.
• Māori. Early writing in Māori did not distinguish vowel length. Some — notably Professor Bruce Biggs[3] — have advocated that double vowels be written to mark long vowel sounds (e.g., Maaori), but he was more concerned that they be marked at all than with the method. The Māori Language Commission (Te Taura Whiri o te Reo Māori) advocates that macrons be used to designate long vowels. The use of the macron is widespread in modern Māori, although sometimes the diaeresis mark is used instead (e.g. "Mäori" instead of "Māori") if the macron is not available for technical reasons [1]. The Māori words for macron are pōtae "hat", or tohuto.
• Tongan. Called the toloi, its usage is similar to that in Māori, including its substitution by a diaeresis.
## Tone
The following languages or alphabets use the macron to mark tones:
• In Pinyin, macrons are used over a, e, i, o, u, ü (ā, ē, ī, ō, ū, ǖ) to indicate the first tone of Mandarin Chinese. The alternative to macron is the number 1 after the syllable, e.g. tā = ta1.
## Other uses
• In French, in comic books that are hand-lettered in all-capitals, sometimes the macron replaces the circumflex[citation needed].
• In some German handwriting the a macron is used to distinguish u from n or instead of the umlaut.
• In some Finnish and Swedish comic books that are hand-lettered, or in handwriting, the macron is used instead of ä or ö, sometimes known colloquially as a "lazy man's umlaut".
• In older handwriting such as the German Kurrentschrift, the macron over an a-e-i-o-u or ä-ö-ü stood for an n, or over an m or an n meant that the letter was doubled. This continued into print in English in the sixteenth century. Over a u at the end of a word, the macron indicated um as a form of scribal abbreviation.
• In Russian handwriting, a lowercase Т looks like a lowercase m, and a macron is often used to distinguish it from Ш, which looks like a lowercase w. Some writers also underline the letter ш to reduce ambiguity further.
• In music, the tenuto marking resembles the macron.
## Non-diacritical usage
• In medical prescriptions and other handwritten notes, macrons mean:
• over c, with, abbreviating the Latin word cum;
• over p, after, abbreviating post;
• over q, every, abbreviating quisque (inflected forms: quoque/quaque);
• over s, without, abbreviating sine;
• over x, except, formed by analogy, and not specifically from any Latin.
• In statistics, mathematics and physics the macron is often used to indicate:
• x̄ a mean (e.g., $\bar{x}$ as the average value of xi)
• In mathematics it may denote:
• the conjugate of a complex number, so that if x = a + ib, then $\overline{x} = a - ib.$
• In mathematics and physics it may denote:
• A vector, so that $\overline x=|x|\hat x$, although boldface and arrows commonly are also used.
• In Old English texts a macron above a letter indicates the omission of an m or n that would normally follow that letter.
## Technical notes
diacritic
character Unicode HTML
Latin
Ā
ā
U+0100
U+0101
Ā
ā
Ē
ē
U+0112
U+0113
Ē
ē
Ī
ī
U+012A
U+012B
Ī
ī
Ō
ō
U+014C
U+014D
Ō
ō
Ū
ū
U+016A
U+016B
Ū
ū
Ȳ
ȳ
U+0232
U+0233
Ȳ
ȳ
Ǣ
ǣ
U+01E2
U+01E3
Ǣ
ǣ
U+1E20
U+1E21
Ḡ
ḡ
diaeresis Ǟ
ǟ
U+01DE
U+01DF
Ǟ
ǟ
Ȫ
ȫ
U+022A
U+022B
Ȫ
ȫ
Ǖ
ǖ
U+01D5
U+01D6
Ǖ
ǖ
U+1E7A
U+1E7B
Ṻ
ṻ
dot above Ǡ
ǡ
U+01E0
U+01E1
Ǡ
ǡ
Ȱ
ȱ
U+0230
U+0231
Ȱ
ȱ
dot below
U+1E38
U+1E39
Ḹ
ḹ
U+1E5C
U+1E5D
Ṝ
ṝ
ogonek Ǭ
ǭ
U+01EC
U+01ED
Ǭ
ǭ
tilde Ȭ
ȭ
U+022C
U+022D
Ȭ
ȭ
acute
U+1E16
U+1E17
Ḗ
ḗ
U+1E52
U+1E53
Ṓ
ṓ
grave
U+1E14
U+1E15
Ḕ
ḕ
U+1E50
U+1E51
Ṑ
ṑ
Cyrillic
Ӣ
ӣ
U+04E2
U+04E3
Ӣ
ӣ
Ӯ
ӯ
U+04EE
U+04EF
Ӯ
ӯ
Greek
U+1FB9
U+1FB1
Ᾱ
ᾱ
U+1FD9
U+1FD1
Ῑ
ῑ
U+1FE9
U+1FE1
Ῡ
ῡ
In Unicode, "combining macron" is a combining character with the code U+0304 (in HTML, ̄ or ̄). This is different from the "macron" at U+00AF ¯, from the "modifier letter macron" at U+02C9 ˉ and from the combining overline at U+0305 ̅. There are several precomposed characters; their HTML/Unicode numbers are as in the table to the right. In LaTeX a macron is created with the command "\=", for example: M\=aori.
The row before the last is the letter Uu with diaeresis (Ü ü) and macron, used in pinyin. The final row is the letter Yy with macron, used sometimes in teaching Old English and Latin.
The basic modern Latin alphabet
Aa Bb Cc Dd Ee Ff Gg Hh Ii Jj Kk Ll Mm Nn Oo Pp Qq Rr Ss Tt Uu Vv Ww Xx Yy Zz
Letters using macron sign
## References
1. ^ P.G.W. Glare (ed.), Oxford Latin Dictionary (Oxford at the Clarendon Press 1990), p. xxiii: Vowel quantities. Normally only long vowels in a metrically indeterminate position are marked.
2. ^ Годечкият Говор от Михаил Виденов,Издателство на българската академия на науките,София, 1978, p. 19: ...характерни за всички селища от годечкия говор....Подобни случай са характерни и за книжовния език-Ст.Стойков, Увод във фонетиката на българския език , стр. 151..
3. ^ Yearbook of the Academy Council - 2000, Royal Society of New Zealand
Basic Latin alphabet
Aa Bb Cc Dd
Ee Ff Gg Hh Ii Jj
Kk Ll Mm Nn Oo Pp
Qq Rr Ss Tt Uu Vv
Ww Xx Yy Zz
E is the fifth letter in the Latin alphabet. Its name in English (pronounced /iː/) is spelled e; the plural is ees, though this is rare.[1] The letter E is the most commonly used letter in the Czech,[2] Danish,[2] Dutch,[2] English,[3] French,[4] German,[5] Hungarian,[2] Latin,[2] Norwegian,[2] Spanish,[6] and Swedish languages.[2]
## History
Egyptian hieroglyph
E’
Proto-Semitic
H
Phoenician
H
Etruscan
E
Greek
Epsilon
Roman/Cyrillic
E
A28 [[File:]] [[File:|64x64px]] File:Alfabeto File:Epsilon uc [[File:|Roman E]]
E is derived from the Greek letter epsilon which is much the same in appearance (Ε, ε) and function. In etymology, the Semitic probably first represented a praying or calling human figure (hillul jubilation), and was probably based on a similar Egyptian hieroglyph that was pronounced and used quite differently. In Semitic, the letter represented /h/ (and /e/ in foreign words), in Greek became Εψιλον (Epsilon) with the value /e/. Etruscans and Romans followed this usage. Arising from the Great Vowel Shift, English usage is rather different, namely /iː/ (derived from /eː/ in "me" or "bee") whereas other words like "bed" are closer to Latin and other languages in usage.
## Usage
Like other Latin vowels, E came in a long and a short variety. Originally, the only difference was in length but later on, short e represented /ɛ/. In other languages that use the letter E or e, it represents various other phonetic values, sometimes with accents to indicate contrasts (e ê é è ë ē ĕ ě ẽ ė ẹ ę ẻ).
Digraphs starting with E are common in many languages to indicate diphthongs and monophthongs, such as EA or EE for /iː/ or /eɪ/ in English, EI for /aɪ/ in German, or EU for /ø/ in French or /ɔɪ/ in German.
At the end of a word, E is very often silent in English (silent e), where old noun inflections have been dropped, although even when silent at the end of a word, it often causes vowels in the word to be pronounced as diphthongs, conventionally called long vowels (compare as a noun rat and as a verb rate).
The letter 'E' is the most common (or highest frequency) letter in the English language (starting off the typographer's phrase ETAOIN SHRDLU) and many other related languages, which has implications in both cryptography and data compression. This makes it a difficult and popular letter to use when writing lipograms. Ernest Vincent Wright's Gadsby (1939), is considered a "dreadful" novel, and that "at least part of Wright's narrative difficulties were caused by language restrictions imposed by the lack of E."[7] Both Georges Perec's novel A Void (La Disparition) (1969) and its English translation by Gilbert Adair omit the letter E and are considered better works.[8]
## Codes for computing
Alternative representations of E
NATO phonetic Morse code Echo ·
In Unicode the capital E is codepoint U+0045 and the lower case e is U+0065.
The ASCII code for capital E is 69 and for lower case e is 101; or in binary 01000101 and 01100101, respectively.
The EBCDIC code for capital E is 197 and for lowercase e is 133.
The numeric character references in HTML and XML are "E" and "e" for upper and lower case, respectively.
In British Sign Language (BSL), the letter 'e' is represented as extended index of right hand touching the tip of index on the left hand. All fingers of left hand should be open.
See E (disambiguation) for uses of the letter E
Similar Latin letters:
• Ɛɛ : Latin epsilon
Similar non-Latin letters:
Similar phonetic symbols:
Special symbols similar to the letter E:
## References
1. ^ "E" Merriam-Webster's Third New International Dictionary of the English Language, Unabridged (1993). Ees is the plural of the name of the letter; the plural of the letter itself is E's, Es, e's, or es.
2. ^ a b c d e f g Kelk, Brian. "Letter frequencies". UK Free Software Network. Retrieved on 2008-06-25.
3. ^ Lewand, Robert. "Relative Frequencies of Letters in General English Plain text". Cryptographical Mathematics. Central College. Retrieved on 2008-06-25.
4. ^ "Frequency of Occurrence of Letters in French". Santa Cruz Public Libraries. Retrieved on 2008-06-25.
5. ^ "Frequency of Occurrence of Letters in German". Santa Cruz Public Libraries. Retrieved on 2008-06-25.
6. ^ "Frequency of Occurrence of Letters in Spanish". Santa Cruz Public Libraries. Retrieved on 2008-06-25.
7. ^ Ross Eckler, Making the Alphabet Dance: Recreational Word Play. New York: St. Martin's Press (1996): 3
8. ^ Eckler (1996): 3. Perec's novel "was so well written that at least some reviewers never realized the existence of a letter constraint."
The Basic modern Latin alphabet
Aa Bb Cc Dd Ee Ff Gg Hh Ii Jj Kk Ll Mm Nn Oo Pp Qq Rr Ss Tt Uu Vv Ww Xx Yy Zz
Letter E with diacritics
ÉéÈèĔĕÊêẾếỀềỄễỂểĚěËëẼẽĖėȨȩḜḝĘęĒēḖḗḔḕẺẻȄȅȆȇẸẹỆệḘḙḚḛɆɇ
Two-letter combinations
Ea Eb Ec Ed Ee Ef Eg Eh Ei Ej Ek El Em En Eo Ep Eq Er Es Et Eu Ev Ew Ex Ey Ez
EA EB EC ED EE EF EG EH EI EJ EK EL EM EN EO EP EQ ER ES ET EU EV EW EX EY EZ
Letter-digit & Digit-letter combinations
E0 E1 E2 E3 E4 E5 E6 E7 E8 E9 0E 1E 2E 3E 4E 5E 6E 7E 8E 9E
`
# Wiktionary
Up to date as of January 15, 2010
## Translingual
The Latin script
Aa Bb Cc Dd Ee Ff Gg Hh Ii Jj Kk Ll Mm Nn Oo Pp Qq Rr Ss Tt Uu Vv Ww Xx Yy Zz
Variations of letter E
Ēē Ềề Ễễ Ểể Ḝḝ Ḗḗ Ḕḕ Ȇȇ Ệệ
Letters using macron sign or underline sign
Ēē Ḡḡ Īī Ōō Ūū Ȳȳ
### Letter
Ē upper case (lower case ē)
1. The letter E with a macron.
# Simple English
The Latin alphabet
Aa Bb Cc Dd
Ee Ff Gg Hh Ii Jj
Kk Ll Mm Nn Oo Pp
Qq Rr Ss Tt Uu Vv
Ww Xx Yy Zz
For the drug sometimes referred to E, see Ecstasy.
E is the fifth (number 5) letter in the English alphabet. | 2019-06-26 03:42:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 3, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6823029518127441, "perplexity": 12387.122023481832}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560628000164.31/warc/CC-MAIN-20190626033520-20190626055520-00313.warc.gz"} |
https://webwork.maa.org/moodle/mod/forum/discuss.php?d=4939&parent=14986 | ## WeBWorK Problems
### Re: MathObjects Methods
by Danny Glin -
Number of replies: 0
The arrow is used to call a method associated with an object (named by a variable), so when you type "$len->length" you are trying to call the method "length" on the object$len. Since the variable $len is undefined you get the error you quoted. What you want to do is call the "length" method on$v, i.e. $v->length evaluates to 3 in your example. If you are trying to store that in$len, then what you want is
$len =$v->length;
In terms of how to search for examples using such things, there isn't really any mechanism to do this from within WeBWorK. If you have command line access to your WeBWorK server you could grep (text search) the OPL for a particular string:
cd /opt/webwork/libraries/webwork-open-problem-library/OpenProblemLibrary
grep -r "\->length" *
(You need the backslash in front of the - because - is a special character in bash.)
In many cases you can just search for a word, but because "length" is a common term the results you're looking for will probably get lost in all the other hits.
If you don't have command-line access to your WeBWorK server (or for convenience), you can download a copy of the OPL from Github onto your local machine, and then search it from there. grep is built-in on Macs and Linux. There is probably a Windows/Powershell equivalent, but I don't know it. | 2022-05-28 20:52:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22468313574790955, "perplexity": 2901.981587919128}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663019783.90/warc/CC-MAIN-20220528185151-20220528215151-00761.warc.gz"} |
https://stats.stackexchange.com/questions/97220/calendar-visualization | # Calendar Visualization
I have a two column table consisting of: date, user_id.
I'd like to visualize a grid with date along the x axis, and user_id along the y.
If date, user_id exists, I'd like to show a grid with a color, say green.
If date, user_id does not exist, I'd like to show a grid in white.
Bonus points if date, user_id shows up multiple points resulting in a darker green color.
What tool, preferably free and online, is best suited to visualize these data in the manner I've described? Follow up question: which other tools are good at visualizing time series data of user events as a calendar? Something like: https://developers.google.com/chart/interactive/docs/gallery/calendar
Two ways to do this are either using a heatmap or the calendar chart/calendar heatmap from the Google Charts API. I am sure that there are other ways to visualize data like this. Note that you will probably need to do a preliminary aggregation to the date, user_id level using mean.
## Heatmap:
This works particularly well when you have multiple variables.
library(gtools)
library(ClassDiscovery)
library(devtools)
#==========================================================
# 1. multiple variables; heatmap
#==========================================================
# generate sequence of dates
vDates = seq.Date(from = as.Date('29-11-2012',
format = '%d-%m-%Y'),
length.out = 203, by = 'day')
# generate the random samples
dfHeatMap = as.matrix(rdirichlet(length(vDates), runif(15)))
row.names(dfHeatMap) = as.character(vDates)
# adjust column labels for neater plotting
vDatesNew = rep(as.Date(NA), length(vDates))
vDatesNew[seq(from = 1, to = 203, by = 10)] =
vDates[seq(from = 1, to = 203, by = 10)]
# adjust row labels for neater plotting
labRow = c(NA, NA, 3, NA, NA, 6, NA, NA, 9,
NA, NA, 12, NA, NA, 15)
# draw the heatmap with aspect control
png('heatmap.png', height = 900, width = 1200, pointsize = 16)
aspectHeatmap(t(dfHeatMap), Rowv = NA, Colv = NA,
col = cm.colors(256), labCol = vDatesNew, labRow = labRow,
margins = c(5, 5), hExp = 1.5, wExp = 4, cex.lab = 2)
dev.off()
## Calendar heatmap:
This uses the Google Charts API for calendar charts. I took some of the configurations for this from here. An almost identical static version of this can be found here.
#==========================================================
# NOTE: this handles only one variable
#==========================================================
dfHeatMap2 = data.frame(heatValue = runif(length(vDates)),
dates = vDates)
print(
gvisCalendar(data=dfHeatMap2, datevar="dates", numvar="heatValue",
options=list(
title="Randomly generated proportion variable",
calendar="{cellSize:10,
yearLabel:{fontSize:20, color:'#444444'},
focusedCellColor:{stroke:'red'}}",
width=1600, height=1200),
), file = 'something.html', tag = 'chart') | 2020-10-23 22:11:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20422105491161346, "perplexity": 7125.852958145133}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107865665.7/warc/CC-MAIN-20201023204939-20201023234939-00084.warc.gz"} |
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# If the volume of a Box is 1,463,000 cubic millimetres, what is the vol
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If the volume of a Box is 1,463,000 cubic millimetres, what is the vol [#permalink]
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23 Jul 2016, 12:27
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If the volume of a Box is 1,463,000 cubic millimetres, what is the volume of the box in cubic meters? (1 millimeter = 0,001 meter)
A) 14,63
B) 1,463
C) 0,1463
D) 0,01463
E) 0,001463
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Re: If the volume of a Box is 1,463,000 cubic millimetres, what is the vol [#permalink]
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23 Jul 2016, 12:32
Avigano wrote:
If the volume of a Box is 1,463,000 cubic millimetres, what is the volume of the box in cubic meters? (1 millimeter = 0,001 meter)
A) 14,63
B) 1,463
C) 0,1463
D) 0,01463
E) 0,001463
Check other Conversion problems to practice from our Special Questions Directory.
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If the volume of a Box is 1,463,000 cubic millimetres, what is the vol [#permalink]
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23 Jul 2016, 15:28
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$$Well,\; square\; meters\; or\; cubic\; meters\; are\; simply\; m*m,$$
$$so\; our\; nice\; "a\; lot\; of\; millions"mm^{3},\; means\; x_{mm}\cdot y_{mm}\cdot z_{mm}\; and\; as\; each\; of\; these\; mm\; has\; a\; conversion\; of\; \frac{1}{1000},$$
$$the\; whole\; pack\; will\; need\; a\; reduction\; of\; 10^{-9}\; ,\; 9\; zeros,\; to\; become\; m^{3}.$$
$$Answer\; \mbox{E}.$$
$$Hope\; it\; helps.\;$$ $$:)$$
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Re: If the volume of a Box is 1,463,000 cubic millimetres, what is the vol [#permalink]
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24 Oct 2016, 16:06
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The trick here maybe to just drop the ugly numbers and transform 1,463,000 cubic mm into 1,000,000 cubic mm just to understand the logic behind.
If volume of a cube equals $$10^6$$ cubic mm and the formula is side * side * side, we can infer that the side measures $$10^2$$ mm, or 100 mm.
And 100 mm = 0.1 mt. Therefore the side measures 0.1 mt.
In volume, this would be represented as 0.1 mt * 0.1 mt * 0.1 mt, which yields 0.001 cubic mt.
The answer choice should have 2 zeroes between the point and the first decimal different to zero.
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Re: If the volume of a Box is 1,463,000 cubic millimetres, what is the vol [#permalink]
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25 Oct 2016, 16:04
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Avigano wrote:
If the volume of a Box is 1,463,000 cubic millimetres, what is the volume of the box in cubic meters? (1 millimeter = 0.001 meter)
A) 14.63
B) 1.463
C) 0.1463
D) 0.01463
E) 0.001463
Since we are converting from cubic millimeters to cubic meters, we have to adjust the conversion.
(1 millimeter)^3 = (0.001 meter)^3
(1 millimeter)^3 = 0.000000001 meter^3
Thus, 1,463,000 cubic millimetres =
0.000000001 x 1,463,000 = 0.001463 cubic meters
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Re: If the volume of a Box is 1,463,000 cubic millimetres, what is the vol [#permalink]
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Re: If the volume of a Box is 1,463,000 cubic millimetres, what is the vol [#permalink] 03 Nov 2017, 15:01
Display posts from previous: Sort by | 2017-12-14 00:46:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4477790296077728, "perplexity": 13502.755048393967}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948532873.43/warc/CC-MAIN-20171214000736-20171214020736-00328.warc.gz"} |
https://themathhelp.com/threads/exponential-equations-3.2834/ | # PracticeExponential Equations 3
#### MarkFL
##### La Villa Strangiato
Staff member
Moderator
Math Helper
Observe that:
$$\displaystyle 10=(5+2\sqrt{6})^1+(5-2\sqrt{6})^1$$
Take it away!!!
pre-trip_rapture
#### pre-trip_rapture
##### Member
Observe that:
$$\displaystyle 10=(5+2\sqrt{6})^1+(5-2\sqrt{6})^1$$
Take it away!!!
How did you get a power of 1?
#### MarkFL
##### La Villa Strangiato
Staff member
Moderator
Math Helper
Because a power of 1 makes the equation true, when we observe that:
$$\displaystyle 5+a+5-a=10$$
#### MarkFL
##### La Villa Strangiato
Staff member
Moderator
Math Helper
And so we write:
$$\displaystyle x^2-5x+5=1$$
$$\displaystyle x^2-5x+4=0$$
$$\displaystyle (x-4)(x-1)=0$$
$$\displaystyle x\in\{1,4\}$$
#### pre-trip_rapture
##### Member
Observe that:
$$\displaystyle 10=(5+2\sqrt{6})^1+(5-2\sqrt{6})^1$$
Take it away!!!
#### MarkFL
##### La Villa Strangiato
Staff member
Moderator
Math Helper
It appears we missed a real solution. Let's look at:
$$\displaystyle \frac{1}{5\pm2\sqrt{6}}\cdot\frac{5\mp2\sqrt{6}}{5\mp2\sqrt{6}}=5\mp2\sqrt{6}$$
So, we see the two bases are each others' reciprocals. Let's begin by writing:
$$\displaystyle (5+2\sqrt{6})^u+(5-2\sqrt{6})^u=10$$
Using our discovery above, we can now write:
$$\displaystyle (5+2\sqrt{6})^u+(5+2\sqrt{6})^{-u}=10$$
If we multiply through by $$v=(5+2\sqrt{6})^u$$ then we get (after arranging in standard form):
$$\displaystyle v^2-10v+1=0$$
$$\displaystyle v=5\pm2\sqrt{6}$$
This means:
$$\displaystyle (5+2\sqrt{6})^u=5\pm2\sqrt{6}$$
And using our initial result above, this means:
$$\displaystyle u=x^2-5x+5=\pm1$$
We've previously solved the positive case, so now the negative:
$$\displaystyle x^2-5x+5=-1$$
$$\displaystyle x^2-5x+6=0$$
$$\displaystyle (x-2)(x-3)=0$$
And so the complete set of real solutions is:
$$\displaystyle x\in\{1,2,3,4\}$$
pre-trip_rapture
#### pre-trip_rapture
##### Member
It appears we missed a real solution. Let's look at:
$$\displaystyle \frac{1}{5\pm2\sqrt{6}}\cdot\frac{5\mp2\sqrt{6}}{5\mp2\sqrt{6}}=5\mp2\sqrt{6}$$
So, we see the two bases are each others' reciprocals. Let's begin by writing:
$$\displaystyle (5+2\sqrt{6})^u+(5-2\sqrt{6})^u=10$$
Using our discovery above, we can now write:
$$\displaystyle (5+2\sqrt{6})^u+(5+2\sqrt{6})^{-u}=10$$
If we multiply through by $$v=(5+2\sqrt{6})^u$$ then we get (after arranging in standard form):
$$\displaystyle v^2-10v+1=0$$
$$\displaystyle v=5\pm2\sqrt{6}$$
This means:
$$\displaystyle (5+2\sqrt{6})^u=5\pm2\sqrt{6}$$
And using our initial result above, this means:
$$\displaystyle u=x^2-5x+5=\pm1$$
We've previously solved the positive case, so now the negative:
$$\displaystyle x^2-5x+5=-1$$
$$\displaystyle x^2-5x+6=0$$
$$\displaystyle (x-2)(x-3)=0$$
And so the complete set of real solutions is:
$$\displaystyle x\in\{1,2,3,4\}$$
Nicely done as always. Are you saying that my simply picture answer is not even close to proving that the LHS = RHS?
#### MarkFL
##### La Villa Strangiato
Staff member
Moderator
Math Helper
When the exponent is 1, the equation is true, but we missed the fact that when the exponent is -1, it is also true. Both values of the exponent lead to 2 values of $$x$$ which make the equation true.
pre-trip_rapture
#### pre-trip_rapture
##### Member
When the exponent is 1, the equation is true, but we missed the fact that when the exponent is -1, it is also true. Both values of the exponent lead to 2 values of $$x$$ which make the equation true.
Staff member | 2020-02-29 03:13:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9560558199882507, "perplexity": 3096.1199370497416}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875148375.36/warc/CC-MAIN-20200229022458-20200229052458-00327.warc.gz"} |
https://answers.ros.org/question/259194/convert-commands-from-prismatic-joint-to-revolute-for-followjointtrajectory-controller-robot-arm-gripper/?sort=oldest | # Convert commands from prismatic joint to revolute for FollowJointTrajectory controller, robot arm gripper
I have a robotic arm (Cyton Gamma 300) with a 2-finger gripper, one joint mimics another (see the picture below).
My goal is to control and execute trajectories on the gripper using MoveIt.
The gripper-related URDF/xacro chunk is as follows (the complete xacro description of the robot is here):
<joint name="gripper_finger1_joint" type="prismatic">
<origin xyz="${gripper_finger1_joint_init_xyz}" rpy="${gripper_finger1_joint_init_rpy}" />
<limit lower="-0.008" upper="0.008" effort="100.0" velocity="1.0" />
<axis xyz="1 0 0" />
</joint>
<!-- Link Description omitted for brevity -->
<joint name="gripper_finger2_joint" type="prismatic">
<!-- Joint Description omitted for brevity -->
<mimic joint="gripper_finger1_joint" multiplier="-1"/>
</joint>
<!-- Link Description omitted for brevity -->
As it can be seen from the description, (i) the joints are defined as prismatic, and as it can be seen from the photo, (ii) the gripper of the real robot is controlled by one 'dynamixel' motor, which is revolute. For the prismatic joint the displacement is specified in meters, whether for the dynamixel joint the rotation angle is defined in Radians.
My goal is to use MoveIt, both with Gazebo and the real robot and be able to pick/place objects with the arm, leveraging available MoveIt capabilities. I am able to plan & execute trajectories for the manipulator (i.e. shoulder/wrist/elbow, 7 dynamixel motors in total), and bring the arm to the target position in space.
However, I cannot execute a trajectory for the gripper properly, since MoveIt reads URDF description and generates a trajectory w.r.t to limits in the URDF, (i.e. it provides displacements in meters, not angle in Radians).
To execute a trajectory in MoveIt I define FollowJointTrajectory controller as follows (complete configuration can be found here):
controller_manager_ns: controller_manager
controller_list:
- name: cyton_gamma_300/arm_controller
action_ns: follow_joint_trajectory
type: FollowJointTrajectory
default: true
joints:
- shoulder_roll_joint
- shoulder_pitch_joint
- elbow_roll_joint
- elbow_pitch_joint
- elbow_yaw_joint
- wrist_pitch_joint
- wrist_roll_joint
- name: cyton_gamma_300/gripper_controller
action_ns: follow_joint_trajectory
type: FollowJointTrajectory
default: true
joints:
- gripper_finger1_joint
For instance, If I want to plan the following trajectory (gripper should come from red to yellow):
I get the following on my /cyton_gamma_300/gripper_controller/follow_joint_trajectory/goal (which is exactly a displacement):
goal:
trajectory:
seq: 0
stamp:
secs: 0
nsecs: 0
frame_id: /world
joint_names: ['gripper_finger1_joint']
points:
-
positions: [0.00961187223614116]
velocities: [0.0]
accelerations: [-0.9530453139061077]
effort: []
time_from_start:
secs: 0
nsecs: 0
-
positions: [0.008]
velocities: [-0.042613731475556624]
accelerations: [-0.9775129354039888]
effort: []
time_from_start:
secs: 0
nsecs: 58159883
-
positions: [0.007838502345021854]
velocities: [-0.058384310708380155]
accelerations: [-0.6298771853005602]
effort: []
time_from_start:
secs: 0
nsecs: 60967905
-
#and so on... the rest is omitted for brevity
Although the trajectory is fine for Gazebo, and will be executed as expected, for the real robot it is a problem since these command would be interpreted as rotation angle, which is not correct.
Question: how to map commands from FollowJointTrajectory to another type (i.e. from prismatic to revolute), and map the results back (i.e. feedback). Is there a standard way of approaching this problem?
P.S. code is here. Thanks in advance!
edit retag close merge delete
Did you manage to figure it out? If yes,please update ,I am facing a similar issue where the robotic arm is placed on a linear actuator.
( 2017-08-09 05:40:55 -0600 )edit
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If I got it correctly, I wouldn't bother with that and declared a pseudo-prizmatic joint for the gripper. Then you can transform linear coordinates of this pseudo-motor to angular for the real robot, using something like
more
I am not sure what are you referring to with a 'pseudo-prismatic' joint. I cannot find this type of joint in the documentation. My question is about finding a standard way of mapping one FollowJointTrajectory controller to another and remapping back results
( 2017-04-15 09:47:21 -0600 )edit | 2021-12-04 01:00:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3324072062969208, "perplexity": 6939.620899322853}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362923.11/warc/CC-MAIN-20211204003045-20211204033045-00396.warc.gz"} |
https://www.math.ubc.ca/~pwalls/math-python/ | Mathematical Python
Mathematical Python is an introduction to mathematical computing including:
• Jupyter notebooks, markdown and $\LaTeX$
• Basic Python programming: datatypes, logic, loops and functions
• Scientific computing with NumPy, SciPy and Matplotlib
• Applications in calculus, linear algebra and differential equations
Notebooks
Mathematical Python is a collection of Jupyter notebooks and are available at:
If you have a UBC CWL:
Prerequisites
• Differential calculus: derivatives of elementary functions, Taylor series and optimization
• Integral calculus: Riemann sums, sequences and series
• Linear algebra: vector and matrix operations, systems of equations, eigenvalues and eigenvectors
• Differential equations: Euler's method for first order equations, linear systems of ODEs
Author
Patrick Walls is an instructor in the Department of Mathematics at the University of British Columbia and teaches mathematical computing, differential equations and vector calculus for mechanical engineering.
Acknowledgements
Thank you ...
Wed 4 Dec 2019 17:20:04 PST | 2021-01-26 18:04:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40645793080329895, "perplexity": 2645.6127702928547}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704803308.89/warc/CC-MAIN-20210126170854-20210126200854-00352.warc.gz"} |
https://webwork.maa.org/moodle/mod/forum/discuss.php?d=3447 | ## WeBWorK Problems
### Empty Set as Student Answer
by Paul Seeburger -
Number of replies: 3
I am trying to edit a set intersection/union problem to allow students to enter the empty set "{ }" as the answer when the intersection is empty. The problem currently asks for "None" to be entered, which I do not find helpful for teaching students the notation of an empty set.
I used the following code to get set up this answer, which displays the correct answer correctly, but when you enter the correct answer as "{}" you are told this is incorrect and "Empty Parentheses" is the feedback.
Context("Interval");
$ans_a = Set(); ANS($ans_a->cmp());
Thanks for any help!
Paul
In reply to Paul Seeburger
### Re: Empty Set as Student Answer
by Davide Cervone -
The following
Context("Interval");
$S = Set(); Context()->texStrings; BEGIN_TEXT $$S$$ = \{$S->ans_rule\}
END_TEXT
Context()->normalStrings;
ANS($S->cmp); correctly accepts {} for me. Can you give the complete problem you are working on? Also, what version of pg and webwork are you using? In reply to Davide Cervone ### Re: Empty Set as Student Answer by Paul Seeburger - Here's the whole problem, Davide. I tried to implement your notation as closely as I could, still with the same results. The empty set '{}' is still not accepted as the correct answer. I'm using ww_version: 2.8, pg_version: 2.8.1. ##DESCRIPTION ## Compound Inequalities ## ##ENDDESCRIPTION # Original Problem Author: Modified by Shafiu Jibrin # from setSets/ur_st_1_2.pg # Location: Northern Arizona University ## DBsubject(Set theory) ## DBchapter(Basic properties and operations) ## DBsection(Union and intersection) ## Institution(The College of Idaho) ## Author(RA Cruz) ## Level(2) ## TitleText1('Essentials of Intermediate Algebra') ## AuthorText1('Blitzer') ## EditionText1('1') ## Section1('4.1') ## Problem1('') ## KEYWORDS('inequalities') ## Date: 2007/10 DOCUMENT(); # This should be the first executable line in the problem. loadMacros( "PGstandard.pl", "PGchoicemacros.pl", "MathObjects.pl", # "PGgraphmacros.pl", # "PGnauGraphics.pl", # "contextInequalitiesAllowStrings.pl", "answerHints.pl", # "CofIdaho_macros.pl" ); TEXT(beginproblem()); ###################################### # Setup @slice = NchooseK(12,9); @A = ($slice[1], $slice[2],$slice[3], $slice[4]); @B = ($slice[5], $slice[6],$slice[8]);
$AiB = "N"; @AuB = ($slice[1], $slice[2],$slice[3], $slice[4],$slice[5], $slice[6],$slice[8]);
for ($k=3;$k>0; $k-=1) { for ($i=0; $i<$k; $i+=1){ if($A[$i]>$A[$k]) {$b = $A[$i];
$A[$i] = $A[$k];
$A[$k] = $b; } } } for ($k=2; $k>0;$k-=1) {
for ($i=0;$i<$k;$i+=1){
if($B[$i]>$B[$k]) {
$b =$B[$i];$B[$i] =$B[$k];$B[$k] =$b;
}
}
}
#for ($k=2;$k>0; $k-=1) { # for ($i=0; $i<$k; $i+=1){ # if($AiB[$i]>$AiB[$k]) { #$b = $AiB[$i];
# $AiB[$i] = $AiB[$k];
# $AiB[$k] = $b; # } # } #} for ($k=6; $k>0;$k-=1) {
for ($i=0;$i<$k;$i+=1){
if($AuB[$i]>$AuB[$k]) {
$b =$AuB[$i];$AuB[$i] =$AuB[$k];$AuB[$k] =$b;
}
}
}
$LEFT_BRACE = '\{';$RIGHT_BRACE = '\}';
Context("Interval");
$ans_a = Set(); ###################################### # Main text Context()->texStrings; BEGIN_TEXT Let $$A= {LEFT_BRACE} A[0], A[1], A[2], A[3] {RIGHT_BRACE}$$ ,$SPACE
$$B= {LEFT_BRACE} B[0], B[1], B[2] {RIGHT_BRACE}$$
$BR Find the following sets in list form. Separate elements with commas. If there are no elements in the set, enter "NONE".$PAR
a) $$A \cap B =$$ \{$ans_a->ans_rule(25) \}$PAR
b) $$A \cup B =$$ \{ans_rule(25)\}
END_TEXT
Context()->normalStrings;
######################################
ANS($ans_a->cmp); Context()->parens->replace('{' => {close => '}', type => 'Set'}); #$ans_a = String("NONE");
#ANS($ans_a->cmp->withPostFilter(AnswerHints( # sub { # my ($correct,$student,$ans) = @_;
# if ($student=~ /\w/) {return$student !~ /[}{]/;}
# } => ["Enter your answer with set notation: { ... }",
# checkCorrect => 1,
# score => 0]
#)));
$ans_b = Set("{$AuB[0],$AuB[1],$AuB[2],$AuB[3],$AuB[4],$AuB[5],$AuB[6]}");
#Answer hints not working
ANS($ans_b->cmp->withPostFilter(AnswerHints( sub { my ($correct,$student,$ans) = @_;
return $student !~ /[}{]/; } => ["Enter your answer with set notation: { ... }", checkCorrect => 1, score => 0] )));$showPartialCorrectAnswers = 1;
######################################
COMMENT('MathObject version');
ENDDOCUMENT();
In reply to Paul Seeburger
### Re: Empty Set as Student Answer
by Davide Cervone -
OK, you left out of your original message the crucial information that you had redefined the open brace using
Context()->parens->replace('{' => {close => '}', type => 'Set'});
which is the source of the problem. You want
Context()->parens->replace('{' => {close => '}', type => 'Set', emptyOK=>1});
to allow empty braces to be allowed.
I'm not sure why you are redefining the braces, ass this new definition (with emptyOK should be exactly what is already in the Interval context, so I'd recommend removing the line entirely. If you way what you are trying to accomplish, I may be able to suggest an alternative. | 2021-10-22 15:28:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7521296143531799, "perplexity": 14288.090972119311}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585516.51/warc/CC-MAIN-20211022145907-20211022175907-00391.warc.gz"} |
https://codereview.stackexchange.com/questions/265811/piping-a-stdtuple-into-a-function | # piping a std::tuple into a function
This is a mythical beast we discussed in a previous question:
template <typename F, class Tuple>
constexpr void operator|(Tuple&& t, F f) noexcept(noexcept(f(std::get<0>(t))))
{
[&]<auto ...I>(std::index_sequence<I...>) noexcept(noexcept(f(std::get<0>(t))))
{
(f(std::get<I>(t)), ...);
}
(std::make_index_sequence<std::tuple_size_v<std::remove_reference_t<Tuple>>>());
}
Usage:
std::forward_as_tuple(1, 2, 3) | [](auto&& v) { std::cout << v << std::endl; };
You could pipe all sorts of things into a function, not just a tuple. Arrays, containers, variants, ..., even structs.
I don't see a great need for this in normal (clear, obvious) code; I believe that most C++ users would expect an ordinary function (in the pattern of std::for_each(), perhaps), rather than overloading the | operator. In the future, | become recognised as a composition operator, but that is not the case in 2021, and so it will be harder for readers to comprehend than the plain function. This is clearly a matter of opinion where reasonable folk may differ, of course, and will likely change as time passes.
I suppose it's arguably similar to | std::views::transform() - except that it can only be a sink of values, because it returns void instead of a tuple of the results. If f() returns void, that may be appropriate, but when f() is a value-returning function, we'll want access to those results (perhaps to pipe into another transform).
Why are we not perfect-forwarding the tuple elements? I'm not convinced that this will work with functions that accept (mutable) references and update the values.
No unit tests are presented. I would expect quite a large suite of tests for a template function such as this, and they should have been in the review request. I certainly wouldn't accept this code into a project unless the tests were in the same submission.
• we could accommodate all of these concerns, I suppose, by optionally returning another tuple. But this is not a serious idea. Aug 7 at 7:55
• Yes, that's what I would recommend - but only when f() applied to all the input types is non-void (remember it can have many overloads). It's probably simpler to just convert the tuple to a std::array<std::variant> and then use a transform using std::visit. Aug 7 at 8:02
• a return value could also indicate, that a conditional pipe is desired, i.e. stop piping when the return value is true or false. Aug 7 at 8:27
• But when you say piping std::tuples is unclear, I don't see any clear/standard ways of doing this. Aug 7 at 8:39
• I believe that most C++ users would expect an ordinary function (in the pattern of std::for_each(), perhaps), rather than overloading the | operator. As I said in the other question, | may in future become recognised as a composition operator, but it is not so in 2021. That's clearly a matter of opinion where reasonable folk may differ, so I don't see any value in arguing the point! Aug 7 at 9:44
# Incorrect noexcept clause
In the noexcept clause, you are only checking if the function applied to the first element of the tuple is noexcept. But overloads that handle the other elements might not be. So we need to check that they are all noexcept. The OP's code in the previous question handled this correctly.
# Use concepts
The problem with your operator|() is that it matches a lot of things that it shouldn't. It would be better to make it match only std::tuples and other things that you can use std::get<>() for the first argument, and function-like things that accept the types in the tuple as the second argument. | 2021-10-27 04:42:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4003469944000244, "perplexity": 1120.5120562623733}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323588053.38/warc/CC-MAIN-20211027022823-20211027052823-00475.warc.gz"} |
https://spie.org/Publications/Proceedings/Paper/10.1117/12.782875 | Share Email Print
### Proceedings Paper
A quasi-two level analytic model for end pumped alkali metal vapor laser
Author(s): G. Hager; J. McIver; D. Hostutler; G. Pitz; G. Perram
Format Member Price Non-Member Price
PDF \$17.00 \$21.00
Paper Abstract
In this paper we describe a quasi-two level analytic model for end pumped Alkali metal vapor lasers. The model is developed by considering the steady state rate equations for the number densities of the, 2S1/2, 2P3/2, and 2P1/2, energy states for the three level laser system. The approximation is then made that the relaxation between the two upper levels, 2P3/2 and 2P1/2, caused by collisions with additive ethane is much faster, in fact infinitely fast, by comparison with any other process in the system including stimulated emission. With this assumption the ratio of the number densities for the upper two levels, 2P3/2 and 2P1/2, is given by its statistical equilibrium value and the mathematical description becomes that of a quasi-two level system from which an analytic solution can be extracted. The analytic model description gives expressions for the threshold pump power and the slope efficiency including intra-cavity losses. Applications of the model and comparisons with the steady state three level model developed by Beach et al. will be presented.
Paper Details
Date Published: 12 May 2008
PDF: 9 pages
Proc. SPIE 7005, High-Power Laser Ablation VII, 700528 (12 May 2008); doi: 10.1117/12.782875
Show Author Affiliations
G. Hager, Air Force Institute of Technology (United States)
J. McIver, Air Force Institute of Technology (United States)
D. Hostutler, Air Force Institute of Technology (United States)
G. Pitz, Air Force Institute of Technology (United States)
G. Perram, Air Force Institute of Technology (United States)
Published in SPIE Proceedings Vol. 7005:
High-Power Laser Ablation VII
Claude R. Phipps, Editor(s) | 2020-04-02 20:21:58 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8031877875328064, "perplexity": 6828.3349189460405}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370507738.45/warc/CC-MAIN-20200402173940-20200402203940-00333.warc.gz"} |
https://nana-music.com/songs/3676 | # 歌詞Where Are You NowJustin Bieber
Bieber Boyd Moore Thomas Wesley Pentz Karl Rubin Brutus Jordan Ware
Bieber Boyd Moore Thomas Wesley Pentz Karl Rubin Brutus Jordan Ware
I need you (the) I need you I need you (the) I need you I need you, you, you, you, you, you I need you (the) I need you I need you (the) I need you I need you, you, you, you, you, you You, you, you I need you the most I gave you the key when the door wasn't open, just admit it See, I gave you faith, turned your doubt into hoping, can't deny it Now I'm all alone and my joys turned to moping Tell me, where are you now that I need you? Where are you now? Where are you now that I need you? Couldn't find you anywhere When you broke down I didn't leave you I was by your side So where are you now that I need you? Where are you now that I need you? Where are you now that I need you? Where are you now that I need you? Where are you now that I need you? I gave you attention when nobody else was paying I gave you the shirt off my back, what you saying? To keep you warm I showed you the game everybody else was playing, that's for sure And I was on my knees when nobody else was praying, oh Lord Where are you now that I need you? Where are you now that I need you? I need you (the) I need you I need you (the) I need you I need you, you, you, you, you, you Where are you now that I need you? I need you (the) I need you I need you (the) I need you I need you, you, you, you, you, you I need you the most Where are you now that I need you? Where are you now that I need you? Where are you now that I need you? I need you the most (I need you the most, I need you the most) | 2020-02-20 04:07:03 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8990495800971985, "perplexity": 3004.690577214182}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875144637.88/warc/CC-MAIN-20200220035657-20200220065657-00545.warc.gz"} |
https://snystrom.github.io/memes-manual/reference/plot_ame_heatmap.html | Plot AME heatmap clustered by similarity in detected motifs
plot_ame_heatmap(
ame,
id = motif_id,
group = NULL,
group_name = NULL,
scale_max = NA
)
## Arguments
ame
ame results data.frame
id
column of motif ids to use (default: motif_id).
group
grouping column if comparing across multiple ame runs (optional, default: NULL).
value
value to display as heatmap intensity. Default: -log10(adj.pvalue). Takes function or column name as input. If set to "normalize", will use normalized rank within group as the heatmap values. **If in doubt**, prefer the -log10(adj.pvalue) plot potentially in conjunction with adjusting scale_max. (See "Normalized rank visualization" section below for more details on how to interpret these data)
group_name
when group = NULL, name to use for input regions. Ignored if group is set.
scale_max
max heatmap value to limit upper-value of scale. Useful if distribution of values vary greatly between groups. Usually a better to tweak this option than to use value = "normalize". The cumulative distribution plot generated by ame_compare_heatmap_methods() can be useful for selecting this value, try to pick a value which captures the largest fraction of hits across all groups while excluding outliers.
## Value
ggplot object
## Details
Normalized rank visualization **NOTE:** The normalized rank visualization eliminates all real values related to statistical significance! Instead, this visualization represents the relative ranks of hits within an AME run, which already pass a significance threshold set during runAME(). This means that even if several motifs have similar or even identical pvalues, their heatmap representation will be a different color value based on their ranked order in the results list. This also means that using the normalized rank visualization will be misleading if there are only a few AME hits; it is only worth using if the number of hits is very large (>100). Both visualizations can be useful and reveal different properties of the data to the user during data exploration. Use ame_compare_heatmap_methods() to help assess differences in the two visualizations. **If in doubt**, prefer the -log10(adj.pvalue) representation.
Common mistake: if value is set to a string that is not "normalize", it will return: "Error: Discrete value supplied to continuous scale". To use a column by name, do not quote the column name.
## Examples
data("example_ame", package = "memes")
# Plot a single category heatmap
plot_ame_heatmap(example_ame\$Decreasing)
# Plot a multi category heatmap
grouped_ame <- dplyr::bind_rows(example_ame, .id = "category")
plot_ame_heatmap(grouped_ame, group = category) | 2022-01-17 15:59:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3161904513835907, "perplexity": 5211.1837135619}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300574.19/warc/CC-MAIN-20220117151834-20220117181834-00441.warc.gz"} |
https://ltwork.net/what-can-you-say-about-silk-fabric-of-the-3-countries-mentioned--727 | # What can you say about silk fabric of the 3 countries mentioned above (Thailand, Cambodia, laos)? how are they alike and different
###### Question:
what can you say about silk fabric of the 3 countries mentioned above (Thailand, Cambodia, laos)? how are they alike and different from other? What country influenced them the most in term of waving silk.
### What ideology in american society is related to liberty for alexis de tocqueville?
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### Use the following scenario to answer questions 6-12 Suppose a company has 5,000 employees, and 41% of
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### Which scenario best describes stiffness?A. Scratching the substance with a penny or paper clipB. Being able to roll an
Which scenario best describes stiffness? A. Scratching the substance with a penny or paper clip B. Being able to roll an object into a sheet C. Not being able to transfer heat D. Being able to distort the shape of an object by bending it... | 2022-10-05 09:12:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3537634313106537, "perplexity": 2193.5110544029017}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337595.1/warc/CC-MAIN-20221005073953-20221005103953-00713.warc.gz"} |
https://mathematica.stackexchange.com/questions/87520/using-mapthread-over-2-lists-different-length-with-criterion-on-minimal-differen | # Using MapThread over 2 lists different length with criterion on minimal differences
Say I have 2 lists
list1 = {-1179.8, -1139.3, -1118.3, -1115.6, -1095.2, -1075.1, -1054.7, 228.1, 249.1, 269.6, 290.6, 292.3, 312.7, 313.2, 333.8, 336.1, 354.2, 377.5, 1087.7, 1106.6, 1151.9}
list2 = {-1104.4, -1071.4, -1071.4, -1067.4, 263.6, 300.6, 311.6, 311.6, 312.6, 1067.6}
and I want to map them 1-1 on one another such that each element from list2 has its partner from list1 and such a partner is always different (no element of list1 is assigned twice). The mapping is defined by minimal difference in absolute value. The remaining elements from list1 will be mapped on 0.
But the function Nearest keeps duplicates.
f[x_, y_] := {x, y}
Partition[Flatten[MapThread[f, {Nearest[list1, list2], list2}]],2]
The desired output for this example is
{{-1179.8, 0}, {-1139.3, 0}, {-1118.3, 0}, {-1115.6, -1104.4},
{-1095.2, -1071.4}, {-1075.1, -1071.4}, {-1054.7, -1067.4}, {228.1, 0},
{249.1, 0}, {269.6, 263.6}, {290.6, 300.6}, {292.3, 311.6},
{312.7, 311.6}, {313.2, 311.6}, {333.8, 312.6}, {336.1, 0},
{354.2, 0}, {377.5, 0 }, {1087.7, 1067.6}, {1106.6, 0}, {1151.9, 0}}
• What's the desired output? :) – Öskå Jul 4 '15 at 9:22
• Why don't you PadRight your list2 and thread over it ? – Sektor Jul 4 '15 at 9:52
• Sorry, it is here. {{-1179.8,0},{-1139.3,0},{-1118.3,0},{-1115.6,-1104.4},{-1095.2,-1071.4},{-1075.1,-1071.4},{-1054.7,-1067.4},{228.1,0},{249.1,0},{269.6,263.6},{290.6,300.6},{292.3,311.6},{312.7,311.6},{313.2,311.6},{333.8,312.6},{336.1,0},{354.2,0},{377.5,0},{1087.7,1067.6},{1106.6,0},{1151.9,0}} – kveta Jul 4 '15 at 10:08
• Can you explain why the first element is {-1179.8,0} and not {-1179.8,-1104.4}? Is it that (1) for every x in list2, there is some element in list1 that is closer to x than -1179.8, and (2) you don't allow duplicates? – march Jul 4 '15 at 13:43
• To march: Absolutely as you wrote. Value -1104.4 is closer to -1115.6 than to -1179.8 – kveta Jul 4 '15 at 14:22
## 2 Answers
I'm not sure I correctly interpreted what you really are looking for, because you mention MapThread that works on elements on the same position in the two lists, whereas in the description you say that for each element of list2 you want the closest from list1. Moreover, this means that once you selected an element from list1, it cannot be considered for further elements in list2, so the final result you get depends on the order on which you work on elements in list2. However, here is a possible solution.
CompareLists[list1_List, list2_List] := Module[{temp, selection},
temp = list1;
Map[
(selection = First@Nearest[temp, #];
temp = DeleteCases[temp, selection];
{selection, #}) &, PadRight[list2, Length[list1]]]]
It's not so elegant but I think it does what you need. Here is the result for the list1 and list2 you provided
CompareLists[list1, list2]
(* {{-1095.2, -1104.4}, {-1075.1, -1071.4}, {-1054.7, -1071.4}, \
{-1115.6, -1067.4}, {269.6, 263.6}, {292.3, 300.6}, {312.7,
311.6}, {313.2, 311.6}, {333.8, 312.6}, {1087.7, 1067.6}, {228.1,
0}, {249.1, 0}, {290.6, 0}, {336.1, 0}, {354.2, 0}, {377.5,
0}, {1106.6, 0}, {-1118.3, 0}, {-1139.3, 0}, {1151.9, 0}, {-1179.8,
0}} *)
• Thanks, it is almost fine. Can be your code expanded to cover minimal value check of the sum of differences instead of just minimal distance to the first assigned element? – kveta Jul 5 '15 at 8:26
• I do not understand exactly your question, however, instead of Nearest (the first row of Map, where "selection" is calculated, you can consider any other function that takes two arguments: the current element of list2 and the whole list1. That function has to return a value taken from the list1. – bobknight Jul 5 '15 at 17:32
Once you select the nearest element for list2[[1]] then other elements from list2 are practically directly assigned. There is one element from list2 not well assigned and it is -1067.4. Its partner -1115.6 is further than -1054.7. When you first reverse the order of initial lists, the desired solution is obtained. This minimize the Total value of differences in the resulting list of 2-tuples.
• This is more of a comment rather than an answer. Consider expanding on it if you wish to really answer the question – Sektor Jul 4 '15 at 21:30
• Right, I would like to. But I need 50 reputations to comment. – bosona Jul 4 '15 at 21:34
• Without checking your assertions, it sounds to me that you're close to an answer. Care to flesh it out, including code so that others can try out your idea? (P.S. You need to put @user for "user" to be notified of your response. Authors of posts are always notified of comments to their posts.) – Michael E2 Jul 4 '15 at 22:17 | 2019-05-24 12:12:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3684595823287964, "perplexity": 4197.5722974387245}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257605.76/warc/CC-MAIN-20190524104501-20190524130501-00081.warc.gz"} |
https://boolesrings.org/jvanname/2017/03/16/ternary-laver-table-calculator/ | # Ternary Laver table calculator (now optimized for efficiency)
At this link, you can find an easy-to-use ternary Laver table calculator which I have just programmed which returns specific information about the output of a ternary Laver table operation. The calculator works for very specific types of ternary Laver tables. Furthermore, at this link, you can find another ternary Laver table calculator that will return the entire (sometimes very large) output of a certain ternary Laver table operation.
The ternary Laver tables are much different than the classical and multigenic Laver tables (I used to call the multigenic Laver tables “generalized Laver tables”) computationally in the following ways:
1. Unlike the classical Laver tables and multigenic Laver tables which are locally finite, the endomorphic Laver tables are infinite.
2. The output of a ternary Laver table operation can grow exponentially with respect to the size of the input.
3. While the fundamental operation on the multigenic Laver tables $(A^{\leq 2^{n}})^{+}$ can be completely described by the final matrix, there does not seem to be any final matrix for the ternary Laver tables. Each ternary Laver table seems to offer unlimited combinatorial complexity.
4. The ternary Laver tables are much more abundant than the multigenic Laver tables. We have very general methods of constructing many ternary Laver tables.
5. While the classical Laver tables and multigenic Laver tables are not suitable platforms for public key cryptosystems, it seems like the ternary Laver tables could be platforms for public key cryptosystems.
And I will probably post the version of the paper on Generalizations of Laver tables (135 pages with proofs and 86 pages without proofs) without proofs in a couple of days. Let me know if the calculator is easy to use or not.
As with the classical and multigenic Laver tables, the ternary Laver tables also produce vivid images. I will post these images soon. | 2017-12-14 20:59:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7032055258750916, "perplexity": 1107.5750422162312}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948550986.47/warc/CC-MAIN-20171214202730-20171214222730-00129.warc.gz"} |
https://mathoverflow.net/questions/124969/an-equalizer-in-commutative-algebras | # An equalizer in commutative algebras
I'm feeling quite a bit embarrassed to ask such a basic thing, but I can't seem to figure it out. Let $R$ be a commutative ring and $A$ be a commutative $R$-algebra. Is the fork $$A \xrightarrow{i} A \otimes_R A \rightrightarrows A \otimes_R A$$ an equalizer, where $i(a) = a \otimes 1$ and the two parallel arrows are given by $f(a \otimes b) = ab \otimes 1$ and $g = id$?
Yes, it is. Let a tensor $t$ be in the equalizer of $f$ and $g$. Then, $f\left(t\right)=g\left(t\right)$. If we write $t$ in the form $\sum\limits_{j\in I} a_j\otimes b_j$ (with $I$ being a finite set, and $a_j$ and $b_j$ being elements of $A$), then this rewrites as $\sum\limits_{j\in I} a_jb_j\otimes 1 = \sum\limits_{j\in I} a_j\otimes b_j$. Thus, $t = \sum\limits_{j\in I} a_j \otimes b_j = \sum\limits_{j\in I} a_j b_j \otimes 1 = i\left(\sum\limits_{j\in I} a_j b_j\right) \in i\left(A\right)$. Thus, the equalizer is contained in $i\left(A\right)$. The reverse inclusion is even more trivial, so the equalizer is actually equal to $i\left(A\right)$. | 2019-08-23 14:58:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9641129970550537, "perplexity": 63.826220266987235}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027318421.65/warc/CC-MAIN-20190823130046-20190823152046-00473.warc.gz"} |
https://socratic.org/questions/58f63a9b7c0149311733c446#408876 | # Question #3c446
Apr 18, 2017
Density
#### Explanation:
The density of mercury $\left(13593 \frac{k g}{m} ^ 3\right)$ is much more than that of ethanol$\left(789 \frac{k g}{m} ^ 3\right)$ and water $\left(1000 \frac{k g}{m} ^ 3\right)$. What this means is that height of mercury required to produce a given pressure is much less than the height of a column of water or ethanol.
Let's do a simple calculation to illustrate the point.
The pressure a manometer normally measures is of the order of atmospheres. $1 a t m o s p h e r e = {10}^{5} P a$ in SI units.
Now, $\mathrm{dg} h = P$ $\implies h = \frac{P}{\mathrm{dg}}$ where
$P$ = pressure
$d$ = density of material used
$g$ = acceleration due to gravity. Let us take it as $10 \frac{m}{s} ^ 2$ for simplicity.
$h$ = height of column
$\therefore$ height of mercury column required to produce/measure a pressure of $1 a t m = {10}^{5} P a$ is:-
$\frac{100000}{13593 \cdot 10} \approx 0.735 m = \textcolor{red}{73.5 c m}$
Similarly height of water column for $1 a t m = {10}^{5} P a$ is:-
$\frac{100000}{1000 \cdot 10} = 10 m = \textcolor{red}{1000 c m}$
and for ethanol is:-
$\frac{100000}{789 \cdot 10} \approx 12.67 m = \textcolor{red}{1267 c m}$
Now you can easily see that the height of mercury column required i.e. $73.5 c m$ is much more feasible to use than heights of $1000 c m$ or $1267 c m$. | 2022-06-30 18:45:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 20, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8392063975334167, "perplexity": 576.2036914229643}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103877410.46/warc/CC-MAIN-20220630183616-20220630213616-00248.warc.gz"} |
https://formulas.tutorvista.com/physics/strain-formula.html | Top
# Strain Formula
The effect of stress is what we call as strain. The strain is the measure of how much deformation has occurred in the body compared to its original shape due to the action of force. It is denoted by $\epsilon$.
Strain Formula is given by
Where,
x = Change in dimension,
L = Original dimension.
There are three types of strain
Longitudinal strain is the ratio of change in length to the original length.
Where, $\Delta$ $l$ = Change in length,
$l$ = Original length.
Shearing strain is the ratio of change in angle to which it is turned to its distance from fixed layer.
Volumetric strain is the ratio of change in volume to the original volume.
Where $\Delta$ V = Change in Volume,
V = Original volume.
Related Calculators Calculate Strain Acceleration Formula Calculator Area of a Circle Formula Calculator Area of a Cylinder Formula Calculator
## Strain Problems
Below are problems based on strain formula which may be helpful to you.
### Solved Examples
Question 1: A rubber band of length 5cm is stretched such that its length increases by 2mm. Calculate the strain.
Solution:
Given: Change in length x = 2mm,
Original length L = 5 cm
Strain is given by $\epsilon$ = $\frac{x}{L}$
= $\frac{2 \times 10^{-3} m}{5 \times 10^{-2}}$
= 4 $\times$ 10-2.
Question 2: A iron bar of 3m long is heated. It stretches by 0.5 mm. Calculate the strain.
Solution:
Given: Change in length x = 0.5 mm,
Original length L = 3m.
Strain is given by $\epsilon$ = $\frac{x}{L}$
= $\frac{5 \times 10^{-3}}{3}$
= 1.67 $\times$ 10-3.
More topics in Strain Formula Young's Modulus Formula Strain Energy Formula
*AP and SAT are registered trademarks of the College Board. | 2019-05-26 20:22:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6174492835998535, "perplexity": 1351.2949117432004}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232259452.84/warc/CC-MAIN-20190526185417-20190526211417-00479.warc.gz"} |
https://www.shaalaa.com/question-bank-solutions/ranjeetha-walks-around-the-circular-park-in-15-minutes-if-she-walks-at-the-rate-of-5-km-hr-how-much-distance-would-she-have-to-travel-at-the-minimum-to-reach-the-centre-of-the-park-from-any-point-time-distance-and-speed-entrance-exam_100698 | # Ranjeetha Walks Around the Circular Park in 15 Minutes If She Walks at the Rate of 5 Km/Hr, How Much Distance Would She Have to Travel, at the Minimum, to Reach the Centre of the Park from Any Point - Mathematics
#### Question
MCQ
Ranjeetha walks around the circular park in 15 minutes. If she walks at the rate of 5 km/hr, how much distance would she have to travel, at the minimum, to reach the centre of the park from any point on its perimeter?
• 100 meter
• 200 meter
• 250 meter
• 300 meter
#### Solution
200 meter
Explanation:
Rajneetha covers around park 2pr in 15 minutes at speed 5 km/hr
Distance covered = S × T
=5xx15/60xx1000 m
= 1250 m
2πr = 1250 m
r=(1250xx7)/(2xx22)= 198.86 ≈ 200 m approx
Concept: Time, Distance and Speed(Entrance Exam)
Is there an error in this question or solution? | 2021-01-19 08:49:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.31448894739151, "perplexity": 2457.4101639482283}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703518201.29/warc/CC-MAIN-20210119072933-20210119102933-00584.warc.gz"} |
https://www.vedantu.com/maths/parallelogram-law | # Parallelogram Law
## What is Parallelogram?
A Parallelogram is a four-sided quadrilateral whose opposite sides are parallel and congruent to each other. Opposite angles of a parallelogram are equal. The parallelogram and a rectangle are near about the same with one distinguishing property that the rectangle has all the angles of 900 and that of parallelogram does not.
In Mathematics, the parallelogram law belongs to elementary Geometry. This law is also known as parallelogram identity. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. In this article, let us study the definition of a parallelogram law, proof, and parallelogram law of vectors in detail.
Parallelogram law of addition states that the sum of the squares of the length of the four sides of a parallelogram equals the sum of the squares of the length of the two diagonals. In Euclidean geometry, it is a must that the parallelogram should have equal opposite sides.
If ABCD is a parallelogram, then AD = BC and AB = DC. Then according to the definition of the parallelogram law, it is stated as
2(AB)2 + 2(BC)2 = (AC)2 + (BD)2.
If a parallelogram is a rectangle, then the law is stated as
2(AB)2 + 2(BC)2 = 2(AC)2
Because in a rectangle, two diagonals are of equal lengths. i.e., (AC=BD)
### Parallelogram Law of vectors
If two vectors say vector p and vector q are acting simultaneously at a point, then it can be represented both in magnitude and direction by the adjacent sides drawn from a point. Therefore, the resultant vector is completely represented both in direction and magnitude by the diagonal of the parallelogram passing through the point.
Consider the above figure,
The vector P represents the side OA and vector Q represents the side OB, respectively.
According to the parallelogram law, the side OC diagonal of the parallelogram represents the resultant vector R.
Vector OA + Vector OB = Vector OC
$\overrightarrow{P}$ + $\overrightarrow{Q}$ = $\overrightarrow{R}$
### Parallelogram Law of Addition of Vectors Procedure
Following are steps for the parallelogram law of addition of vectors are:
• Draw a vector using a suitable scale in the direction of the vector
• Draw the second vector using the same scale from the tail of the first vector
• Treat these vectors as the adjacent sides and complete the parallelogram
• Now, the diagonal represents the resultant vector in both magnitude and direction
### Parallelogram Law Proof
Step 1: Let AD=BC = p, AB = DC = q, and ∠ BAD = α
Step 2: Using the law of cosines in the BAD, we get
p2+ q2 – 2pqcos(α) = BD2 ——-(1)
Step 3: We know that in a parallelogram, the adjacent angles are supplementary so it sums up 1800. So
Step 4: Now, again use the law of cosines in the ADC
p2 + q2 – 2pqcos(180 – α) = AC2 ——-(2)
Step 5: Apply trigonometric identity cos(180 – x) = – cos x in step (2)
p2 + q2 + 2pqcos(α) = AC2
Step 6: Now, take the sum of the squares of the diagonals adding equation 1 and 2
BD2 + AC2 =p2 + q2 – 2pqcos(α) + p2 + q2 + 2pqcos(α)
BD2 + AC2 =2p2 + 2q2 ——-(3)
BD2 + AC2 = 2(AB)2 + 2( BC)2
Hence, the parallelogram law is proved.
1. What are Vectors?
A Physical quantity that has both magnitude and a direction is said to be a vector. In a line, the length of a line is a magnitude and the arrow is its direction. The start point is its tail and the endpoint is its head.
An increase and decrease in temperature is the best example of a vector, it has both magnitude and direction. There are ten types of vectors and they are
1. Zero vector
2. Unit Vector
3. Position Vector
4. Co-initial Vector
5. Like and Unlike Vectors
6. Co-planar Vector
7. Collinear Vector
8. Equal Vector
9. Displacement Vector
10. Negative of a Vector
2. What is The Triangle Laws Of Vector Addition
Suppose, we have two vectors A and B as shown.
To add this two vectors simply place the head of one vector over the tail of the other vector as shown.
Now join the other end points of both the vectors together as shown,
The result of the addition of given vectors is given by vector C which represents the sum of vectors A and B.
i.e. C = A + B
Vector addition is commutative in nature i.e. C = A + B = B + A
Similarly if we have to subtract both the vectors using the triangle law then we simply reverse the direction of any vector and add it to another one as shown. | 2020-09-27 20:37:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7863139510154724, "perplexity": 560.3408477350168}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600401578485.67/warc/CC-MAIN-20200927183616-20200927213616-00656.warc.gz"} |
https://core-econ.org/the-economy/book/text/leibniz-03-05-01.html | Leibniz
3.5.1 Optimal allocation of free time: MRT meets MRS
Alexei wants to get as high an exam grade as possible whilst sacrificing the least possible amount of free time. We have seen diagrammatically that he maximizes his utility by choosing the point where an indifference curve is tangential to the feasible frontier, at which the marginal rate of substitution (MRS) is equal to the marginal rate of transformation (MRT). In this Leibniz we show how to formulate Alexei’s decision mathematically as a constrained optimization problem, and solve it to find the optimal combination of grade and free time.
Alexei’s optimal choice of free time and exam grade is illustrated in Figure 1 below. It combines his feasible set and indifference curves. The optimum is achieved at the point E on the feasible frontier, where the frontier has the same slope as the indifference curve.
Alexei’s optimal choice of free time and exam grade.
Figure 1 Alexei’s optimal choice of free time and exam grade.
Alexei’s utility function is $U(t,\ y)$: utility depends positively on hours of free time t and the exam grade y. He wishes to maximize his utility given the constraint imposed by his feasible set of grades and free time. As in Leibniz 3.4.1, if the production function is $y=f(h)$, where h is hours of study, the equation of the feasible frontier is:
Thus Alexei’s problem is to choose t and y to maximize $U(t,\ y)$ subject to the constraint $y=f(24 - t)$.
constrained optimization problem
Problems in which a decision-maker chooses the values of one or more variables to achieve an objective (such as maximizing profit) subject to a constraint that determines the feasible set (such as the demand curve).
This is an example of what is known in mathematics as a problem of constrained optimization. Sometimes in this sort of problem the constraint is written as an inequality: $y \leq f(24 - t)$, which can be interpreted as saying that his choice must lie in the feasible set. But because his utility depends positively on t and y, we know that he will want to choose a point on the frontier. So we can write the constraint as an equation, which makes the problem easier to solve mathematically.
One way to solve Alexei’s problem is to use the constraint to substitute for y in terms of t in the utility function. Then utility is expressed as a function of the single variable t:
which may be maximized with respect to t by equating its derivative to zero. This derivative is the total derivative of utility with respect to t, which may be calculated in the usual way via the chain rule:
The term $\dfrac{dy}{dt}$ on the right-hand side is calculated by differentiating the production function $y= f(24- t)$:
by the composite function rule, so:
This equation says that as we move along the feasible frontier in the direction of increasing t, the net effect on utility is the result of the direct effect of more free time, which is of course positive, together with the negative indirect effect of a lower exam grade.
At the point that solves Alexei’s maximization problem, $\dfrac{dU}{dt}=0$. So at this point:
This has an obvious interpretation in terms of the two effects on utility mentioned in the preceding paragraph: at the optimum point, the positive effect of a little more free time and the negative effect of a slightly lower exam grade balance each other out.
Rearranging the last equation, we see that:
at the optimum point. The left-hand side is the absolute value of the slope of the feasible frontier, which we called the marginal rate of transformation (MRT) in Leibniz 3.4.1, and as we saw in Leibniz 3.2.1, the right-hand side is the absolute value of the slope of the indifference curve, which we called the marginal rate of substitution (MRS). Thus, at the optimum point, the slopes are equal as in Figure 1. In other words,
This is known as a first-order condition for optimization, since it was obtained by equating a first derivative (in this case the total derivative $\frac{dU}{dt}$) to zero. Because much of economics concerns constrained optimization, you will find similar conditions in later Leibniz supplements.
Remember that we want to find the values of t and y that maximize Alexei’s utility. So far we have shown that the values of t and y we are looking for must satisfy the first-order condition. To solve the problem fully and find the optimal values, we need to note that they must also lie on the feasible frontier. So we have a pair of simultaneous equations:
which must be satisfied by t and y. In the next section, we will derive these equations for particular utility and production functions, and solve them to find the optimal values of t and y.
Read more: Sections 8.1 to 8.3 for maximization, and Section 14.2 for the distinction between total and partial derivative, of Malcolm Pemberton and Nicholas Rau. 2015. Mathematics for economists: An introductory textbook, 4th ed. Manchester: Manchester University Press
Optimal allocation of free time: an example
We now illustrate the principles of the previous section with specific production and utility functions.
budget constraint
An equation that represents all combinations of goods and services that one could acquire that exactly exhaust one’s budgetary resources.
The constrained optimization problem has two parts: the objective function, which describes the utility Alexei wants to maximize, and the constraint, which in this case is Alexei’s production function for exam grades.
We assume as in Leibniz 3.2.1 that Alexei has a Cobb-Douglas utility function:
where a and b are positive constants. (We use a and b rather than $\alpha$ and $\beta$ because $\alpha$ will be used in the production function.)
As in Leibniz 3.1.1, we assume that the relationship describing how Alexei turns hours of study h into an exam grade y is
where A and $\alpha$ are positive constants and $\alpha \lt 1$. Also, as before, we may write this production function in terms of hours of free time t, since $h = 24 -t$. Doing this, and assuming for simplicity that $A=1$, we see that:
This is the equation of the feasible frontier.
Alexei’s problem is to choose t and y to maximize $t^a y^b$, subject to the constraint:
The previous section gives us two ways of solving this problem: we may either use the substitution method or apply the formula. We will demonstrate both, and confirm that they give us the same answer.
Applying the formula
By ‘the formula’ we mean the first-order condition $\text{MRT} = \text{MRS}$. We know that the solution to the problem must satisfy this condition, so we calculate the MRT from the feasible frontier, and the MRS from the utility function, and equate the two.
As we saw in Leibniz 3.4.1, the MRT is the absolute value of the slope of the feasible frontier. Using the equation above for the frontier:
Also, we showed in Leibniz 3.2.1 that the MRS is the ratio of the marginal utilities. The marginal utilities are found by differentiating the utility function:
So the MRS is given by:
Equating MRT to MRS and multiplying through by $\frac{t}{y}$,
Solving this equation for t, we see that $t = \frac{24}{(1+c)}$, where $c=\frac{\alpha b}{a}$. Substituting this into the production function we obtain the full solution to Alexei’s problem:
These are the values of t and y that give Alexei the highest utility he can achieve within the feasible set.
Using the substitution method
The method consists of substituting the constraint into the objective function to make it a function of just one variable and maximizing that function. Substituting the constraint $y = \left(24- t \right)^\alpha$ into the utility function gives us an expression for utility as a function of t alone:
To maximize U, we calculate $dU/dt$ and equate it to zero. Using the product rule,
As in the general analysis of the previous section, this expresses the net effect on utility of an increase in t as the result of a positive direct effect and the negative effect of a lower exam grade.
Setting $dU/dt$ equal to zero and dividing the resulting equation by $t^{a-1}(24 - t )^{\alpha b}$, we see that $a = \frac{\alpha b t}{24 – t}$. Rearranging,
This is the same equation for t that we obtained using $\text{MRT} = \text{MRS}$, and the rest of the solution is as before.
Read more: Sections 8.1 to 8.3 of Malcolm Pemberton and Nicholas Rau. 2015. Mathematics for economists: An introductory textbook, 4th ed. Manchester: Manchester University Press. | 2019-11-22 21:34:17 | {"extraction_info": {"found_math": true, "script_math_tex": 27, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8542765974998474, "perplexity": 385.3499706837086}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496671548.98/warc/CC-MAIN-20191122194802-20191122223802-00099.warc.gz"} |
http://blog.zarang.com/ | ## Automatic Tagging
September 23rd, 2013
Challenge
Your challenge, should you choose to accept it, is to implement an efficient and effective supervised algorithm that allows short text documents (comprising of a title heading and a paragraph or two of body text) to be automatically tagged. That is, devise a method that assigns one or more tags $\{T_1,T_2,...T_M\}$ to each of the given documents, $\{D_1, D_2, ...., D_N\}$.
To make the challenge more interesting the training set is pretty large.
• $N$ ~ 6 million,
• $M$ ~ 40,000.
• Total of 17M tags.
• Each question has 1-5 tags, typically 2-3.
• Most common tag is used ~400k times.
## Startling word discovery
May 25th, 2012
My eight year-old daughter came home the other day and wrote the word 'startling' on a piece of paper and put it in front of my face. I have come to learn that this is her way of saying,
"Daddy, let's play this word game that I learnt at school!
Oh, and by the way, let's skip the boring part where I tell you the rules.
Instead let's just play and when you make a mistake I'll tell you."
I'll be a bit kinder and explains the rules at the start: You have to pick a letter so that the remaining letters still form a word. (So in the first round, I could have picked the letter 't', or 'l', which would result in the words 'starling' or 'starting', respectively.)
You win the game if you correctly identify a sequence of letters to eliminate at each state, so that your last valid word is only a single letter long. For example, here is one way you can produce a winning sequence:
startling, starting, staring, string, sting, sing, sin, in, I.
She then said that her teacher had told the class that this was the longest word where you can go all the way down to 1-letter. Being as curious as I am, I thought I would try to verify this. Furthermore, I was intrigued as to what other words had this characterisitic. | 2015-02-01 05:32:44 | {"extraction_info": {"found_math": true, "script_math_tex": 4, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 4, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7497854232788086, "perplexity": 1083.7616050579722}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422122127848.98/warc/CC-MAIN-20150124175527-00183-ip-10-180-212-252.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/141119/how-strong-are-large-cardinal-properties-of-ord | # How strong are large cardinal properties of Ord?
Ordinal numbers are generalizations of natural numbers. In this sense the "proper class" of all ordinals ($Ord$) is very similar to "infinite" set of all natural numbers ($\omega$). In the other direction we know that many large cardinal axioms are generalizations of the properties of $\omega$ and without assumption of uncountablity one can prove (in $ZFC$) that $\omega$ is a large cardinal in many types. For example $\omega$ is a strongly compact cardinal because $\mathcal{L}_{\omega , \omega}$ is a compact logic. This fact shows that the nature of $\omega$ is very adequate to be a large cardinal. Now a natural question arises for the $Ord$ like this:
Question (1): Is the nature of $Ord$ adequate to be a large cardinal? More precisely if $A$ be a large cardinal type, then how strong is the statement: "$Ord$ is a large cardinal of type $A$"?
It seems that because of the similarity between $\omega$ and $Ord$ these statements must be very close to a provable statement in $ZFC$ or be a "very weak large cardinal axiom". For example it is provable in $ZFC$ that "$Ord$ is strongly inaccessible" and the statement "$Ord$ is Mahlo" is weaker than existence of an "uncountable" Mahlo cardinal (see the Cantor's upper attic for more details). Even there are some problems during investigating large cardinal properties of $Ord$, for example in defining some types of large cardinality for $Ord$ particularly in large cardinals based on elementary embedding definitions. So:
Question (2): Is there an equivalent definition for any large cardinal axiom $A$ which the statement "$Ord$ is a large cardinal of type $A$" be meaningful?
Question (3): Is there a large cardinal axiom like $A$ such that: $ZFC\vdash \neg~(Ord~is~a~large~cardinal~of~type~A)$
-
Here is one way to formalize your concept a little more tightly, which provides the answers to your questions. For any large cardinal property $P$, let's take the phrase "$\Ord$ is $P$" to be the theory asserting $\sigma$, for any sentence $\sigma$ that ZFC proves is true in $V_\delta$ under the assumption that $\delta$ has property $P$ in $V$.
With this formalization, "$\Ord$ is $P$" asserts that the universe is just like we would expect, if we were living inside $V_\delta$ for an actual $P$ cardinal $\delta$. For example, "$\Ord$ is measurable" implies that there are a proper class of weakly compact cardinals, since this is true in $V_\delta$, whenever $\delta$ is measurable, and "$\Ord$ is supercompact" implies that there are many partially supercompact cardinals, with nice limit properties. For example, they would form a stationary class in the sense that every definable class club would contain one of them. This notion seems to capture what one would want to mean by saying $\Ord$ has property $P$ as a purely first-order theory about sets.
With this idea, the point I would like to make is that assuming $\Ord$ is $P$ is essentially equivalent to assuming that what you have is $V_\delta$, where $\delta$ has property $P$ in a larger universe.
Theorem. For any large cardinal property $P$, a model of set theory $M$ satisfies "$\Ord$ is $P$" if and only if $M\prec V_\delta^N$ for some taller model of set theory $N$ with a cardinal $\delta$ having property $P$ in $N$.
Proof. The backward direction is immediate, since $\delta$ having property $P$ implies that $V_\delta$ satisfies every assertion of $\Ord$ is $P$. For the forward direction, suppose $M$ satisfies $\Ord$ is $P$. Let $T$ be the theory consisting of $\ZFC$, plus the assertion "$\delta$ is $P$", using a new constant symbol $\delta$, plus the assertions $\varphi^{V_\delta}$, for any $\varphi$ in the elementary diagram of $M$, using constants for elements of $M$. This theory is finitely consistent, since otherwise there would be finitely many assertions in it that are contradictory, and so there would be a statement $\varphi$ true in $M$ that provably could not hold in $V_\delta$ for any cardinal $\delta$ with property $P$. But that would contradict our assumption that $M\models\Ord$ is $P$.
If $N$ is any model of the theory, then $\delta$ has property $P$ in $N$, and we get $M\prec V_\delta^N$, because $V_\delta^N$ satisfies the elementary diagram of $M$. Another way to say this is that there is an elementary embedding $j:M\to V_\delta^N$, mapping every element of $M$ to the interpretation of its constant in $N$. QED
Thus, if one is inclined to assume $\Ord$ is $P$, then why not go ahead and make the full move to a model with an actual $P$ cardinal $\delta$, such that our old world looks exactly like $V_\delta$ in this new world. In particular, under this terminology, the theory $\ZFC+\Ord$ is $P$ is equiconsistent with $\ZFC+\exists \delta$ with property $P$.
Corollary. The following theories are equiconsistent:
1. $\ZFC+\Ord$ is $P$.
2. $\ZFC+\exists \kappa$ with property $P$.
Lastly, I would like to point out that there is some variance in the literature about what "$\Ord$ is $P$" should mean. For example, one often finds the phrase "$\Ord$ is Mahlo" to mean only the weaker assertion, that every definable closed unbounded class of cardinals contains a regular cardinal. This is what one finds, for example, at Cantor's Attic. But this is strictly weaker in consistency strength than ZFC+$\exists\kappa$ Mahlo, since this latter theory implies the consistency of the former, as it is true in $V_\kappa$ whenever $\kappa$ is Mahlo.
-
This answer, however, does not address the part of your question about properties of Ord generalizing properties of $\omega$, which is an interesting topic about which people have said quite a lot. Perhaps someone will post an answer along those lines. – Joel David Hamkins Sep 3 '13 at 15:37
Something doesn't sit right with this corollary. In every model of $\sf ZFC$ we have that $\sf Ord$ is strongly inaccessible. – Asaf Karagila Sep 3 '13 at 16:24
I don't agree that it is right to say that ZFC proves that Ord is inaccessible, but rather only that it is "definably inaccessible" or "definably regular", in the sense that it proves only that no definable class is a singularizing function. This is strictly weaker than actual regularity, as witnessed by the singular worldly cardinals. ZFC proves really only that "Ord is worldly". – Joel David Hamkins Sep 3 '13 at 18:15
My view is that the slogan "ZFC proves that Ord is inaccessible" is just a metaphor, given to understand what we are trying to get at with the replacement axiom. A more careful view leads to the definition I give here, and with this view we shouldn't really say that ZFC proves Ord is inaccessible. – Joel David Hamkins Sep 3 '13 at 19:02
@Goldstern: Some of my best friends are not well-founded! :-) – Asaf Karagila Oct 17 '14 at 2:03 | 2015-11-27 23:11:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9260942339897156, "perplexity": 194.9288979286706}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398450659.10/warc/CC-MAIN-20151124205410-00291-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://analog-electronics.eu/Structured-Electronic-Design/Errata_1.html | # Errata edition 1¶
1. Page 34, Table 2.1 the word ‘power’ in the table cells needs to be replaced with ‘V/I’
2. Page 39: Equation 2.15 should be:
$A_{i}=\frac{I_{\ell }}{I_{s}}=\frac{1}{A\frac{Z_{\ell }}{Z_{s}}+B\frac{1}{Z_{s}}+CZ_{\ell }+D}.$
3. Page 44: Caption of Figure 2.22 should be:
Network models of amplifiers that have at least one nonzero transmission-1 matrix parameter.
1. Left: High-level model for amplifiers having $$A\neq 0$$.
1. Right: Voltage-controlled voltage source (VCVS): $$B=C=D=0$$.
2. Left: High-level model for amplifiers having $$B\neq 0$$.
1. Right: Voltage-controlled current source (VCCS): $$A=C=D=0$$.
2. Left: High-level model for amplifiers having $$C\neq 0$$.
1. Right: Current-controlled voltage source (CCVS): $$A=B=D=0$$.
2. Left: High-level model for amplifiers having $$D\neq 0$$.
1. Right: Current-controlled current source (CCCS): $$A=B=C=0$$.
4. Page 141: Equation 4.176 should be:
$\frac{g_m}{I_{DS}}=\frac{1}{n\,U_T}$
5. Chapter 5:
The CS stage operating point and device characteristics should be simulated using the library file CMOS18-0.lib. The figures in the book are generated with this library.
6. Page 163:
1. Text in example 5.1 below th listing of the circuit file should be:
$$V_{GS_{Q}}=616.34139176mV$$
2. Netlist file at the bottom of the page should be CSbiased0_9V-10uA.cir.
7. Page 166:
1. Netlist file at the top of the page should be CSbiased0_9V-10uAViIo.cir.
2. Netlist file at the bottom of the page should be CSbiased0_9V-10uAViVo.cir.
8. Page 342: Equation 10.31 should be:
$A_v=\frac{V_{\ell}}{V_s}=\frac{R_a+R_b}{R_b}$
9. Page 388: Typo in sidenote 8:
‘Rourh’ should be ‘Routh’
10. Page 425: Error in references to figures: text below Figure 12.26 should be:
1. … (see Figure 12.27B)
2. … (see Figure 12.27A)
11. Page 427: References to example 7.3 should be replaced with references to example 11.3.
12. Page 431: Reference to example 7.3 should be replaced with reference to example 11.3.
13. Page 437: Reference to example 8.10 should be replaced with reference to example 12.10.
14. Page 439: Reference to example 8.8 should be replaced with reference to example 12.8.
15. Page 440: Reference to example 8.8 should be replaced with reference to example 12.8.
16. Page 443: Caption of Figure 12.50 should be:
Small-signal equivalent circuit of a current-driven basic amplification stage driving an RC load.
17. Page 444:
1. Reference to example 7.3 should be replaced with reference to example 11.3.
2. Text below Figure 12.51: If the poles are well separated, etc, should be:
If the poles are well separated ($$C_3 \gg C_2$$) and ($$R_2 \gg R_3$$), their frequencies can be estimated as discussed in section 18.5.3: $$p_2\approx -\frac{1}{(R_2+R_3)C_3}$$ and $$p_3\approx -\frac{1}{R_3C_2}$$.
18. Page 445: Numeric error in Example 12.15:
$$R_z=6.046$$ k $$\Omega$$
This should also be modified in the circuit file. It results in different frequencies of the poles and zeros listed on page 446. The new dominant pole is created at -0.89Hz.
19. Page 446: Reference to example 8.15 should be replaced with reference to example 12.15.
20. Page 447: Reference to example 8.15 should be replaced with reference to example 12.15.
21. Page 448:
1. Reference to example 8.15 should be replaced with reference to example 12.15.
2. Expression in Figure 12.55 should be: $$\omega_z=\frac{1}{R_bC}$$
22. Page 454: Reference to example 4, and reference to example 7.4 should be replaced with reference to example 11.4. | 2019-05-27 01:29:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8967028260231018, "perplexity": 2726.4133248870094}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232260358.69/warc/CC-MAIN-20190527005538-20190527031538-00445.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/37112-diagonal-matrix.html | 2. Remember that when you diagonalize a matrix B, the diagonal values of the resulting diagonalized matrix will be the eigenvalues of B, and that a matrix P such that $P^{-1}BP$ is diagonal will have as it's columns the eigenvectors corresponding to those eigenvalues. Thus, we require that B have real eigenvectors. As our matrix is real, real eigenvalues are a necessary condition for real eigenvectors.
Note that $\det(I\lambda-B)=0$ gives $(\lambda-a)^2+b^2=0$. | 2013-06-19 19:44:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8981813788414001, "perplexity": 155.20564235650045}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368709101476/warc/CC-MAIN-20130516125821-00020-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://api.queryxchange.com/q/21_2361404/is-every-bounded-linear-functional-continous-and-how-does-this-affect-the-definition-of-the-weak-topology/ | # Is every bounded linear functional continous, and how does this affect the definition of the weak topology?
by eurocoder Last Updated July 17, 2017 10:20 AM
I know that a linear operator is bounded if and only if it is continuous. But what about a bounded linear functional? Is this just a special case of a linear operator, and hence it is also bounded if and only if it is continuous?
But if that is the case, it would seem to make the definition of the weak topology redundant:
The weak topology on $X$ is the weakest topology $U\subset2^X$ with respect to which every bounded linear functional $\Lambda:X\to \mathbb{R}$ is continuous.
If every bounded linear functional is continuous due to being bounded, then why would continuity even be required in the definition of the weak topology?
Tags : | 2017-09-22 06:25:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8829940557479858, "perplexity": 75.05487855966734}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818688671.43/warc/CC-MAIN-20170922055805-20170922075805-00472.warc.gz"} |
https://googology.wikia.org/wiki/Alternating_factorial | 10,965 Pages
The alternating factorial of a number $$n$$ is $$\sum^n_{m = 1} (-1)^{n - m} \cdot m!$$, or the alternating sum of all the factorials up to $$n$$. For example, the alternating factorial of 5 is $$1! - 2! + 3! - 4! + 5!=101$$.[1]
It was Miodrag Živković who proved in 1999 that there are only a finite number of primes that can be expressed as the alternating factorial of a number $$n$$. In particular, the prime 3,612,703 divides all sufficiently large alternating factorial numbers.
The first few values n for which are (probable) primes are 3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, 2653, 3069, 3943, 4053, 4998, 8275, 9158, 11164, 43592, 59961, ... (OEIS A001272; extending Guy 1994, p. 100). | 2021-08-01 21:13:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.881469190120697, "perplexity": 221.5144186375738}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154219.62/warc/CC-MAIN-20210801190212-20210801220212-00208.warc.gz"} |
https://chemistry.stackexchange.com/questions/31378/neutralisation-and-end-point-ph | # Neutralisation and end point pH
So I just finished my final paper yesterday and I tried to attempt this 8 mark question. I am not sure how to do it.
$25\,\mathrm{ml}$ of $0.5\,\%~\mathrm{w/v}$ lactic acid ($\ce{C3H6O3}$, molecular mass $90.1\,\mathrm{g\,mol^{-1}}$, $\mathrm{p}K_{\mathrm{a}} = 3.86$ was neutralised by $13.1\,\mathrm{ml}$ of $0.1010\,\mathrm{M}$ monobasic base. Estimate the end point $\mathrm{pH}$ and calculate the amount of $\ce{C3H6O3}$ in % of the initial one.
(I can’t remember the exact words used in the paper for the % part, but I think it is asking for like % purity?)
I calculated the $\mathrm{pH}$ using the formula $\mathrm{pH} = 0.5 (\mathrm{p}K_\mathrm{a} - \log{M})$ but I think it is wrong. Any help would be appreciated.
• At the end point, you have a solution of sodium lactate (weak base). The concentration of sodium lactate is given by the equation: $$C'=\frac{C_b\, V_b{eq}}{V_a +V_b{eq}}$$ Where $C_b$ is the concentration of the base,$V_b{eq}$ is the voulume of base at the end point and $V_a$is the voulume of lactic acid. $$C'=\frac{0.1010 \times 13.1}{25 + 13.1}= 0.03473 \,\ce{mol/L}$$ The $\ce{pH}$ of the solution at the end point is given by the equation: $$\ce{pH}= \frac{1}{2}(\ce{pK_w +pK_a -pC'})$$ $$\ce{pH}= \frac{1}{2}(14 +3.86 +\log 0.03473 )=8.20$$
• To calculate the purity percentage of lactic acid, we calculate the number of moles of pure lactic acid: At the end point $$n_a=n_b$$ This means: $$n_a=C_b\,V_b= 0.1010\times 13.1 \times 10^{-3 }=0.001323 \mathrm {moles}$$ The mass of pure lactic acid is: $$m=0.001323 \times 90.1= 0.1192\, \mathrm{g}$$ Or the mass of $0.5 \% w/v$ lactic acid is: $$m'=\frac {0.5 \times 25}{100}= 0.125 \,\mathrm{g}$$ The purity percentage of lactic acid $P \%$: $$P\%=\frac{m\times 100}{m'}=95.36\%$$ | 2019-08-23 13:45:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9240770936012268, "perplexity": 630.3961801303725}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027318421.65/warc/CC-MAIN-20190823130046-20190823152046-00224.warc.gz"} |
https://blog.jverkamp.com/2011/10/25/wombat-ide-a-bunch-of-new-procedures/ | # Wombat IDE - A bunch of new procedures
I’ve added a bunch of new procedures today, mostly to improve compatibility with Chez Scheme and the like. All together, I’ve added these functions:
• cpu-time - get a timestamp to use in timing function runtime
~ (cpu-time)
1.339726427264E9
~ (let ([start (cpu-time)])
(long-running-function)
(- (cpu-time) start))
0.7339999675750732
• filter - keep only elements that match a given predicate
~ (filter even? '(8 6 7 5 3 0 9))
(8 6 0)
• fold-left - perform formulaic recursive procedures on lists from, see the linked Wikipedia article for more details
~ (fold-left cons '() '(8 6 7 5 3 0 9))
(((((((() . 8) . 6) . 7) . 5) . 3) . 0) . 9)
~ (fold-left + 0 '(8 6 7 5 3 0 9))
38
• fold-right - similar to fold-left with the arguments reversed
~ (fold-right cons '() '(8 6 7 5 3 0 9))
(8 6 7 5 3 0 9)
~ (fold-right + 0 '(8 6 7 5 3 0 9))
38
• any? - tests if any item in a list matches a given predicate
~ (any? even? '(8 6 7 5 3 0 9))
#t
• all? - tests if all items in a list match a given predicate
~ (all? even? '(8 6 7 5 3 0 9))
#f
Hopefully at least some of these will be useful. The current version is 1.297.15. | 2019-08-17 13:06:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2152133733034134, "perplexity": 4489.309597791712}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027313259.30/warc/CC-MAIN-20190817123129-20190817145129-00518.warc.gz"} |
https://encyclopediaofmath.org/index.php?title=Sharing&diff=next&oldid=19146 | Difference between revisions of "Sharing"
imputation (in the theory of games)
A distribution of the overall gain of all players in a cooperative game which satisfies the rationality condition. Formally, if for a game with a set $J=\{1,\ldots,n\}$ of players a characteristic function $v(J)$ is defined, a sharing is a vector $x=(x_1,\ldots,x_n)$ such that $\sum_{i=1}^n$; $x_i\geq v(i)$, $i=1,\ldots,n$. | 2021-09-26 10:05:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8611286878585815, "perplexity": 354.4112974230238}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057857.27/warc/CC-MAIN-20210926083818-20210926113818-00521.warc.gz"} |
http://www.drew-lewis.com/publication/2019-01-01-Co-tame-polynomial-automorphisms | # Co-tame polynomial automorphisms
Published in International Journal of Algebra and Computation, 2019
Recommended citation:
E. Edo & D. Lewis. Co-tame polynomial automorphisms, International Journal of Algebra and Computation, 29(5), (2019), 803-825.
### Abstract
A polynomial automorphism of $\mathbb{A}^n$ over a field of characteristic zero is called co-tame if, together with the affine subgroup, it generates the entire tame subgroup. We prove some new classes of automorphisms of $\mathbb{A}^n$, including nonaffine $3$-triangular automorphisms, are co-tame. Of particular interest, if $n=3$, we show that the statement “Every m-triangular automorphism is either affine or co-tame” is true if and only if $m=3$; this improves upon positive results of Bodnarchuk (for $m=2$, in any dimension $n$) and negative results of the authors (for $m=6$, $n=3$). The main technical tool we introduce is a class of maps we term translation degenerate automorphisms; we show that all of these are either affine or co-tame, a result that may be of independent interest in the further study of co-tame automorphisms. | 2022-10-01 09:04:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8174352049827576, "perplexity": 765.5245386403407}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335573.50/warc/CC-MAIN-20221001070422-20221001100422-00258.warc.gz"} |
https://crecelocal.com/value-driven-kinj/factorization-questions-with-answers-000198 | eg {1, 2, 3, 6, 9, 18} is the set of factors of 18. expand factorise 5a(a − 2) 5a2 − 10a ‘Gnidnapxe’ is the reverse of ‘expanding’. I don't know where to start... 5. There are 4 methods: common factor, difference of two squares, trinomial/quadratic expression and completing the square. How can i factor f(x) = 2x^2 + x - 6 2. challenge question -- Factor the polynomial completely 3. How to factor these equations? Test your understanding with practice problems and step-by-step solutions. Question 10. Factorise x² + 2x - 8 Factoring practice Factor the following polynomials (as fully as possible). Factorise the following polynomials. Perhaps you can learn from the questions someone else has already asked. Use MathJax to format equations. ... Get a free answer to a quick problem. Answers (x - 2) (x + 5) (x - 4) (x + 6) (x - 3) (x - 6) (x - 4) (2 x + 3) It is as factored as it gets. You may speak with a member of our customer support team by calling 1-800-876-1799. Practice: Prime factorization. FACTORING POLYNOMIALS 1) First determine if a common monomial factor (Greatest Common Factor) exists. Please be sure to answer the question. Our online prime factorization trivia quizzes can be adapted to suit your requirements for taking some of the top prime factorization quizzes. The number 48 may be written as a product in a number of di erent ways: 48 = 3 16 = 4 12 = 2 24 Factoring is also the opposite of Expanding: In the following two polynomials, find the value of ‘a’ if x – a is a factor … (a) 16x 4 – 81 (b) (a – b) 2 + 4ab Solution: (a) 16x 4 – 81 = (4x 2) 2 – (9)2 = (4x 2 + 9)(4x 2 – 9) = (4x 2 + 9)[(2x) 2 – (3) 2] = (4x 2 + 9)(2x + 3) (2x – 3) (b) (a – b) 2 + 4ab = a 2 – 2ab + b 2 + 4ab = a 2 + 2ab + b 2 = (a + b) 2. We have a wealth of resources to improve students’ confidence answering all types of expanding and factorising maths questions. Factoring worksheets: Factor to prime factors (0-100) Below are six versions of our grade 6 math worksheet on factoring numbers less than 100 to their prime factors. Common divisibility examples. This question is of factorization. There are 4 methods: common factor, difference of two squares, trinomial/quadratic expression and completing the square. N5 Maths Exam Questions & Answers by Topic Thanks to the SQA and authors for making the excellent resources below freely available. How to factor this expression? Similar: Factor numbers to prime factors (0-500) Greatest common factor of 2 numbers (2-50) Improve your math knowledge with free questions in "Prime factorization" and thousands of other math skills. Which of the following is not a prime factorization? basically what are asking me to factor is : #ax^2 + bx +c#. Factorization, GCD, LCM: Prime Factorization These worksheets require trees to determine the prime factorization of a number, including showing expanded and exponential forms. Seven children came to my daughter's birthday party and I have twenty-eight treats I can hand out. Up Next. Prime Factorization Trees: Easy Difficulty Products Flow rates are measured in mL/hr (milliliters per hour). 16, 24 5. Factors and Multiples for Grade 4 : Common factors, greatest common factor, finding the GCF, several problems for practicing, … Download [233.44 KB] Identifying Prime and Composite Numbers : Questions like Determine if the number 31, 98, 76 … is a Prime(P) or Composite(C) number. A comprehensive database of prime factorization quizzes online, test your knowledge with prime factorization quiz questions. Provide details and share your research! Prime factorization exercise. Phone support is available Monday-Friday, 9:00AM-10:00PM ET. a) 20 = … Find an Online Tutor Now Choose an expert and meet online. Good question . Questions & answers have been split up by topic for your ease of reference. Least common multiple. You can now earn points by answering the unanswered questions listed. THE FACTORS ARE (-12), (-3) SUM = COEFFICIENT OF x =( -15) product = (COEFFICIENT of x^2 #*# COEFFICIENT of constant)= 36 Now you have to find factors which have a sum of -15 and a product of 36 . 12, 18 2. Factorize the following: 1. And a "Factor Tree" can help: find any factors of the number, then the factors of those … Factoring Practice I. 6X^2+X-1 2. 6y(2y - 3) - 2y + 3 [we can do this because 6y(2y - 3) is the same as 12y² - 18y] Now, make the last two expressions look like the expression in the bracket: 6y(2y - 3) -1(2y - 3) The answer is (2y - 3)(6y - 1) Example. Most questions answered within 4 hours. 4X^2-11X-3 3. Common divisibility examples. A factor of a given number is another number that will divide into the given number with no remainder. Be aware of opposites: Ex. It’s ‘factorising’, you dummy! These worksheets are pdf files.. Improve your math knowledge with free questions in "Prime factorization" and thousands of other math skills. Next lesson. 4X^2+4X-3 Worksheets > Math > Grade 6 > Factoring > Factoring numbers to prime factors (0-100). In the following two polynomials. Sometimes it helps to look at a simpler case before venturing into the abstract. Factoring multiple-term expressions is a pretty big part of algebra, so you should expect to find some questions on it in the Praxis Core exam. Factoring Algebra Algebra 1 … Answer the following questions on prime factorization. The first two terms, 12y² and -18y both divide by 6y, so 'take out' this factor of 6y. I forgot how to factor! For example, $$2x - 6{x}^{2}$$ can be factorised as follows: $2x - 6{x}^{2} = 2x(1 - 3x)$ 10, 35 3. Our mission is to provide a free, world-class education to anyone, anywhere. We have factor maths worksheets suitable for all abilities, and they are all supplied with answers to assess how well your child or pupil is doing, and highlight areas for revision. 4. 28, 49 6. This quiz will test your knowledge on the ability to solve IV flow rate drip factors gtt/min.In nursing school, you will have to learn how to calculate how much of a intravenous medication will be given via a flow rate. 1. You will need to get assistance from your school if you are having problems entering the answers into your online assignment. 7. RELATED TOPICS. Making statements based on opinion; back them up with references or personal experience. (i) When x 3 + 3x 2 – kx + 4 is divided by (x – 2), the remainder is k. Find the value of k. Question 11. Find the value of ‘a’ if x + a is a factor of each of the two: Question 12. Common factors (EMAH) Factorising based on common factors relies on there being factors common to all the terms. Factorization Questions and Answers (857 questions and answers). Here is a set of practice problems to accompany the Factoring Polynomials section of the Preliminaries chapter of the notes for Paul Dawkins Algebra course at Lamar University. Hence The following are not prime factorization. In the following practice questions, you go in one direction to find the full factorization of an expression, and then in the opposite direction to … The product is a multiplication of the factors. (g) 2x+ 3y (h) 16x y28x2y + 9y Question 2 (Simple Factorisation into double brackets) Factorise each of the following expressions. Factor trees may be used to find the GCF of difficult numbers. a) 20 = 2×10 , b) 14 = 2×7 , c) 64 = 4 3, d) 120 = 2 3 × 15 Solution Prime factorization involves only prime numbers. (a) x2+ 3x+ 2 (b) x2+ 5x+ 6 (c) x2+ 10x+ 21 (d) x2+ 8x+ 16 (e) x2+ 4x+ 4 (f) x2+ 9x+ 20 (g) x2+ 13x+ 30 (h) x2+ 3x 10 (i) x2+ 4x 5 … Here are some questions other visitors have asked on our free math help message board. Factorising Exercises Question 1 Factorise each of the following expressions. #4x^2 – 15x + 9.# First you need to make column like this. But avoid … Asking for help, clarification, or responding to other answers. 27, 63 Section 1 Finding Factors Factorizing algebraic expressions is a way of turning a sum of terms into a product of smaller ones. MathJax reference. here is an eg. worked examples Common divisibility examples. Factorise: In previous grades, we factorised by taking out a common factor and using difference of squares. Factoring (called "Factorising" in the UK) is the process of finding the factors: It is like "splitting" an expression into a multiplication of simpler expressions. Clear, easy to follow, step-by-step worked solutions to all N5 Maths Questions below are available in the Online Study Pack. You are allowed to answer only once per question. Factorising an expression is to write it as a product of its factors. This is the currently selected item. Factorization Questions and Answers - Math Discussion The worksheets are available in both html and PDF formats (both are easy to print), and they come with an answer key on the second page of the file. 8, 30 4. Factorisation math tests for GCSE maths, Factorisation simple equations, Factorisation of polynomial equations, factorisation of quadratic polynomial - expanding brackets, collecting like terms, factorising Question 2. No packages or subscriptions, pay only for the time you need. (d) 7x2yz 28y (e) 9x2y + 3xy (f) x+ x2+ x3. 1. Prime factorization exercise. (a-b) and (b-a) These may become the same by factoring -1 from one of them. Greatest Common Factor (GCF) Find the GCF of the numbers. Factor Tree. 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Calling 1-800-876-1799 ( e ) 9x2y + 3xy ( f ) x+ x2+ x3 4x^2 – +! | 2021-04-11 22:07:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.32874584197998047, "perplexity": 1831.3806322326848}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038065492.15/warc/CC-MAIN-20210411204008-20210411234008-00134.warc.gz"} |
https://hero.handmade.network/forums/code-discussion/t/29-handmade_penguin_sdl_port_discussion#149 | David Gow
10 posts
Handmade Penguin is a tutorial for porting Handmade Hero to Linux using the SDL library.
I thought this thread might be a good place for people to post any questions they have about it (or SDL/Linux development in general), as well as report bugs, comments, etc.
To start off, I'll announce Chapter 11: Making Graphics Portable. I had to write half of this one twice thanks to a wayward power outage, so there might be a few half-finished sentances or similar lying around. :/
— David
Stefan Koch
22 posts
I think it's a great thing you are dooing.
Thumbs up!
David Gow
10 posts
Chapter 12: Platform-independent Sound Output is now out.
If no-one objects, I'll no longer be doing both the ring buffer and SDL_QueueAudio() versions of the source code in future, unless there's a particularly interesting reason to look at it. Speak up now if this upsets you!
— David
David Vereb
1 posts
Edited by David Vereb on
I was starting a list of comments and typos. The only real-code typo I saw so far was this. There were a few missing letters in the text here and there - I'll send you a full list when I'm caught up to the main stream. :D
David Gow
10 posts
Fixed, thanks!
I'm looking forward to your comments: these were all written pretty quickly, so I'm sure there are lots of mistakes.
— David
Dan
21 posts / 2 projects
None
Edited by Dan on
Not sure if I should make new thread with this, but I am having issue with playing sound. It seems that no sound is playing out of my speakers.
This is my callback. This callback is being called by SDL, so I am not forgetting an SDL_PauseAudio(0) or anything like that.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 static void SDLAudioCallBack(void* userData, uint8_t* stream, int len) { int16_t* currSample = (int16_t*)stream; uint32_t tone = 256; uint32_t period = 48000 / tone; uint32_t halfPeriod = period / 2; for (size_t i = 0; i < (len / sizeof(*currSample)); i+=2) { int16_t halfSample = ((i / halfPeriod) % 2) ? 1 : -1; currSample[i] = halfSample * 3000; //left channel currSample[i+1] = halfSample * 3000; //right channel } }
This is my audio spec
1 2 3 4 5 6 7 8 9 10 SDL_AudioSpec desiredAudio = {}; desiredAudio.channels = 2; desiredAudio.samples = 4096; desiredAudio.freq = SOUND_FREQ; desiredAudio.format = AUDIO_S16LSB; desiredAudio.callback = SDLAudioCallBack; if (SDL_OpenAudio(&desiredAudio, NULL) < 0) { printSDLErrorAndExit(); }
Any help would be greatly appreciated! Thanks!
Also, in addition, it seems that _rdtsc() or __rdtsc() does not seem to be in x86intrin.h on Mac or Linux Mint for whatever reason (perhaps I am compiling with the wrong flags? I am compiling in C using clang, not g++. Maybe it's not in clang's std library?). As a workaround, I found that this implimentation seems to work:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 static inline uint64_t __rdtsc(void) { uint32_t eax = 0, edx; __asm__ __volatile__("cpuid;" "rdtsc;" : "+a" (eax), "=d" (edx) : : "%rcx", "%rbx", "memory"); __asm__ __volatile__("xorl %%eax, %%eax;" "cpuid;" : : : "%rax", "%rbx", "%rcx", "%rdx", "memory"); return (((uint64_t)edx << 32) | eax); }
Just as a suggestion: If it turns out that these functions really are missing on these platforms and this implementation is good enough, would you want to update the tutorial with this implementation?
Thanks!
David Gow
10 posts
I'm afraid I'm totally stumped: your sound code works perfectly here. The two follow-up questions I'd ask are:
1. If you're using PulseAudio, does your game show up in the volume control?
2. Does setting the environment variable SDL_AUDIODRIVER=alsa help? If you're using a non-default alsa device, you may also need AUDIODEV=<device>.
As to the intrinsics, it seems that clang only added support for _rdtsc in version 3.5, so you'll need to upgrade if you want to use the intrinsics. Otherwise, the inline assembly implementation will work fine. I've quickly noted this in the chapter, but I might discuss it further if I get time.
Best of luck,
— David
David Gow
10 posts
Chapter 13: Platform-indpendent User Input is out.
I've also upgraded the server it's hosted on, so if anything dies, do let me know.
— David
Dan
21 posts / 2 projects
None
Found the issue. Thanks David! It seems that the correct audio device wasn't selected. I was confused because I was watching the Handmade hero archive before with no problems. Maybe for some reason, in Linux Mint, the correct audio device was assigned to firefox but not the rest of my programs or something.
David Gow
10 posts
Chapter 14: A Trip Down Memory Lane is now up.
The top-secret behind-the-scenes info:
• The site now has a proper SSL certificate. Let me know if I broke the Apache config and you can no longer connect.
• I spent far too long coming up with the title for this one, so I apologise for it being a bit later than usual.
— David
David Gow
10 posts
Chapter 15: Platform-indpendent Debug File I/O is up.
I've included a little bit on using the mmap() for mapping files into memory as a bonus!
— David
David Gow
10 posts
Sorry for the delay; Chapter 16 "Bits and Pieces" of Handmade Penguin is now up.
This is the longest chapter yet, and goes into the top secret techniques, passed down from Linux developer to Linux developer, for getting your binaries to work across lots of different distributions.
Please let me know if anything doesn't work, or doesn't make sense, or eats your pets and/or beloved family members. This sort of compatibility is made of hacks built on top of other, more horrible, hacks, so the result is always going to be a bit ugly.
Thanks, and apologies for the wait,
— David
David Gow
10 posts
Fear not, Chapter 17 is here.
With some luck, the delays that plagued the last couple of chapters are over. I have lots of notes on my notepad, so it's just typing up and debugging to go.
— David
David Gow
10 posts
It's Chapter 18 time for Handmade Penguin.
We look at getting refresh rates, sleeping with SDL_Delay() and synchronising to the vertical blank.
For the interested, here's a little bit of the raw data from one of my SDL_Delay() timing experiments. You can read more about how timers on linux work here.
Enjoy,
— David
SDL_Delay(1) takes 1.095611 ms
SDL_Delay(1) takes 1.079751 ms
SDL_Delay(1) takes 1.083415 ms
SDL_Delay(1) takes 1.085282 ms
SDL_Delay(1) takes 1.088205 ms
SDL_Delay(1) takes 1.086309 ms
SDL_Delay(1) takes 1.069637 ms
SDL_Delay(1) takes 1.068966 ms
SDL_Delay(1) takes 1.068637 ms
SDL_Delay(1) takes 1.081438 ms
SDL_Delay(1) takes 1.078134 ms
SDL_Delay(1) takes 1.085287 ms
SDL_Delay(1) takes 1.085804 ms
SDL_Delay(1) takes 1.085721 ms
SDL_Delay(1) takes 1.024315 ms
SDL_Delay(1) takes 1.081754 ms
SDL_Delay(1) takes 1.082592 ms
SDL_Delay(1) takes 1.085334 ms
SDL_Delay(1) takes 1.086600 ms
SDL_Delay(1) takes 1.085623 ms
SDL_Delay(1) takes 1.085648 ms
SDL_Delay(1) takes 1.085545 ms
SDL_Delay(1) takes 1.085480 ms
SDL_Delay(1) takes 1.082745 ms
SDL_Delay(1) takes 1.085969 ms
Neo Ar
165 posts / 1 project
riscy.tv host | 2022-08-16 03:55:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2985025644302368, "perplexity": 5267.980421181645}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572220.19/warc/CC-MAIN-20220816030218-20220816060218-00149.warc.gz"} |
https://brilliant.org/problems/where-is-the-error/ | # Where is the error?
I am performing the following steps to prove $-1 = 1:$
1. $-1 = i^{2}$
2. $-1=i \times i$
3. $-1 = \sqrt{-1} \times \sqrt{-1}$
4. $-1 =\sqrt{-1 \times -1}$
5. $-1 =\sqrt{1}$
6. $-1=1.$
In which step does the error first appear?
× | 2022-06-25 04:38:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 7, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6395285129547119, "perplexity": 424.81653403266307}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103034170.1/warc/CC-MAIN-20220625034751-20220625064751-00695.warc.gz"} |
http://mathhelpforum.com/pre-calculus/57371-f-x-ce-kt.html | # Math Help - F(x) = ce^kt
1. ## F(x) = ce^kt
Find the function f(t) = ce^kt that passes through the given points in the plane.
a) (0,5) and (1,1)
b) (2,4) and (3,7)
Okay i don't know how the teacher did this in class today. Teach me the ways.
2. Substitute and solve simultaneously for c and k. eg. for the first question you have
$
5 = ce^{k(0)}$
and
$
1 = ce^{k(1)}$
3. so you have y=5 and y=1
4. Originally Posted by theflyingcow
Find the function f(t) = ce^kt that passes through the given points in the plane.
a) (0,5) and (1,1)
b) (2,4) and (3,7)
Okay i don't know how the teacher did this in class today. Teach me the ways.
a) Think of these two points in the form of (t, f(t)). So plugging in the first point we get $5=ce^{k*0}=ce^0 = c(1) = c$. So c=5. Now use that and the 2nd point. $1=5e^{k*1}$ Divide by 5, then take the natural log of both sides, $\ln( \frac{1}{5} )= k\ln(e)=k$. So now you have k.
5. so k= ln(1/5)
where do I go now
6. Ok you need to think about your function, F(t), and what every letter represents. I solved for c and k. They are both constants defined by the two points in part (a). For part (b) they will change. e is a constant number 2.71... which is Euler's constant. If you are unfamiliar with this, you should look it up. Now you wrote in the title of this thread, F(x), not F(t). Think about this. If F is a function dependent on x, why are there no x's in the equation? If x isn't the variable, what is? t is. Or you need to change it to F(x)=ce^{kx}. Either way, but you need to be consistent.
So, you have c and k (constants) and e (always a constant), so this problem is completed.
7. but we use two points and the overall k = (ln 1/5) / 1.
8. Originally Posted by theflyingcow
but we use two points and the overall k = (ln 1/5) / 1.
I don't get this post.
Yes, two points are needed to SOLVE for c and k, which are constants. What do you mean by the overall k?
9. the k for the second point sorry
so the function is y = 5e^((ln 1/5) / 1)*t)
10. Originally Posted by theflyingcow
the k for the second point sorry
so the function is y = 5e^((ln 1/5) / 1)*t)
Yes, but you don't need to write anything /1 . It's understood that anything divided by one is itself and it's always omitted.
I will say you'll need to use the method that badgerigar described in his post for part (b). | 2014-09-19 12:18:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8571131229400635, "perplexity": 581.0081075451926}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657131304.74/warc/CC-MAIN-20140914011211-00147-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://math.stackexchange.com/questions/1808970/finite-generation-of-global-sections-of-a-tensor-product | # Finite generation of global sections of a tensor product
Let $X$ be a scheme (quasi-projective over a field if it helps). Let $\mathcal F$ and $\mathcal G$ be $\mathcal O_X$-modules and $U$ an open subset of $X$ such that $\mathcal F(U)$ and $\mathcal G(U)$ are $\mathcal O_X(U)$-modules of finite type. Let $\mathcal F \otimes_{\mathcal O_X}\mathcal G$ be the tensor product sheaf. Is $(\mathcal F \otimes_{\mathcal O_X}\mathcal G)(U)$ an $\mathcal O_X(U)$-module of finite type?
• You certainly like this kind of question! Anyway, I think you can do something dumb like take $X = U = \mathbb P^1_k$, $\mathcal{F} = \bigoplus_{n\in\mathbb Z} \mathcal{O}(-1)$, $\mathcal{G} = \mathcal{O}(1)$, so coherence seems helpful. – Hoot Jun 1 '16 at 22:25
• Even though $F$ is not coherent, that answers completely what I had in mind. Thank you (I will edit the question removing the coherence so that it is completely answered by your comment and accept it if you want to write it as an answer). – A.G Jun 1 '16 at 22:35 | 2019-07-21 21:01:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8771337866783142, "perplexity": 190.40343256706723}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195527204.71/warc/CC-MAIN-20190721205413-20190721231413-00014.warc.gz"} |
https://www.gradesaver.com/textbooks/math/calculus/calculus-early-transcendentals-8th-edition/chapter-16-section-16-5-curl-and-divergence-16-5-exercise-page-1110/24 | ## Calculus: Early Transcendentals 8th Edition
$curl (F+G)=curl F+curl G$
A vector field $F$ is conservative if and only if $curl F=0$ Let us consider that $F=ai+b j+c k$ Then, we have $curl F=[c_y-b_z]i+[a_z-c_z]j+[b_x-a_y]k$ Plug $F=a_1i+b_1j+c_1z; G=a_2i+b_2j+c_2k$ This implies that $curl (F+G)=curl [(a_1+a_2)i+(b_1+b_2)j+(b_3+c_3)k$ Now, use the distributive property of the cross product. This gives: $curl (F+G)=\nabla \times F+\nabla \times G=curl F+curl G$ | 2019-11-13 04:22:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9308105707168579, "perplexity": 78.55546024040642}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496665985.40/warc/CC-MAIN-20191113035916-20191113063916-00195.warc.gz"} |
https://www.physicsforums.com/threads/what-can-i-do-to-prepare-for-college-and-a-career.908913/ | # What can I do to prepare for college and a career?
Hello all, I created an account just to make this post but I've been lurking for a while.
I'm currently a Junior in high school and I have some rather ambitious goals. While it's not very likely I will ever achieve all of them, I'd like to at least get half-way there. I'm currently in AP Physics, and not to brag as I know many others struggle, but the class is quite easy for me. I am also fluent in a few programming languages, those being Python and C++, which I've known for quite a while and it wasn't until recently that I found out programming is quite useful for college courses in Physics and advanced mathematics.
I have quite a lot of goals, and I am not too sure how to obtain them. I've spoken with the counselor at my school and he said the best that I can do for help is talk to a professor, but I don't have any way to talk to one. My long-term goal is to achieve a Ph. D in Physics, preferably for theoretical physics or astrophysics. This will be tough for me as I must rely on student loans, which I'm not sure I can even qualify for. My family cannot support me due to some large financial issues.
I'm not too sure what jobs will be available to me if I ever do get a Ph. D, or a Masters for that matter. I have realized that the job market for Physics is quite saturated, and to get a research position is like winning the lottery. With that said, is there any advice for me going forward? (Job possibilities, college/financial advice, etc).
Choppy
My long-term goal is to achieve a Ph. D in Physics, preferably for theoretical physics or astrophysics. This will be tough for me as I must rely on student loans, which I'm not sure I can even qualify for. My family cannot support me due to some large financial issues.
First off I would encourage you to keep learning about what interests your, particularly at this stage of the game. In high school it's a good idea to build a strong foundation in mathematics and the sciences and then specialize as you go further. It sounds like you're on that track.
With regard to money - a few points. You're not necessarily doomed to a single option of loans. First of all, how much money are you earning now? If you don't have a part-time job, it's a good idea to get one. Even saving up a few thousand dollars to take a bite out of those first year expenses can help immensely. Starting now can also help you to build up experience, which can help you to get better-than-minimum wage jobs in the four months out of the year that you're not in school later on. Second, if you're really doing that well in AP physics, make sure that you push yourself to get into the academic scholarship GPA range. One thing that I've observed about scholarships is that they tend to snowball. Finally, you get paid to go to graduate school - or at least supported financially. It's not a lot, but usually students make enough money to support themselves through a PhD that they don't have to take out any further loans while they do it. Oh - one more. Shop around. Consider cost of living where you want to go. Prices can vary dramatically and the quality of education and opportunities associated with the school name are not necessarily directly proportional.
As you learn, keep an open mind too. There's more to physics than theoretical astrophysics.
Dr. Courtney
Gold Member
2020 Award
I recommend against borrowing too much money.
Even if your parents cannot write checks to your school to help fund your education, there are a few things they may be able to do: let you live at home as long as possible and keep you on their health insurance. Living away from home adds on the order of $50,000 to one's college debt over four years. Living at home means much lower debt. Even if you don't live close enough to compute to a good school with a physics major, odds are you can combine distance learning and commuting for two years or so to a school you do live close enough to reach 60-80 credit hours so you are only two years away before transferring to a better school to finish your BS degree. Also, if you are near as good as you think, odds are pretty good you will qualify for merit-based scholarships at in-state schools with physics programs. I'd focus on a decent school in your state. Private and out-or-state schools usually add$50,000-\$100,000 to your expenses over 4 years compared with in-state schools.
To best position yourself for merit-based scholarships, keep your grades up and take the ACT or SAT as soon as possible to see where you stand in that regard. You should also figure out how you can become involved in research as soon as possible. Getting into a local group may be challenging, but a winning project in your local ISEF-affiliated science fair (or other high school science venue) IS attainable.
See:
https://www.physicsforums.com/insights/secrets-successful-science-projects/
Take my science project advice seriously. Students I've mentored in science projects have won 1st place in their category at state science fairs 11 of 14 times and second place at state 2 of 14 times. When they reach college, they do very well with scholarships.
Last edited:
CrysPhys
Take my science project advice seriously. Student's I've mentored in science projects have won 1st place in their category at state science fairs 11 of 14 times and second place at state 2 of 14 times. When they reach college, they do very well with scholarships.
Absolutely agree. And science projects will help get you into college. When you write your personal essay, you won't be writing BS about your passion for physics, you will be relating concrete examples demonstrating your passion for physics.
Dr. Courtney | 2021-06-18 06:51:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1790590137243271, "perplexity": 721.2038226016988}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487635724.52/warc/CC-MAIN-20210618043356-20210618073356-00081.warc.gz"} |
https://www.gamedev.net/forums/topic/493898-general-projectioncamera-help/ | # OpenGL general projection/camera help
This topic is 3866 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
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I am fresh out of beginners programming (game maker) and I need some help with opengl's projection system. What I need is a camera that has movement and direction to it, not a camera that sits and has the world move around it. What type of projection should I use gluPerspective or the other one. What would I have to use to get a simple movement system setup. Would it have anything to do with gluLookAt? I am I bit lost and kind find many answers on my own. Some help would go a long way. Edit: Ok currently I am using gluPerspective(45,width/height,1,100000). Thats about all I could figure out, well that and how to get the world to move around it using glTranslate. Now I need to leave how to keep the world in one set location and move the camera around. Does anyone have any idea how this can be done? [Edited by - snowfell on May 10, 2008 11:36:34 AM]
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when you load the projection matrix you set the camera for projection view, thats what gluPerspective() and glFrustum() are for, they setup the andle of view and respective ratio and also the near and far camera planes.
now to setup the camera position, that is calculated on the modelview matrix. and you can use gluLookAt() for that as you said.
a simple example would be:
void render(){ glMatrixMode( GL_MODELVIEW ); //load the modelview matrix mode calculations glLoadIdentity(); //load the identity //set camera position glTranslated( 0.0, 0.0, 100.0 ); //camera will be at 0, 0, -100 world coordinates, as glulookat rotates its position gluLookAt( 0.0, 0.0, 0.0, 0.0, 0.0, -5.0, 0.0, 1.0, 0.0 ); //set it on the new origin and looking 5 unites farther into the screen (also sets the up view vector) //render your scene glPushMtarix(); //save current matrix into the stack (...) glPopMatrix();}
my comments might not be entirely correct (im also pretty new to opengl) but it can help a bit.
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Actually, if I recall correctly, standard setting of the camera position and orientation (via glTranslate and the like) can be done with the projection matrix - in fact, if I'm not much mistaken, that sounds like what you're asking for.
I'm not sure about gluLookAt; I don't think that I've used it in ages, personally, but I do have a vague memory that that call, specifically, might be intended to be used with the modelview matrix selected.
However, why not move the world? It amounts to the same thing, I believe - it just operates the other way around (that is, instead of translating to {-1, -2, 4}, you'd translate to {1, 2, -4}). Unless you're doing something unusual, then I don't think that it should introduce any difficult problems.
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What I do is I make a class that holds 3 vectors; Position, Rotation and View Direction. I then have 4 functions for movement and 4 for updating, For movement: Left, right, forward and backward then for Updating the vectors mentioned earlier and another to update them all.
Left, right, forward and backward change the position vector and then the position updating functions calls glTranslatef with Position's values.
I have the mouse edit the rotation then the rotation updating function calls glRotatef with Rotation's values.
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Well I understand that projection draws everything on a hor and vert axis (y being vert, x and z being hor). Now what I don't understand is how to get motion and direction out of the camera. So far I have been able to make a weak 3d engine using glut. I have been able to render a simple floor and and put colored markers at each corner.
So this is the bare bone engine I wiped together:
void projection(int width, int height)
{
glViewport(0,0,width,height);
glMatrixMode(GL_PROJECTION);
gluPerspective(45.0f,width/height,1.0f,10000.0f);
gluLookAt(cord_x,cord_y+5,cord_z,0.0,0.0,dir,0.0,zdir,0.0);
}
void display()
{
float room_width=255;
float room_height=255;
glClear(GL_COLOR_BUFFER_BIT|GL_DEPTH_BUFFER_BIT);
glMatrixMode(GL_MODELVIEW);
glBegin(GL_POLYGON);
glColor3d(.25,.25,.25);
glVertex3f(0,0,0); glVertex3f(room_width,0,0);
glVertex3f(room_width,0,room_height); glVertex3f(0,0,room_height);
glEnd();
//drawing...
}
Is their anything I can change to include a motion system? Or are there any improvement I could add to what I have done so far.
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If you have the view direction you can multiply that by a distance and add/subtract it to the current position to move in that direction.
If you don't have the view direction here's how to work it out:
float Cos_x, PI = 3.141592654; Vector A; A.x = cos((Rotation_y + 90.0f) * (PI/180)); A.z = -sin((Rotation_y + 90.0f) * (PI/180)); Cos_x = cos(Rotation_x * (PI/180)); ViewDir.x = A.x * Cos_x; ViewDir.z = A.z * Cos_x; ViewDir.y = sin(Rotation_x * (PI/180));
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Quote:
Original post by snowfellvoid projection(int width, int height){...glMatrixMode(GL_PROJECTION);...gluLookAt(cord_x,cord_y+5,cord_z,0.0,0.0,dir,0.0,zdir,0.0);}
You should not put the view matrix into the projection matrix. Instead, it should be put into model-view matrix.
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Quote:
Original post by MrPickleIf you have the view direction you can multiply that by a distance and add/subtract it to the current position to move in that direction.If you don't have the view direction here's how to work it out:*** Source Snippet Removed ***
Ok so where might I put this and its variables? I have seen codes very similar to this in my game making past. Another rather important question I have is where do most of the projection codes go, and why (In their own function or what)?
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In OpenGL, you have three predefined matrices, i.e. model-view matrix, projection matrix and texture matrix.
Model-view matrix, as suggested by its name, is to handle model and viewing. Just for the naming reason, you should handle the viewing using model-view matrix. Besides, there are good some reasons behind why model and viewing get together. It is because both are usually translation and rotation only. Also, it is easier to make specular lighting correct... ...
http://www.sjbaker.org/steve/omniv/projection_abuse.html
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Ok I was able to read through that page but it didn't really explain projection order well. I was just wondering what a effective 3d projection system might look like (as in where gluLookAt and other important codes should be place and how to be used). I have a simple engine going and so far I have a general outline to this, but I need some structure to improve it. If it would help I could post what I have done so far, but only if it could help get some answers.
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× | 2018-12-16 08:17:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19101282954216003, "perplexity": 1632.7016249586857}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376827596.48/warc/CC-MAIN-20181216073608-20181216095608-00229.warc.gz"} |
https://www.physicsforums.com/threads/loop-and-allied-qg-bibliography.7245/page-128 | # Loop-and-allied QG bibliography
1. Nov 19, 2017
### atyy
https://arxiv.org/abs/1708.07445
Towards the map of quantum gravity
Jakub Mielczarek, Tomasz Trześniewski
(Submitted on 24 Aug 2017 (v1), last revised 5 Oct 2017 (this version, v2))
In this paper we point out some possible links between different approaches to quantum gravity and theories of the Planck scale physics. In particular, connections between Loop Quantum Gravity, Causal Dynamical Triangulations, Ho\v{r}ava-Lifshitz gravity, Asymptotic Safety scenario, Quantum Graphity, deformations of relativistic symmetries and nonlinear phase space models are discussed. The main focus is on quantum deformations of the Hypersurface Deformations Algebra and Poincar\'{e} algebra, nonlinear structure of phase space, the running dimension of spacetime and nontrivial phase diagram of quantum gravity. We present an attempt to arrange the observed relations in the form of a graph, highlighting different aspects of quantum gravity. The analysis is performed in the spirit of a mind map, which represents the architectural approach to the studied theory, being a natural way to describe the properties of a complex system. We hope that the constructed graphs (maps) will turn out to be helpful in uncovering the global picture of quantum gravity as a particular complex system and serve as a useful guide for the researchers.
https://arxiv.org/abs/1708.07716
Extended Phase Space Analysis of Interacting Dark Energy Models in Loop Quantum Cosmology
Hmar Zonunmawia, Wompherdeiki Khyllep, Nandan Roy, Jibitesh Dutta, Nicola Tamanini
(Submitted on 25 Aug 2017)
The present work deals with the dynamical system investigation of interacting dark energy models (quintessence and phantom) in the framework of Loop Quantum Cosmology by taking into account a broad class of self-interacting scalar field potentials. The main reason for studying potentials beyond the exponential type is to obtain additional critical points which can yield more interesting cosmological solutions. The stability of critical points and the asymptotic behavior of the phase space are analyzed using dynamical system tools and numerical techniques. We study two class of interacting dark energy models and consider two specific potentials as examples: the hyperbolic potential and the inverse power-law potential. We found a rich and interesting phenomenology including the avoidance of big rip singularities due to loop quantum effects, smooth and non-linear transitions from matter domination to dark energy domination and finite periods of phantom domination with dynamical crossing of the phantom barrier.
https://arxiv.org/abs/1708.08667
A non-polynomial gravity formulation for Loop Quantum Cosmology bounce
Stefano Chinaglia, Aimeric Colleaux, Sergio Zerbini
(Submitted on 29 Aug 2017 (v1), last revised 5 Sep 2017 (this version, v2))
Recently the so-called mimetic gravity approach has been used to obtain corrections to Friedmann equation of General Relativity similar to the ones present in loop quantum cosmology. In this paper, we propose an alternative way to derive this modified Friedmann equation via the so-called non-polynomial gravity approach, which consists in adding geometric non-polynomial higher derivative terms to Hilbert-Einstein action, which are nonetheless polynomials and lead to second order differential equation in Friedmann-Lema\^itre-Robertson-Walker spacetimes. Our explicit action turns out to be a realization of the Helling proposal of effective action with infinite number of terms. The model is investigated also in presence of non vanishing cosmological constant and a new exact bounce solution is found and studied.
https://arxiv.org/abs/1709.03242
Noncommutativity in Effective Loop Quantum Cosmology
Abraham Espinoza-García (UPIIG-IPN, México), Efraín Torres-Lomas (UG, México)
(Submitted on 11 Sep 2017 (v1), last revised 12 Sep 2017 (this version, v2))
We construct two noncommutative extensions of the Loop Quantum Cosmology effective scheme for the open FLRW model with a standard scalar field with quadratic potential. Firstly, noncommutativity is implemented in the configuration sector only (among the holonomy variable and the matter degree of freedom). We show that this type of noncommutativity seems to retain key features of the Loop Quantum Cosmology paradigm for a free field; however, when considering the addition of a quadratic potential,this compatibility weakens regarding the trajectories followed by the scalar field. Secondly, noncommutativity is implemented in the momentum sector (among the momentum associated to the holonomy variable and the momentum associated to the matter field). In the free case, the only effect of this noncommutativity is that of making the volume function to grow faster, retaining key features of the Loop Quantum Cosmology paradigm. We show that, when considering a quadratic potential, this second kind of noncommutativity is more favored than the first one in regard to the trajectories followed by the scalar field.
https://arxiv.org/abs/1709.06331
Von-Neumann Stability and Singularity Resolution in Loop Quantized Schwarzschild Black Hole
Alec Yonika, Gaurav Khanna, Parampreet Singh
(Submitted on 19 Sep 2017)
Though loop quantization of several spacetimes has exhibited existence of a bounce via an explicit evolution of states using numerical simulations, the question about the black hole interior has remained open. To answer this question, it is important to first understand the stability of the quantum Hamiltonian constraint. We take first steps towards addressing these issues for a loop quantization of the Schwarzschild interior. The von-Neumann stability analysis is performed using separability of solutions as well as a full two dimensional quantum difference equation. This results in a condition which translates to stability for black holes which have a very large mass compared to the Planck mass. In addition, stability analysis leads to a constraint on the localization of the allowed states. With the caveat of using kinematical norm, Gaussian states are evolved using the quantum difference equation and singularity resolution is obtained. Bounce is found for one of the triad variables, but for the other triad variable singularity resolution amounts to a non-singular passage through the zero volume. States are found to be peaked at the classical trajectory for a long time before and after the singularity resolution, and retain their semi-classical character across the zero volume.
https://arxiv.org/abs/1709.08370
Random Invariant Tensors
Youning Li, Muxin Han, Dong Ruan, Bei Zeng
(Submitted on 25 Sep 2017)
Invariant tensors are states in the (local) SU(2) tensor product representation but invariant under global SU(2) action. They are of importance in the study of loop quantum gravity. A random tensor is an ensemble of tensor states. An average over the ensemble is carried out when computing any physical quantities. The random tensor exhibits a phenomenon of concentration of measure', saying that for any bipartition, the expected value of entanglement entropy of its reduced density matrix is asymptotically the maximal possible as the local dimension goes to infinity. This is also true even when the average is over the invariant subspace instead of the whole space for 4−valent tensors, although its entropy deficit is divergent. One might expect that for n≥5, n−valent random invariant tensor would behavior similarly. However, we show that, the expected entropy deficit of reduced density matrix of such n−valent random invariant tensor from maximum, is not divergent but a finite number. Under some special situation, the number could be even smaller than half a bit, which is the deficit of random pure state over the whole Hilbert space from maximum.
https://arxiv.org/abs/1709.08511
Intertwiner Entanglement on Spin Networks
Etera R. Livine
(Submitted on 25 Sep 2017)
In the context of quantum gravity, we clarify entanglement calculations on spin networks: we distinguish the gauge-invariant entanglement between intertwiners located at the nodes and the entanglement between spin states located on the network's links. We compute explicitly these two notions of entanglement between neighboring nodes and show that they are always related to the typical ln(2j+1) term depending on the spin j living on the link between them. This ln(2j+1) contribution comes from looking at non-gauge invariant states, thus we interpret it as gauge-breaking and unphysical. In particular, this confirms that pure spin network basis states do not carry any physical entanglement, so that true entanglement and correlations in loop quantum gravity comes from spin or intertwiner superpositions.
https://arxiv.org/abs/1709.08989
Simplicity constraints: a 3d toy-model for Loop Quantum Gravity
Christoph Charles
(Submitted on 26 Sep 2017)
In Loop Quantum Gravity, tremendous progress has been made using the Ashtekar-Barbero variables. These variables, defined in a gauge-fixing of the theory, correspond to a parametrization of the solutions of the so-called simplicity constraints. Their geometrical interpretation is however unsatisfactory as they do not constitute a space-time connection. It would be possible to resolve this point by using a full Lorentz connection or, equivalently, by using the self-dual Ashtekar variables. This leads however to simplicity constraints or reality conditions which are notoriously difficult to implement in the quantum theory.
We explore in this paper the possibility of imposing such constraints at the quantum level in the context of canonical quantization. To do so, we define a simpler model, in 3d, with similar constraints by extending the phase space to include an independent vielbein. We define the classical model and show that a precise quantum theory by gauge-unfixing can be defined out of it, completely equivalent to the standard 3d euclidean quantum gravity.
We discuss possible future explorations around this model as it could help as a stepping stone to define full-fledged covariant Loop Quantum Gravity.
https://arxiv.org/abs/1709.09806
The emergence of 3+1D Einstein gravity from topological gravity
Zheng-Cheng Gu
(Submitted on 28 Sep 2017)
Quantum field theory successfully explains the origin of all fundamental forces except gravity due to the renormalizability and ultraviolet(UV) completion puzzles. The ADS/CFT correspondence conjecture might naturally resolve the above two puzzles for ADS space gravity. In this paper, we propose a topological scenario to resolve the above two puzzles for generic cases(e.g., with or without cosmological constant term). First, we propose a 3+1D topological (quantum) gravity theory which is perturbatively renormalizable and potentially UV complete, this step can be regarded as a straightforward generalization of Edward Witten's Chern-Simons theory proposal for 2+1D topological gravity. Then, we show that Einstein-Cartan equation and classical space-time naturally emerge from topological (quantum) gravity via loop condensation. The second step is a unique feature in 3+1D and it might even naturally explain why our space-time is four dimensional. Experimentally measurable low energy predictions are also discussed.
https://arxiv.org/abs/1710.04015
Cosmological Coherent State Expectation Values in LQG I. Isotropic Kinematics
Andrea Dapor, Klaus Liegener
(Submitted on 11 Oct 2017)
This is the first paper of a series dedicated to LQG coherent states and cosmology. The concept is based on the effective dynamics program of Loop Quantum Cosmology, where the classical dynamics generated by the expectation value of the Hamiltonian on semiclassical states is found to be in agreement with the quantum evolution of such states. We ask the question of whether this expectation value agrees with the one obtained in the full theory. The answer is in the negative. This series of papers is dedicated to detailing the computations that lead to that surprising result. In the current paper, we construct the family of coherent states in LQG which represent flat (k=0) Robertson-Walker spacetimes, and present the tools needed to compute expectation values of polynomial operators in holonomy and flux on such states. These tools will be applied to the LQG Hamiltonian operator (in Thiemann regularization) in the second paper of the series. The third paper will present an extension to k≠0 cosmologies and a comparison with alternative regularizations of the Hamiltonian.
https://arxiv.org/abs/1710.04473
Entanglement entropy and correlations in loop quantum gravity
Alexandre Feller, Etera R. Livine
(Submitted on 12 Oct 2017)
Black hole entropy is one of the few windows toward the quantum aspects of gravitation and its study over the years have highlighted the holographic nature of gravity. At the non-perturbative level in quantum gravity, promising explanations are being explored in terms of the entanglement entropy between regions of space. In the context of loop quantum gravity, this translates into the analysis of the correlations between regions of the spin network states defining the quantum state of geometry of space. In this paper, we explore a class of states, motivated by results in condensed matter physics, satisfying an area law for entanglement entropy and having non-trivial correlations. We highlight that entanglement comes from holonomy operators acting on loops crossing the boundary of the region.
https://arxiv.org/abs/1710.06195
On the volume simplicity constraint in the EPRL spin foam model
(Submitted on 17 Oct 2017)
We propose a quantum version of the quadratic volume simplicity constraint for the EPRL spin foam model. It relies on a formula for the volume of 4-dimensional polyhedra, depending on its bivectors and the knotting class of its boundary graph. While this leads to no further condition for the 4-simplex, the constraint becomes non-trivial for more complicated boundary graphs. We show that, in the semi-classical limit of the hypercuboidal graph, the constraint turns into the geometricity condition observed recently by several authors.
https://arxiv.org/abs/1711.04991
Anomaly free cosmological perturbations with generalised holonomy correction in loop quantum cosmology
Yu Han, Molin Liu
(Submitted on 14 Nov 2017)
In the spatially flat case of loop quantum cosmology, the connection k¯ is usually replaced by the μ¯ holonomy sin(μ¯k)μ¯ in the effective theory. In this paper, instead of the μ¯ scheme, we use a generalised, undertermined function g(k¯,p¯) to represent the holonomy and by using the approach of anomaly free constraint algebra we fix all the counter terms in the constraints and find the restriction on the form of g(k¯,p¯), then we derive the gauge invariant equations of motion of the scalar, tensor and vector perturbations and study the inflationary power spectra with generalised holonomy corrections.
https://arxiv.org/abs/1711.05693
Connecting Loop Quantum Gravity and String Theory via Quantum Geometry
Deepak Vaid
(Submitted on 15 Nov 2017)
We argue that String Theory and Loop Quantum Gravity can be thought of as describing different regimes of a single unified theory of quantum gravity. LQG can be thought of as providing the pre-geometric exoskeleton out of which macroscopic geometry emerges and String Theory then becomes the \emph{effective} theory which describes the dynamics of that exoskeleton. The core of the argument rests on the claim that the Nambu-Goto action of String Theory can be viewed as the expectation value of the LQG area operator evaluated on the string worldsheet.
https://arxiv.org/abs/1711.05967
A Renormalizable SYK-type Tensor Field Theory
Joseph Ben Geloun, Vincent Rivasseau
(Submitted on 16 Nov 2017)
In this paper we introduce a simple field theoretic version of the Carrozza-Tanasa-Klebanov-Tarnopolsky (CTKT) "uncolored" holographic tensor model. It gives a more familiar interpretation to the previously abstract modes of the SYK or CTKT models in terms of momenta. We choose for the tensor propagator the usual Fermionic propagator of condensed matter, with a spherical Fermi surface, but keep the CTKT interactions. Hence our field theory can also be considered as an ordinary condensed matter model with a non-local and non-rotational invariant interaction. Using a multiscale analysis we prove that this field theory is just renormalizable to all orders of perturbation theory in the ultraviolet regime.
https://arxiv.org/abs/1711.06085
Gravity Induced Non-Local Effects in the Standard Model
S. O. Alexeyev, X. Calmet, B. N. Latosh
(Submitted on 16 Nov 2017)
We show that the non-locality recently identified in quantum gravity using resummation techniques propagates to the matter sector of the theory. We describe these non-local effects using effective field theory techniques. We derive the complete set of non-local effective operators at order NG2 for theories involving scalar, spinor, and vector fields. We then use recent data from the Large Hadron Collider to set a bound on the scale of space-time non-locality and find M⋆>3×10−11 GeV.
Last edited: Nov 20, 2017
2. Dec 3, 2017
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https://arxiv.org/abs/1711.08482
AdS2 holography and the SYK model
Gábor Sárosi
(Submitted on 22 Nov 2017)
These are lecture notes based on a series of lectures presented at the XIII Modave Summer School in Mathematical physics aimed at PhD students and young postdocs. The goal is to give an introduction to some of the recent developments in understanding holography in two bulk dimensions, and its connection to microscopics of near extremal black holes. The first part reviews the motivation to study, and the problems (and their interpretations) with holography for AdS2 spaces. The second part is about the Jackiw-Teitelboim theory and nearly-AdS2 spaces. The third part introduces the Sachdev-Ye-Kitaev model, reviews some of the basic calculations and discusses what features make the model exciting.
https://arxiv.org/abs/1711.08470
Propagators for gauge-invariant observables in cosmology
Markus B. Fröb, William C. C. Lima
(Submitted on 22 Nov 2017)
We make a proposal for gauge-invariant observables in perturbative quantum gravity in cosmological spacetimes, building on the recent work of Brunetti et al. [JHEP 08 (2016) 032]. These observables are relational, and are obtained by evaluating the field operator in a field-dependent coordinate system. We show that it is possible to define this coordinate system such that the non-localities inherent in any higher-order observable in quantum gravity are causal, i.e., the value of the gauge-invariant observable at a point x only depends on the metric and inflation perturbations in the past light cone of x. We then construct propagators for the metric and inflaton perturbations in a gauge adapted to that coordinate system, which simplifies the calculation of loop corrections, and give explicit expressions for relevant cases: matter- and radiation-dominated eras and slow-roll inflation.
https://arxiv.org/abs/1711.09270
Loop Quantum Cosmology Corrected Gauss-Bonnet Singular Cosmology
K. Kleidis, V.K. Oikonomou
(Submitted on 25 Nov 2017)
In this work we investigate which Loop Quantum Cosmology corrected Gauss-Bonnet F(G) gravity can realize two singular cosmological scenarios, the intermediate inflation and the singular bounce scenarios. The intermediate inflation scenario has a Type III sudden singularity at t=0, while the singular bounce has a soft Type IV singularity. By using perturbative techniques, we find the holonomy corrected F(G) gravities that generate at leading order the aforementioned cosmologies and we also argue that the effect of the holonomy corrections is minor to the power spectrum of the primordial curvature perturbations of the classical theory.
https://arxiv.org/abs/1711.09941
Ryu-Takayanagi Formula for Symmetric Random Tensor Networks
Goffredo Chirco, Daniele Oriti, Mingyi Zhang
(Submitted on 27 Nov 2017)
We consider the special case of Random Tensor Networks (RTN) endowed with gauge symmetry constraints on each tensor. We compute the R\enyi entropy for such states and recover the Ryu-Takayanagi (RT) formula in the large bond regime. The result provides first of all an interesting new extension of the existing derivations of the RT formula for RTNs. Moreover, this extension of the RTN formalism brings it in direct relation with (tensorial) group field theories (and spin networks), and thus provides new tools for realizing the tensor network/geometry duality in the context of background independent quantum gravity, and for importing quantum gravity tools in tensor network research.
https://arxiv.org/abs/1711.10861
The time-dependent mass of cosmological perturbations in the hybrid and dressed metric approaches to loop quantum cosmology
Beatriz Elizaga Navascués, Daniel Martín de Blas, Guillermo A. Mena Marugán
(Submitted on 29 Nov 2017)
Loop quantum cosmology has recently been applied in order to extend the analysis of primordial perturbations to the Planck era and discuss the possible effects of quantum geometry on the cosmic microwave background. Two approaches to loop quantum cosmology with admissible ultraviolet behaviour leading to predictions that are compatible with observations are the so-called hybrid and dressed metric approaches. In spite of their similarities and relations, we show in this work that the effective equations that they provide for the evolution of the tensor and scalar perturbations are somewhat different. When backreaction is neglected, the discrepancy appears only in the time-dependent mass term of the corresponding field equations. We explain the origin of this difference, arising from the distinct quantization procedures. Besides, given the privileged role that the Big Bounce plays in loop quantum cosmology, e.g. as a natural instant of time to set initial conditions for the perturbations, we also analyze the positivity of the time-dependent mass when this bounce occurs. We prove that the mass of the tensor perturbations is positive in the hybrid approach when the kinetic contribution to the energy density of the inflaton dominates over its potential, as well as for a considerably large sector of backgrounds around that situation, while this mass is always nonpositive in the dressed metric approach. Similar results are demonstrated for the scalar perturbations in a sector of background solutions that includes the kinetically dominated ones, namely, the mass then is positive for the hybrid approach, whereas it typically becomes negative in the dressed metric case. More precisely, this last statement is strictly valid when the potential is quadratic for values of the inflaton mass that are phenomenologically favored.
https://arxiv.org/abs/1711.10943
The loop quantum cosmology bounce as a Kasner transition
Edward Wilson-Ewing
(Submitted on 29 Nov 2017)
For the Bianchi type I space-time (vacuum or with a massless scalar field), the loop quantum cosmology bounce can be viewed as a rapid transition between two classical solutions, with a simple transformation rule relating the Kasner exponents of the two epochs. This transformation rule can be extended to other Bianchi space-times under the assumption that during the loop quantum cosmology bounce the contribution of the spatial curvature to the Hamiltonian constraint is negligible compared to the kinetic terms. For the vacuum Bianchi type IX space-time there are transformation rules for how each of the parameters characterizing the Kasner epochs change during the bounce. This provides a quantum gravity extension to the Mixmaster dynamics of general relativity, and may have interesting implications for the Belinski-Khalatnikov-Lifshitz conjecture.
Last edited: Feb 18, 2018
3. Dec 23, 2017
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https://arxiv.org/abs/1712.03677
Covariance and Anomaly-freedom in symmetry-reduced self dual models of Loop Quantum Gravity
Jibril Ben Achour, Suddhasattwa Brahma
(Submitted on 11 Dec 2017)
In effective models of loop quantum gravity (LQG), the curvature of the connection in the Hamiltonian constraint is regularised based on the holonomy of the connection, prior to quantization. At this very first step, whether the holonomy-corrected system of "first-class" constraints form a closed algebra such that they still act as generators of gauge transformations, and consequently eliminate the same number of spurious degrees of freedom, is a crucial question which needs to be clarified before dealing with quantum dynamics. In the real Ashtekar-Barbero framework, such holonomy-corrected models have typically a deformed notion of covariance when no local degrees of freedom are involved, and fail to be (gauge-)covariant in models exhibiting local physical degrees of freedom. Recently discovered no-go results in models involving non-perturbative inhomogeneity challenge the possibility of including holonomy modifications in realistic scenarios depicting gravitational collapse of scalar matter or cylindrical gravitational waves. Moreover, it is known that the inclusion of the μ¯-scheme, which implements a coarse-graining procedure at the effective level, leads to additional difficulties in such models. In this article, we show how such conclusions can be by-passed when working in the self dual formulation, i.e. we investigate the fate of covariance in holonomy-corrected models of LQG based on the original self dual Ashtekar formulation. We consider two systems of particular interest: spherically symmetric gravity minimally coupled to a scalar field and (unpolarized) Gowdy cosmology. Both have local degrees of freedom and, therefore, represent midisuperspace models beyond what has been studied in the LQG literature.
https://arxiv.org/abs/1712.06918
On the distribution of the eigenvalues of the area operator in loop quantum gravity
J. Fernando Barbero, Juan Margalef-Bentabol, Eduardo J. S. Villaseñor
(Submitted on 19 Dec 2017)
We study the distribution of the eigenvalues of the area operator in loop quantum gravity concentrating on the part of the spectrum relevant for isolated horizons. We first show that the approximations relying on integer partitions are not sufficient to obtain the asymptotic behaviour of the eigenvalue distribution for large areas. We then develop a method, based on Laplace transforms, that provides a very accurate solution to this problem. The representation that we get is valid for any area and can be used to obtain its asymptotics in the large area limit.
https://arxiv.org/abs/1712.07266
Cosmological evolution as squeezing: a toy model for group field cosmology
Eugene Adjei, Steffen Gielen, Wolfgang Wieland
(Submitted on 19 Dec 2017)
We present a simple model of quantum cosmology based on the group field theory (GFT) approach to quantum gravity. The model is formulated on a subspace of the GFT Fock space for the quanta of geometry, with a fixed volume per quantum. In this Hilbert space, cosmological expansion corresponds to the generation of new quanta. Our main insight is that the evolution of a flat FLRW universe with a massless scalar field can be described on this Hilbert space as squeezing, familiar from quantum optics. As in GFT cosmology, we find that the three-volume satisfies an effective Friedmann equation similar to the one of loop quantum cosmology, connecting the classical contracting and expanding solutions by a quantum bounce. The only free parameter in the model is identified with Newton's constant. We also comment on the possible topological interpretation of our squeezed states. This paper can serve an introduction into the main ideas of GFT cosmology without requiring the full GFT formalism; our results can also motivate new developments in GFT and its cosmological application.
4. Feb 3, 2018
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https://arxiv.org/abs/1801.00273
A new bound on polymer quantization via an opto-mechanical setup
M. Khodadi, K. Nozari, S. Dey, A. Bhat, Mir Faizal
(Submitted on 31 Dec 2017)
The existence of a minimal measurable length as a characteristic length in the Planck scale is one of the main features of quantum gravity and has been widely explored in the context. Various different deformations of spacetime have been employed successfully for the purpose. However, polymer quantization approach is a relatively new and dynamic field towards the quantum gravity phenomenology, which emerges from the symmetric sector of the loop quantum gravity. In this article, we extend the standard ideas of polymer quantization to find a new and tighter bound on the polymer deformation parameter. Our protocol relies on an opto-mechanical experimental setup that was originally proposed in Ref.\cite{ref:Igor} to explore some interesting phenomena by embedding the minimal length into the standard canonical commutation relation. We extend this scheme to probe the \emph{polymer length} deformed canonical commutation relation of the center of mass mode of a mechanical oscillator with a mass around the Planck scale. The method utilizes the novelty of exchanging the relevant mechanical information with a high intensity optical pulse inside an optical cavity. We also demonstrate that our proposal is within the reach of the current technologies and, thus, it could uncover a decent realization of quantum gravitational phenomena thorough a simple table-top experiment.
https://arxiv.org/abs/1801.00768
Emergent de Sitter epoch of the quantum Cosmos
Mehdi Assanioussi, Andrea Dapor, Klaus Liegener, Tomasz Pawłowski
(Submitted on 2 Jan 2018)
The quantum nature of the Big Bang is reexamined in the framework of Loop Quantum Cosmology. The strict application of a regularization procedure to the Hamiltonian, originally developed for the Hamiltonian in loop quantum gravity, leads to a qualitative modification of the bounce paradigm. Quantum gravity effects still lead to a quantum bounce connecting deterministically large classical Universes. However, the evolution features a large epoch of de Sitter Universe, with emergent cosmological constant of Planckian order, smoothly transiting into a flat expanding Universe.
https://arxiv.org/abs/1801.01479
Black Holes as Quantum Gravity Condensates
Daniele Oriti, Daniele Pranzetti, Lorenzo Sindoni
(Submitted on 4 Jan 2018)
We model spherically symmetric black holes within the group field theory formalism for quantum gravity via generalised condensate states, involving sums over arbitrarily refined graphs (dual to 3d triangulations). The construction relies heavily on both the combinatorial tools of random tensor models and the quantum geometric data of loop quantum gravity, both part of the group field theory formalism. Armed with the detailed microscopic structure, we compute the entropy associated with the black hole horizon, which turns out to be equivalently the Boltzmann entropy of its microscopic degrees of freedom and the entanglement entropy between the inside and outside regions. We recover the area law under very general conditions, as well as the Bekenstein-Hawking formula. The result is also shown to be generically independent of any specific value of the Immirzi parameter.
https://arxiv.org/abs/1801.03027
Characteristic Time Scales for the Geometry Transition of a Black Hole to a White Hole from Spinfoams
Marios Christodoulou, Fabio D'Ambrosio
(Submitted on 9 Jan 2018)
Quantum fluctuations of the metric provide a decay mechanism for black holes, through a transition to a white hole geometry. Old perplexing results by Ambrus and H\'aj\'i\v{c}ek and more recent results by Barcel\'o, Carballo--Rubio and Garay, indicate a characteristic time scale of this process that scales linearly with the mass of the collapsed object. We compute the characteristic time scales involved in the quantum process using Lorentzian Loop Quantum Gravity amplitudes, corroborating these results but reinterpreting and clarifying their physical meaning. We first review and streamline the classical set up, and distinguish and discuss the different time scales involved. We conclude that the aforementioned results concern a time scale that is different from the lifetime, the latter being the much longer time related to the probability of the process to take place. We recover the exponential scaling of the lifetime in the mass, as expected from na\"ive semiclassical arguments for the probability of a tunneling phenomenon to occur.
https://arxiv.org/abs/1801.03353
Bohmian quantum gravity and cosmology
Nelson Pinto-Neto, Ward Struyve
(Submitted on 10 Jan 2018)
Quantum gravity aims to describe gravity in quantum mechanical terms. How exactly this needs to be done remains an open question. Various proposals have been put on the table, such as canonical quantum gravity, loop quantum gravity, string theory, etc. These proposals often encounter technical and conceptual problems. In this chapter, we focus on canonical quantum gravity and discuss how many conceptual problems, such as the measurement problem and the problem of time, can be overcome by adopting a Bohmian point of view. In a Bohmian theory (also called pilot-wave theory or de Broglie-Bohm theory, after its originators de Broglie and Bohm), a system is described by certain variables in space-time such as particles or fields or something else, whose dynamics depends on the wave function. In the context of quantum gravity, these variables are a space-time metric and suitable variable for the matter fields (e.g., particles or fields). In addition to solving the conceptual problems, the Bohmian approach yields new applications and predictions in quantum cosmology. These include space-time singularity resolution, new types of semi-classical approximations to quantum gravity, and approximations for quantum perturbations moving in a quantum background.
https://arxiv.org/abs/1801.06017
Spin networks on adiabatic quantum computer
Jakub Mielczarek
(Submitted on 18 Jan 2018)
The article is addressing a possibility of implementation of spin network states on adiabatic quantum computer. The discussion is focused on application of currently available technologies and analyzes a concrete example of D-Wave machine. A class of simple spin network states which can be implemented on the Chimera graph architecture of the D-Wave quantum processor is introduced. However, extension beyond the currently available quantum processor topologies is required to simulate more sophisticated spin network states, which may inspire development of new generations of adiabatic quantum computers. A possibility of simulating Loop Quantum Gravity is discussed and a method of solving a graph non-changing scalar (Hamiltonian) constraint with the use of adiabatic quantum computations is proposed.
https://arxiv.org/abs/1801.07313
Towards Cosmological Dynamics from Loop Quantum Gravity
Bao-Fei Li, Parampreet Singh, Anzhong Wang
(Submitted on 22 Jan 2018 (v1), last revised 1 Feb 2018 (this version, v2))
We present a systematic study of the cosmological dynamics resulting from an effective Hamiltonian, recently derived in loop quantum gravity using Thiemann's regularization and earlier obtained in loop quantum cosmology (LQC) by keeping the Lorentzian term explicit in the Hamiltonian constraint. We show that quantum geometric effects result in higher than quadratic corrections in energy density in comparison to LQC causing a non-singular bounce. Dynamics can be described by the Hamilton's or the Friedmann-Raychaudhuri equations, but the map between the two descriptions is not one-to-one. A careful analysis resolves the tension on symmetric versus asymmetric bounce in this model, showing that the bounce must be asymmetric and symmetric bounce is physically inconsistent, in contrast to the standard LQC. In addition, the current observations only allow a scenario where the pre-bounce branch is asymptotically de Sitter, similar to a quantization of the Schwarzschild interior in LQC, and the post-bounce branch yields the classical general relativity. For a quadratic potential, we find that a slow-roll inflation generically happens after the bounce, which is quite similar to what happens in LQC.
Last edited: Feb 17, 2018
5. Feb 17, 2018
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https://arxiv.org/abs/1802.02382
Space and Time in Loop Quantum Gravity
Carlo Rovelli
(Submitted on 7 Feb 2018)
Quantum gravity is expected to require modifications of the notions of space and time. I discuss and clarify how this happens in Loop Quantum Gravity.
https://arxiv.org/abs/1802.02661
Gravitation in terms of observables
Rodolfo Gambini, Jorge Pullin
(Submitted on 7 Feb 2018)
In the 1960's, Mandelstam proposed a new approach to gauge theories and gravity based on loops. The program for gauge theories was completed for Yang--Mills theories by Gambini and Trias in the 1980's. Gauge theories could be understood as representations of certain group: the group of loops. The same formalism could not be implemented at that time for the gravitational case. Here we would like to propose an extension to the case of gravity. The resulting theory is described in terms of loops and open paths and can provide the underpinning for a new quantum representation for gravity distinct from the one used in loop quantum gravity or string theory. In it, space-time points are emergent entities that would only have quasi-classical status. The formulation may be given entirely in terms of Dirac observables that form a complete set of gauge invariant functions that completely define the Riemannian geometry of the spacetime. At the quantum level this formulation will lead to a reduced phase space quantization free of any constraints.
6. Mar 11, 2018
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https://arxiv.org/abs/1802.06251
Radial gauge fixing of first order gravity
Emanuele Alesci, Costantino Pacilio, Daniele Pranzetti
(Submitted on 17 Feb 2018)
We consider the first order connection formulation of 4D general relativity in the radial gauge. We show how the partial gauge fixing of the phase space canonical coordinates leads to the appearance of second class constraints in the theory. We employ the gauge unfixing procedure in order to successfully complete the Dirac treatment of the system. While equivalent to the inversion of the Dirac matrix, the gauge unfixing allows us to work directly with the reduced phase space and the ordinary Poisson bracket. At the same time, we explicitly derive the new set of residual first class constraints preserving the partial gauge fixing, which are linear combinations of the original constraints, and these turn out to contain nonlinear terms. While providing an explicit example of how to consistently recast general relativity in a given partial gauge, the main motivation of this classical analysis is the application of the Quantum Reduced Loop Gravity program to a Schwarzschild black hole geometry.
https://arxiv.org/abs/1802.07033
The constraint algebra in Smolins' G→0 limit of 4d Euclidean Gravity
(Submitted on 20 Feb 2018)
Smolin's generally covariant GNewton→0 limit of 4d Euclidean gravity is a useful toy model for the study of the constraint algebra in Loop Quantum Gravity. In particular, the commutator between its Hamiltonian constraints has a metric dependent structure function. While a prior LQG like construction of non-trivial anomaly free constraint commutators for the model exists, that work suffers from two defects. First, Smolin's remarks on the inability of the quantum dynamics to generate propagation effects apply. Second, the construction only yields the action of a single Hamiltonian constraint together with the action of its commutator through a continuum limit of corresponding discrete approximants; the continuum limit of a product of 2 or more constraints does not exist. Here, we incorporate changes in the quantum dynamics through structural modifications in the choice of discrete approximants to the quantum Hamiltonian constraint. The new structure is motivated by that responsible for propagation in an LQG like quantization of Paramaterized Field Theory and significantly alters the space of physical states. We study the off shell constraint algebra of the model in the context of these structural changes and show that the continuum limit action of multiple products of Hamiltonian constraints is (a) supported on an appropriate domain of states (b) yields anomaly free commutators between pairs of Hamiltonian constraints and (c) is diffeomorphism covariant. Many of our considerations seem robust enough to be applied to the setting of 4d Euclidean gravity.
https://arxiv.org/abs/1802.09114
Loop Quantum Corrected Einstein Yang-Mills Black Holes
Mason Protter, Andrew DeBenedictis
(Submitted on 26 Feb 2018)
In this paper we study the homogeneous interiors of black holes possessing SU(2) Yang-Mills fields subject to corrections inspired by loop quantum gravity. The systems studied possess both magnetic and induced electric Yang-Mills fields. We consider the system of equations both with and without Wilson loop corrections to the Yang-Mills potential. The structure of the Yang-Mills Hamiltonian along with the restriction to homogeneity allows for an anomaly free effective quantization. In particular we study the bounce which replaces the classical singularity and the behavior of the Yang-Mills fields in the quantum corrected interior, which possesses topology R×S2. Beyond the bounce the magnitude of the Yang-Mills electric field asymptotically grows monotonically. This results in an ever expanding R sector even though the two-sphere volume is asymptotically constant. The results are similar with and without Wilson loop corrections on the Yang-Mills potential.
https://arxiv.org/abs/1803.00332
Geometry Transition in Covariant Loop Quantum Gravity
Christodoulou Marios
(Submitted on 1 Mar 2018)
In this manuscript we present a calculation of a physical observable in a non-perturbative quantum gravitational physical process from covariant Loop Quantum Gravity. The process regards the transition of a trapped region to an anti--trapped region, treated as a quantum geometry transition akin to gravitational tunneling. Figuratively speaking, this is a quantum transition of a black hole to a white hole. The physical observables are the characteristic timescales in which the process takes place.
After an introduction, we begin with two chapters that review, define and extend main tools relevant to Lorentzian spinfoams and their semiclassical limit. We then dedicate a chapter to the classical exterior spacetime, which provides the setup for the problem. In the last two chapters, we arrive at an explicit, analytically well-defined and finite expression for a transition amplitude describing this process and use the semiclassical approximation to estimate the relevant amplitudes for an arbitrary choice of boundary conditions. We conclude that the transition is predicted to be allowed by LQG, with a characteristic duration that is linear in the mass, when the process takes place. The probability for the process to take place is exponentially suppressed but non-zero, resulting to a long lifetime.
Comments: PhD thesis submitted for the degree of Doctor in Theoretical and Mathematical Physics. Defended at the Center for Theoretical Physics/CNRS/Aix-Marseille University, the 23rd of October 2017. The manuscript is written in English and begins with a short summary in French
https://arxiv.org/abs/1803.01119
Effective line elements and black-hole models in canonical (loop) quantum gravity
Martin Bojowald, Suddhasattwa Brahma, Dong-han Yeom
(Submitted on 3 Mar 2018)
Canonical quantization is often used to suggest new effects in quantum gravity, in the dynamics as well as the structure of space-time. Usually, possible phenomena are first seen in a modified version of the classical dynamics, for instance in an effective Friedmann equation, but there should also be implications for a modified space-time structure. Quantum space-time effects, however, are often ignored in this setting because they are not obvious: they require a careful analysis of gauge transformations and the anomaly problem. It is shown here how modified space-time structures and effective line elements can be derived unambiguously, provided an off-shell anomaly-free system of modified constraints exists. The resulting effective line elements reveal signature change as an inescapable consequence of non-classical gauge transformations in the presence of holonomy modifications. The general framework is then specialized to black-hole models in loop quantum gravity. In contrast to previous studies, a self-consistent space-time structure is taken into account, leading to a new picture of black-hole interiors.
https://arxiv.org/abs/1803.01152
Loop quantum deformation of a Schwarzschild black hole: an effective metric
Jibril Ben Achour, Frédéric Lamy, Hongguang Liu, Karim Noui
(Submitted on 3 Mar 2018)
We consider the modified Einstein equations obtained in the framework of effective loop quantum gravity for spherically symmetric space-times. When one takes into account (only point-wise holonomy) quantum corrections, the deformation of Einstein equations is parametrized by a function f(x) of one variable . We solve explicitly these equations for static black holes and find the effective metric in the region inside the black hole for any f(x). When f(x) is the usual function used in loop quantum gravity, the effective metric presents strong similarities with the Reissner-Nordstrom metric (with a regular trapped region): it tends to the expected Schwarzschild metric when one approaches the outer horizon, and the inner horizon replaces the original Schwarzschild singularity. We discuss the possibility to extend the solution outside the trapped region, and possible phenomenological consequences of our results.
https://arxiv.org/abs/1803.02577
The Bronstein hypercube of quantum gravity
Daniele Oriti
(Submitted on 7 Mar 2018 (v1), last revised 8 Mar 2018 (this version, v2))
We argue for enlarging the traditional view of quantum gravity, based on "quantizing GR", to include explicitly the non-spatiotemporal nature of the fundamental building blocks suggested by several modern quantum gravity approaches (and some semi-classical arguments), and to focus more on the issue of the emergence of continuum spacetime and geometry from their collective dynamics. We also discuss some recent developments in quantum gravity research, aiming at realising these ideas, in the context of group field theory, random tensor models, simplicial quantum gravity, loop quantum gravity, spin foam models.
Last edited: Mar 11, 2018
7. Mar 29, 2018
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https://arxiv.org/abs/1803.04374
Spacetime is as spacetime does
Vincent Lam, Christian Wuthrich
(Submitted on 12 Mar 2018)
Theories of quantum gravity generically presuppose or predict that the reality underlying relativistic spacetimes they are describing is significantly non-spatiotemporal. On pain of empirical incoherence, approaches to quantum gravity must establish how relativistic spacetime emerges from their non-spatiotemporal structures. We argue that in order to secure this emergence, it is sufficient to establish that only those features of relativistic spacetimes functionally relevant in producing empirical evidence must be recovered. In order to complete this task, an account must be given of how the more fundamental structures instantiate these functional roles. We illustrate the general idea in the context of causal set theory and loop quantum gravity, two prominent approaches to quantum gravity.
https://arxiv.org/abs/1803.06963
Interpreting Theories without a Spacetime
Sebastian De Haro, Henk De Regt
(Submitted on 19 Mar 2018)
In this paper we have two aims: first, to draw attention to the close connexion between interpretation and scientific understanding; second, to give a detailed account of how theories without a spacetime can be interpreted, and so of how they can be understood.
In order to do so, we of course need an account of what is meant by a theory without a spacetime': which we also provide in this paper.
We describe three tools, used by physicists, aimed at constructing interpretations which are adequate for the goal of understanding. We analyse examples from high-energy physics illustrating how physicists use these tools to construct interpretations and thereby attain understanding. The examples are: the 't Hooft approximation of gauge theories, random matrix models, causal sets, loop quantum gravity, and group field theory.
https://arxiv.org/abs/1803.09653
Mimetic Loop Quantum Cosmology
Jaume de Haro, Llibert Aresté Saló, Supriya Pan
(Submitted on 26 Mar 2018)
Considering as usual that the underlying geometry of our universe is well described by the spatially flat Friedmann-Lemaitre-Robertson-Walker line element, we show that the background of holonomy corrected Loop Quantum Cosmology (LQC) is equivalent to a simple modified version of the mimetic gravity. We also analyze the scalar and tensor perturbations of this modified mimetic model from which we find that, at the level of scalar perturbations, the modified mimetic model is exactly equivalent to the LQC while at the level of tensor perturbations, the modified mimetic gravity is indistinguishable from the General Relativity.
https://arxiv.org/abs/1803.10289
Emergence of Spacetime in a restricted Spin-foam model
Sebastian Steinhaus, Johannes Thürigen
(Submitted on 27 Mar 2018)
The spectral dimension has proven to be a very informative observable to understand the properties of quantum geometries in approaches to quantum gravity. In loop quantum gravity and its spin foam description, it has not been possible so far to calculate the spectral dimension of spacetime. As a first step towards this goal, here we determine the spacetime spectral dimension in the simplified spin foam model restricted to hypercuboids. Using Monte Carlo methods we compute the spectral dimension for state sums over periodic spin foam configurations on infinite lattices. For given periodicity, i.e. number of degrees of freedom, we find a range of scale where an intermediate spectral dimension between 0 and 4 can be found, continuously depending on the parameter of the model. Under an assumption on the statistical behaviour of the Laplacian we can explain these results analytically. This allows us to take the thermodynamic limit of large periodicity and find a phase transition from a regime of effectively 0-dimensional to 4-dimensional spacetime. At the point of phase transition, dynamics of the model are scale invariant which can be seen as restoration of diffeomorphism invariance of flat space. Considering the spectral dimension as an order parameter for renormalization we find a renormalization group flow to this point as well. Being the first instance of an emergence of 4-dimensional spacetime in a spin foam model, the properties responsible for this result seem to be rather generic. We thus expect similar results for more general, less restricted spin foam models.
8. Apr 4, 2018
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https://arxiv.org/abs/1803.10807
Hamiltonian structure and connection-dynamics of Weyl gravity
Qian Chen, Yongge Ma
(Submitted on 28 Mar 2018)
A crucial property of Weyl gravity is its conformal invariance. It is shown how this gauge symmetry is exactly reflected by the two constraints in the Hamiltonian framework. Since the spatial 3-metric is one of the configuration variables. The phase space of Weyl gravity can be extended to include internal gauge freedom by triad formalism. Moreover, by a canonical transformation, we obtain a new Hamiltonian formulation of Weyl gravity with an SU(2) connection as one of its configuration variables. This connection dynamical formalism lays a foundation to quantize Weyl gravity nonperturbatively by applying the method of loop quantum gravity.
https://arxiv.org/abs/1803.10858
Is the average of timelike singularities really spacelike?
Eugenio Bianchi, Hal M. Haggard
(Submitted on 28 Mar 2018)
Due to quantum fluctuations, a non-rotating black hole should be the average over an ensemble of black hole geometries with angular momentum. This observation invites the question: Is the average of timelike singularities really spacelike? We use the Bekenstein-Hawking entropy formula to introduce a microcanonical ensemble for spin fluctuations and argue that the onset of quantum gravity is always spacelike. We also hint at the possibility of an observational test.
https://arxiv.org/abs/1803.10809
Volume and Boundary Face Area of a Regular Tetrahedron in a Constant Curvature Space
Omar Nemoul, Noureddine Mebarki
(Submitted on 23 Mar 2018)
An example of the volume and boundary face area of a curved polyhedron for the case of regular spherical and hyperbolic tetrahedron is discussed. An exact formula is explicitly derived as a function of the scalar curvature and the edge length. This work can be used in loop quantum gravity and Regge calculus in the context of a non-vanishing cosmological constant.
https://arxiv.org/abs/1804.00012
Effective universality in quantum gravity
Astrid Eichhorn, Peter Labus, Jan M. Pawlowski, Manuel Reichert
(Submitted on 30 Mar 2018)
We investigate the asymptotic safety scenario for a scalar-gravity system. This system contains two avatars of the dynamical Newton coupling, a gravitational self-coupling and a scalar-graviton coupling. We uncover an effective universality for the dynamical Newton coupling on the quantum level: its momentum-dependent avatars are in remarkable quantitative agreement in the scaling regime of the UV fixed point. For the background Newton coupling, this effective universality is not present, but qualitative agreement remains.
https://arxiv.org/abs/1804.00960
Singularity from star collapse, torsion and asymptotic safety of gravity
Abhishek Majhi
(Submitted on 3 Apr 2018)
A star of mass greater than the Chandrasekhar limit is believed to undergo a gravitational collapse to form a singularity, owing to Hawking-Penrose singularity theorem which is based on the Raychaudhuri equation in the absence of torsion. We argue that the spin-aspect of matter can lead to the evasion of singularity, caused by its mass-aspect, via torsion in asymptotically safe gravity.
https://arxiv.org/abs/1804.01003
An area rescaling ansatz and black hole entropy from loop quantum gravity
Abhishek Majhi
(Submitted on 3 Apr 2018)
Considering the possibility of renormalization' of the gravitational constant on the horizon, leading to a dependence on the level of the associated Chern-Simons theory, a rescaled area spectrum is proposed for the non-rotating black hole horizon in loop quantum gravity. The statistical mechanical calculation leading to the entropy provides a unique choice of the rescaling function for which the Bekenstein-Hawking area law is yielded without the need to choose the Barbero-Immirzi parameter (γ). γ is determined by studying the limit in which the renormalized' gravitational constant on the horizon asymptotically approaches the bare' value. Unlike the usual, much criticized, practice of choosing γ just for the sake of the entropy matching the area law, its value is now rather determined by a physical consistency requirement.
9. Apr 10, 2018
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https://arxiv.org/abs/1804.02184
The emergence of space and time
Christian Wuthrich
(Submitted on 6 Apr 2018)
Research in quantum gravity strongly suggests that our world in not fundamentally spatiotemporal, but that spacetime may only emerge in some sense from a non-spatiotemporal structure, as this paper illustrates in the case of causal set theory and loop quantum gravity. This would raise philosophical concerns regarding the empirical coherence and general adequacy of theories in quantum gravity. If it can be established, however, that spacetime emerges in the appropriate circumstances and how all its relevant aspects are explained in fundamental non-spatiotemporal terms, then the challenge is fully met. It is argued that a form of spacetime functionalism offers the most promising template for this project.
https://arxiv.org/abs/1804.02428
A predictive framework for quantum gravity and black hole to white hole transition
Robert Oeckl (CCM-UNAM)
(Submitted on 6 Apr 2018)
The apparent incompatibility between quantum theory and general relativity has long hampered efforts to find a quantum theory of gravity. The recently proposed positive formalism for quantum theory purports to remove this incompatibility. We showcase the power of the positive formalism by applying it to the black hole to white hole transition scenario that has been proposed as a possible effect of quantum gravity. We show how the characteristic observable of this scenario, the bounce time, can be predicted within the positive formalism, while a traditional S-matrix approach fails at this task. Our result also involves a conceptually novel use of positive operator valued measures.
https://arxiv.org/abs/1804.02262
Cosmological consequences of Quantum Gravity proposals
Marco de Cesare
(Submitted on 6 Apr 2018)
In this thesis, we study the implications of Quantum Gravity models for the dynamics of spacetime and the ensuing departures from classical General Relativity. The main focus is on cosmological applications, particularly the impact of quantum gravitational effects on the dynamics of a homogenous and isotropic cosmological background. Our interest lies in the consequences for the evolution of the early universe and singularity resolution, as well as in the possibility of providing an alternative explanation for dark matter and dark energy in the late universe.
The thesis is divided into two main parts, dedicated to alternative (and complementary) ways of tackling the problem of Quantum Gravity. The first part is concerned with cosmological applications of background independent approaches to Quantum Gravity, both in the context of loop quantisation and in quantum geometrodynamics. Particularly relevant in this work is the Group Field Theory approach, which we use to study the effective dynamics of the emergent universe from a full theory of Quantum Gravity (i.e. without symmetry reduction).
In the second part, modified gravity theories are introduced as tools to provide an effective description of quantum gravitational effects, e.g. by introducing new degrees of freedom and symmetries. Particularly relevant in this respect is local conformal invariance, which finds a natural realisation in the framework of Weyl geometry. We build a modified theory of gravity based on such symmetry principle, and argue that new fields in the extended gravitational sector may play the role of dark matter. New degrees of freedom are also natural in models with varying fundamental constants', which we examine critically.
Finally, we discuss prospects for future work and point at directions for the derivation of realistic cosmological models from Quantum Gravity candidates.
https://arxiv.org/abs/1804.02560
Quantum gravity for piecewise flat spacetimes
Aleksandar Mikovic, Marko Vojinovic
(Submitted on 7 Apr 2018)
We describe a theory of quantum gravity which is based on the assumption that the spacetime structure at small distances is given by a piecewise linear (PL) 4-manifold corresponding to a triangulation of a smooth 4-manifold. The fundamental degrees of freedom are the edge lengths of the triangulation. One can work with finitely many edge lengths, so that the corresponding Regge path integral can be made finite by using an appropriate path-integral measure. The semi-classical limit is computed by using the effective action formalism, and the existence of a semi-classical effective action restricts the choice of the path-integral measure. The classical limit is given by the Regge action, so that one has a quantum gravity theory for a piecewise-flat general relativity. By using the effective action formalism we show that the observed value of the cosmological constant can be recovered from the effective cosmological constant. When the number of 4-simplices in the spacetime triangulation is large, then the PL effective action is well approximated by a quantum field theory effective action with a physical cutoff determined by the smallest edge length.
10. Jul 13, 2018
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https://arxiv.org/abs/1804.00023
Renormalization in symmetry restricted spin foam models with curvature
Benjamin Bahr, Giovanni Rabuffo, Sebastian Steinhaus
(Submitted on 30 Mar 2018 (v1), last revised 17 Apr 2018 (this version, v2))
We study the renormalization group flow of the Euclidean Engle-Pereira-Rovelli-Livine and Freidel-Krasnov (EPRL-FK) spin foam model in its asymptotic limit. The vertex amplitude is deformed to include a cosmological constant term. The state sum is reduced to describe a foliated spacetime whose spatial slices are flat, isotropic and homogeneous. The model admits a non-vanishing extrinsic curvature whereas the scale factor can expand or contract at successive time steps.
The reduction of degrees of freedom allows a numerical evaluation of certain geometric observables on coarser and finer discretizations. Their comparison defines the renormalization group (RG) flow of the model in the parameters (α,Λ,G). We first consider the projection of the RG flow along the α direction, which shows a UV-attractive fixed point. Then, we extend our analysis to two- and three-dimensional parameter spaces. Most notably, we find the indications of a fixed point in the (α,Λ,G) space showing one attractive and two repulsive directions.
https://arxiv.org/abs/1804.04147
White-hole dark matter and the origin of past low-entropy
Carlo Rovelli, Francesca Vidotto
(Submitted on 11 Apr 2018 (v1), last revised 21 Apr 2018 (this version, v2))
Recent results on the end of black hole evaporation give new weight to the hypothesis that a component of dark matter could be formed by remnants of evaporated black holes: stable Planck-size white holes with a large interior. The expected lifetime of these objects is consistent with their production at reheating. But remnants could also be pre-big bang relics in a bounce cosmology, and this possibility has strong implications on the issue of the source of past low entropy: it could realise a perspectival interpretation of past low entropy. The ideas briefly presented in this essay are developed in forthcoming papers.
https://arxiv.org/abs/1805.08257
Probing the Shape of Quantum Surfaces: the Quadrupole Moment Operator
Christophe Goeller, Etera R. Livine
(Submitted on 21 May 2018)
The standard toolkit of operators to probe quanta of geometry in loop quantum gravity consists in area and volume operators as well as holonomy operators. New operators have been defined, in the U(N) framework for intertwiners, which allow to explore the finer structure of quanta of geometry. However these operators do not carry information on the global shape of the intertwiners. Here we introduce dual multipole moments for continuous and discrete surfaces, defined through the normal vector to the surface, taking special care to maintain parametrization invariance. These are raised to multipole operators probing the shape of quantum surfaces. Further focusing on the quadrupole moment, we show that it appears as the Hessian matrix of the large spin Gaussian approximation of coherent intertwiners, which is the standard method for extracting the semi-classical regime of spinfoam transition amplitudes. This offers an improvement on the usual loop quantum gravity techniques, which mostly focus on the volume operator, in the perspective of modeling (quantum) gravitational waves as shape fluctuations waves propagating on spin network states.
https://arxiv.org/abs/1804.08643
Loop quantum gravity and the continuum
Wolfgang Wieland
(Submitted on 23 Apr 2018)
In this paper, we will make an attempt to clarify the relation between three-dimensional euclidean loop quantum gravity with vanishing cosmological constant and quantum field theory in the continuum. We will argue, in particular, that in three spacetime dimensions the discrete spectra for the geometric boundary observables that we find in loop quantum gravity can be understood from the quantisation of a conformal boundary field theory in the continuum without ever introducing spin networks or triangulations of space. At a technical level, the starting point is the Hamiltonian formalism for general relativity in regions with boundaries at finite distance. At these finite boundaries, we choose specific Robin boundary conditions (the boundary is a minimal surface) that are derived from a boundary field theory for an SU(2) boundary spinor, which is minimally coupled to the spin connection in the bulk. The resulting boundary equations of motion define a conformal field theory with vanishing central charge. We will quantise this boundary field theory and show that the length of a one-dimensional cross section of the boundary has a discrete spectrum. In addition, we will introduce a new class of coherent states, study the quasi-local observables that generate the quasi-local Virasoro algebra and discuss some strategies to evaluate the partition function of the theory.
https://arxiv.org/abs/1805.08644
On the Hamiltonian operator in loop quantum gravity
Cong Zhang, Jerzy Lewandowski, Yongge Ma
(Submitted on 22 May 2018 (v1), last revised 23 May 2018 (this version, v2))
Although the physical Hamiltonian operator can be constructed in the deparameterized model of loop quantum gravity coupled to a scalar field, its property is still unknown. This open issue is attacked in this paper by considering an operator H^v representing the square of the physical Hamiltonian operator acting nontrivially on two-valent spin networks. The Hilbert space Hv preserved by the graphing changing operator H^v is consist of spin networks with a single two-valent non-degenerate vertex. The matrix element of H^v are explicitly worked out in a suitable basis. It turns out that the operator H^v is essentially self-adjoint, which implies a well-defined physical Hamiltonian operator in Hv for the deparameterized model.
https://arxiv.org/abs/1804.11101
The Tensor Track V: Holographic Tensors
Nicolas Delporte, Vincent Rivasseau
(Submitted on 30 Apr 2018)
We review the fast developing subject of tensor models for the NAdS2/NCFT1 holographic correspondence. We include a brief review of the Sachdev-Ye-Kitaev (SYK) model and then focus on the associated quantum mechanical tensor models (GW and CTKT). We examine their main features and how they compare with SYK. To end, we discuss different extensions: the large D limit of matrix-tensor models, the large N expansion of symmetric/antisymmetric tensors, the use of probes, the construction of a bilocal action for tensors, some attempts to extend the above models to higher dimensions and a proposal to break the tensor symmetry.
https://arxiv.org/abs/1805.01619
Functional Renormalization Group analysis of rank 3 tensorial group field theory: The full quartic invariant truncation
Joseph Ben Geloun, Tim A. Koslowski, Daniele Oriti, Antonio D. Pereira
(Submitted on 4 May 2018)
In this paper we consider the complete momentum-independent quartic order truncation for the effective average action of a real Abelian rank 3 tensorial group field theory. This complete truncation includes non-melonic as well as double-trace interactions. In the usual functional renormalization group perspective, the inclusion of more operators that belong to the underlying theory space corresponds to an improvement of the truncation of the effective average action. We show that the inclusion of non-melonic and double-trace operators in the truncation brings subtleties. In particular, we discuss the assignment of scaling dimensions to the non-melonic sector and how the inclusion of double-trace operators considerably changes the results for critical exponents when they are not included. We argue that this is not a particular problem of the present model by comparing the results with a pure tensor model. We discuss how these issues should be investigated in future work.
https://arxiv.org/abs/1805.03099
The separate universe framework in group field theory condensate cosmology
Florian Gerhardt, Daniele Oriti, Edward Wilson-Ewing
(Submitted on 8 May 2018)
We use the separate universe framework to study cosmological perturbations within the group field theory formalism for quantum gravity, based on multi-condensate quantum states. Working with a group field theory action for gravity minimally coupled to four scalar fields that can act as a set of relational clock and rods, we argue that these multi-condensate states correspond to cosmological space-times with small long-wavelength scalar perturbations. Equations of motion for the cosmological perturbations are derived, which in the classical limit agree with the standard results of general relativity and also include quantum gravity corrections that become important when the space-time curvature approaches the Planck scale.
https://arxiv.org/abs/1805.03224
Pre-big-bang black-hole remnants and the past low entropy
Carlo Rovelli, Francesca Vidotto
(Submitted on 8 May 2018)
Dark matter could be composed by black-hole remnants formed before the big-bang era in a bouncing cosmology. This hypothetical scenario has major implications on the issue of the arrow of time: it would upset a common attribution of past low entropy to the state of the geometry, and provide a concrete realisation to the perspectival interpretation of past low entropy.
https://arxiv.org/abs/1805.03872
Small black/white hole stability and dark matter
Carlo Rovelli, Francesca Vidotto
(Submitted on 10 May 2018)
We show that the expected lifetime of white holes formed as remnants of evaporated black holes is consistent with their production at reheating. We give a simple quantum description of these objects and argue that a quantum superposition of black and white holes with large interiors is stable, because it is protected by the existence of a minimal eigenvalue of the area, predicted by Loop Quantum Gravity. These two results support the hypothesis that a component of dark matter could be formed by small black hole remnants.
https://arxiv.org/abs/1806.00456
Towards a dual spin network basis for (3+1)d lattice gauge theories and topological phases
Clement Delcamp, Bianca Dittrich
(Submitted on 1 Jun 2018)
Using a recent strategy to encode the space of flat connections on a three-manifold with string-like defects into the space of flat connections on a so-called 2d Heegaard surface, we propose a novel way to define gauge invariant bases for (3+1)d lattice gauge theories and gauge models of topological phases. In particular, this method reconstructs the spin network basis and yields a novel dual spin network basis. While the spin network basis allows to interpret states in terms of electric excitations, on top of a vacuum sharply peaked on a vanishing electric field, the dual spin network basis describes magnetic (or curvature) excitations, on top of a vacuum sharply peaked on a vanishing magnetic field (or flat connection). This technique is also applicable for manifolds with boundaries. We distinguish in particular a dual pair of boundary conditions, namely of electric type and of magnetic type. This can be used to consider a generalization of Ocneanu's tube algebra in order to reveal the algebraic structure of the excitations associated with certain 3d manifolds.
https://arxiv.org/abs/1807.03066
Numerical methods for EPRL spin foam transition amplitudes and Lorentzian recouping theory
Pietro Dona, Giorgio Sarno
(Submitted on 9 Jul 2018)
The intricated combinatorial structure and the non-compactness of the Lorentz group have always made the computation of SL(2,C) EPRL spin foam transition amplitudes a very hard and resource demanding task. With \texttt{sl2cfoam} we provide a C-coded library for the evaluation of the Lorentzian EPRL vertex amplitude. We provide a tool to compute the Lorentzian EPRL 4-simplex vertex amplitude in the intertwiner basis and some utilities to evaluate SU(2) invariants, booster functions and SL(2,C) Clebsch-Gordan coefficients. We discuss the data storage, parallelizations, time, and memory performances and possible future developments.
https://arxiv.org/abs/1807.03334
An introduction to the SYK model
(Submitted on 9 Jul 2018)
These notes are a short introduction to the Sachdev-Ye-Kitaev model. We discuss: SYK and tensor models as a new class of large N quantum field theories, the near-conformal invariance in the infrared, the computation of correlation functions, generalizations of SYK, and applications to AdS/CFT and strange metals.
https://arxiv.org/abs/1807.02501
Tensor networks as path integral geometry
Ashley Milsted, Guifre Vidal
(Submitted on 6 Jul 2018)
In the context of a quantum critical spin chain whose low energy physics corresponds to a conformal field theory (CFT), it was recently demonstrated [A. Milsted G. Vidal, arXiv:1805.12524] that certain classes of tensor networks used for numerically describing the ground state of the spin chain can also be used to implement (discrete, approximate versions of) conformal transformations on the lattice. In the continuum, the same conformal transformations can be implemented through a CFT path integral on some curved spacetime. Based on this observation, in this paper we propose to interpret the tensor networks themselves as a path integrals on curved spacetime. This perspective assigns (a discrete, approximate version of) a geometry to the tensor network, namely that of the underlying curved spacetime.
11. Aug 17, 2018
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https://arxiv.org/abs/1807.06098
Spin-foam model for gravity coupled to massless scalar field
Marcin Kisielowski, Jerzy Lewandowski
(Submitted on 16 Jul 2018)
A spin-foam model is derived from the canonical model of Loop Quantum Gravity coupled to a massless scalar field. We generalized to the full theory the scheme first proposed in the context of Loop Quantum Cosmology by Ashtekar, Campiglia and Henderson, later developed by Henderson, Rovelli, Vidotto and Wilson-Ewing.
https://arxiv.org/abs/1807.06354
Hamiltonian analysis of the BFCG formulation of General Relativity
Aleksandar Mikovic, Miguel A. Oliveira, Marko Vojinovic
(Submitted on 17 Jul 2018)
We perform the complete Hamiltonian analysis of the BFCG action for General Relativity. We determine all the constraints of the theory and classify them into the first-class and the second-class constraints. We also show how the canonical formulation of BFCG General Relativity reduces to the Einstein-Cartan and triad canonical formulations. The reduced phase space analysis also gives a 2-connection which is suitable for the construction of a spin-foam basis which will be a categorical generalization of the spin-network basis from Loop Quantum Gravity.
https://arxiv.org/abs/1807.06848
Deformations of Lorentzian Polyhedra: Kapovich-Millson phase space and SU(1,1) Intertwiners
Etera R. Livine
(Submitted on 18 Jul 2018)
We describe the Lorentzian version of the Kapovitch-Millson phase space for polyhedra with N faces. Starting with the Schwinger representation of the su(1,1) Lie algebra in terms of a pair of complex variables (or spinor), we define the phase space for a space-like vectors in the three-dimensional Minkowski space R1,2. Considering N copies of this space, quotiented by a closure constraint forcing the sum of those 3-vectors to vanish, we obtain the phase space for Lorentzian polyhedra with N faces whose normal vectors are space-like, up to Lorentz transformations. We identify a generating set of SU(1,1)-invariant observables, whose flow by the Poisson bracket generate both area-preserving and area-changing deformations. We further show that the area-preserving observables form a glN(R) Lie algebra and that they generate a GLN(R) action on Lorentzian polyhedra at fixed total area. That action is cyclic and all Lorentzian polyhedra can be obtained from a totally squashed polyhedron (with only two non-trivial faces) by a GLN(R) transformation. All those features carry on to the quantum level, where quantum Lorentzian polyhedra are defined as SU(1,1) intertwiners between unitary SU(1,1)-representations from the principal continuous series. Those SU(1,1)-intertwiners are the building blocks of spin network states in loop quantum gravity in 3+1 dimensions for time-like slicing and the present analysis applies to deformations of the quantum geometry of time-like boundaries in quantum gravity, which is especially relevant to the study of quasi-local observables and holographic duality.
https://arxiv.org/abs/1807.10704
Gravitational Fluctuations as an Alternative to Inflation
Herbert W. Hamber, Lu Heng Sunny Yu
(Submitted on 27 Jul 2018)
In this work we explore an explanation for the galaxy power spectrum P(k) based on the non-perturbative quantum field-theoretical treatment of Einstein gravity, instead of one based on inflation models. In particular the power spectral index, which represents the slope on the P(k) graph, can be related to critical scaling exponents derived from the Wilson renormalization group analysis, and one finds that the derived value fits favorably with the Sloan Digital Sky Survey telescope data. We then make use of the transfer functions, based only on the Boltzmann equations which describe states out of equilibrium, and Einstein's General Relativity, to extrapolate the power spectrum to the Cosmic Microwave Background (CMB) regime and find that the results fits rather well with current data. Our approach contrasts with the conventional explanation which uses inflation to generate the scale invariant Harrison-Zel'dovich spectrum on CMB scales, and uses the transfer function to extrapolate it to galaxy regime. The results we present here only assumes quantum field theory and Einstein's Gravity, and hence provides a competing explanation of the power spectrum, without relying on the assumptions usually associated with inflationary models.
https://arxiv.org/abs/1808.00207
Quantum fields in the background spacetime of a loop quantum gravity black hole
Flora Moulin, Killian Martineau, Julien Grain, Aurélien Barrau
(Submitted on 1 Aug 2018)
The description of black holes in loop quantum gravity is a hard and tricky task. In this article, we focus on a minisuperspace approach based on a polymerization procedure. We consider the resulting effective metric and study the propagation of quantum fields in this background. The cross sections for scalar particles and fermions are explicitly calculated. The radial equation of motion is also derived in full generality, beyond the specifically considered metric.
https://arxiv.org/abs/1808.00673
From Euclidean to Lorentzian Loop Quantum Gravity via a Positive Complexifier
(Submitted on 2 Aug 2018 (v1), last revised 5 Aug 2018 (this version, v2))
We construct a positive complexifier, differentiable almost everywhere on the classical phase space of real triads and SU(2) connections, which generates a Wick Transform from Euclidean to Lorentzian gravity everywhere except on a phase space set of measure zero. This Wick transform assigns an equal role to the self dual and anti-self dual Ashtekar variables in quantum theory. We argue that the appropriate quantum arena for an analysis of the properties of the Wick rotation is the diffeomorphism invariant Hilbert space of Loop Quantum Gravity (LQG) rather than its kinematic Hilbert space. We examine issues related to the construction, in quantum theory, of the positive complexifier as a positive operator on this diffeomorphism invariant Hilbert space. Assuming the existence of such an operator, we explore the possibility of identifying physical states in Lorentzian LQG as Wick rotated images of physical states in the Euclidean theory. Our considerations derive from Thiemann's remarkable proposal to define Lorentzian LQG from Euclidean LQG via the implementation in quantum theory of a phase space Wick rotation' which maps real Ashtekar-Barbero variables to Ashtekar's complex, self dual variables.
https://arxiv.org/abs/1808.01252
A review on Loop Quantum Gravity
Pablo Antonio Moreno Casares
(Submitted on 3 Aug 2018)
The aim of this dissertation is to review Loop Quantum Gravity', explaining the main structure of the theory and indicating its main open issues. We will develop the two main lines of research for the theory: the canonical quantization (first two chapters) and spin foams (third). The final chapter will be devoted to studying some of the problems of the theory and what things remain to be developed. In chapter 3 we will also include an example of a simple calculation done in the frame of LQG: Schwarzschild black hole entropy.
https://arxiv.org/abs/1808.01744
The no-boundary wave function for loop quantum cosmology
Suddhasattwa Brahma, Dong-han Yeom
(Submitted on 6 Aug 2018)
Proposing smooth initial conditions is one of the most important tasks in quantum cosmology. On the other hand, the low-energy effective action, appearing in the semiclassical path integral, can get nontrivial quantum corrections near classical singularities due to specific quantum gravity proposals. In this article, we combine the well-known no-boundary proposal for the wavefunction of the universe with quantum modifications coming from loop quantum cosmology (LQC). Remarkably, we find that the restriction of a slow-roll' type potential in the original Hartle-Hawking proposal is considerably relaxed due to quantum geometry regularizations. Interestingly, the same effects responsible for singularity-resolution in LQC also end up expanding the allowed space of smooth initial conditions leading to an inflationary universe.
12. Oct 10, 2018
### julian
https://arxiv.org/abs/1808.00535
Pure states statistical mechanics: On its foundations and applications to quantum gravity
Fabio Anza
(Submitted on 1 Aug 2018)
The project concerns the interplay among quantum mechanics, statistical mechanics and thermodynamics, in isolated quantum systems. The underlying goal is to improve our understanding of the concept of thermal equilibrium in quantum systems. First, I investigated the role played by observables and measurements in the emergence of thermal behaviour. This led to a new notion of thermal equilibrium which is specific for a given observable, rather than for the whole state of the system. The equilibrium picture that emerges is a generalization of statistical mechanics in which we are not interested in the state of the system but only in the outcome of the measurement process. I investigated how this picture relates to one of the most promising approaches for the emergence of thermal behaviour in isolated quantum systems: the Eigenstate Thermalization Hypothesis. Then, I applied the results to study some equilibrium properties of many-body localised systems. Despite the localization phenomenon, which prevents thermalization of subsystems, I was able to show that we can still use the predictions of statistical mechanics to describe the equilibrium of some observables. Moreover, the intuition developed in the process led me to propose an experimentally accessible way to unravel the interacting nature of many-body localised systems. Second, I exploited the "Concentration of Measure" phenomenon to study the macroscopic properties of the basis states of Loop Quantum Gravity. These techniques were previously used to explain why the thermal behaviour in quantum systems is such an ubiquitous phenomenon, at the macroscopic scale. I focused on the local properties, their thermodynamic behaviour and interplay with the semiclassical limit. This was motivated by the necessity to understand, from a quantum gravity perspective, how and why a classical horizon exhibits thermal properties.
https://arxiv.org/abs/1808.03472
Towards conditions for black-hole singularity-resolution in asymptotically safe quantum gravity
(Submitted on 10 Aug 2018)
We explore the fate of the curvature singularity of Schwarzschild (deSitter) black holes in asymptotically safe quantum gravity. Specifically, we upgrade the classical spacetime by including the running of the Newton coupling and cosmological constant. In this setting, the antiscreening character of the gravitational interaction can remove the singularity, yet a nonzero value of the cosmological constant in the ultraviolet appears to reintroduce it. We find hints that a finite value of the cosmological constant in the infrared is compatible with singularity resolution provided that the cosmological constant is driven to zero fast enough in the ultraviolet. We compare the corresponding bounds on the critical exponents to the literature.
https://arxiv.org/abs/1808.05842
On the possibility of laboratory evidence for quantum superposition of geometries
Marios Christodoulou, Carlo Rovelli
(Submitted on 17 Aug 2018)
We analyze the recent proposal of measuring a quantum gravity phenomenon in the lab by entangling two particles gravitationally. We give a generally covariant description of this phenomenon, where the relevant effect turns out to be a quantum superposition of proper times. We point out that measurement of this effect would count as evidence for quantum superposition of spacetime geometries. This interpretation addresses objections appeared in the literature. We observe that the effect sheds light on the Planck mass, and argue that it is very plausibly a real effect.
https://arxiv.org/abs/1808.06974
Detailed background dynamics and trans-planckian effects in loop quantum cosmology
Killian Martineau
(Submitted on 21 Aug 2018)
Cosmology appears as the most promising way to test and constrain quantum gravity theories. Loop quantum gravity is among the most advanced attempts to perform a non-perturbative quantization of general relativity. Its cosmological counterpart, loop quantum cosmology, has clear predictions both for the cosmological background and for the perturbations. In particular, the initial Big Bang singularity is replaced by a bounce due to quantum geometry effects. In this proceeding I will focus on new results obtained in loop quantum cosmology: i) the prediction of the duration of inflation as a function of all the unknown parameters of the model and ii) new primordial power spectra obtained with modified dispersion relations accounting for trans-planckian effects.
https://arxiv.org/abs/1808.08857
A status report on the phenomenology of black holes in loop quantum gravity: Evaporation, tunneling to white holes, dark matter and gravitational waves
Aurélien Barrau, Killian Martineau, Flora Moulin
(Submitted on 27 Aug 2018)
The understanding of black holes in loop quantum gravity is becoming increasingly accurate. This review focuses on the possible experimental or observational consequences of the underlying spinfoam structure of space-time. It adresses both the aspects associated with the Hawking evaporation and the ones due to the possible existence of a bounce. Finally, consequences for dark matter and gravitational waves are considered.
https://arxiv.org/abs/1808.09216
Abelian 2+1D Loop Quantum Gravity Coupled to a Scalar Field
Christoph Charles
(Submitted on 28 Aug 2018)
In order to study 3d loop quantum gravity coupled to matter, we consider a simplified model of abelian quantum gravity, the so-called U(1)^3 model. Abelian gravity coupled to a scalar field shares a lot of commonalities with parameterized field theories. We use this to develop an exact quantization of the model. This is used to discuss solutions to various problems that plague even the 4d theory, namely the definition of an inverse metric and the role of the choice of representation for the holonomy-flux algebra.
https://arxiv.org/abs/1808.09765
Phase transitions in group field theory: The Landau perspective
Andreas G. A. Pithis, Johannes Thürigen
(Submitted on 29 Aug 2018)
In various approaches to quantum gravity continuum spacetime is expected to emerge from discrete geometries through a phase transition. In group field theory, various indications for such a transition have recently been found but a complete understanding of such a phenomenon remains an open issue. In this work, we investigate the critical behavior of different group field theory models in the Gaussian approximation. Applying the Ginzburg criterion to quantify field fluctuations, we find that this approximation breaks down in the case of three-dimensional Euclidean quantum gravity as described by the dynamical Boulatov model on the compact group SU(2). This result is independent of the peculiar gauge symmetry and specific form of nonlocality of the model. On the contrary, we find that the Gaussian approximation is valid for a rank-1 GFT on the noncompact sector of fields on SL(2,R) related to Lorentzian models. Though a nonperturbative analysis is needed to settle the question of phase transitions for compact groups, the results may also indicate the necessity to consider group field theory on noncompact domains for phase transitions to occur.
https://arxiv.org/abs/1808.09971
Volume of 4-polytopes from bivectors
Benjamin Bahr
(Submitted on 29 Aug 2018)
In this article we prove a formula for the volume of 4-dimensional polytopes, in terms of their face bivectors, and the crossings within their boundary graph. This proves that the volume is an invariant of bivector-coloured graphs in $S^3$.
https://arxiv.org/abs/1808.10225
Phenomenology of Quantum Reduced Loop Gravity in the isotropic cosmological sector
Emanuele Alesci, Aurélien Barrau, Gioele Botta, Killian Martineau, Gabriele Stagno
(Submitted on 30 Aug 2018)
Quantum reduced loop gravity is designed to consistently study symmetry reduced systems within the loop quantum gravity framework. In particular, it bridges the gap between the effective cosmological models of loop quantum cosmology and the full theory, addressing the dynamics before the minisuperspace reduction. This mostly preserves the graph structure and SU(2) quantum numbers. In this article, we study the phenomenological consequences of the isotropic sector of the theory, the so-called emergent bouncing universe model. In particular, the parameter space is scanned and we show that the number of inflationary e-folds is almost always higher than the observational lower-bound. We also compute the primordial tensor power spectrum and study its sensitivity upon the fundamental parameters used in the model.
https://arxiv.org/abs/1808.10469
Group field theory and its cosmology in a matter reference frame
Steffen Gielen
(Submitted on 30 Aug 2018 (v1), last revised 25 Sep 2018 (this version, v2))
While the equations of general relativity take the same form in any coordinate system, choosing a suitable set of coordinates is essential in any practical application. This poses a challenge in background-independent quantum gravity, where coordinates are not a priori available and need to be reconstructed from physical degrees of freedom. We review the general idea of coupling free scalar fields to gravity and using these scalars as a "matter reference frame." The resulting coordinate system is harmonic, i.e. it satisfies harmonic (de Donder) gauge. We then show how to introduce such matter reference frames in the group field theory approach to quantum gravity, where spacetime is emergent from a "condensate" of fundamental quantum degrees of freedom of geometry, and how to use matter coordinates to extract physics. We review recent results in homogeneous and inhomogeneous cosmology, and give a new application to the case of spherical symmetry. We find tentative evidence that spherically symmetric group field theory condensates defined in this setting can reproduce the near-horizon geometry of a Schwarzschild black hole.
https://arxiv.org/abs/1809.00313
Cosmological perturbations with inverse-volume corrections in loop quantum cosmology
Yu Han
(Submitted on 2 Sep 2018)
Although the cosmological perturbations with inverse-volume corrections from loop quantum cosmology have been studied using the anomaly free algebra approach in many literatures, there still remains an important issue that some counter terms in the perturbed constraints cannot be uniquely fixed on the spatially flat FRW background, which causes ambiguities in the perturbation equations. In this paper we show that this problem can be overcome by extending the anomaly free algebra to spatially closed FRW background. We find that a consistent deformed algebra can be obtained in the spatially closed case, and each counter term can be uniquely fixed in terms of the inverse-volume correction functions, then by taking the large $r_o$ limit, we recover the anomaly free Hamiltonian on the spatially flat background, using this Hamiltonian we obtain the gauge invariant cosmological perturbations for scalar, vector and tensor modes in the spatially flat case. Moreover, we also derive the quantum corrected Mukhanov equations, from which the scalar and tensor spectral indices with inverse-volume corrections are given. The results obtained in this paper show some differences with those in previous literatures.
https://arxiv.org/abs/1809.00556
A change of perspective: switching quantum reference frames via a perspective-neutral framework
Augustin Vanrietvelde, Philipp A Hoehn, Flaminia Giacomini, Esteban Castro-Ruiz
(Submitted on 3 Sep 2018)
Treating reference frames fundamentally as quantum systems is inevitable in quantum gravity and also in quantum foundations once considering laboratories as physical systems. Both fields thereby face the question of how to describe physics relative to quantum reference systems and how the descriptions relative to different such choices are related. Here, we exploit a fruitful interplay of ideas from both fields to begin developing a unifying approach to transformations among quantum reference systems that ultimately aims at encompassing both quantum and gravitational physics. In particular, using a gravity inspired symmetry principle, which enforces physical observables to be relational and leads to an inherent redundancy in the description, we develop a perspective-neutral structure, which contains all frame perspectives at once and via which they are changed. We show that taking the perspective of a specific frame amounts to a fixing of the symmetry related redundancies in both the classical and quantum theory and that changing perspective corresponds to a symmetry transformation. We implement this using the language of constrained systems, which naturally encodes symmetries. Within a simple one-dimensional model, we recover some of the quantum frame transformations of arXiv:1712.07207, embedding them in a perspective-neutral framework. Using them, we illustrate how entanglement and classicality of an observed system depend on the quantum frame perspective. Our operational language also inspires a new interpretation of Dirac and reduced quantized theories as perspective-neutral and perspectival quantum theories, respectively. In this light, we suggest a new take on the relation between a `quantum general covariance' and the diffeomorphism symmetry in quantum gravity.
https://arxiv.org/abs/1809.00913
Observation of thermal Hawking radiation at the Hawking temperature in an analogue black hole
Juan Ramón Muñoz de Nova, Katrine Golubkov, Victor I. Kolobov, Jeff Steinhauer
(Submitted on 4 Sep 2018 (v1), last revised 14 Sep 2018 (this version, v2))
We measure the correlation spectrum of the Hawking radiation emitted by an analogue black hole and find it to be thermal at the Hawking temperature implied by the analogue surface gravity. The Hawking radiation is in the regime of linear dispersion, in analogy with a real black hole. Furthermore, the radiation inside of the black hole is seen to be composed of negative-energy partners only. This work confirms the prediction of Hawking's theory regarding the value of the Hawking temperature, as well as the thermality of the spectrum. The thermality of Hawking radiation is the root of the information paradox. The correlations between the Hawking and partner particles imply that the analogue black hole has no analogue firewall.
https://arxiv.org/abs/1809.01747
Glimpses of Space-Time Beyond the Singularities Using Supercomputers
Parampreet Singh
(Submitted on 5 Sep 2018)
A fundamental problem of Einstein's theory of classical general relativity is the existence of singularities such as the big bang. All known laws of physics end at these boundaries of classical space-time. Thanks to recent developments in quantum gravity, supercomputers are now playing an important role in understanding the resolution of big bang and black hole singularities. Using supercomputers, explorations of the very genesis of space and time from quantum geometry are revealing a novel picture of what lies beyond classical singularities and the new physics of the birth of our universe.
https://arxiv.org/abs/1809.01908
A Local Resolution of the Problem of Time
Edward Anderson
(Submitted on 6 Sep 2018 (v1), last revised 24 Sep 2018 (this version, v2))
We here announce and outline a solution of this major and longstanding foundational problem, dealing with all seven of its heavily-interrelated local facets.
13. Oct 10, 2018
### julian
https://arxiv.org/abs/1809.03172
Non-adiabatic Evolution of Primordial Perturbations and non-Gaussinity in Hybrid Approach of Loop Quantum Cosmology
Qiang Wu, Tao Zhu, Anzhong Wang
(Submitted on 10 Sep 2018)
While loop quantum cosmology (LQC) predicts a robust quantum bounce of the background evolution of a Friedmann-Robertson-Walker (FRW) spacetime prior to the standard slow-roll inflation, whereby the big bang singularity is resolved, there are several different quantization procedures to cosmological perturbations, for instance, {\em the deformed algebra, dressed metric, and hybrid quantizations}. This paper devotes to study the quantum bounce effects of primordial perturbations in the hybrid approach. The main discrepancy of this approach is the effective positive mass at the quantum bounce for the evolution of the background that is dominated by the kinetic energy of the inflaton field at the bounce, while this mass is always nonpositive in the dressed metric approach. It is this positivity of the effective mass that violates the adiabatic evolution of primordial perturbations at the initial moments of the quantum bounce. With the assumption that the evolution of the background is dominated by the kinetic energy of the inflaton at the bounce, we find that the effective potentials for both scalar and tensor perturbations can be well approximately described by a P\"{o}schl-Teller (PT) potential, which allows us to find analytical solutions of perturbations, and from these analytical expressions we are able to study the non-adiabatic evolution of primordial perturbations in details. In particular, we derive their quantum bounce effects and investigate their observational constraints. In addition, the impacts of quantum bounce effects on the non-Gaussinity and their implication on the explanations of observed power asymmetry in CMB have also been explored.
https://arxiv.org/abs/1809.03884
Perturbations in Hybrid Loop Quantum Cosmology: Continuum Limit in Fourier Space
Beatriz Elizaga Navascués, Guillermo A. Mena Marugán
(Submitted on 11 Sep 2018)
We analyze the passage to a continuum limit of the mode spectrum of primordial perturbations around flat cosmological spacetimes in hybrid Loop Quantum Cosmology, showing that this limit can be reached even if one starts by considering a finite fiducial cell as spatial slice. We focus our attention on regimes in which the background cosmology follows the effective dynamics of Loop Quantum Cosmology, although we comment on extensions of our arguments beyond this regime, as well as to some formalisms other than the hybrid approach. Whereas the perturbed system can be described in an invariant way under changes of the fiducial volume using the standard variables of the improved prescription for Loop Quantum Cosmology, we show that the desired continuum limit can be established by means of scaling transformations of the physical volume when this volume grows unboundedly. These transformations lead to a model with a continuum of modes and independent of any scale of reference for the physical volume. For the sake of comparison, we also consider an alternative road to the continuum in Fourier space that has been employed in geometrodynamics and is based on the use of scaling transformations of the fiducial volume, together with variables that are independent of them.
https://arxiv.org/abs/1809.04465
Anomaly freedom in perturbative models of Euclidean loop quantum gravity
Jian-Pin Wu, Martin Bojowald, Yongge Ma
(Submitted on 12 Sep 2018)
Euclidean gravity provides an interesting test system for an analysis of cosmological perturbations in an effective Hamiltonian constraint with holonomy modifications from loop quantum gravity. This paper presents a discussion of scalar modes, with a specific form of the holonomy modification function derived from a general expansion in a connection formulation. Compared with some previous models, the constraint brackets are deformed in a different and more restricted way. A general comparison of anomaly-free brackets in various effective and operator versions shows overall consistency between different approaches.
https://arxiv.org/abs/1809.05093
Switching quantum reference frames in the N-body problem and the absence of global relational perspectives
Augustin Vanrietvelde, Philipp A Hoehn, Flaminia Giacomini
(Submitted on 13 Sep 2018)
Given the importance of quantum reference systems to both quantum and gravitational physics, it is pertinent to develop a systematic method for switching between the descriptions of physics relative to different choices of quantum reference systems, which is valid in both fields. Here, we continue with such a unifying approach, begun in arxiv:1809.00556, whose key ingredients is a gravity-inspired symmetry principle, which enforces physics to be relational and leads, thanks to gauge related redundancies, to a perspective-neutral structure which contains all frame choices at once and via which frame perspectives can be consistently switched. Formulated in the language of constrained systems, the perspective-neutral structure turns out to be the constraint surface classically and the gauge invariant Hilbert space in the Dirac quantized theory. By contrast, a perspective relative to a specific frame corresponds to a gauge choice and the associated reduced phase and Hilbert space. Quantum reference frame switches thereby amount to a symmetry transformation. In the quantum theory, they require a transformation that takes one from the Dirac to a reduced quantum theory and we show that it amounts to a trivialization of the constraints and a subsequent projection onto the classical gauge fixing conditions. We illustrate this method in the relational N-body problem with rotational and translational symmetry. This model is particularly interesting because it features the Gribov problem so that globally valid gauge fixing conditions are impossible which, in turn, implies also that globally valid relational frame perspectives are absent in both the classical and quantum theory. These challenges notwithstanding, we exhibit how one can systematically construct the quantum reference frame transformations for the three-body problem.
https://arxiv.org/abs/1809.08083
Time in quantum theory, the Wheeler-DeWitt equation and the Born-Oppenheimer approximation
Alexander Yu. Kamenshchik, Alessandro Tronconi, Tereza Vardanyan, Giovanni Venturi
(Submitted on 21 Sep 2018)
We compare two different approaches to the treatment of the Wheeler-DeWitt equation and the introduction of time in quantum cosmology. One approach is based on the gauge-fixing procedure in theories with first-class constraints, while the other uses the Born-Oppenheimer method. We apply both to a very simple cosmological model and observe that they give similar predictions. We also discuss the problem of time in non-relativistic quantum mechanics and some questions concerning the correspondence between classical and quantum theories.
https://arxiv.org/abs/1809.08277
Hiding the cosmological constant
S. Carlip
(Submitted on 21 Sep 2018)
Perhaps the expectations of quantum field theory are right, and the universe really does have a very large cosmological constant. I show that if one does not assume homogeneity or an arrow of time at the Planck scale, a large class of initial data for general relativity exhibits expansions and shears that are enormous at small scales, but quickly average to zero macroscopically. For an infinite subset of this data, the averaged spatial curvature is also small, and has a vanishing time derivative. Subsequent evolution is more complex, but I argue that quantum fluctuations should preserve these properties. The resulting picture is a version of Wheeler's "spacetime foam," in which the cosmological constant produces high curvature at the Planck scale but is hidden at observable scales.
https://arxiv.org/abs/1809.09659
A quantum gravity extension to the Mixmaster dynamics
Edward Wilson-Ewing
(Submitted on 25 Sep 2018)
In the loop quantum cosmology effective dynamics for the vacuum Bianchi type I and type IX space-times, a non-singular bounce replaces the classical singularity. The bounce can be approximated as an instantaneous transition between two classical vacuum Bianchi I solutions, with simple transition rules relating the solutions before and after the bounce: the evolution of the mean logarithmic scale factor is reversed, while the shape parameters are unaffected. As a result, the loop quantum cosmology effective dynamics for the vacuum Bianchi IX space-time can be approximated by a sequence of classical vacuum Bianchi I solutions, following the usual Mixmaster transition maps in the classical regime, and undergoing a bounce with this new transition rule in the Planck regime.
https://arxiv.org/abs/1809.09874
The Vacuum State of Primordial Fluctuations in Hybrid Loop Quantum Cosmology
Beatriz Elizaga Navascués, Daniel Martín de Blas, Guillermo A. Mena Marugán
(Submitted on 26 Sep 2018)
We investigate the role played by the vacuum of the primordial fluctuations in hybrid Loop Quantum Cosmology. We consider scenarios where the inflaton potential is a mass term and the unperturbed quantum geometry is governed by the effective dynamics of Loop Quantum Cosmology. In this situation, the phenomenologically interesting solutions have a preinflationary regime where the kinetic energy of the inflaton dominates over the potential. For these kind of solutions, we show that the primordial power spectra depend strongly on the choice of vacuum. We study in detail the case of adiabatic states of low order and the non-oscillating vacuum introduced by Mart\'in de Blas and Olmedo, all imposed at the bounce. The adiabatic spectra are typically suppressed at large scales, and display rapid oscillations with an increase of power at intermediate scales. In the non-oscillating vacuum, there is power suppression for large scales, but the rapid oscillations are absent. We argue that the oscillations are due to the imposition of initial adiabatic conditions in the region of kinetic dominance, and that they would also be present in General Relativity. Finally, we discuss the sensitivity of our results to changes of the initial time and other data of the model.
https://arxiv.org/abs/1810.00949
On the Empirical Consequences of the AdS/CFT Duality
Radin Dardashti, Richard Dawid, Sean Gryb, Karim Thébault
(Submitted on 27 Sep 2018)
We provide an analysis of the empirical consequences of the AdS/CFT duality with reference to the application of the duality in a fundamental theory, effective theory and instrumental context. Analysis of the first two contexts is intended to serve as a guide to the potential empirical and ontological status of gauge/gravity dualities as descriptions of actual physics at the Planck scale. The third context is directly connected to the use of AdS/CFT to describe real quark-gluon plasmas. In the latter context, we find that neither of the two duals are confirmed by the empirical data.
https://arxiv.org/abs/1810.00238
The BKL scenario, infrared renormalization, and quantum cosmology
Martin Bojowald
(Submitted on 29 Sep 2018)
A discussion of inhomogeneity is indispensable to understand quantum cosmology, even if one uses the dynamics of homogeneous geometries as a first approximation. While a full quantization of inhomogeneous gravity is not available, a broad framework of effective field theory provides important ingredients for quantum cosmology. Such a setting also allows one to take into account lessons from the Belinski-Khalatnikov-Lifshitz (BKL) scenario. Based on several new ingredients, this article presents conditions on various parameters and mathematical constructions that appear in minisuperspace models. Examples from different approaches demonstrate their restrictive nature.
https://arxiv.org/abs/1810.01259
A relational Hamiltonian for group field theory
Edward Wilson-Ewing
(Submitted on 2 Oct 2018)
Using a massless scalar field as a clock variable, the Legendre transform of the group field theory Lagrangian gives a relational Hamiltonian. In the classical theory, it is natural to define 'equal relational time' Poisson brackets, where 'equal time' corresponds to equal values of the scalar field clock. The quantum theory can then be defined by imposing 'equal relational time' commutation relations for the fundamental operators of the theory, with the states being elements of a Fock space with their evolution determined by the relational Hamiltonian operator. A particularly interesting family of states are condensates, as they are expected to correspond to the cosmological sector of group field theory. For the relational Hamiltonian considered in this paper, the coarse-grained dynamics of a simple type of condensate states agree exactly with the Friedmann equations in the classical limit, and also include quantum gravity corrections that ensure the big-bang singularity is replaced by a bounce.
https://arxiv.org/abs/1810.01671
Cosmological Implications of the Bekenstein Bound
Tom Banks, Willy Fischler
(Submitted on 3 Oct 2018 (v1), last revised 9 Oct 2018 (this version, v2))
A brief review of "Holographic Space-Time" in the light of the seminal contributions of Jacob Bekenstein.
https://arxiv.org/abs/1810.01880
Black hole entropy and the Bekenstein bound
Raphael Bousso
(Submitted on 3 Oct 2018)
I share some memories and offer a personal perspective on Jacob Bekenstein's legacy, focussing on black hole entropy and the Bekenstein bound. I summarize a number of fascinating recent developments that grew out of Bekenstein's pioneering contributions, from the Ryu-Takayanagi proposal to the Quantum Null Energy Condition.
https://arxiv.org/abs/1810.02828
How perturbative is quantum gravity?
Astrid Eichhorn, Stefan Lippoldt, Jan M. Pawlowski, Manuel Reichert, Marc Schiffer
(Submitted on 5 Oct 2018)
We explore asymptotic safety of gravity-matter systems, discovering indications for a near-perturbative nature of these systems in the ultraviolet. Our results are based on the dynamical emergence of effective universality at the asymptotically safe fixed point. Our findings support the conjecture that an asymptotically safe completion of the Standard Model with gravity could be realized in a near-perturbative setting.
https://arxiv.org/abs/1810.04153
How to switch between relational quantum clocks
Philipp A Hoehn, Augustin Vanrietvelde
(Submitted on 9 Oct 2018)
Every clock is a physical system and thereby ultimately quantum. A naturally arising question is thus how to describe time evolution relative to quantum clocks and, specifically, how the dynamics relative to different quantum clocks are related. This is a particularly pressing issue in view of the multiple choice facet of the problem of time in quantum gravity, which posits that there is no distinguished choice of internal clock in generic general relativistic systems and that different choices lead to inequivalent quantum theories. Exploiting a recent unifying approach to switching quantum reference systems (arXiv:1809.00556, arXiv:1809:05093), we exhibit a systematic method for switching between different clock choices in the quantum theory. We illustrate it by means of the parametrized particle, which, like gravity, features a Hamiltonian constraint. We explicitly switch between the quantum evolution relative to the non-relativistic time variable and that relative to the particle's position, which requires carefully regularizing the zero-modes in the so-called time-of-arrival observable. While this toy model is simple, our approach is general and, in particular, directly amenable to quantum cosmology. It proceeds by systematically linking the reduced quantum theories relative to different clock choices via the clock-choice-neutral Dirac quantized theory, in analogy to coordinate changes on a manifold. This method overcomes the multiple choice problem here, showing that it is actually a multiple choice feature of the complete relational quantum theory, taken as the conjunction of Dirac and reduced quantized theories. Precisely this conjunction permits to consistently switch between different temporal reference systems, which is a prerequisite for a quantum notion of general covariance.
14. Jan 13, 2019
### *now*
https://arxiv.org/abs/1802.04264
White Holes as Remnants: A Surprising Scenario for the End of a Black Hole
Eugenio Bianchi, Marios Christodoulou, Fabio D'Ambrosio, Hal M. Haggard, Carlo Rovelli
(Submitted on 12 Feb 2018 (v1), last revised 17 Mar 2018 (this version, v2))
Quantum tunneling of a black hole into a white hole provides a model for the full life cycle of a black hole. The white hole acts as a long-lived remnant, solving the black-hole information paradox. The remnant solution of the paradox has long been viewed with suspicion, mostly because remnants seemed to be such exotic objects. We point out that (i) established physics includes objects with precisely the required properties for remnants: white holes with small masses but large finite interiors; (ii) non-perturbative quantum-gravity indicates that a black hole tunnels precisely into such a white hole, at the end of its evaporation. We address the objections to the existence of white-hole remnants, discuss their stability, and show how the notions of entropy relevant in this context allow them to evade several no-go arguments. A black hole's formation, evaporation, tunneling to a white hole, and final slow decay, form a unitary process that does not violate any known physics.
https://arxiv.org/abs/1811.00532
Statistical equilibrium of tetrahedra from maximum entropy principle
Goffredo Chirco, Isha Kotecha, Daniele Oriti
(Submitted on 1 Nov 2018)
Discrete formulations of (quantum) gravity in four spacetime dimensions build space out of tetrahedra. We investigate a statistical mechanical system of tetrahedra from a many-body point of view based on non-local, combinatorial gluing constraints that are modelled as multi-particle interactions. We focus on Gibbs equilibrium states, constructed using Jaynes' principle of constrained maximisation of entropy, which has been shown recently to play an important role in characterising equilibrium in background independent systems. We apply this principle first to classical systems of many tetrahedra using different examples of geometrically motivated constraints. Then for a system of quantum tetrahedra, we show that the quantum statistical partition function of a Gibbs state with respect to some constraint operator can be reinterpreted as a partition function for a quantum field theory of tetrahedra, taking the form of a group field theory.
https://arxiv.org/abs/1811.03667
Light Cone Black Holes
Tommaso De Lorenzo, Alejandro Perez
(Submitted on 8 Nov 2018)
When probed with conformally invariant matter fields, light cones in Minkowski spacetime satisfy thermodynamical relations which are the analog of those satisfied by stationary black holes coupled to standard matter fields. These properties stem from the fact that light cones are conformal Killing horizons stationary with respect to observers following the radial conformal Killing fields in flat spacetime. The four laws of light cone thermodynamics relate notions such as (conformal) temperature, (conformal) surface gravity, (conformal) energy and a conformally invariant notion related to area change. These quantities do not admit a direct physical interpretation in flat spacetime. However, they become the usual thermodynamical quantities when Minkowski is mapped, via a Weyl transformation, to a target spacetime where the conformal Killing field becomes a proper Killing field. In this paper we study the properties of such spacetimes. The simplest realisation turns out to be the Bertotti-Robinson solution, which is known to encode the near horizon geometry of near extremal and extremal charged black holes. The analogy between light cones in flat space and black hole horizons is therefore strengthened. The construction works in arbitrary dimensions; in two dimensions one recovers the Jackiv-Teitelboim black hole of dilaton gravity. Other interesting realisations are also presented.
https://arxiv.org/abs/1811.08007
Quantum insights on Primordial Black Holes as Dark Matter
Francesca Vidotto
(Submitted on 19 Nov 2018)
A recent understanding on how quantum effects may affect black-hole evolution opens new scenarios for dark matter, in connection with the presence of black holes in the very early universe. Quantum fluctuations of the geometry allow for black holes to decay into white holes via a tunnelling. This process yields to an explosion and possibly to a long remnant phase, that cures the information paradox. Primordial black holes undergoing this evolution constitute a peculiar kind of decaying dark matter, whose lifetime depends on their mass M and can be as short as M2. As smaller black holes explode earlier, the resulting signal have a peculiar fluence-distance relation. I discuss the different emission channels that can be expected from the explosion (sub-millimetre, radio, TeV) and their detection challenges. In particular, one of these channels produces an observed wavelength that scales with the redshift following a unique flattened wavelength-distance function, leaving a signature also in the resulting diffuse emission. I conclude presenting the first insights on the cosmological constraints, concerning both the explosive phase and the subsequent remnant phase.
https://arxiv.org/abs/1811.11744
Holographic description of boundary gravitons in (3+1) dimensions
Seth K. Asante, Bianca Dittrich, Hal M. Haggard
(Submitted on 27 Nov 2018)
Gravity is uniquely situated in between classical topological field theories and standard local field theories. This can be seen in the the quasi-local nature of gravitational observables, but is nowhere more apparent than in gravity's holographic formulation. Holography holds promise for simplifying computations in quantum gravity. While holographic descriptions of three-dimensional spacetimes and of spacetimes with a negative cosmological constant are well-developed, a complete boundary description of zero curvature, four-dimensional spacetime is not currently available. Building on previous work in three-dimensions, we provide a new route to four-dimensional holography and its boundary gravitons. Using Regge calculus linearized around a flat Euclidean background with the topology of a solid hyper-torus, we obtain the effective action for a dual boundary theory which describes the dynamics of the boundary gravitons. Remarkably, in the continuum limit and at large radii this boundary theory is local and closely analogous to the corresponding result in three-dimensions. The boundary effective action has a degenerate kinetic term that leads to singularities in the one-loop partition function that are independent of the discretization. These results establish a rich boundary dynamics for four-dimensional flat holography.
https://arxiv.org/abs/1812.01542
On the possibility of experimental detection of the discreteness of time
Marios Christodoulou, Carlo Rovelli
(Submitted on 4 Dec 2018 (v1), last revised 8 Dec 2018 (this version, v2))
The Bose-Marletto-Vedral experiment tests a non-relativistic quantum effect due to a gravitational interaction. It has received attention because it may soon be within observational reach in the lab. We observe here that: (i) in relativistic language the experiment tests an interference effect between proper-time intervals; (ii) the relevant difference of proper times is of the order of the Planck time if the masses of the particles in the experiment are of the order of the Planck mass (micrograms); (iii) the experiment might open a window on the structure of time at the Planck scale: if time differences are discrete at this scale ---as quantum gravity research may suggest--- the Planckian discreteness of time could show up as quantum levels of a measurable entanglement entropy.
https://arxiv.org/abs/1812.05127
Quantum gravity and black hole spin in gravitational wave observations: a test of the Bekenstein-Hawking entropy
Eugenio Bianchi, Anuradha Gupta, Hal M. Haggard, B. S. Sathyaprakash
(Submitted on 12 Dec 2018)
Black hole entropy is a robust prediction of quantum gravity with no observational test to date. We use the Bekenstein-Hawking entropy formula to determine the probability distribution of the spin of black holes at equilibrium in the microcanonical ensemble. We argue that this ensemble is relevant for black holes formed in the early universe and predicts the existence of a population of black holes with zero spin. Observations of such a population at LIGO, Virgo, and future gravitational wave observatories would provide the first experimental test of the statistical nature of black hole entropy.
https://arxiv.org/abs/1812.06193
Tullio Regge's legacy: Regge calculus and discrete gravity
John W. Barrett, Daniele Oriti, Ruth M. Williams
(Submitted on 14 Dec 2018)
The review paper "Discrete Structures in Physics", written in 2000, describes how Regge's discretization of Einstein's theory has been applied in classical relativity and quantum gravity. Here, developments since 2000 are reviewed briefly, with particular emphasis on progress in quantum gravity through spin foam models and group field theories. | 2019-02-23 03:17:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6044048070907593, "perplexity": 746.1637095999428}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550249434065.81/warc/CC-MAIN-20190223021219-20190223043219-00576.warc.gz"} |
https://www.physicsforums.com/threads/mechanics-of-materials-ii-mohrs-circle.166554/ | # Mechanics of Materials II - Mohr's Circle
1. Apr 19, 2007
### Double A
1. The problem statement, all variables and given/known data
For an element [Stress Block], determine the range of values of $$\tau_{xy}$$ for which the maximum tensile stress is equal to or less than 60 MPa.
Given in the provided figure:
$$\sigma_x$$ = -120 MPa
$$\sigma_y$$ = -60 MPa
2. Relevant equations
$$\sigma_{ave} = \frac{\sigma_x + \sigma_y}{2}$$
$$R = \tau_{max} = \sqrt{(\frac{\sigma_x - \sigma_y}{2})^2 + \tau_{xy}^2}$$
$$\sigma_{max,min} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{(\frac{\sigma_x - \sigma_y}{2})^2 + \tau_{xy}^2}$$
$$\tan{2\theta_p} = \frac{2\tau_{xy}}{\sigma_x - \sigma_y}$$
3. The attempt at a solution
I have drawn a representation of Mohr's Circle using the provided data. I am confused with the statment saying "tensile stress" when the provided stresses are in compression. They represent the shear stress in the positive direction. I am also unclear about how to approch this beyond my Mohr's circle figure.
I'm not sure if this is correct but I tried this:
$$\sigma_{max} = \frac{\sigma_x + \sigma_y}{2} + \sqrt{(\frac{\sigma_x - \sigma_y}{2})^2 + \tau_{xy}^2}$$
Solve for $$\tau_{xy}$$ and inputing known values:
$$\tau_{xy} = \pm$$59.9 MPa
Last edited: Apr 19, 2007 | 2017-05-24 13:48:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.569446861743927, "perplexity": 1102.891652487419}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607846.35/warc/CC-MAIN-20170524131951-20170524151951-00489.warc.gz"} |
https://www.datacamp.com/community/blog/writing-functions-in-python | Official Blog
data analysis
# Writing Functions in Python
You will learn to do the following: define functions without parameters, define functions with single parameters, and define functions that return a single value.
## Course Excerpt: Writing Functions in Python
Below is an excerpt--video and transcript--from the first chapter of the Python Data Science Toolbox I course. It discusses writing functions in Python. Here is the full chapter, including interactive exercises.
If you're looking for a hands-on tutorial on Python functions, consider DataCamp's Python Functions Tutorial.
## Introduction to User-Defined Functions
Welcome to the course! My name is Hugo Bowne-Anderson and I am a Data Scientist at DataCamp. In this course, the first of the Python Data Science toolbox courses, you'll learn to write your very own functions and you'll have the opportunity to apply these newfound skills to questions that commonly arise in Data Science contexts.
Specifically, in this video and in the interactive exercises that follow it, you will learn to do the following:
• define functions without parameters,
• define functions with single parameters, and
• define functions that return a single value.
In the next section, you'll learn how to pass multiple arguments to functions, as well as return multiple values from them.
Let's begin!
Let's check out Python's built-in function str(), which accepts an object such as a number and returns a string object. You can assign a call to str() to a variable to store its return value.
While built-in Python functions are cool, as a Data Scientist, you'll need functions that have functionality specific to your needs. Fortunately, you can define your own functions in Python!
We'll now see how to define functions via an example, a function that squares a number. The function name square will be perfect for this. To define the function, We begin with the keyword def, followed by the function name square; this is then followed by a set of parentheses and a colon. This piece of code is called a "function header".
To complete the function definition, let's write the function body by squaring a value, say 4, and printing the output.
Right now, our square function does not have any parameters within the parentheses (). We will add them later. Now, whenever this function is called, the code in the function body is run. In this case, new_value is assigned the value of 4 ** 2 and then printed out.
You can call the function as you do with pre-built functions: square(). This should yield the value, 16.
What if you wanted to square any other number besides 4, though? To add that functionality, you add a parameter to the function definition in between the parentheses. Here you see that we've added a parameter value and in the new function body, the variable new_value takes the square of value, which is then printed out. We can now square any number that we pass to the function square as an argument.
A quick word on parameters and arguments:
• When you define a function, you write parameters in the function header.
• When you call a function, you pass arguments into the function.
The function square now accepts a single parameter and prints out its squared value. But what if we don't want to print that value directly and instead we want to return the squared value and assign it to some variable? You can have your function return the new value by adding the return keyword, followed by the value to return. Now we can assign to a variable num the result of the function call as you see here.
There's another essential aspect of writing functions in Python: docstrings. Docstrings are used to describe what your function does, such as the computations it performs or its return values. These descriptions serve as documentation for your function so that anyone who reads your function's docstring understands what your function does, without having to trace through all the code in the function definition.
Function docstrings are placed in the immediate line after the function header and are placed in between triple quotation marks. An appropriate Docstring for our function square is 'Returns the square of a value'.
You've now just learned the basics of defining your own functions! Now it's your turn. In the next few exercises, you will try your hand at defining and using your own functions
## Multiple Parameters and Return Values
Welcome back! You're doing great at defining your own functions. Good job!
At this point, you already know how to define your own functions and even return values from them. What you'll learn next is how to pass multiple arguments to functions, as well as return not just one, but multiple values from them.
Let's tweak the square() function we've been working on a little bit more. Suppose that, instead of simply squaring a value, we'd like to raise a value to the power of another value that's also passed to the function. We can do this by having our function accept two parameters instead of just one. You should also change your function name AND docstrings to reflect this new behavior. raise_to_power is an appropriate function name.
Notice that there are now two parameters in the function header instead of one, value1 and value2. In the lines after that, the behavior of the overall function was also changed by raising value1 to the power of value2.
You can call the function by passing in two arguments because the function has two parameters, as declared in the function header. The order in which the arguments are passed corresponds to the order of the parameters in the function header. This means that when we call raise_to_power(2, 3), when the function is executed, 2 would be assigned to value1 and 3 to value2. Looking at the function body, this means that the computation value1 to the power of value2 translates to 2 to the power of 3. This function call then returns the value 8.
You can also make your function return multiple values. You can do that by constructing objects known as tuples in your functions.
A tuple is like a list, in that it can contain multiple values. There are some differences, however:
• Firstly, unlike a list, a tuple is immutable, that is, you cannot modify the values in a tuple once it has been constructed.
• Secondly, while lists are defined using square brackets [], tuples are constructed using a set of parentheses ().
Here we construct a tuple containing 3 elements.
You can also unpack a tuple into several variables in one line. Doing so means that you assign to the variables a, b, and c the tuple values, in the order that they appear in the tuple. Additionally, you can also access individual tuple elements like you do with lists. Doing this here accesses the second element of the tuple. Why is that? Recall that with lists, you can use zero-indexing to access list elements. You can do the same thing with tuples! Pretty cool, right?
Let's now modify the behavior of your raise() function. Instead of returning just the value of value1 raised to the power of value2, let's also return the value of value2 raised to the power of value1. You thus need to make raise() return two values instead of one. We can use what we now know of Tuples to do this!
We first change the name of our function and the docstring to reflect the new behavior of our function. We then, in the function body, construct a tuple consisting of the values we want the function to return and, also in the function body, we return the tuple!
Calling the function constructed demonstrates that it does exactly what we want! Now it's your turn to play with writing function that accept multiple arguments and return multiple values. Enjoy! | 2022-01-16 10:03:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4011201560497284, "perplexity": 728.1950589066067}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320299852.23/warc/CC-MAIN-20220116093137-20220116123137-00548.warc.gz"} |
http://news.datascience.org.ua/2019/02/25/guide-to-simple-rails-features/ | # Guide to simple Rails features
Guide to simple Rails featuresMicah ShuteBlockedUnblockFollowFollowingFeb 20I recently made my first Ruby on Rails app — here are some resources and strategies I used to make it work, including:OAuth login/signup (Google)Authorization helpers and controller methods ( current_user, logged_in, authorize(user), render 403 errors)Polymorphic ActiveRecord relationships (follow users, message, and react to posts)Complex ActiveRecord class methods (most popular post)Setting up Bootstrap (incorporate with Rails’ forms)Rendering Markdown from user input ( kramdown setup, rouge syntax highlighting, custom sanitize)Google Authentication[Google OAuth gem](https://github.
com/zquestz/omniauth-google-oauth2)You can add a google_uid column and a google_refresh_token to the user table if you need a unique property with which to query a user, or you want to use more advanced features of Google's API (and allow an offline refresh of the token).
Because I have a uniqueness validation of a user's email and I am only using Google for authentication, I didn't need those columns but added them anyway just in case I had a design change in the future.
In my schema, I made a user's email validate uniqueness so that logging in with Google would check for an existing account with that email.
→ Note: To make this method foolproof, it would be important to have a user verify their email if the account is made traditionally (ie without OAuth).
Also, you need to have an action dedicated to handling the OAuth callback.
This action needs to be associated with the callback route you specify in the Google API page (the link to that page is on the GitHub of the Google OAuth gem linked above).
My custom route for the callback that I defined in my routes.
rb file was: get 'auth/google_oauth2/callback, to: 'sessions#googleAuth'My callback action in my SessionsController utilized a custom class method written in the User model, to which it passed the return from the OAuth process, like so:From there, you just check to see if the user is valid, and log them in if it is, and otherwise display an error.
→ Note: The RandomPasswordStrategy is just a class I made that uses the sysrandom/securerandom method SecureRandom.
hex(64) combined with random ordering of required symbols and numbers (as required by the password policy) to create a complex, valid random password.
Authorization Methods, Error PagesI put multiple private methods in my ApplicationController to abstract away authentication throughout my app, making it easier to authenticate a user in the correct manner and display an appropriate message to them if there was an issue.
I wanted logged out users to be able to see most of the app but restrict certain aspects to only be accessible to users with accounts.
Also, I wanted to ensure that only the owner can edit/delete their posts and publish new posts under their name.
Additionally, I wanted some pages to display differently based on whether or not the user is logged in.
To help perform these tasks in a modular and simple way, I added two methods toApplicationController so they are available to all Controllers (which inherit from ApplicationController), as well as adding them as helper_methods to make them available in the Views if necessary:These are primarily used as decision-makers on what is to be shown on certain pages, or as building blocks to more advanced authorization methods.
If an unauthorized user tries to access a page you don't want them to, it is appropriate to show a 403 error page.
This is easily accomplished by creating a public/403.
html.
erb file, and calling it via another method in ApplicationControllerWhich can accept a custom flash message as a parameter, and which serves as a good building block for other methods, like this:If authorize is called without a user parameter, it will only display a 403 error page if the user is not logged in.
If it is called with a user parameter (which is not null), the 403 will be called unless that specific user is logged in.
It also returns a boolean to indicate success or failure.
→ Note: If you need to use current_user in the controller after calling authorize, you will get an error if the authorization fails.
An easy way to fix this is to use an if-else block with the authorization check rather than using it as a standalone function at the top of your controller action.
This way, any code which calls current_user will not be run unless authorization succeeds, and if it fails, once the controller action is complete it will render the 403 error page.
Following Users, Sending Messages, Reacting to Posts, Scope MethodsFor some relationships, ActiveRecord relations aren't as simple as something likehas_many :classes, through: :user_classes.
Three situations particularly I had to get more creative to make work:Following other Users:Many to many relationships, i.
e.
a user is following many users and is followed by many users.
The answer to the problem is just a simple join table, but it's just slightly different than a standard join table, as it joins a table to itself, and requires a few extra words in the ActiveRecord class methods.
The join table I used was:And then to make the User model work correctly:MessagingThis was pretty similar to implementing following/followers, except the model Message is the join table, and is also the model that we are interested in, so it is simply just:Reacting to PostsThere were a few solutions to this problem that are viable, but I decided to make a database table reaction_types which has a many-to-many polymorphic relationship to anything that is reactable, in my case topics and posts.
reaction was the join table betweenreactables and the reaction_types.
I did this instead of reactions holding the reaction type via a string in order to minimize errors and to allow for flexibility – for example, the easy expansion of allowed reaction-types in the future.
Doing it this way made reactions a 3-way join table between user, reactables, and reaction_types.
reaction_types have to be seeded to the types you want to be available to the user.
I chose like, dislike, genius, and report.
In the same fashion using polymorphic relationships, you can create tags that can be attached to taggable objects, and post replies to objects which are postable.
Advanced ActiveRecord class methodsWhen I was trying to make the correct ActiveRecord class methods to retrieve specific "statistics" for a user dashboard or to find popular topics for my homepage, I found that the best resource for me was to see other advanced and specific class methods and use those examples as a reference.
I will put some of mine here in the hopes that others can use it as a reference.
To create a "spotlight topic" of the day for my homepage, take the most positively reacted topic in the past 24 hours:Find theTopic which was written by a certain User with the most reactions of a specific type.
You can use this to add to their dashboard, so they can see what their most liked post is, for example:Setting up Bootstrap[This post](https://medium.
freecodecamp.
org/add-bootstrap-to-your-ruby-on-rails-project-8d76d70d0e3b) walks through how to set up your rails app with bootstrap in just a few minutes.
There is a complication with bootstrap that won't be ideal right 'out of the box': while rails will give your form_for fields the fields_with_errors class automatically when there are issues with the form inputs, Bootstrap will not respond to it.
Instead, it responds to the class is-invalid.
– [This blog](https://jasoncharnes.
com/bootstrap-4-rails-fields-with-errors/) shows a good working solution to make form errors work well with bootstrap.
It just includes adding a file and code to your config/initializers folder, and basically all it does is change the default field-with-errors class to is-invalid, which is what bootstrap uses to indicate a field which was filled out incorrectly.
Rendering MarkdownI was between the two libraries redcarpet and kramdown.
I ended up choosing [kramdown](https://github.
com/gettalong/kramdown) because:It is more recently committed and seems to be maintained moreIts popularity is growing while redcarpet is decreasingIt allows integration with MathJax which renders LaTex equations.
Getting basic integration was relatively simple.
It got a little more complicated to achieve the following 2 goals:→ Syntax highlighting for code→ Being able to sanitize the user's input while allowing all of the Markdown features to implement correctly.
I used [rouge](https://github.
com/jneen/rouge) for syntax highlighting.
I'll show you the code below to get it working with Kramdown, but the hard piece of information to find was how to get the proper CSS files that make the highlighting happen.
Here's a good [stack overflow answer](https://stackoverflow.
com/questions/43905103/kramdown-rouge-doesnt-highlight-syntax) that's hard to find.
Basically, once you tell kramdown that you want to use rouge (and you have installed both the kramdown gem and the rouge gem), you can run rougify help style in your terminal, and you can see all of the custom CSS files you can add to your app/assets/stylesheets path.
To get the raw CSS, type rougify style followed by one of the allowed styles in your terminal.
For example: rougify style colorful.
The CSS will be printed in your terminal.
You could also runrougify style colorful > .
/app/assets/stylesheets/rouge_style.
cssNow it’s relatively easy to render markdown.
I made a helper method to make it extra simple:Custom sanitize method:As you can see here, I am not using raw or .
html_safe to render this HTML data because ultimately it came from a user and cannot be trusted.
However, the safe sanitize method disallows certain things I want to be rendered, such as tables.
You can get proper sanitation AND your desired HTML tags by manually appending to the whitelisted tags allowed through sanitize by doing something like what is shown below.
General tipsSet up the app with PostgreSQL [Here's a blog post](https://medium.
com/@micah.
shute/setting-up-windows-subsystem-for-linux-wsl-6346ff23b8bb) I wrote about setting up WSL, but there's a section at the bottom that covers using pgAdmin and setting up a Rails app with PostgreSQLSave all secret keys as environment variables I used [Figaro](https://github.
com/laserlemon/figaro) to make it easy.
Avoid the N+1 problem This occurs when you get a collection of models, of which you want to query and show nested models.
If you iterate over your models n times to do this, you are making n+1 database calls which can be quite expensive with large amounts of data, especially when requesting over a network.
This is easily fixed by using the includes method when making your initial query.
[Check this site out](https://guides.
rubyonrails.
org/active_record_querying.
html) and go to the section called "Solution to N + 1 queries problem" for more info.
Clean up the database automatically when objects are destroyed When you appropriately add dependent: :destroy to your ActiveRecord class methods (examples can be seen in code snippets above), you are telling ActiveRecord to destroy these relations upon the destruction of the model.
As you can see above, I implemented this for reactions within post.
rb.
This tells rails that when a post is destroyed, all user reactions to it should also be destroyed.
This prevents disconnected likes and dislikes, etc, from floating around in your database for no reason.
Also, note this should be a one-sided relationship.
You would not want a post to be destroyed if a user destroys their one reaction to it.
Adding custom files and classes There are going to be things you want your app to do that are outside of MVC.
This means you are going to want to add files outside of the standard model, view and controller directories.
Here are some resources with advice on how to do this:→ [A StackOverflow answer](https://stackoverflow.
com/questions/15260984/guidelines-for-where-to-put-classes-in-rails-apps-that-dont-fit-anywhere)→ [A GitHub gist post](https://gist.
github.
com/maxim/6503591)Personally, I used a lib directory inside of my app directory because it is eagerly loaded in production and lazy loaded in development.
I did not need to alter any configuration or environment files for the classes within files in that directory were referenced.
. | 2019-07-17 03:04:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17281092703342438, "perplexity": 2254.9907869881818}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525009.36/warc/CC-MAIN-20190717021428-20190717043428-00127.warc.gz"} |
http://mathhelpforum.com/algebra/226430-solve-inequality.html | # Math Help - Solve the Inequality
1. ## Solve the Inequality
Solve the inequality and express the solution in terms of intervals whenever possible.
(a) [ (x + 1) / (2x - 3) ] > 2
(b) x(2x + 3) >= 5
2. ## Re: Solve the Inequality
have you made any attempts? Do you have any work to show us?
3. ## Re: Solve the Inequality
When I solve for x in (a), I get x > 7/3. I think that the interval is (7/3, +infinite).
When I solve for x in (b), I get x >= 1, x >= -5/2. I cannot figure out how to write the interval. Maybe, it is, [1, +infinite), [-5/2, +infinite).
4. ## Re: Solve the Inequality
Originally Posted by joshuaa
Solve the inequality and express the solution in terms of intervals whenever possible.
(a) [ (x + 1) / (2x - 3) ] > 2
$\dfrac{x+1}{2x-3}>2\\\dfrac{x+1}{2x-3}-2>0\\\dfrac{-3x+7}{2x-3}>0$
5. ## Re: Solve the Inequality
Originally Posted by joshuaa
When I solve for x in (a), I get x > 7/3. I think that the interval is (7/3, +infinite).
When I solve for x in (b), I get x >= 1, x >= -5/2. I cannot figure out how to write the interval. Maybe, it is, [1, +infinite), [-5/2, +infinite).
I'm seeing a different answer so let's take a look.
$\frac{x+1}{2x-3}>2$
if $(2x-3)>0 \Rightarrow x>\frac{3}{2}$ this becomes
$x+1>4x-6$
$7>3x$
$\frac{7}{3}>x$
so this yields $\frac{3}{2} < x < \frac{7}{3}$
if $2x-3<0 \Rightarrow x<\frac{3}{2}$
then you have to flip the inequality sign since you are multiplying by a negative number.
$x+1<4x-6$
$7 < 3x$
$\frac{7}{3} < x$
clearly x can't be both less than $\frac{3}{2}$ and greater than $\frac{7}{3}$ so there are no solutions for $x < \frac{3}{2}$
thus the final answer for (a) is $\frac{3}{2} < x < \frac{7}{3}$ or $\left(\frac{3}{2}, \frac{7}{3}\right)$
for (b)
$x(2x+3) \geq 5$
$2x^2+3x-5 \geq 0$
$(2x+5)(x-1) \geq 0$
so these factors are either both postive or both negative or 0
$2x+5 \geq 0 \Rightarrow x \geq -\frac{5}{2}$
$x-1 \geq 0 \Rightarrow x \geq 1$
these two result in $x \geq 1$
$2x+5 \leq 0 \Rightarrow x \leq -\frac{5}{2}$
$x-1 \leq 0 \Rightarrow x \leq 1$
these two result in $x \leq -\frac{5}{2}$
so the intervals the inequality is satisfied on are
$x \leq -\frac{5}{2}$ and $1 \leq x$ or $\left(-\infty, -\frac{5}{2}\right] \cup \left[1, \infty\right)$
6. ## Re: Solve the Inequality
I am a little confused. In (a), how did you get, (2x - 3) > 0?
When you get the factors of x, why do you solve the inequality twice? Like this, (2x - 3) > 0, (2x - 3) < 0.
7. ## Re: Solve the Inequality
when you multiply the sides of an inequality by a negative number you must switch which way the inequality points. So you have to consider both cases.
In this case we multiplied by (2x-3) so we had to deal with the case when this is positive vs. when it is negative separately.
8. ## Re: Solve the Inequality
Originally Posted by joshuaa
I am a little confused. In (a), how did you get, (2x - 3) > 0?
When you get the factors of x, why do you solve the inequality twice? Like this, (2x - 3) > 0, (2x - 3) < 0.
$\dfrac{x+1}{2x-3}>2\\\dfrac{x+1}{2x-3}-2>0\\\dfrac{-3x+7}{2x-3}>0$
I always strongly discourage students from "multiplying" sides.
I encourage that they work by comparing all to zero, as I did above.
In the last line you need $\dfrac{+}{+}~\text{ or }~\dfrac{-}{-}$
9. ## Re: Solve the Inequality
Plato,
I have seen your 1st reply, and I am confused.
Usually, if we want to factor, we get, for example, (x + 5) (x - 4) > 0. Then I say, x > -5, x > 4.
If I have the same factors, but with division, like this, (x + 5) / (x - 4) > 0, can I say, x > -5, x > 4?
Also, if this is true, why I did not directly say in example (a), (x + 1) > 2, (2x - 3) > 2?
romsek
Now, I got it why we did the inequality twice, but confused about, how directly, you got that (2x - 3) > 0?
10. ## Re: Solve the Inequality
It has to be either positive or negative since division by 0 isn't defined. I tried both. (2x-3) < 0 lead to a solution that couldn't exist thus it must be that (2x-3) > 0.
11. ## Re: Solve the Inequality
Thank you a Lot. It is clear now, but I need some practice to master this subject and eliminate confusion.
12. ## Re: Solve the Inequality
Originally Posted by joshuaa
Usually, if we want to factor, we get, for example, (x + 5) (x - 4) > 0. Then I say, x > -5, x > 4.
I don't think that you understand the logic of problem solving.
In the case $(x+5)(x-4)>0$ so we need $(+)(+)\text{ OR }(-)(-)$.
If $(x+5)>0\text{ AND }(x-4)>0$
$x>-5\text{ AND }x>4$
$\text{so }x>4$.
If $(x+5)<0\text{ AND }(x-4)<0$
$x<-5\text{ AND }x<4$
$\text{so }x<-5$.
What is the union of those two?
No Union.
14. ## Re: Solve the Inequality
I am sorry. Maybe, (-infinite, -5) U (4, +infinite).
15. ## Re: Solve the Inequality
Originally Posted by joshuaa
Thank you a Lot. It is clear now, but I need some practice to master this subject and eliminate confusion.
I'd pay attention to what Plato is saying. If he doesn't like multiplying out in problems like this there's certainly a good reason for it.
Page 1 of 2 12 Last | 2014-12-22 17:08:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8872027397155762, "perplexity": 837.3516928241506}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802775517.52/warc/CC-MAIN-20141217075255-00150-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://www.rdocumentation.org/packages/base/versions/3.6.2/topics/unique | # unique
0th
Percentile
##### Extract Unique Elements
unique returns a vector, data frame or array like x but with duplicate elements/rows removed.
Keywords
manip, logic
##### Usage
unique(x, incomparables = FALSE, …)# S3 method for default
unique(x, incomparables = FALSE, fromLast = FALSE,
nmax = NA, …)# S3 method for matrix
unique(x, incomparables = FALSE, MARGIN = 1,
fromLast = FALSE, …)# S3 method for array
unique(x, incomparables = FALSE, MARGIN = 1,
fromLast = FALSE, …)
##### Arguments
x
a vector or a data frame or an array or NULL.
incomparables
a vector of values that cannot be compared. FALSE is a special value, meaning that all values can be compared, and may be the only value accepted for methods other than the default. It will be coerced internally to the same type as x.
fromLast
logical indicating if duplication should be considered from the last, i.e., the last (or rightmost) of identical elements will be kept. This only matters for names or dimnames.
nmax
the maximum number of unique items expected (greater than one). See duplicated.
arguments for particular methods.
MARGIN
the array margin to be held fixed: a single integer.
##### Details
This is a generic function with methods for vectors, data frames and arrays (including matrices).
The array method calculates for each element of the dimension specified by MARGIN if the remaining dimensions are identical to those for an earlier element (in row-major order). This would most commonly be used for matrices to find unique rows (the default) or columns (with MARGIN = 2).
Note that unlike the Unix command uniq this omits duplicated and not just repeated elements/rows. That is, an element is omitted if it is equal to any previous element and not just if it is equal the immediately previous one. (For the latter, see rle).
Missing values ("NA") are regarded as equal, numeric and complex ones differing from NaN; character strings will be compared in a “common encoding”; for details, see match (and duplicated) which use the same concept.
Values in incomparables will never be marked as duplicated. This is intended to be used for a fairly small set of values and will not be efficient for a very large set.
When used on a data frame with more than one column, or an array or matrix when comparing dimensions of length greater than one, this tests for identity of character representations. This will catch people who unwisely rely on exact equality of floating-point numbers!
##### Value
For a vector, an object of the same type of x, but with only one copy of each duplicated element. No attributes are copied (so the result has no names).
For a data frame, a data frame is returned with the same columns but possibly fewer rows (and with row names from the first occurrences of the unique rows).
A matrix or array is subsetted by [, drop = FALSE], so dimensions and dimnames are copied appropriately, and the result always has the same number of dimensions as x.
##### Warning
Using this for lists is potentially slow, especially if the elements are not atomic vectors (see vector) or differ only in their attributes. In the worst case it is $$O(n^2)$$.
##### References
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.
duplicated which gives the indices of duplicated elements.
rle which is the equivalent of the Unix uniq -c command.
##### Aliases
• unique
• unique.default
• unique.data.frame
• unique.matrix
• unique.array
##### Examples
library(base) # NOT RUN { x <- c(3:5, 11:8, 8 + 0:5) (ux <- unique(x)) (u2 <- unique(x, fromLast = TRUE)) # different order stopifnot(identical(sort(ux), sort(u2))) length(unique(sample(100, 100, replace = TRUE))) ## approximately 100(1 - 1/e) = 63.21 unique(iris) # }
Documentation reproduced from package base, version 3.6.2, License: Part of R 3.6.2
### Community examples
Looks like there are no examples yet. | 2021-03-09 05:00:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19822852313518524, "perplexity": 3143.1780130363427}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178385984.79/warc/CC-MAIN-20210309030723-20210309060723-00638.warc.gz"} |
http://mathhelpforum.com/differential-equations/87822-making-substitution-differential-equation.html | # Math Help - Making a substitution in a differential equation.
1. ## Making a substitution in a differential equation.
The variables x and y are related by the differential equation:
$\frac{dy}{dx} = \frac{x^2 - y^2}{xy}$
Use the substitution y=xz, where z is a function of x, to obtain the differential equation:
$x\frac{dz}{dx} = \frac{1-az^2}{z}$
My questions sheet has 2 instead of the a, but I get a=1, which is correct?
Cheers.
2. Originally Posted by Nyoxis
The variables x and y are related by the differential equation:
$\frac{dy}{dx} = \frac{x^2 - y^2}{xy}$
Use the substitution y=xz, where z is a function of x, to obtain the differential equation:
$x\frac{dz}{dx} = \frac{1-az^2}{z}$
My questions sheet has 2 instead of the a, but I get a=1, which is correct?
Cheers.
The a should be a 2.
$x \frac{dz}{dx} + z = \frac{x^2 - x^2z^2}{x^2z}$
and isloating z' gives
$x \frac{dz}{dx} = \frac{1 - 2z^2}{z}$.
3. Originally Posted by danny arrigo
The a should be a 2.
$x \frac{dz}{dx} + z = \frac{x^2 - x^2z^2}{x^2z}$
Ah I see where I went wrong, I forgot to use the product rule when differentiating y=xz, thanks for the help. | 2015-06-02 21:00:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9653432965278625, "perplexity": 669.883645268942}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1433195036266.4/warc/CC-MAIN-20150601214356-00050-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/flux-of-circuit-made-from-metal-bar-on-parallel-rails.807257/ | # Flux of Circuit made from Metal Bar on Parallel Rails
Tags:
1. Apr 7, 2015
### LunaFly
1. The problem statement, all variables and given/known data
A simple model of a railgun is a metal bar which runs on two long parallel rails. The two rails are connected to a charged capacitor with capacitance C and a resistor with resistance R. After charging, the capacitor can discharge through the circuit. You may assume that the rails and sliding bar are perfect conductors, and that the bar moves without any friction. The two rails are cylinders with radius r with centers separated by a distance d. Assume that the sliding bar is a distance x along the rails where x>>d>>r.
(a) At some instant, the circuit has current I running through it. If the sliding bar is fixed in position, what is the total magnetic flux through the middle of the circuit? You may assume that the rails are long enough that fringe effects can be ignored and the field from the rails can be approximated to be that of infinite wires. You may also ignore the contributions from the sliding bar and the capacitor at the end of the circuit.
2. Relevant equations
∫B⋅da = ΦB
3. The attempt at a solution
I tried to model the wires as infinite, then use the principle of superposition to add up the magnetic fields in the center of the circuit. I assumed the current flows counter-clockwise through the circuit. I also assigned my coordinate system with the x-axis along the bottom rail, z-axis vertical (across the circuit, in the same direction as the bar), and the y-axis pointing out of the plane of the page. From this approach I found the magnetic field enclosed in the circuit to be:
B = μoI/(2π) * (1/z + 1/(d-z))
pointing out of the page. The 1/z term is the contribution from the bottom rail, and the 1/(d-z) term is the contribution from the top rail. My issue is when I integrate this expression over the area of the circuit to find the flux, I end up with a ln(0) term.
I am not sure how to approach this problem now. I've tried an Amperian loop as well but I was having a difficult time finding a shape that I could integrate over.
2. Apr 7, 2015
### BiGyElLoWhAt
Hmmm... I see your problem. Honestly, what I would be tempted to do is this: assume the rails carry current through the center of the rail (as opposed to the surface) and then integrate from the surface (r) to the surface of the other rail (r+d) and then bam, ln(0) turns into ln(r). I'm gonna think about it some more to see if there's a better way (I'm sure there is) so don't take that and run just yet.
Biot Svart law seems ideal to me, unless someone can provide a reason why it isn't.
3. Apr 7, 2015
### BiGyElLoWhAt
I'm not thinking of anything else at the moment. One other thing you need to do, though, is make sure you account for the other 2 sides of the circuit. You have top and bottom, but what about left and right?
4. Apr 7, 2015
### LunaFly
Thanks for the reply. I too came to the conclusion of integrating from the surface of the bottom of the rail to the surface of the top rail. This gets rid of the problem of including an infinite field in the integral. Using this approach I end up with a flux of:
Φ= μo*I*x/(2π) * ln (d/r -1)
This seems okay. As far as the other 2 sides of the circuit go, the problem statement says we can ignore the contributions from the sides of the circuit, so we have a free pass there.
The formula for the magnetic field from the infinite wire comes from the Biot Savart law so it is somewhat used in the problem. I don't see how we could apply it directly to this system without separating the wires and finding the same formula used above, but that doesn't mean there isn't a way! Do you see anything we could do with it? Thanks again.
5. Apr 7, 2015
### BiGyElLoWhAt
That's actually a particular application of the Biot-Savart law. The original is $\int_a^b \frac{\mu_0 I \vec{\ell}\times d\vec{r}}{4\pi r^2}$ I was just mentioning it to let you know you were on the right track.
$\frac{\mu_0 I}{2\pi r}$ is a solution of the Biot-Savart law for the field of an infinite wire at a distance r. | 2017-10-24 04:28:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6862068772315979, "perplexity": 301.3034292840875}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187828134.97/warc/CC-MAIN-20171024033919-20171024053919-00048.warc.gz"} |
https://stats.stackexchange.com/questions/341936/average-of-precision-and-true-negative-rate-as-an-accuracy-metric | # Average of Precision and True negative rate as an accuracy metric
When attempting to construct a classifier for a somewhat imbalanced data set, I was led to the question of measuring the performance of the classifier. One of the first things I thought of was to take the average of precision and specificity:
$$\frac{(\text{true positive rate} + \text{true negative rate)}}{2}$$
It seems quite an intuitive measure to me. Moreover one can take a weighted average depending on how important the prediction rate for each class is. Also it generalizes in a straightforward way to multiple classes. One can incorporate it into a 2-class cost matrix by setting the off diagonals to:
$$\frac{(N-N_1)}{N}, \frac{(N-N_2)}{N}$$
However, I could not find any reference online to any such measure - there appears to be no name for it! There are all sorts of combinations of other measures, but not this one. I was wondering if someone can point me to any discussion of it.
Edit: I originally had misunderstood the definition of specificity as $TN \over TN+FN$ when actually it is defined as $TN \over TN+FP$. I have corrected the title. As far as I can find, there is no name for "true-negative rate".
What you are proposing is the (weighted or not) arithmetic mean of precision and specificity.
Two of the most common (confusion matrix) metrics being used are:
• F1-score. This is essentially the harmonic mean of precision and recall.
• Fβ-score. A weighted version of the above. It measures the effectiveness of retrieval with respect to a user who attaches β times as much importance to recall as precision.
• Geometric Mean Score. It corresponds to the geometric mean of sensitivity and specificity and is used mostly in imbalanced problems.
Your question essentially comes down to:
Why are the harmonic and geometric means preferred to the arithmetic mean for averaging confusion matrix metrics?
An excellent answer concerning these three means and their diffenences can be found here. Also, a discussion on which mean to use and when can be found here.
• Thanks for the information. According to the answer at [this link][1] the average of the per class accuracy is not a "proper score function" - but I'm not convinced. I have to read up on some of the classification terminology and get back. [1]: stats.stackexchange.com/questions/251970/… – skm Apr 24 '18 at 9:23
• Nothing mentioned above is a proper accuracy score. And are you certain it was appropriate to cast this problem as a classification problem and not one of risk estimation? Describe the problem more fully. And note that imbalance is to be expected, so it's not clear why you mentioned that. – Frank Harrell Apr 24 '18 at 12:16 | 2019-07-23 22:32:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8112844824790955, "perplexity": 406.5717166685757}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195529737.79/warc/CC-MAIN-20190723215340-20190724001340-00405.warc.gz"} |
http://math.stackexchange.com/questions/151158/questions-about-this-proof-of-prim-kruskal-algorithm | Could someone please help me understand lines 5, 6 and 7 in the below proof for Prim's / Kruskal's greedy algorithm of finding minimum spanning tree:
Suppose that either algorithm produces a tree T. Let there exist another spanning tree S with a smaller total weight. Let e be an edge of smallest weight which lies in T but not S.
If we add e to S, we obtain a cycle, from Equivalent Definitions for Tree.
5 - This cycle contains an edge e' which is in S but not T, otherwise T would not be a tree.
6 - So, we replace e' in S with e from T, and obtain a new spanning tree S'.
7 - From the method of construction of T, it follows that the weight of e can not exceed that of e'.
So the total weight of S' does not exceed the total weight of T. Also, S' has one more edge in common with T than S has. We repeat the above procedure, and repeatedly change edges of S for edges of T, and each time the weight of the intermediate graph does not exceed that of T. Thus the weight of T does not exceed that of S, contradicting the definition of S. Hence T must be a minimum spanning tree.
My questions deal with points 5, 6, 7:
5 - If T has e', why is it not a tree?
6- Why does replace e' by e create a new spanning tree and not a new graph?
7- What make sure that w(e) <= w(e')?
Thank you.
-
Line $5$: It's a cycle. If all its edges are in $T$, that means $T$ has a cycle, and trees don't have cycles.
Line $6$: A spanning tree is a graph.
Line $7$: Since $e'$ is not in $T$, it was not added because its two vertices were already connected. But if $w(e)\gt w(e')$, then this connection could not have been through $e$, since $e$ wouldn't have been added yet, so the cycle (minus $e'$) would constitute a second connection between the two vertices of $e'$, thus forming a cycle, whereas $T$ is a tree and has no cycles.
-
Thank you. I have 2 more questions: a) why does the edge swapping between S and T seem to follow the greedy algorithm. b) do you how why the author of this proof had come up with the "edge swapping" argument. – Cody May 29 '12 at 15:12
@Cody: Unfortunately I don't understand your first question and I don't have an answer to the second one. – joriki May 29 '12 at 18:38
If T has every e' in the cycle, then in particular is has a cycle. Trees don't have cycles. Thus it wouldn't be a tree.
It is a new graph. The graph happens to be a tree. In fact, because they chose the edge from a cycle, which has the property that every vertex has degree 2, removing 1 edge does not separate any vertex. So it is a spanning tree.
Line 7 alludes the the construction of the trees. You didn't include this here, but in all likelihood it probably has a step like "Of the available edges E, choose e of minimal weight and add it to the tree." It's probably a bit more complicated, but it's something to that effect.
-
I like that we both wrote "Trees don't have cycles." :-) – joriki May 29 '12 at 14:11
Yes - that's very funny. – mixedmath May 29 '12 at 14:22 | 2014-10-02 06:41:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7416198253631592, "perplexity": 458.2483426191942}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1412037663718.7/warc/CC-MAIN-20140930004103-00284-ip-10-234-18-248.ec2.internal.warc.gz"} |
https://mtrend.vn/question/i-tim-loi-sai-va-sua-lai-1-when-he-still-worked-for-ofam-he-was-coming-up-with-different-ideas-t-929/ | ## I.Tìm lỗi sai và sửa lại: 1. When he still worked for Oxfam, he was coming up with different ideas to help needy people. 2. The volunteers went to a n
Question
I.Tìm lỗi sai và sửa lại:
1. When he still worked for Oxfam, he was coming up with different ideas to help needy people.
2. The volunteers went to a nearby school on a Sunday morning, picked up a food package, and delivered them to an elderly person.
II.
1. You should have taken those books back to the library.
=>Those books …………………………………………………………………………
2. We cannot exchange articles
→ Articles…………………………………………………………………………………………………..
3. You mustn’t move c; he’s too ill.
→ This man…………………………………………………………………………………………………..
4. When they have widened this street, the roar of the traffic will keep residents awake all night.
→ When they have widened this street,……………………………………………………………………..
5.They ought to have reported the accident to the police.
→ The accident…………………………………………………………………………………………………..
6. You must dry-clean it.
→ It…………………………………………………………………………………………………..
7. We often talk during our break time. (spending)
→ While we ……………………………………………………………………
8. He often bothers us during our learning lessons. (learning)
→ While we …………………………………………………………………………
9. They usually make noise during my taking days off. (taking)
→ While I …………………………………………………………………………
10. She frequently dozes during her working. (working)
→ While she ………………………………………………………………………
in progress 0
2 tháng 2021-10-06T09:40:01+00:00 2 Answers 1309 views 0
1. $_Study well_$
I.Tìm lỗi sai và sửa lại:
1. When he still worked for Oxfam, he was coming up with different ideas to help needy people.
-> came
2. The volunteers went to a nearby school on a Sunday morning, picked up a food package, and delivered them to an elderly person.
-> it
II.
1. You should have taken those books back to the library.
=>Those books …… should have been taken back to the library.……………………………………………………………………
2. We cannot exchange articles
→ Articles……..cannot be exchanged……………………………………………………………………………………………
3. You mustn’t move c; he’s too ill.
→ This man………………………..is too ill to move…………………………………………………………………………
4. When they have widened this street, the roar of the traffic will keep residents awake all night.
→ When they have widened this street,……….. residents will be kept awake all night by the roar of the traffic ( Bị động )…………………………………………………………..
. 5.They ought to have reported the accident to the police.
→ The accident…………………… should have been reported to the police………………………………………………………………………………
6. You must dry-clean it.
→ It must be dry-cleaned…………………………………………………………………………………………………..
7. We often talk during our break time. (spending)
→ While we ……are spending our break time, we often talk………………………………………………………………
8. He often bothers us during our learning lessons. (learning)
→ While we ………are learning lessons , he often bothers us…………………………………………………………………
9. They usually make noise during my taking days off. (taking)
→ While I ………am taking days off, they usually make noise…………………………………………………………………
10. She frequently dozes during her working. (working)
→ While she …………she is working, she frequently dozes……………………………………………………………
2. I.Tìm lỗi sai và sửa lại
1. still worked => was still working
2. them => it
II. Viết lại câu
1.Those books should have been taken back to the library.
2. Articles cannot be exchanged
3. This man is too ill to move
4. When they have widened this street, residents will be kept awake all night by the roar of the traffic
5. The accident should have been reported to the police.
6. It must be dry-cleaned
7. While we are spending our break time, we often talk
8. While we are learning, he often bothers us
9. While I am taking days off, they usually make noise
10. While she is working, she frequently dozes | 2022-01-22 08:24:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4451582133769989, "perplexity": 13963.80707028712}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303779.65/warc/CC-MAIN-20220122073422-20220122103422-00568.warc.gz"} |
https://en.wikisource.org/wiki/Page:Popular_Science_Monthly_Volume_82.djvu/311 | # Page:Popular Science Monthly Volume 82.djvu/311
307
THE PROGRESS OF SCIENCE
THE PROGRESS OF SCIENCE
Scarcely a mouth passes without the occurrence of one or more events disquieting to those who would make our universities the homes of scientific research, creative scholarship and social progress. Such circumstances do not usually become known, for it is to the private advantage of those concerned that they be hushed. Strange as it may seem at first sight, the state universities are on the whole making progress in the direction of greater academic freedom and dignity, while the private corporations tend to exhibit the reactionary tendencies of their boards and administrative officers. If, however, the people learn the importance to the nation of maintaining their universities on a high plane, all is well. It is easy to tax corporations which become antisocial into innocuousness. Indeed each university will find its own level by its own weight. Harvard and Columbia are still our richest institutions and probably still maintain their leadership in advanced work and public service; but they are losing ground relatively to the state universities and perhaps even in comparison with their own positions ten years ago. It would surprise most people to see the list of those who have recently declined to consider chairs at these two universities.
It is the high traditions of Harvard which give significance to the curious circular recently sent from the controller's office to those whom one university president habitually calls "the instructional force." The circular is accompanied by four large pages of instructions and a schedule containing some 180 blank spaces to be filled and is couched in jargon about "prorating salaries to the various classified functions," and the like. The professors and instructors are informed that
(2a X 3a) ${\displaystyle {{\ce {+}}}}$ (2b X 3b) ${\displaystyle {{\ce {+}}}}$ (2c X 3c) ${\displaystyle {{\ce {+}}}}$ (2d X 3d) ${\displaystyle {{\ce {+}}}}$ (2e X 3e) ${\displaystyle {{\ce {=}}}}$ total hours of regular exercises per course.
They are told that
Preparation for lectures should include only that time which was taken during the half-year for lectures delivered in this period. It should not include time spent in the general collection of materials.
Surely the only correct answer to the question how many hours a day a professor spends on his work and in preparation is twenty-four. This circular was naturally resented by members of the faculty and was partially, but somewhat grudgingly, withdrawn, the president stating that it was "issued under a misunderstanding," presumably a misunderstanding of the sentiments of the faculty.
This Harvard incident is serio-comic. At Wesleyan there has occurred within I the same past month a wholly serious breach of academic decency. The professor of economics and social science, who has served the university and the public with distinction for twenty years, made some remarks in regard to the observance of the sabbath, which found their way into the newspapers. The president wrote inquiring whether he was correctly reported, and on being told what he had said, asked for his resignation. This was promptly sent, and the president relieved him from his duties at once. The five letters passed in the same day, and the president must have acted without adequate consultation or consideration. It is as extraordinary as it is ominous that in our present academic system the liberty of speech of a professor and the fate of his wife and children should be dependent on the will of an official. In this case the professor was speaking within his own professional field, and not even to students of the university | 2018-05-27 19:14:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 5, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24198201298713684, "perplexity": 2259.523234105181}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794870082.90/warc/CC-MAIN-20180527190420-20180527210420-00080.warc.gz"} |
https://quantummechanics.ucsd.edu/ph130a/130_notes/node83.html | ## Time Development of a Gaussian Wave Packet *
So far, we have performed our Fourier Transforms at and looked at the result only at . We will now put time back into the wave function and look at the wave packet at later times. We will see that the behavior of photons and non-relativistic electrons is quite different.
Assume we start with our Gaussian (minimum uncertainty) wavepacket at . We can do the Fourier Transform to position space, including the time dependence.
We write explicitly that depends on . For our free particle, this just means that the energy depends on the momentum. For a photon, , so , and hence . For an non-relativistic electron, , so , and hence .
To cover the general case, lets expand around the center of the wave packet in k-space.
We anticipate the outcome a bit and name the coefficients.
For the photon, and . For the NR electron, and .
We see that the photon will move with the velocity of light and that the wave packet will not disperse, because .
For the NR electron, the wave packet moves with the correct group velocity, , but the wave packet spreads with time. The RMS width is .
A wave packet naturally spreads because it contains waves of different momenta and hence different velocities. Wave packets that are very localized in space spread rapidly.
Jim Branson 2013-04-22 | 2020-10-25 08:04:43 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8247565031051636, "perplexity": 550.0322736097896}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107888402.81/warc/CC-MAIN-20201025070924-20201025100924-00391.warc.gz"} |
https://www.physicsforums.com/threads/sliding-and-toppling.22882/ | # Sliding and toppling.
1. Apr 27, 2004
### jimmy p
Ok, I have a maths question. This is a little different because I dont want the answer, I want the formula so I can work out the answer (and future answers) myself. lol it's not very good self-learning with a book that doesnt tell you much. Anyway here is the question.
A solid uniform cube of side 4cm and weight 60N is situated on a rough horizontal plane. The coefficient of friction between the cube and the plane is 0.4. A force, P, acts in the middle of one of the edges of the top of the cube, at right angles to it and at angle theta to the upward vertical.
In the cses when the value theta is (a) 60 degrees (b) 80 degrees, find
(i) the force P needed to make the cube slide, assuming it does not topple;
(ii) the force P needed to make the cube topple, assuming it does not slide.
So I have tried many variations using sin, cos and tan but cannot seem to get the right answer. Can someone tell me the formula so i can work these out for myself?
2. Apr 28, 2004
### Staff: Mentor
Don't think in terms of finding "the formula". Instead, learn the basic physics involved and then you can figure out all the formulas you need no matter what.
As always in these kinds of problems, consider all the forces acting on the object. I count four: (1) the applied force P, which acts along the angle you described (θ from vertical), (2) the weight of the cube, (3) the friction of the plane against the cube, and (4) the normal force of the plane up on the cube.
Draw a picture!
In this case you need to overcome the static friction to get the cube to slide. Consider the limiting case where the force P is as big as it can be without causing translation. The cube is in equilibrium:
The sum of the forces in the y-direction must equal zero, so:
P cosθ -mg + N = 0
The sum of the forces in the x-direction must also equal zero, so:
P sinθ -μN = 0
Use these equations to figure out P, given θ.
In this case we want it to topple (tip over) not translate. So this time we consider torques about a pivot point. Consider the case where the force P just barely causes the cube to start to tip. At that point the torques about the edge must equal zero, so:
mgL/2 - P cosθ(L/2) - P sinθL = 0
("L" is the length of the side of the cube.) I'll leave it for you to figure out what I did to get that equation. | 2017-03-26 13:33:21 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8243840932846069, "perplexity": 481.21850944289093}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189239.16/warc/CC-MAIN-20170322212949-00566-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://computergraphics.stackexchange.com/questions/9294/inverted-normals-in-raytracer/9296#9296 | # Inverted Normals in Raytracer
I am having the following issue. I am working on a ray tracer for school and I am trying to render a sphere where the radiance of a intersection point is the dot product of the ray direction and the normal at the point of intersection. This works fine if return the inverted normal ( i.e. a vector going from the intersection point to the center of the sphere but not the other way around as it should be ) and I can't figure out what I messed up. Anyone have a intuitive clue on what I could have messed up? Can't really post code here as it's a school task but feel free to ask for further information and I am trying to provide as best as I can. (left: inverted normals, right: right normals)
The issue is, that you considered that the rays coming from the camera to be the light carrying rays. Instead, the rays bounce off the surface, and return to the light source (which you have decided to position at the camera). This means that your light direction is not $$\pmb{d}$$ but rather $$-\pmb{d}$$, then $$\cos\theta = - \pmb{d} \cdot \pmb{n}$$, provided that both vectors are normalized.
Also, you really want to try to retirn the correct normal, as it is, your current code will return the wrong normal if the camera is inside the sphere. So your normal should be: n = dot(n,d)<0 ? n : -n;, where n is the normal returned by your surface, and d is the ray direction for the ray that intersected the surface. | 2021-10-18 17:08:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4451881945133209, "perplexity": 253.3332031844404}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585204.68/warc/CC-MAIN-20211018155442-20211018185442-00588.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/132478-groups.html | # Math Help - Groups
1. ## Groups
hi i have this question set me and i have no idea could someone help me please:
Let G be a p-group for some prime p. Assume that G acts on a finite set X and let X^G={x memeber of X | g.x =g for all g memeber of G}
show that order X^G=order X mod p
Thanks Rich
2. Originally Posted by bobisback
hi i have this question set me and i have no idea could someone help me please:
Let G be a p-group for some prime p. Assume that G acts on a finite set X and let X^G={x memeber of X | g.x = x for all g memeber of G}
show that order X^G=order X mod p
Thanks Rich
i fixed the typo you made in the definition of $X^G.$ well, the idea is exactly the same as the one that we use to prove that the center of every finite p-group is non-trivial:
consider the partition $\{Gx_1, \cdots , Gx_m \}$ for $X.$ now $x_j \in X^G$ if and only if $|Gx_j|=1.$ thus $|X|=\sum_{j=1}^m |Gx_j|=|X^G| + \sum_{x_j \notin X^G} |Gx_j|.$ but by the orbit-stabilizer theorem we
have $|Gx_j|=[G:G_{x_j}]$ and since $x_j \notin X^G$ implies that $|Gx_j| > 1$ and so $[G:G_{x_j}] > 1,$ we'll get $p \mid [G:G_{x_j}]$ for all $x_j \notin X^G$ and the result follows. | 2016-05-29 06:30:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 12, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9632053971290588, "perplexity": 422.01078362069836}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049278389.62/warc/CC-MAIN-20160524002118-00238-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/8298-evaluating-definite-integral-print.html | # evaluating a definite integral
• Dec 1st 2006, 08:34 PM
thedoge
evaluating a definite integral
How would I evaluate this integral using the 2nd Fundamental Theorem of Calculus? Thanks in advance.
http://img218.imageshack.us/img218/4903/problemeu5.jpg
• Dec 1st 2006, 10:14 PM
Jameson
Ahh I love these questions.
Let $\frac{2i}{n}=x$. Thus $dx=\frac{2}{n}$.
Now rewrite to an integral integral with this substitution.
$\int_{L} 1 +x +x^2dx$
Now for the bounds.
For the first bound let $i=0$ and for the second let $i=n$. So the bounds are from 0 to 2.
Got it? :)
• Dec 2nd 2006, 08:08 AM
topsquark
Quote:
Originally Posted by Jameson
Ahh I love these questions.
Let $\frac{2i}{n}=x$. Thus $dx=\frac{2}{n}$.
Now rewrite to an integral integral with this substitution.
$\int_{L} 1 +x +x^2dx$
Now for the bounds.
For the first bound let $i=0$ and for the second let $i=n$. So the bounds are from 0 to 2.
Got it? :)
Shouldn't that second bound be $\infty$?
-Dan
• Dec 2nd 2006, 11:18 AM
Jameson
I don't believe so. Why would it be inifinity? On these problems i=0, then i=n. Is there another way you do it?
• Dec 2nd 2006, 11:36 AM
TD!
For the integral, the limits of x are indeed 0 to 2.
• Dec 2nd 2006, 02:12 PM
thedoge
Yea that explanation makes sense Jameson, but I'm still not sure how this would be solved once reaching that point. How'd you determine the bounds were 0 and 2?
• Dec 2nd 2006, 02:46 PM
ThePerfectHacker
Quote:
Originally Posted by thedoge
How would I evaluate this integral using the 2nd Fundamental Theorem of Calculus? Thanks in advance.
http://img218.imageshack.us/img218/4903/problemeu5.jpg
The standard form the the Riemann Ingegral is,
$\lim_{n\to \infty} \sum_{k=1}^n f(a+k\Delta x)\Delta x$
In this case,
$\Delta x=\frac{2}{n}$
Thus,
$b-a=2$
But there is not "a" term,
$a=0$ thus, $b=2$.
And the function is,
$f(x)=1+x+x^2$
Thus we need to find,
$\int_0^2 1+x+x^2 dx$
Which we can easily evaluate by Fundamental theorem because this function is continous on the closed interval.
• Dec 2nd 2006, 05:58 PM
topsquark
Quote:
Originally Posted by topsquark
Shouldn't that second bound be $\infty$?
-Dan
:o Momentary brain fart!
-Dan | 2017-12-18 14:53:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 19, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9871044754981995, "perplexity": 2493.2647547288393}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948617816.91/warc/CC-MAIN-20171218141805-20171218163805-00579.warc.gz"} |
https://www.ic.sunysb.edu/Class/phy141md/doku.php?id=phy131studiof17:lectures:chapter3&rev=1501868505&do=diff | # Differences
This shows you the differences between two versions of the page.
— phy131studiof17:lectures:chapter3 [2017/08/04 13:41] (current) Line 1: Line 1: + ~~SLIDESHOW~~ + + ====== Chapter 3 - Vectors ====== + + ===== Vectors and scalars ===== + + Vector quantities with number, **direction** and units: + + * Displacement $\vec{r}$ [m] + * Velocity $\vec{v}$ [ms<sup>-1] + * Acceleration $\vec{a}$ [ms<sup>-2] + + Scalar quantities number and units only + + * Distance traveled [m] + * Speed [ms<sup>-1] + + ===== Graphical representation of vectors and components ===== + + It is frequently useful to draw two dimensional vectors as arrows, and to split them in to components that lie along the coordinate axes. The choice of coordinate axes is up to you..but choosing the right ones will make the problem easier or harder. + + We can take a look at the acceleration due to gravity as vector using a phone accelerometer, using this [[http://www.ic.sunysb.edu/class/phy141md/iphone/gvector.html|tool]] (click on the link from your phone's browser). + + ===== Adding and subtracting vectors ===== + + {{vectoraddsubstract.png}} + + + ===== Unit Vectors ===== + + {{unitvectors.png}} + + It can be useful to express vector quantities in terms of [[http://en.wikipedia.org/wiki/Unit_vector|unit vectors]]. These are dimensionless vectors of length = 1 that point along the coordinate axes. They are usually denoted with carets (hats), i.e. $(\hat{i},\hat{j},\hat{k})$ + + For example: + + $\vec{v}\,\mathrm{ms^{-1}}=v_{x}\,\mathrm{ms^{-1}}\,\hat{i}+v_{y}\,\mathrm{ms^{-1}}\,\hat{j}+v_{z}\,\mathrm{ms^{-1}}\,\hat{k}$ + + or + + $\vec{r}\,\mathrm{m}=x\,\mathrm{m}\,\hat{i}+y\,\mathrm{m}\,\hat{j}+z\,\mathrm{m}\,\hat{k}$ + + ===== Vectors and motion ===== + + + $\vec{r_{1}}=x_{1}\,\hat{i}+y_{1}\,\hat{j}+z_{1}\,\hat{k}$ + + $\vec{r_{2}}=x_{2}\,\hat{i}+y_{2}\,\hat{j}+z_{2}\,\hat{k}$ + + $\Delta\vec{r}=\vec{r_{2}}-\vec{r_{1}}=(x_{2}-x_{1})\,\hat{i}+(y_{2}-y_{1})\,\hat{j}+(z_{2}-z_{1})\,\hat{k}$ + + Average velocity: $\vec{v_{ave}}=\frac{\Delta\vec{r}}{\Delta t}$ + + Instantaneous velocity: $\vec{v}=\frac{d\vec{r}}{dt}=\frac{dx}{dt}\,\hat{i}+\frac{dy}{dt}\,\hat{j}+\frac{dz}{dt}\,\hat{k}=v_{x}\,\hat{i}+v_{y}\,\hat{j}+v_{z}\,\hat{k}$ + + Average acceleration: $\vec{a_{ave}}=\frac{\Delta\vec{v}}{\Delta t}$ + + Instantaneous acceleration: $\vec{a}=\frac{d\vec{v}}{dt}=\frac{dv_x}{dt}\,\hat{i}+\frac{dv_y}{dt}\,\hat{j}+\frac{dv_z}{dt}\,\hat{k}=\frac{d^{2}x}{dt^2}\,\hat{i}+\frac{d^{2}y}{dt^2}\,\hat{j}+\frac{d^{2}z}{dt^2}\,\hat{k}$ + ===== Vectors - Components ===== + + {{vectorcomponentadd.png}} + + The angles $\theta_{1}$ and $\theta_{2}$ are defined with respect to the positive $x$ axis, ie. $\theta_{1}$ is negative and $\theta_{2}$ is positive. + + | $v_{1x}=v_{1}\cos\theta_{1}$ <html>    | $v_{2x}=v_{2}\cos\theta_{2}$ <html>    | + | $v_{1y}=v_{1}\sin\theta_{1}$ <html>    | $v_{2y}=v_{2}\sin\theta_{2}$ <html>    | + | | | + | | | + | $v_{Rx}=v_{1x}+v_{2x}$ <html>    | $\tan\theta_{R}=\frac{v_{Ry}}{v_{Rx}}$ <html>    | + | $v_{Ry}=v_{1y}+v_{2y}$ <html>    | $v_{R}=\sqrt{v_{Rx}^2+v_{Ry}^2}$ <html>    | + + + + + + ===== Vectors - Multiplication by a scalar ===== + + Multiplication of a vector by a scalar can change the magnitude, but not the direction of the vector, ie. each component of the vector is multiplied by the scalar in the same way. | 2020-08-04 09:01:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6960957050323486, "perplexity": 1294.261018979427}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735867.23/warc/CC-MAIN-20200804073038-20200804103038-00309.warc.gz"} |
https://homework.cpm.org/category/ACC/textbook/ccaa/chapter/7%20Unit%206/lesson/CC2:%207.2.2/problem/7-112 | ### Home > CCAA > Chapter 7 Unit 6 > Lesson CC2: 7.2.2 > Problem7-112
7-112.
Solve the proportions using any strategy you choose. Show all of your steps. Homework Help ✎
1. $\frac { 35 } { 70 } = \frac { x } { 100 }$
Use cross multiplication.
$35·100=70x\\ \quad \ 3500 = 70x$
$x= \frac{3500}{70}=50$
1. $\frac { 12 } { 33 } = \frac { m } { 11 }$
Use the inverse to undo.
$\left(\frac{11}{1}\right)\frac{12}{33}=\frac{m}{11}\left(\frac{11}{1}\right)$
$m=4$
1. $\frac { x } { 15 } = \frac { 12 } { 75 }$
Use a Giant One to find the equivalent ratio.
$\frac{x}{15}\left(\frac{?}{?}\right)=\frac{12}{75}$
$\frac{x}{15}\left(\frac{5}{5}\right)=\frac{12}{75}$
$x=2.4$
1. $\frac { 4 } { 32 } = \frac { 10.5 } { x }$
Try using one of the methods presented in either part (a), (b), or (c). | 2020-02-20 18:58:57 | {"extraction_info": {"found_math": true, "script_math_tex": 11, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5616040229797363, "perplexity": 4868.884354962117}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145260.40/warc/CC-MAIN-20200220162309-20200220192309-00268.warc.gz"} |
https://cstheory.stackexchange.com/questions/25047/candidates-for-one-way-function | # Candidates for One-Way Function
Why are the candidates for one-way functions so few?
Today, almost all candidates are based on elementary mathematics, except Goldreich's candidate 2000 and ... (?!).
Why one can not generate several candidates by using advanced mathematics, for example using complex structures of combinatorics?
• huh? isnt every NP complete problem a candidate for a one-way function? maybe there is some other core question here? – vzn Jun 27 '14 at 20:56
• @vzn: NP-complete problems are hard in worst cases, while one-way functions (OWFs) are hard on average. For this reason, most NP-complete problems cannot be used (or at least, we don't know how to use them) as a basis for OWFs. To further complicate things, not every hard-on-average problem can be used as a OWF! See Impagliazzo's Five Worlds for more information. – M.S. Dousti Jun 28 '14 at 8:43
• right. anyway "candidate" means also roughly "conjectured" and isnt it an open conjecture that every NP complete problem could "somehow" be used as a OWF? anyway it would seem that this question comes down to the same ubiquitous difficulty in the field of proving lower bounds... referred to in Arora/Barak as "complexity theory's Waterloo". (also tightly coupled with basic open questions in average case complexity theory about how to find uniformly hard distributions for NP complete problems...) in other words it cuts to the heart of key open conjectures in TCS close to P=?NP – vzn Jun 28 '14 at 15:19
• The premise in the question seems faulty. For instance, $f(k) = \text{AES}(k,0)$ (the AES encryption of the all-zeros plaintext under the key $k$) is a good candidate for a one-way function. I'd hesitate to say that it is based on elementary mathematics; it's not really based on any kind of mathematics, exactly. That also makes it clear that there are many candidate one-way function (the candidates aren't few at all); there are at least as many candidates as proposed block ciphers, and there are many proposed block ciphers out there. – D.W. Jan 12 '18 at 22:13
Here is a "canned" answer that might be useful, but has no cryptographic depth (hopefully we get answers with depth as well).
What makes for a good candidate OWF? The naive answer tends to boil down to "something that looks hard to invert to me", but the expert's response is usually more like "something that many smart people have tried to invert but failed" (or something whose invertability would imply that of such a function). From this perspective, it is worse for the problem to be more obscure (fewer have tried it), and it may be worse that the function is more complicated (this obfuscates whether it is truly difficult or you just cannot see the solution yourself).
To put the intuition another way, a common "bad intuition" is that if a problem looks more complicated or difficult to define/understand, then it is more likely to be computationally difficult to solve. Theoretical crypto does not accept this premise. The primary evidence we have for computational hardness is a history of failed attempts, which means good candidates should be simple, well-known functions with long histories.
As for your last question, the are several candidates for combinatorial one-way functions. This paper by Kojevnikov and Nikolenko lists three combinatorial complete one-way functions that are based on the tiling problem of Levin, semi- Thue systems, and Post Correspondence problem ( complete means those functions are one-way if one-way functions do exist).
Update: A more interesting candidate was given by Gligoroski. He proposed candidate combinatorial one-way function based on Latin squares (Quasigroups).
• Another candidate using latin squares has been presented by Danilo Gligoroski. – Arash Ahadi Jun 27 '14 at 19:30
the conjecture that (roughly) "every NP complete function could somehow be used to create one way functions" has not been disproven so there is not really a shortage of "candidates" in at least that sense. an interesting recent formulation of this is by Cook et al, On the One-Way Function Candidate Proposed by Goldreich where they analyze the complexity of Goldreich's function by reducing it to SAT and doing empirical SAT experiments which give circumstantial evidence of its hardness (along with theoretical evidence also).
also existence of one way functions is tightly coupled to core long open problems in complexity theory eg the P=?NP problem. which is in turn tightly coupled with proving lower bounds in complexity theory, referred to by Arora/Barak in Complexity theory: a modern approach as complexity theory's Waterloo.
here is one ref by leading researchers on this angle that may be helpful:
• What "conjecture" are you talking about? The "somehow" statement is so fluffy that it's meaningless. I don't see how Cook et al relates to it at all, it's about a specific candidate OWF and not about a general scheme to construct OWFs from NP-complete languages. The Akavia et al. results are relevant and give a specific meaning to something like your "conjecture" -- existence of an efficient reduction from a worst-case NP-complete problem to inverting an OWF on average -- but their results are negative, i.e. they rule out some reductions. – Sasho Nikolov Jun 29 '14 at 23:23
• admittedly sketchy, some imagination & connecting-the-dots & further assembly reqd. cook ref demonstrates how "reverse-engineering"/breaking a OWF seems to be universally reducible to solving a SAT construction (ie case study as circumstantial evidence). the Akavia paper shows the general idea (of OWFs based on NP hardness) is at least under consideration. – vzn Jun 30 '14 at 1:12
• Reducing inverting OWF to solving SAT is an easy exercise. You probably want the reverse direction, but Akavia et al. give some evidence that this is unlikely. If anything is known about the Akavia et al. formalization of the "conjecture", it is that it is likely to be wrong. – Sasho Nikolov Jun 30 '14 at 1:52
• ok! anyway the basic idea, loosely aligned with the refs & in contrast to the basic premise of the question, is that knowledge in this area seems to be limited based on existing/known theoretical "machinery/technology" which is not up to the task of answering definitively (and in many ways this is tied to a/ the central quandary of the field), & conceivably there could be a large undiscovered universe of OWFs, nothing known really rules it out, and that a seeming "lack of candidates" may be more related to a human-oriented lack of imagination or ingenuity at this point. – vzn Jun 30 '14 at 4:18
• filling in more bkg thinking. the Cook paper seems to show in particular (and by analogy, in general) that OWFs can be thought of as generators for "always hard" instances of SAT. by properties/defn of NP completeness there exist P-time 1-1 instance conversions between SAT and every other NP complete problem. so it seems any "secure" OWF can also be a generator for "always hard" instances of any NP complete problem and each one seems a "different version/variant" of the OWF.... reminiscent of Berman-Hartmanis conjecture – vzn Jul 3 '14 at 15:27 | 2019-05-27 11:02:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.703460693359375, "perplexity": 1165.0454100295049}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232262369.94/warc/CC-MAIN-20190527105804-20190527131804-00435.warc.gz"} |
https://www.physicsforums.com/threads/differential-eq-implicit-sol.257714/ | # Differential eq (implicit sol)
1. Sep 20, 2008
### Qyzren
1. The problem statement, all variables and given/known data
i have a differential equation.
∂h/∂t + [g sin a h²/v]∂h/∂x = 0. where h = h(x,t).
i need to show by substitution that the (implicit) general solution for h is h = f(x - (g sin a h²t/v)) where f is an arbitrary differential function of a single variable.
3. The attempt at a solution
∂h/∂x = ∂f/∂x
∂h/∂t = ∂f/∂t * (-g sin a h²/v)
subbing in gives
∂f/∂t*(-g sin a h²/v) + (g sin a h²/v)*∂f/∂x = 0
cancelling g sin a h²/v
-∂f/∂t + ∂f/∂x = 0
so now i have to show... ∂f/∂t = ∂f/∂x ???
seems like i'm going in circles...
any help will be appreciated
2. Sep 20, 2008
### xaos
try:
h(x,t)=f(y(x,t))
3. Sep 20, 2008
### bdforbes
I'm not sure where the brackets should be, so let me just confirm that this is the problem:
$$\frac{\partial h}{\partial t} + \frac{gsin(a)h^2}{v}\frac{\partial h}{\partial x} = 0$$
$$h=f(x-\frac{gsin(a)h^{2}t}{v})$$
4. Sep 20, 2008
### bdforbes
Okay I worked through it, that must be the correct form. Your problem is that you're interpreting f as a function of two variables. As xaos said, write h=f(y(x,t)). Then:
$$\frac{\partial h}{\partial t} = f'(y)\frac{\partial}{\partial t}(y)$$
5. Sep 20, 2008
### HallsofIvy
Staff Emeritus
f is a function of a single variable so "$\partial f/\partial x$" and "$\partial f/\partial t$" are meaningless. Having that "h" inside f makes it complicated: More correctly
$$\partial h/\partial x= f'(x- (g sin(ah^2t/v))(1- 2aght/v)(-g cos(ah^2t/v))(2aht/v)\partial h/\partial x$$
and
$$\partial h/\partial t= f'(x- (g sin(ah^2t/v)(-2aght/v)(-gcos(ah^2t/v))((2aht/v)\partial h/\partial t+ ah^2/v)$$ | 2017-02-23 07:57:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7336804866790771, "perplexity": 3928.6452412807257}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171162.4/warc/CC-MAIN-20170219104611-00542-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://openstudy.com/updates/4e5012780b8b958804aa6fd6 | ## Adorkable Group Title 7t^9 Divided by t^6 . Help me out please? 2 years ago 2 years ago
1. LagrangeSon678 Group Title
you have to use this rule of exponenets : a^m/a^n=a^m-n
2. jim_thompson5910 Group Title
$\large \frac{7t^9}{t^6}$ $\large 7t^{9-6}$ $\large 7t^3$ So $\large \frac{7t^9}{t^6}=7t^3$
3. Outkast3r09 Group Title
Or you can use the multiplication law
4. Outkast3r09 Group Title
in that case you know that 1/t^6 = t^-6 so you get 7t^9*t^-6 (when multiplying two of the same base terms withexponents you add so it's be $7t^(9+-6) = 7t^3$
5. jim_thompson5910 Group Title
type \ [ 7^{-6} = \frac{1}{7^6} \ ] to get $\large 7^{-6} = \frac{1}{7^6}$
6. jim_thompson5910 Group Title
oh and remove all spaces from the code above
7. Outkast3r09 Group Title
gotcha... why don't they have a fraction button in the menu >_<
8. jim_thompson5910 Group Title
everything you'll need can be found in the equation editor by clicking on the "equation" button below
9. jim_thompson5910 Group Title
good question, didn't realize it wasn't there
10. Outkast3r09 Group Title
haha yeah it'd be useful
11. jim_thompson5910 Group Title
for more functions, look up LaTex since this is the language it's using | 2014-07-22 17:41:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5356654524803162, "perplexity": 10090.46217687425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997860453.15/warc/CC-MAIN-20140722025740-00171-ip-10-33-131-23.ec2.internal.warc.gz"} |
https://robotics.stackexchange.com/questions/10556/quadrature-encoder-signal-from-dc-motor-is-very-noisy/10559 | # Quadrature encoder signal from dc motor is very noisy
I'm starting out with robotics, got my first DC gear motor with quadrature encoder (https://www.pololu.com/product/2824):
I ultimately plan to hook it up to a motor driver connected to a Tiva Launchpad. However, since I'm a noob, and curious, I am starting by just playing with it with my breadboard, oscilloscope, and voltage source. E.g., when I plug in the motor power lines into my (variable) voltage source the axis spins nicely as expected between 1 and 12 V.
The problems start when I try to check how the encoder works. To do this, first I plug a a 5V source into the encoder GND/Vcc, and then try to monitor the encoder output.
While the motor is running, I check the Yellow (encoder A output) cable (referencing it to the green (encoder GND) cable). I made a video that shows a representative output from one of the lines (no USB on my old oscilloscope so I took a video of it using my phone).
As you would see at the video, the output doesn't look anything like the beautiful square waves you typically see in the documentation. Instead, it is an extremely degraded noisy sin wave (at the correct frequency for the encoder). The amplitude of the sin is not constant, but changes drastically over time. Strangely, sometimes it "locks in" and looks like the ideal square wave, for about a second or two, but then it gets all wonky again.
Both of the lines (encoder A and B output) act this way, and they act this way at the same time (e.g., they will both lock in and square up at the same time, for those brief glorious moments of clarity). Both of my motors are the same, so I don't think it's that I have a bad motor.
I have also checked using Vcc=12V, but it made no difference other than changing the amplitude of the output.
• Are you just connecting the "Encoder A Output" (for example) directly to the Oscilloscope input? If so I would suggest also adding a pull-up resistor (maybe 5k or 10k ohms) from each output to "Encoder Vcc". This might be necessary to get the proper output from the encoders, if they are Hall-effect types. – Andy Aug 30 '16 at 6:24
• If you look at the encoder board on the motor, you will see it already has pull up resistors and even some filtering caps. I would suggest fiddling a little with a sensor. Try to bend it towards the magnet and see if it helps. – mactro Aug 30 '16 at 8:50
• Oh crap I reversed Encoder Vcc/Encoder Ground last night when I "fiddled with it". And that's when it stopped working. Seems to be close now. Oh, and one of my connections was fubar in my breadboard so I was only getting ~50 mV through from 5V. Not sure how that is possible...but I checked and there it was. Dear lord I am such a noob (if you can't tell...programmer here, not hardware guy). – neuronet Aug 30 '16 at 13:29
• Umm, while I hate to admit this, I have to do it. I haven't done electronics in a few years, and never did it much. I forgot the top/bottom power rails on a breadboard are separate. There was no bad connection on my breadboard. I was a freaking idiot (Napolean Dynamite voice). – neuronet Aug 30 '16 at 13:46
• Thank you for updating - it's always good for the rest of us to know where the missing link was! – Andy Aug 30 '16 at 14:04
Conceptually everything was set up correctly, but a number of basic mistakes affected the signal. Here is the set-up which should be used to record the signal from one of the encoder outputs:
A set up like this should result in a a clean signal if your motor/encoder is not broken. Once it's up and running, seeing the signal is simple. It's as easy as checking the square wave coming off of any common function generator: fiddle with your oscilloscope until it looks good (or just hit autoset and any good oscilloscope will do it for you, or read the excellent answer by Chuck for more help on that).
The underlying problems seem to stem from an accumulation of rust when it comes to dealing with real, wire and breadboard, electronics projects:
1. The top/bottom breadboard power rails are typically separated, so you cannot assume continuity between them. Always keep a multimeter handy and do simple continuity tests before doing deeper troubleshooting. When stuff's busted, never assume continuity. Test for continuity.
2. Be careful not to confuse motor ground with encoder ground, motor ground is likely to be electrically very noisy,m which will corrupt your encoder signal.
• I'll add to your comment about testing for continuity - whenever you get a printed circuit board (PCB) that you ordered, BEFORE you solder anything to it, check continuity between power and ground, power and signal buses, ground and signal buses, and the individual lines in all the signal buses. Some manufacturers, cheap ones especially, may not etch the board correctly and this will bridge two buses together. Similarly, if your power, ground, or signal buses are very long (service a large number of components), spot check between the "source" and each device to check. – Chuck Aug 31 '16 at 18:32
• Once you solder devices onto the board, the impedances of those devices (filter capacitor, etc.) makes it really hard to determine if there's really a short between V+ and GND or similar. I would have saved weeks of my life if this had been my practice from the start. – Chuck Aug 31 '16 at 18:34
• Thanks for your answer neuronet. I've edited it to be a little nicer, since we expect people to be as nice to themselves as we expect them to be to others. *8') – Mark Booth Jan 8 '19 at 12:04
I followed the link to your Reddit post and, after checking the datasheet for your motor, I agree with what some people are saying there, but I'll expand their answers and hopefully give you some insight into what I think is happening.
I hope this doesn't come off as mean since I'm not trying to be, but how good are you with the oscilloscope? I could see if the scope is configured wrong, it will not be displaying properly. If you have it set to trigger on one of the rising edges and have the sample frequency high enough to get multiple samples per square wave, they should all overlap and look nice and stable.
I think u have to read it at a frequency specified on the encoder data sheet.
I think what is happening, that the others are commenting on, is that you are not sampling the output at a high enough frequency. You haven't commented as to what the motor speed is when you monitor the encoder output, but let's assume it's the full unloaded output speed of 200rpm.
The encoder is setup to output 3,200 counts per revolution of the output shaft. At 200rpm, that means you're getting $(200*3200) = 640,000 \mbox{ counts/min}$ which is equivalent to $10.6 \mbox{ kHz}$.
As explained in this video on aliasing, when you under-sample a signal you get false signals - signals that, as far as your instruments are concerned, are real but don't actually exist. They "appear" because of instrument inadequacies.
Finally, as explained in this video on oscilloscope sampling rates, the person says,
Note by the way also the trigger is not working. This is maybe your first clue if you're playing with your scope and you can't get it to trigger right, "What's going on here, why isn't it triggering correctly?" It's triggering on the aliased signal.
Your video is very blurry, so it's very hard to read what anything is set to on your scope, but it kinda looks like, at the bottom, it says, "<10Hz". Also, at the top, it looks like it says, "M Pos: 270.0ms". This all makes it seem like your scope is set way too slow to be able to see the signal correctly.
At 10.6 kHz, the signal period is (1/10666) = 0.000094 seconds, or 94$\mu$s. An oscilloscope setting of 100$\mu$s per division should get the actual signal when the motor is at full speed.
tl;dr - Set the scope to 100$\mu$s per division. You're reading an aliased signal, not the real signal, which is why it keeps jumping around like that.
• Unfortunately, my errors were way more basic than this (see comments above), but this is a good answer so I'll upvote it, and won't delete my question because this is useful. The engineer where I work also went right to this as a problem. The problem was so basic I literally wasn't getting the right signals, and once I did I got the osc settings right pretty quickly (despite my incompetence with wires, I can do oscilloscope settings). – neuronet Aug 30 '16 at 13:43
• Frankly I was about to delete my question, because it was based on such basic stupid mistakes, but then your answer popped up, so now here we are. A monument to my amateurishness with basic breadboard electronics. – neuronet Aug 30 '16 at 13:52
• @neuronet - Glad you figured it out! If you leave the question open, it gets put in a "question purgatory" where it gets bumped occasionally and never marked as resolved. If you could, please post an answer (you can answer your own question) and, after the time limit expires you can accept it as the answer. I think it's maybe a day or two until you can accept it. This will mark the question as resolved (and help future visitors find the correct answer quickly!) – Chuck Aug 30 '16 at 13:53
• Thanks, will do. Must run to work but will post when I get a chance. Will try to actually post something useful for people who want to monitor their quad encoders on their tabletops like I did, so at least there will be a little tutorial on that. Dear lord I'm so embarrassed. – neuronet Aug 30 '16 at 14:03
• @neuronet - Eh I wouldn't worry about it too much. Hopefully you didn't spend too much time on it. I've done terrible, terrible simple things like this before, too - wondering why a value isn't changing when I'm not actually referencing that value, wondering why a file looks so strange when I loaded the wrong file, wondering where my glasses are when I'm holding them, etc. Mistakes happen! :-P – Chuck Aug 30 '16 at 15:05 | 2021-05-18 11:18:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40721067786216736, "perplexity": 1247.7214783051725}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989819.92/warc/CC-MAIN-20210518094809-20210518124809-00311.warc.gz"} |
https://www.ias.ac.in/describe/article/pram/079/04/0703-0717 | • Searches for physics beyond the Standard Model at the Tevatron
• # Fulltext
https://www.ias.ac.in/article/fulltext/pram/079/04/0703-0717
• # Keywords
Fermilab; CDF; D0; searches.
• # Abstract
During the last 10 years, the Fermilab Tevatron has produced $p \bar{p}$ collision at a centre-of-mass energy of 1.96 TeV that the CDF and DO Collaborations have scrutinized looking for new physics in a wide range of final states. Here, recent updates of new physics searches are reported using a data sample of up to 9 fb-1
• # Author Affiliations
1. INFN Laboratori Nazionali di Frascati, via E. Fermi 40, 00044 Frascati (RM), Italy
• # Pramana – Journal of Physics
Current Issue
Volume 93 | Issue 6
December 2019
• # Editorial Note on Continuous Article Publication
Posted on July 25, 2019 | 2019-11-21 10:31:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18725551664829254, "perplexity": 5559.219935325083}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670770.21/warc/CC-MAIN-20191121101711-20191121125711-00302.warc.gz"} |
https://gluon-cv.mxnet.io/build/examples_action_recognition/dive_deep_i3d_kinetics400.html | # 4. Dive Deep into Training I3D mdoels on Kinetcis400¶
This is a video action recognition tutorial using Gluon CV toolkit, a step-by-step example. The readers should have basic knowledge of deep learning and should be familiar with Gluon API. New users may first go through A 60-minute Gluon Crash Course. You can Start Training Now or Dive into Deep_.
## Start Training Now¶
Note
Feel free to skip the tutorial because the training script is self-complete and ready to launch.
Download Full Python Script: train_recognizer.py
For more training command options, please run python train_recognizer.py -h Please checkout the model_zoo for training commands of reproducing the pretrained model.
### Network Structure¶
First, let’s import the necessary libraries into python.
from __future__ import division
import argparse, time, logging, os, sys, math
import numpy as np
import mxnet as mx
import gluoncv as gcv
from mxnet import gluon, nd, init, context
from mxnet import autograd as ag
from mxnet.gluon import nn
from mxnet.gluon.data.vision import transforms
from gluoncv.data.transforms import video
from gluoncv.data import Kinetics400
from gluoncv.model_zoo import get_model
from gluoncv.utils import makedirs, LRSequential, LRScheduler, split_and_load, TrainingHistory
Here we pick a widely adopted model, I3D-InceptionV1, for the tutorial. I3D (Inflated 3D Networks) is a widely adopted 3D video classification network. It uses 3D convolution to learn spatiotemporal information directly from videos. I3D is proposed to improve C3D model by inflating from 2D models. We can not only reuse the 2D models’ architecture (e.g., ResNet, Inception), but also bootstrap the model weights from 2D pretrained models. In this manner, training 3D networks for video classification is feasible and getting much better results.
# number of GPUs to use
num_gpus = 1
ctx = [mx.gpu(i) for i in range(num_gpus)]
# Get the model i3d_inceptionv1_kinetics400 with 400 output classes, without pre-trained weights
net = get_model(name='i3d_inceptionv1_kinetics400', nclass=400)
net.collect_params().reset_ctx(ctx)
print(net)
### Data Augmentation and Data Loader¶
Data augmentation for video is different from image. For example, if you want to randomly crop a video sequence, you need to make sure all the video frames in this sequence undergo the same cropping process. We provide a new set of transformation functions, working with multiple images. Please checkout the video.py for more details. Most video data augmentation strategies used here are introduced in [Wang15].
transform_train = transforms.Compose([
# Fix the input video frames size as 256×340 and randomly sample the cropping width and height from
# {256,224,192,168}. After that, resize the cropped regions to 224 × 224.
video.VideoMultiScaleCrop(size=(224, 224), scale_ratios=[1.0, 0.875, 0.75, 0.66]),
# Randomly flip the video frames horizontally
video.VideoRandomHorizontalFlip(),
# Transpose the video frames from height*width*num_channels to num_channels*height*width
# and map values from [0, 255] to [0,1]
video.VideoToTensor(),
# Normalize the video frames with mean and standard deviation calculated across all images
video.VideoNormalize([0.485, 0.456, 0.406], [0.229, 0.224, 0.225])
])
With the transform functions, we can define data loaders for our training datasets.
# Batch Size for Each GPU
per_device_batch_size = 5
# Number of data loader workers
num_workers = 0
# Calculate effective total batch size
batch_size = per_device_batch_size * num_gpus
# Set train=True for training the model.
# new_length indicates the number of frames we use as input.
# new_step indicates we skip one frame to sample the input data.
train_dataset = Kinetics400(train=True, new_length=32, new_step=2, transform=transform_train)
print('Load %d training samples.' % len(train_dataset))
shuffle=True, num_workers=num_workers)
### Optimizer, Loss and Metric¶
# Learning rate decay factor
lr_decay = 0.1
# Epochs where learning rate decays
lr_decay_epoch = [40, 80, 100]
optimizer = 'sgd'
# Set parameters
optimizer_params = {'learning_rate': 0.01, 'wd': 0.0001, 'momentum': 0.9}
# Define our trainer for net
trainer = gluon.Trainer(net.collect_params(), optimizer, optimizer_params)
In order to optimize our model, we need a loss function. For classification tasks, we usually use softmax cross entropy as the loss function.
loss_fn = gluon.loss.SoftmaxCrossEntropyLoss()
For simplicity, we use accuracy as the metric to monitor our training process. Besides, we record metric values, and will print them at the end of training.
train_metric = mx.metric.Accuracy()
train_history = TrainingHistory(['training-acc'])
### Training¶
After all the preparations, we can finally start training! Following is the script.
Note
In order to finish the tutorial quickly, we only train for 3 epochs on a tiny subset of Kinetics400, and 100 iterations per epoch. In your experiments, we recommend setting epochs=100 for the full Kinetics400 dataset.
epochs = 3
lr_decay_count = 0
for epoch in range(epochs):
tic = time.time()
train_metric.reset()
train_loss = 0
# Learning rate decay
if epoch == lr_decay_epoch[lr_decay_count]:
trainer.set_learning_rate(trainer.learning_rate*lr_decay)
lr_decay_count += 1
# Loop through each batch of training data
for i, batch in enumerate(train_data):
# Extract data and label
with ag.record():
output = []
for _, X in enumerate(data):
X = X.reshape((-1,) + X.shape[2:])
pred = net(X)
output.append(pred)
loss = [loss_fn(yhat, y) for yhat, y in zip(output, label)]
# Backpropagation
for l in loss:
l.backward()
# Optimize
trainer.step(batch_size)
# Update metrics
train_loss += sum([l.mean().asscalar() for l in loss])
train_metric.update(label, output)
if i == 100:
break
name, acc = train_metric.get()
# Update history and print metrics
train_history.update([acc])
print('[Epoch %d] train=%f loss=%f time: %f' %
(epoch, acc, train_loss / (i+1), time.time()-tic))
# We can plot the metric scores with:
train_history.plot()
Due to the tiny subset, the accuracy number is quite low. You can Start Training Now on the full Kinetics400 dataset.
If you would like to explore more recent models (e.g., SlowFast), feel free to read the next tutorial on SlowFast.
### References¶
Wang15
Limin Wang, Yuanjun Xiong, Zhe Wang, and Yu Qiao. “Towards Good Practices for Very Deep Two-Stream ConvNets.” arXiv preprint arXiv:1507.02159 (2015).
Total running time of the script: ( 0 minutes 0.000 seconds)
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https://web2.0calc.com/questions/help-me_17340 | +0
# Help Me
+1
332
2
+549
If a, b, and c are positive integers less than 13 such that
\begin{align*} 2ab+bc+ca&\equiv 0\pmod{13}\\ ab+2bc+ca&\equiv 6abc\pmod{13}\\ ab+bc+2ca&\equiv 8abc\pmod {13} \end{align*}
then determine the remainder when a+b+c is divided by 13.
Jul 21, 2019
#1
+30322
+4
Here's a rough and ready way to tackle this:
Notice that if b = 2a and c = 3a the left hand side of the first equation becomes 4a2 + 6a2 + 3a2 = 13a2
In other words it is a multiple of 13 and hence satisfies the first equation.
For this situation, if a, b and c are all positive integers less than 13, then a can only be one of 1, 2, 3 or 4.
Trying these in turn we find that only a = 3 (hence b = 6 and c = 9) satisfy the second and third equations.
Hence a + b + c = 5 mod(13)
Jul 21, 2019
edited by Alan Jul 21, 2019
#2
+25237
+4
If a, b, and c are positive integers less than 13 such that
\large{\begin{align*} 2ab+bc+ca&\equiv 0\pmod{13}\\ ab+2bc+ca&\equiv 6abc\pmod{13}\\ ab+bc+2ca&\equiv 8abc\pmod {13} \end{align*}}
then determine the remainder when a+b+c is divided by 13.
$$\begin{array}{|lrcll|} \hline & 2ab+bc+ca &\equiv& 0\pmod{13} \quad |\quad : (abc)\\ (1) & \dfrac{2}{c} + \dfrac{1}{a} + \dfrac{1}{b} &\equiv& 0 \pmod{13} \\ \hline & ab +2bc+ ca &\equiv& 6abc\pmod{13} \quad |\quad : (abc)\\ (2) & \dfrac{1}{c} + \dfrac{2}{a} + \dfrac{1}{b} &\equiv& 6 \pmod{13} \\ \hline & ab + bc+2ca &\equiv& 8abc\pmod{13} \quad |\quad : (abc)\\ (3) & \dfrac{1}{c} + \dfrac{1}{a} + \dfrac{2}{b} &\equiv& 8 \pmod{13} \\ \hline \end{array}$$
$$\begin{array}{|lrcll|} \hline 3*(1)-(2)-(3): & 3*\left( \dfrac{2}{c} + \dfrac{1}{a} + \dfrac{1}{b}\right) \\ & -\left( \dfrac{1}{c} + \dfrac{2}{a} + \dfrac{1}{b}\right) \\ & -\left( \dfrac{1}{c} + \dfrac{1}{a} + \dfrac{2}{b}\right) \\ & &\equiv& 3*0 - 6 - 8 \pmod{13} \\ & \dfrac{6}{c} + \dfrac{3}{a} + \dfrac{3}{b} \\ & -\dfrac{1}{c} - \dfrac{2}{a} - \dfrac{1}{b} \\ & -\dfrac{1}{c} - \dfrac{1}{a} - \dfrac{2}{b} \\ & &\equiv& -14 \pmod{13} \\ & \dfrac{4}{c} &\equiv& -14 \pmod{13} \\ & \dfrac{4}{c} &\equiv& 13-14 \pmod{13} \\ & \dfrac{4}{c} &\equiv& -1 \pmod{13} \quad | \quad +1 \\ & \dfrac{4}{c}+1 &\equiv& 0 \pmod{13} \quad | \quad *c \\ & 4+c &\equiv& 0 \pmod{13} \quad | \quad -4 \\ & c &\equiv& -4 \pmod{13} \\ & c &\equiv& 13-4 \pmod{13} \\ & \mathbf{ c } &\equiv& \mathbf{ 9 \pmod{13} } \\ \hline \end{array}$$
$$\begin{array}{|lrcll|} \hline 3*(3)-(2)-(1): & 3*\left( \dfrac{1}{c} + \dfrac{1}{a} + \dfrac{2}{b}\right) \\ & -\left( \dfrac{1}{c} + \dfrac{2}{a} + \dfrac{1}{b}\right) \\ & -\left( \dfrac{2}{c} + \dfrac{1}{a} + \dfrac{1}{b}\right) \\ & &\equiv& 3*8 - 6 - 0 \pmod{13} \\ & \dfrac{3}{c} + \dfrac{3}{a} + \dfrac{6}{b} \\ & -\dfrac{1}{c} - \dfrac{2}{a} - \dfrac{1}{b} \\ & -\dfrac{2}{c} - \dfrac{1}{a} - \dfrac{1}{b} \\ & &\equiv& 18 \pmod{13} \\ & \dfrac{4}{b} &\equiv& 18-13 \pmod{13} \\ & \dfrac{4}{b} &\equiv& 5 \pmod{13} \quad | \quad -5 \\ & \dfrac{4}{b}-5 &\equiv& 0 \pmod{13} \quad | \quad *b \\ & 4-5b &\equiv& 0 \pmod{13} \quad | \quad *(-1) \\ & 5b &\equiv& 0 \pmod{13} \quad | \quad +4 \\ & 5b &\equiv& 4 \pmod{13} \quad | \quad : 5 \\ & b &\equiv& 4*\dfrac{1}{5} \pmod{13} \\ & && \boxed{\dfrac{1}{5} \pmod{13} \\ \equiv 5^{\varphi(13)-1}\pmod{13} \\ \equiv 5^{12-1}\pmod{13} \\ \equiv 5^{11}\pmod{13} \\ \equiv 48828125\pmod{13} \\ \equiv 8\pmod{13} } \\ & b &\equiv& 4*8 \pmod{13} \\ & b &\equiv& 32 \pmod{13} \\ & b &\equiv& 32-2*13 \pmod{13} \\ & b &\equiv& 6 \pmod{13} \\ & \mathbf{ b } &\equiv& \mathbf{ 6 \pmod{13} } \\ \hline \end{array}$$
$$\begin{array}{|rcll|} \hline 2ab+bc+ca &\equiv& 0\pmod{13} \quad & | \quad b=6, \ c=9 \\ 2a*6+6*9+9*a &\equiv& 0\pmod{13} \\ 21a+54 &\equiv& 0\pmod{13} \quad & | \quad 21\equiv 8\pmod{13},\ \quad 54\equiv 2\pmod{13} \\ 8a +2 &\equiv& 0\pmod{13} \quad & | \quad :2 \\ 4a+1 &\equiv& 0 \pmod{13} \quad & | \quad -1 \\ 4a &\equiv& -1 \pmod{13} \quad | \quad : 4 \\ a &\equiv& (-1)*\dfrac{1}{4} \pmod{13} \\ && \boxed{\dfrac{1}{4} \pmod{13} \\ \equiv 4^{\varphi(13)-1}\pmod{13} \\ \equiv 4^{12-1}\pmod{13} \\ \equiv 4^{11}\pmod{13} \\ \equiv 4194304\pmod{13} \\ \equiv 10\pmod{13} } \\ a &\equiv& (-1)*10 \pmod{13} \\ a &\equiv& -10 \pmod{13} \\ a &\equiv& 13-10 \pmod{13} \\ \mathbf{ a } &\equiv& \mathbf{ 3 \pmod{13} } \\ \hline \end{array}$$
$$\begin{array}{|rcll|} \hline && \mathbf{a+b+c \pmod{13}} \\ &\equiv& 3+6+9 \pmod{13} \\ &\equiv& 18 \pmod{13} \\ &\equiv& 18-13 \pmod{13} \\ &\equiv& \mathbf{ 5 \pmod{13} } \\ \hline \end{array}$$
Jul 21, 2019 | 2020-07-08 00:12:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7026008367538452, "perplexity": 1130.304390534751}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655896169.35/warc/CC-MAIN-20200708000016-20200708030016-00333.warc.gz"} |
http://www.soulphysics.org/2009/01/group-structural-realism-part-2/ | # Group Structural Realism (Part 2)
Part 1 | Part 2 | Part 3 | Part 4
Wigner’s Legacy.
Yuval Ne’eman and Shlomo Sternberg have recorded an old particle physicist’s adage:
Ever since the fundamental paper of Wigner on the irreducible representations of the Poincaré group, it has been a (perhaps implicit) definition in physics that an elementary particle ‘is’ an irreducible representation of the group, G, of ‘symmetries of nature’. (Ne’eman and Sternberg 1991, 327.)
This idea captures much of the physical basis of GSR. Let’s discuss a bit about how one arrives at such a view.
Despite their abstractness, irreducible unitary representations do seem to satisfy our intuitions about elementary particles. Jonathan Bain points out two such intuitions: (1) an elementary particle should be uniquely labeled by a mass and a spin parameter (that is, by the eigenvalues of a total 4-momentum and a total 4-angular momentum operator); and (2) a particle should be invariant up to the group of spacetime symmetries, in order to satisfy “our intuitions concerning the continuity of particle identity through time” (Bain 2000, 402fn). One also wants that, (3) an elementary particle cannot be ‘decomposed’ into further particles; and (4) a particle should be associated with a set of observables that describe its possible states.
One can now observe: Wigner showed that the irreducible unitary representations of the Poincaré group do indeed satisfy (1) and (3) because of irreducibility; they satisfy (2) because they represent the Poincaré group; and (4) follows from the fact that they are unitary.
Although this metaphysical picture of ‘particles as representations’ is often attributed to Wigner, he does not seem to have advocated it in print. The famous 1939 paper (PDF) that Ne’emann and Sternberg refer to actually sets out only to identify the values of physical magnitudes (the so-called ‘quantum numbers’) with parameters labeling group representations, namely, which represent the group of spacetime symmetries. By classifying all the irreducible unitary representations of the Lorentz group, Wigner was able to identify all the possible labels of mass, spin and parity. This provided a deep connection between a symmetry group of nature, and the measurable properties of a quantum system.
A simple textbook example will help illustrate this connection. Take a familiar physical property like angular momentum. Quantum theory assigns a fixed value to some aspects of angular momentum, such as the total angular momentum of an isolated system. Others aspects, such as ‘angular momentum in the z-direction,’ are (prior to measurement) assigned a spectrum of values. How can concrete physical magnitudes like these be properties of a symmetry group?
In the case of angular momentum (ignoring spin, to simplify the example), one begins with the group SO(3) of continuous rotations about a point. The faithful irreducible representations of SO(3) turn out to be representable by groups of complex-valued matrices of odd dimension (2j+1), where j is a positive integer. If desired, a given representation can be thought of as acting on, say, the state space of an electron shell around a Hydrogen atom. However, the imagery of this individual object isn’t required for our construction. Instead, we can skip directly to defining the total angular momentum j = (n − 1)/2, in terms of the dimension n of the representation. The angular momentum operators can then be picked out as elements of the representation, and angular momentum in the z-direction can then be defined and shown to have the usual integer-valued spectrum, {−j, -j+1, …, 0, …, j-1, j}.
In summary: angular momentum is recovered, with all its expected properties, from facts about the symmetry group, and no assumptions about the state ψ of an individual object. Indeed, the construction seems to achieve precisely what Eddington hoped: “In fundamental investigations the conception of group-structure appears quite explicitly as the starting point; and nowhere in the subsequent development do we admit material not derived from group structure” (Eddington 1958, 147). That such a development is possible is a fact about the physics. But it is also what paves the way to a reasonable structuralist metaphysics. Wigner’s approach is just what is needed to allow the group structural realist to speak safely of properties like angular momentum, without recourse to an ontology of individual objects.
Wigner’s legacy provides us with a very interesting strategy. One can speak perfectly naturally about physical quantities, having begun the construction of quantum theory from a symmetry group. A measurable quantity like angular momentum j is of course derived from a representation space, and one can speak freely about its invariance under the action on that space. To the advocate of GSR, this is not a problem. GSR simply holds that the metaphysically most significant feature of this space is that it provides a copy of the rotation group – not that it refers to the possible states of an individual object.
Construed this way, GSR seems to lead to some surprisingly informative consequences:
Let’s think about what it would mean if spacetime had a symmetry group other than the Poincaré group. This new group would have different representations, and would thus allow for different properties of quantum systems. On Ne’eman and Sternberg’s definition, this would mean that there are different ‘particles.’ In fact, that is exactly what Bargmann (1954) and Lévy-Leblond (1967) were able to show: the Galilei group gives rise to a theory of ‘Galilei particles,’ which are different (in particular, with respect to the ‘mass’ parameter) than the usual ‘Poincaré particles.’
Let’s take another case: what would it mean for nature to admit more symmetries than just those of spacetime? According to GSR, this larger group would provide richer representations, and hence more properties for quantum particles. This is just what is suggested by the study of so-called ‘internal’ symmetries. For example, Gell-Mann’s (1961) adoption of the symmetry group SU(3) led him to organize a new taxonomy of hadrons (as they are now called) according to the irreducible representations of the new symmetry group.
Of course, in building up quantum theory as it gets used in practice, many other mathematical objects besides groups come into play: vector spaces, commutation relations, Hermitian forms, and on and on. What GSR postulates is that, out of all these tools, group structure is of central metaphysical importance. Other realists might propose a different foundation for the theory, perhaps by arguing (with Geoffrey Sewell) that, “theories of such systems should be based on the algebraic structure of their observables, rather than on particular representations thereof” (Sewell 2002, 18). So why choose GSR over all these other options? Here the two overarching aims of structural realism come into play: groups are thought to do a better job of providing a general programmatic account of science, or of solving specific problems in the interpretation of scientific theory.
We’ll discuss why this is in the next post.
Soul Physics is authored by Bryan W. Roberts. Thanks for subscribing.
Want more Soul Physics? Try the Soul Physics Tweet. | 2019-01-17 00:28:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8668654561042786, "perplexity": 744.6443279730689}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583658662.31/warc/CC-MAIN-20190117000104-20190117022104-00192.warc.gz"} |
http://openstudy.com/updates/4f18e6d0e4b04992dd21e02f | ## anonymous 4 years ago Find the limit:
1. anonymous
$(\int\limits_{0}^{x}(3t-1)^{10} dt)/3x$
2. anonymous
as x approaches 0 I'm confused because the question has two variables: x and t. How should I approach this?
3. watchmath
0
4. Mr.Math
Evaluate the limit in the top first.
5. watchmath
sorry, I didn't see 3x :)
6. watchmath
Use L'Hospital rule
7. Mr.Math
Oh right. L'Hopital's rule is the best choice: $\frac{d}{dx}\int\limits_0^x(3t-1)^{10}dt=(3x-1)^{10}.$
8. Mr.Math
The derivative of the bottom is obviously 3. Hence the limit is $$\frac{3(0)-1)^{10}}{3}=\frac{1}{3}$$.
9. anonymous
This makes sense, but how did you know to use l'hopital's rule?
10. Mr.Math
If we plug x=0, you will get $$\frac{0}{0}$$. If you plug x in the top, you get an integral from 0 to 0, which results in a value of 0. I think it's very obvious in the denominator.
11. anonymous
oh okay, thanks a ton! | 2016-10-27 09:10:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9649283289909363, "perplexity": 1513.178653999857}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988721174.97/warc/CC-MAIN-20161020183841-00439-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.tensorflow.org/versions/r1.8/api_docs/python/tf/contrib/graph_editor/keep_t_if_possible_handler | # tf.contrib.graph_editor.keep_t_if_possible_handler
tf.contrib.graph_editor.keep_t_if_possible_handler(
info,
t
)
See the guide: Graph Editor (contrib) > Module: transform
Transform a tensor into itself (identity) if possible.
This handler transform a tensor into itself if the source and destination graph are the same. Otherwise it will create a placeholder. This handler is typically used to transform a hidden input tensors.
#### Args:
• info: Transform._TmpInfo instance.
• t: tensor whose input must be transformed into a place holder.
#### Returns:
The tensor generated by the newly created place holder. | 2018-08-15 14:39:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.261316180229187, "perplexity": 4873.802014585077}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221210133.37/warc/CC-MAIN-20180815141842-20180815161842-00664.warc.gz"} |
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Accuracy of Pitch - Arvind Narayanan's journal [entries|archive|friends|userinfo]
## Arvind Narayanan's journal
Research Blog | Web page
Accuracy of Pitch [Jan. 8th, 2006|09:14 pm]
Arvind Narayanan
[ Tags | music ]
I've been doing a bit of Carnatic-related coding recently. Here's a couple of graphs comparing the accuracy of pitch of a beginner and an expert, who shall be nameless. The range is one octave - the lower sa is at 0 and the upper sa is at 12. The difference is obvious. Surprisingly, the expert has some room for improvement too. Computers can greatly enhance Carnatic learning, but most practitioners are too tradition-bound to go near them.
Pitch graph - beginner
Pitch graph - expert
Another thing this graph shows is that the claim made by many people that the equally tempered scale is not suitable for Carnatic is baseless - the deviation from perfect pitch of even the expert (2-4 hertz) is greater than the difference between the 22-shruti scale and the equally tempered 12-note scale (at most 1 hertz).
From: (Anonymous)2006-01-08 10:59 am (UTC) (Link)
I am Sandeep. Why should one go for the equally tempered scale at all ( why not 1, 16/15,... )?
Also do you have similar results for hindustAni? There are some hindustAni guys who tell me that only in hindustAni music do people actually sing the 22 notes perfectly; they have given some arguments based on the sequences of notes in rAgas that, say, multAni and miyAn kI tODi. I cannot distinguish between, say trishruti RShabhaM, catushruti RShabhaM and
2^(2/12). Another thing is, I have heard that shyAma shAstri began his
"birAna varAlicci" with trishruti RShabham. Should this be true, it seems
possible that there existed guys who had an ear fine enough to distinguish between such frequencies. May be one should hunt in traditional homes or so for people who claim they can sing these and cross-check. Similarly I saw in one of sAmbamUrtis books some ancient Indian text mentions "experiments" to illustrate the 22 shrutis but haven't bothered to procure those or check.
Also do you know how constant the tALam cycles are, atleast for
expert singers and mRdangists? As an aside I think it is laziness/inertia and not tradition that prevents musicians from approaching computers. I would like people to use time checking devices right from beginning to ensure that their kAla-pramANa is impeccable and also get their 22 shrutis right from the beginning, if possible.
Thanks a lot man.
From:
2006-01-08 11:36 am (UTC)
### kAlapramANa
Have a look here.
From: 2006-01-08 11:46 am (UTC) (Link)
We want to use equi-tempered for the obvious reason that instrumentalists can use the same instrument no matter what the base pitch of the vocalist. In ancient Tam land, they used the yazh with the tonic scale. They went on adding strings to the yazh to keep up with the different possibilities to the point where no one could keep track of the mess any more. This is the same reason that equi-tempered is used EVERYWHERE IN THE WORLD. I don't want to rehash those reasons.
As for Hindustani - I have heard all knowledgeable people say that hindustani singers have much better accuracy of pitch than carnatic. I am positive that no human cannot sing the 22 notes perfectly. Actually "22 shrutis" is bogus - there are infinitely many shrutis in the tonic scale (for want of a better name) and different people choose a different subset of the shrutis and call them the 22 shrutis. Anyway, the difference between a tonic P and an equi-tempered P is 0.2 hertz! No one can distinguish the two. But other notes may have as much as a 1 hz difference, which can possibly be sung differently by hindustani singers. It is slightly possible that some highly accomplished Carnatic singers can also do that.
From: 2006-01-08 12:01 pm (UTC) (Link)
I am clueless about mridangam; mssnlayam should know much better.
Your idea of "experiments" is interesting - finding the pitch resolution of the human ear. Wikipedia says:
Frequency resolution of the ear is, in the middle range, about 2 Hz. That is, changes in pitch larger than 2 Hz can be perceived. However, even smaller pitch differences can be perceived through other means. For example, the interference of two pitches can often be heard as a (low-)frequency difference pitch. This effect of phase variance upon the resultant sound is known as 'beating'. (http://en.wikipedia.org/wiki/Psychoacoustics)
I guess what one needs to measure is the resolution when two notes are sung not at the same time but one after the other (i.e, can one tell the difference between S followed by tri-R and S followed by chathur-R). I'm guessing the resolution must be somewhere around 1 hz.
P.S my source for history of tam music is Rangaramanuja Iyengar's book.
From: (Anonymous)2006-01-08 12:09 pm (UTC) (Link)
I am Sandeep. Well, these 22 shrutis are those that approximate the 12 notes, and most of them are in small integer ratios. And how exactly does one conclude that notes of 1 hz difference can possibly be sung differently but the 22 shrutis cannot be reproduced perfectly? And can you give any reference to the claim that everywhere else in the world people use the equally tempered scale? And in carnatic music we use instruments like violin, vINa etc. - for these and almost any instrument that can play gamakams, I doubt if it is at all possible to accurately reproduce differences of the order of 1hz ( think of what small distance the finger will have to move through ). As it is violinists etc. don't find any difficulty tuning their instrument to different shrutis and junta don't seem to notice the $\epsilon$ differences that may occur. That being the case, I don't see how shifting to equally tempered scales will be of any help whatsoever. Also there is this issue that if the ears were actually fine enough to notice such differences then the equally tempered notes not being small integral multiples of each other should produce very close harmonics simultaneously ( for instance twice the madhyasthAyi pan~camam and thrice the madhyasthAyi ShaDjam will play from the tanpura or whatever ) which would be jarring. Thanks.
From: 2006-01-08 09:38 pm (UTC) (Link)
My reference for the ubiquity of the equitempered scale is from some book I read long ago. Clearly it was an exaggeration to say that all world music uses it; but it is definitely very widespread. For example wikipedia says that arab music uses the 24-tone equi-tempered scale.
From:
2006-01-08 11:03 am (UTC)
### Methodology?
Could you please provide more information about the methodology? What did you measure? Did you make them sing each note and measure the pitch?
One more issue. My understanding is that the Carnatic scale is not equally-tempered, i.e. the frequencies are not spaced equally apart on the log scale. The notes in Western music and Carnatic music have different frequencies (though they are close to each other).
From:
2006-01-08 11:53 am (UTC)
### Re: Methodology?
I would say that there is no "Carnatic scale". The scale is defined by the pitch at which you sing, and your instruments are tuned to produce. My point here is that pitch variations of even expert carnatic singers are so much that it is not possible to pinpointedly say that it follows one scale or the other. The errors are much greater than the difference between scales.
As for instruments, I'm not knowledgeable about their construction, but as far as I can see they must necessarily be equi-tempered, otherwise you need a colossal number of strings (see my post above).
BTW, wikipedia outrightly claims that modern carnatic uses the chromatic (equitempered) scale.
As for methodology, as you said I took a sample of each of them singing each note (flat, no gamakam) and measured the pitch. | 2018-05-21 13:11:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4916784465312958, "perplexity": 2116.7809987531305}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794864186.38/warc/CC-MAIN-20180521122245-20180521142245-00168.warc.gz"} |
https://www.educator.com/mathematics/pre-algebra/fung/solving-equations-with-variables-on-both-sides.php | Nancy Fung
Solving Equations with Variables on Both Sides
Slide Duration:
Section 1: Variables and Algebraic Expressions
Numerical Expressions
13m 41s
Intro
0:00
What You'll Learn and Why
0:08
Topics Overview
0:09
Vocabulary
0:22
Order of Operations
0:26
Numerical Expression
1:03
Simplify
1:27
Simplifying an Expression
1:44
Example 1: Simplify the Expression
1:45
Simplifying an Expression
3:26
Example 2: Simplify the Expression
3:27
Using an Expression to Solve a Problem
4:29
Example 3: Babysitting
4:33
Using an Expression to Solve a Problem
6:14
Example 4: Shopping
6:17
Extra Example 1: Simplify the Expression
7:35
Extra Example 2: Simplify the Expression
8:55
Extra Example 3: Finding Total Cost
10:02
Extra Example 4: Finding Total Cost
11:44
Algebraic Expressions
9m 11s
Intro
0:00
What You'll Learn and Why
0:05
Topics Overview
0:06
Vocabulary
0:16
Variable
0:19
Algebraic Expression
0:30
Evaluate
0:51
Modeling an Algebraic Expression
1:16
Model the Expression 2x + 4
1:17
Evaluating an Algebraic Expression
1:47
Evaluate 3x - 7 for x = 8.2
1:48
Evaluating an Algebraic Expression
2:45
Evaluate (3.7 + x) ÷ 2 for x = 9.6
2:46
Using a Table to Evaluate an Expression
4:10
Example: Pairs of Shoes
4:13
Extra Example 1: Evaluate the Expression
5:46
Extra Example 2: Evaluate the Expression
6:06
Extra Example 3: Perimeter of a Rectangle
6:46
Extra Example 4: Finding Income
7:40
Writing Expressions
8m 24s
Intro
0:00
What You'll Learn and Why
0:05
Topics Overview
0:06
Vocabulary
0:15
0:26
Subtraction
1:04
Multiplication
2:06
Division
2:31
Translating Words to Expressions
3:02
Example: 9 Less than Twice a Number
3:08
Writing an Algebraic Expression
3:58
Example: Cost of Bowling
4:07
Extra Example 1: Writing Expressions
5:14
Extra Example 2: Writing Expressions
6:06
Extra Example 3: Writing Expressions
7:11
Extra Example 4: Writing Expressions
7:58
Estimating for Reasonableness
10m 20s
Intro
0:00
What You'll Learn and Why
0:05
Topics Overview
0:06
Vocabulary
0:16
Compatible Numbers
0:17
Estimating by Rounding
1:08
Estimate 36 + 6 + 58
1:30
Estimate 94 - 35 - 42
2:29
Estimating with Compatible Numbers
3:02
Estimate 297 ÷ 17
3:17
Estimate 9 Times 38
3:39
Estimating for Reasonableness
4:13
Example: Total Cost of the Items
4:15
Extra Example 1: Estimating with Compatible Numbers
6:02
Extra Example 2: Estimating by Rounding
7:05
Extra Example 3: Estimating for Reasonableness
7:27
Extra Example 4: Estimating for Reasonableness
9:15
Properties of Numbers
12m 1s
Intro
0:00
What You'll Learn and Why
0:05
Topics Overview
0:06
Vocabulary
0:16
0:21
1:46
Vocabulary Cont.
3:08
3:09
Recognizing Properties
3:55
Examples: Which Property is Illustrated?
3:58
Using Properties of Numbers
5:07
Using Mental Math to Find the Total Cost
5:18
Using Properties of Numbers
6:29
Using Mental Math to Simplify
6:30
Extra Example 1: Using Properties of Numbers
8:29
Extra Example 2: Using Properties of Numbers
9:06
Extra Example 3: Using Properties of Numbers
10:02
Extra Example 4: Using Properties of Numbers
10:32
Distributive Property
9m 40s
Intro
0:00
What You'll Learn and Why
0:05
Topics Overview
0:06
Vocabulary
0:15
Distributive Property
0:16
Using the Distributive Property
1:32
Example: 8 ( 9 + 11 )
1:35
Using the Distributive Property
2:58
Example: 7 ( 12 - h )
2:59
Example: ( m + 2 ) 5
3:20
Distributive Property in Mental Math
3:34
Example: Finding Total Cost
3:38
Extra Example 1: Summer Job
4:55
Extra Example 2: Total Cost
6:03
Extra Example 3: Fundraiser
7:36
Extra Example 4: Tomato Plants
8:24
Section 2: Integers and Exponents
Integers and Absolute Value
11m 35s
Intro
0:00
What You'll Learn and Why
0:06
Topics Overview
0:10
Vocabulary
0:21
Opposites
0:27
Integers
1:03
Absolute Value
1:23
Vocabulary
2:21
Number Line
2:23
Finding Absolute Value
2:35
Example: Absolute Value
2:36
More Examples
4:05
Example: Absolute Value of 5
4:06
Example: Absolute Value of Negative 2
4:15
Comparing Integers
4:29
Boiling Points of Elements
4:34
Comparing Integer Examples
6:00
Example 1: Comparing Integers
6:04
Example 2: Comparing Integers
6:17
Example 3: Comparing Integers
6:30
Comparing Integer Examples
6:49
Comparing Temperature
6:53
Extra Example 1: Simplify Absolute Value
8:13
Extra Example 2: Simplify Absolute Value
9:01
Extra Example 3: Simplify Absolute Value
9:29
Extra Example 4: Simplify Absolute Value
10:49
10m 20s
Intro
0:00
What You'll Learn and Why
0:05
Topics Overview
0:06
Vocabulary
0:26
Integers
0:30
0:55
Example : -3 + -7
1:01
Example : +11 + 8
1:06
Example : 2 + (+6)
1:21
1:43
Example: -3 + -7
1:48
Example: -21 + -3
2:05
Example: -11 + (-4)
2:41
3:01
Using a Number Line: -8 + 10
3:52
Using a Number Line: 4 + (-6)
4:36
5:39
Using Absolute Value: -18 + 7
5:48
6:33
7:23
Extra Example 3: Money Problem
8:46
Extra Example 4: Measurement Problem
9:15
Subtracting Integers
12m 1s
Intro
0:00
What You'll Learn and Why
0:06
Topics Overview
0:07
Vocabulary
0:21
Integers
0:25
Opposites
0:47
Rules for Multiplying Signs
1:18
Using a Number Line
2:00
Example: 2 - 5
2:25
Other Examples
2:52
Using Number Line: 10 - (-13)
3:02
Rewriting Absolute Value
4:51
Extra Example 1: Subtracting Integers
5:48
Extra Example 2: Temperature
7:26
Extra Example 3: Depth
8:51
Extra Example 4: Change in Yards
11:09
Multiplying Integers
14m 28s
Intro
0:00
What You'll Learn and Why
0:05
Topics Overview
0:06
Vocabulary
0:43
Integers
0:47
Opposites
1:03
1:41
Example 1: Using a Number Line
1:43
More Examples
3:28
Example 2: Using a Number Line
3:30
Using a Number Line
4:45
Example 3: Using a Number Line
4:46
Example 4: Using a Number Line
5:59
7:20
Arithmetic
7:35
Algebra
8:00
Rules for Multiplying Different Signs Integers
8:17
Arithmetic
8:29
Algebra
8:58
Multiplying Integer Examples
9:20
Examples of Multiplying Integers
9:21
Using Multiplication of Integers to Solve a Problem
10:07
Elevation
10:12
Temperature
11:21
Determine the Sign of the Product
12:19
Example 5: Determine the Sign
12:20
Example 6: Determine the Sign
12:50
Extra Example 1: Product of Three Negative Numbers
13:07
Extra Example 2: Product of Four Negative Numbers
13:45
Extra Example 3: Product of Five Negative Numbers
13:58
Extra Example 4: Product of 103 Negative Numbers
14:13
Dividing Integers
20m 18s
Intro
0:00
What You'll Learn and Why
0:06
Topics Overview
0:07
Vocabulary
0:27
Quotient
0:30
Rules for Dividing Integers
0:49
1:03
1:36
Rules for Dividing Integers, cont.
2:06
Arithmetic (Different Signs Integers)
2:14
Algebraic (Different Signs Integers)
2:41
Dividing Integer Examples
3:24
Dividing Integers: 14 ÷ 7
3:30
Dividing Integers: 45 ÷ (-9)
3:37
Dividing Integer Examples
3:51
Dividing Integers: (-105) ÷ (-15)
3:55
Dividing Integers: (-42) ÷ 6
5:07
Average Rate of Change
5:17
Using Integers to Represent the Situation
5:25
Example: Spend $360 in 6 Days 5:40 Example: Runs 1000 Feet in 4 Minutes 6:30 Average Rate of Change Word Problems 7:27 Example: Average Decrease in Value 7:32 Average Rate of Change Word Problems 9:19 Example: Average Increase in Stock 9:23 Average Rate of Change Word Problems 10:46 Example: Average Increase in Speed 10:51 Dividing Integers 12:00 Odd Number of Negatives 12:03 Even Number of Negatives 12:49 Order of Operations and Sign of Final Answer 13:50 Example: -120 ÷ (-5) ÷ -4 13:56 Extra Example 1: Order of Operations 14:48 Extra Example 2: Evaluate the Expression 15:29 Extra Example 3: Rate of Change 17:18 Extra Example 4: Rate of Evaporation 19:22 Positive Exponents 20m 5s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:26 Factor 0:32 Exponent 1:16 Vocabulary 1:57 Base 1:58 Power 2:18 Writing Expressions with Exponents 2:31 Example 1: Writing Expressions with Exponents 2:36 Example 2: Writing Expressions with Exponents 3:00 Writing Expressions with Exponents 3:20 Example 3: Writing Expressions with Exponents 3:25 Example 4: Writing Expressions with Exponents 3:53 Simplifying Power 4:06 Example 5: Simplifying Power 4:14 Example 6: Simplifying Power 5:03 Simplifying Power 6:06 Example 7: Simplifying Power 6:09 Example 8: Simplifying Power 6:50 Order of Operations 7:24 PEMDAS 7:26 Order of Operations 8:32 Multiplying/Dividing and Adding/Subtracting 8:34 Evaluating Expressions with Exponents 10:07 Example 9: Evaluating Expressions with Exponents 10:11 Example 10: Evaluating Expressions with Exponents 11:07 Extra Example 1: Evaluate 12:33 Extra Example 2: Evaluate 13:42 Extra Example 3: Height of the Rocket 15:00 Extra Example 4: Number of Cells 16:38 Section 3: Equations and Applications Solving Addition Equations 21m 47s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:18 Equation 0:21 Isolate 1:00 Inverse Operations 1:18 Subtraction Property of Equality 1:59 Arithmetic 2:12 Algebraic 2:53 Inverse Operations 3:32 Example: 38 + x = 42 3:40 Using Substitution to Check Answer 4:43 Inverse Operations 5:19 Example: y + 7.3 = 9.1 5:22 Using Substitution to Check Answer 5:53 Draw a Model 6:26 Weight Gain 6:42 Draw a Model 8:20 Mountain Climber 8:23 Examples by Writing Equations 10:25 Calculating Profit: Sweat Shirt 10:30 Examples by Writing Equations 11:37 Calculating Profit: Car Dealer 11:38 Extra Example 1: Inverse Operation 14:21 Extra Example 2: Inverse Operation 15:37 Extra Example 3: Real Estate 17:23 Extra Example 4: Birth Date 20:06 Solving Subtraction Equations 19m 34s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:07 Vocabulary 0:23 Addition Property of Equality: Arithmetic 0:31 Addition Property of Equality: Algebraic 1:14 Subtraction Property of Equality: Arithmetic 1:54 Subtraction Property of Equality: Algebraic 2:19 Solving an Equation by Adding 3:05 Example: b - 2 = 2 3:22 Example: 23 - j = 12 4:00 Solving an Equation by Adding 5:29 Example: a - 7.9 = 17.9 5:32 Example: -5.6 + x = 10.2 6:33 Solving an Equation by Writing an Equation 7:42 Example: Bank Withdrawal 7:48 Solving an Equation by Writing an Equation 9:21 Example: Temperature 9:23 Extra Example 1: Solving Subtraction Equations 11:50 Extra Example 2: Solving Subtraction Equations 12:46 Extra Example 3: Money 13:40 Extra Example 4: Selling Price 16:01 Solving Multiplication and Division Equations 26m 11s Intro 0:00 What You'll Learn and Why 0:06 Topics Overview 0:07 Vocabulary 0:19 Division Property of Equality: Arithmetic 0:27 Division Property of Equality: Algebraic 1:05 Multiplication Property of Equality: Arithmetic 1:38 Multiplication Property of Equality: Algebraic 2:07 Vocabulary 2:49 Inverse Operations 2:53 Solve the Equation Using Division 3:09 Example: 8x = 56 3:12 Example: -6y = 42 3:59 Solve the Equation Using Division 4:47 Example: 0.9c = 1.89 4:53 Solve the Equation Using Division 6:11 Example: Saving Money 6:17 Example: Soccer Team 8:14 Solve the Equation Using Multiplication 9:56 Example: a/7 = 9 10:04 Example: t/1.7 = 6 10:52 Solve the Equation Using Multiplication 12:09 Example: y/-45 = 3.2 12:17 Example: -p = 14 13:13 Solve the Equation Using Multiplication 14:10 Example: Distant 14:16 Extra Example 1: Solve the Equation 15:58 Extra Example 2: Solve the Equation 17:25 Extra Example 3: Height of an Elephant 20:55 Extra Example 4: Money 23:07 Solving Two-Step Equations 19m 10s Intro 0:00 What You'll Learn and Why 0:06 Topics Overview 0:07 Two-Step Equation Involvement 0:19 Solving Two-Step Equations 0:41 Example: 8y - 11 = 32 0:45 Example: 32 = t/5 + 8 2:55 Solving Two-Step Equations 4:49 Example: Recommended Daily Intake 4:59 Solving Two-Step Equations 7:01 Example: Cost of Each Ride 7:02 Extra Example 1: Solving Two-Step Equations 10:13 Extra Example 2: Solving Two-Step Equations 12:54 Extra Example 3: Length of Phone Call 13:56 Extra Example 4: Cost of Owning a Pet 16:40 Square Roots 12m 16s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:16 Perfect Square 0:20 Square Root 0:46 Square Roots 1:13 Every Positive Number has Two Square Roots 1:14 Square Roots of a Number are Opposites 1:40 Square Root Symbol 1:54 Positive Square Root of a Number 1:56 Compare: √25 and -√25 2:08 Find the Square Root 2:50 Example: Square Root of 81 2:52 Example: Square Root of 121 3:13 Estimating Square Roots 3:29 Example: Square Root of 23 3:35 Example: Square Root of 390 4:13 Example: Negative Square Root of 125 4:50 Estimating Square Roots 5:27 Estimating Length 5:31 Simplifying Square Roots 7:05 Example: Square Root of 36 7:08 Example: Simplifying Square Roots 7:47 Extra Example 1: Estimate the Length 8:21 Extra Example 2: Simplify the Expression 9:05 Extra Example 3: Estimate the Length 9:50 Extra Example 4: Simplify the Expression 10:34 Pythagorean Theorem 14m 49s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:36 Right Triangle 0:39 Legs 0:57 Hypotenuse 1:02 Pythagorean Theorem 1:11 Arithmetic Example 1:12 Algebra Example 2:41 Find the Length of the Hypotenuse 3:04 Example 1: Hypotenuse of a Triangle 3:07 Example 2: Hypotenuse of a Triangle 4:30 Find the Length of the Hypotenuse 6:18 Example 3: Hypotenuse of a Right Triangle 6:19 Extra Example 1: Square Roots 8:41 Extra Example 2: Perimeter 9:43 Extra Example 3: Length of Screen 11:58 Extra Example 4: Length of Wire 13:14 Using the Pythagorean Theorem 16m 15s Intro 0:00 What You'll Learn and Why 0:07 Topics Overview 0:08 Vocabulary 0:30 The Pythagorean Theorem 0:32 Find the Length of a Leg 1:14 Example 1: Length of Ramp 1:19 Example: 2 Length of Platform 4:22 Identifying a Right Triangle 6:13 Example 3: Determine Right Triangle 6:14 Example 4: Determine Right Triangle 8:08 Extra Example 1: Find the Missing Leg Length 10:04 Extra Example 2: Length of Ladder 11:39 Extra Example 3: Determine Given Lengths 13:04 Extra Example 4: Length of Pole 14:20 Section 4: Rational Numbers Prime Factorization 16m 40s Intro 0:00 What You'll Learn and Why 0:06 Topics Overview 0:07 Vocabulary 0:23 Factor 0:24 Composite Number 1:04 Vocabulary 1:17 Prime Number 1:18 Prime Factorization 1:57 Divisibility Rules 2:54 Divisibility Rules for 2, 3, 4 ,5 ,6, 9 and 10 2:55 Finding Factors 4:56 Possible Arrangements 4:59 Finding Factors 6:06 How Many Oranges? 6:07 Prime or Composite 6:43 Prime or Composite: 48 6:47 Prime or Composite: 53 7:09 Prime or Composite: 57 7:35 Prime Factorization 8:16 Prime Factorization of 42 8:23 Prime Factorization of 84 8:47 Find the Number 9:23 Find the Number with the Given Prime Factorization 9:24 Extra Example 1: Prime or Composite 11:04 Extra Example 2: Prime Factorization of 72 12:23 Extra Example 3: Marching Arrangements 12:56 Extra Example 4: Flowers Arrangements 14:50 Greatest Common Divisor 14m 16s Intro 0:00 What You'll Learn and Why 0:07 Topics Overview 0:08 Vocabulary 0:41 Factor 0:43 Common Divisor 1:00 Greatest Common Divisor (GCD)/ Greatest Common Factor (GCF) 1:16 Find the GCD by Listing Divisors 1:34 GCD of 27 and 36 1:46 GCD of 18 and 49 2:52 Prime Factorization to Find GCD 3:30 GCD of 42 and 72 3:42 GCD of 21 and 63 4:46 GCD in Word Problems 5:30 Greatest Number of Police Officers 5:32 GCD in Word Problems 7:15 Cutting Two Pipes 7:16 Extra Example 1: GCD of 32, -24, 40 8:08 Extra Example 2: How Many Groups? 9:41 Extra Example 3: GCD of Two Prime Numbers 11:34 Extra Example 4: How Many Children? 12:26 Equivalent Fractions 11m 22s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:21 Equivalent Fractions 0:23 Simplest Form 1:15 Creating Equivalent Fractions 1:59 Method to Create Equivalent Fractions 2:00 Equivalent Fractions 2:18 Write Two Fractions Equivalent to 5/8 2:22 Write Two Fractions Equivalent to 2/14 3:12 GCD to Simplify 3:51 Find the GCD of 24 and 32 4:03 Write 24/32 in Simplest Form 4:43 Write -27/45 in Simplest Form 5:08 Writing a Fraction 6:04 Example: What Fraction of the Vehicles are Trucks? 6:11 Writing a Fraction 7:45 Example: What Fraction of the Seats are Empty? 7:55 Extra Example 1: Pizza 9:00 Extra Example 2: Driving Time 9:28 Extra Example 3: Simplest Form 10:29 Extra Example 4: Basketball Team 10:52 Equivalent Forms of Rational Numbers 20m 47s Intro 0:00 What You'll Learn and Why 0:06 Topics Overview 0:08 Vocabulary 0:30 Rational Number 0:31 Terminating Decimal 0:53 Repeating Decimal 1:04 Converting Decimals to Fractions 1:22 Write 0.47 as a Fraction 1:26 Write 0.48 as a Fraction 2:03 Write 0.245 as a Fraction 3:21 Converting Decimals to Fractions 4:20 Write 0.08 as a Fraction 4:30 Write 0.8 as a Fraction 4:53 Converting Fractions to Decimals 5:26 Write 1/2 as a Decimal 5:30 Write 6/33 as a Decimal 6:12 Write -9/5 as a Decimal 7:39 Converting Fractions to Decimals in Word Problems 8:19 Batting Average 8:23 Converting Fractions to Decimals in Word Problems 11:22 Cooking Festival 11:26 Extra Example 1: Write 0.038 as a Fraction 14:45 Extra Example 2: Write -13/7 as a Decimal 15:35 Extra Example 3: Batting Average 16:38 Extra Example 4: Rational Number 19:55 Comparing and Ordering Rational Numbers 20m 21s Intro 0:00 What You'll Learn and Why 0:07 Topics Overview 0:08 Vocabulary 0:29 Least Common Multiple (LCM) 0:30 Least Common Denominator (LCD) 1:50 Ordering Rational Numbers 2:45 Numbers as Decimals 2:46 Numbers as Fractions 5:36 Compare Each Pair of Numbers 8:10 Compare 3/4 and 4/5 8:11 Compare 3/11 and 1/6 8:44 Comparing rational Numbers in Word Problems 9:19 Cookies or French Fries? 9:22 Extra Example 1: Least to Greatest (Decimals) 11:32 Extra Example 2: Least to Greatest (Fractions) 13:35 Extra Example 3: Music Notes 15:54 Extra Example 4: Chocolate or Fruit 17:16 Section 5: Rational Number Operations Adding Rational Numbers 15m 4s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:19 Least Common Multiple (LCM) 0:20 Least Common Denominator (LCD) 0:28 Adding Fractions with Unlike Denominators 1:22 Example: 3/4 + 2/5 1:28 Example: -3/5 + 1/7 2:29 Adding Different Forms of Rational Numbers 3:23 Example: Change to Fractions 3:31 Example: Change to Decimals 5:14 Extra Example 1: Adding Different Forms of Numbers 7:02 Extra Example 2: Exercising 10:06 Extra Example 3: Adding Different Forms of Numbers 11:20 Extra Example 4: Cooking Recipe 13:47 Subtracting Rational Numbers 14m 40s Intro 0:00 What You'll Learn and Why 0:06 Topics Overview 0:07 Vocabulary 0:19 Least Common Denominator (LCD) 0:20 Subtracting with Unlike Denominators 0:41 Example: 5/9 - 3/5 0:44 Example: 3/4 - 7/8 1:23 Subtracting Rational Numbers 1:59 Example: 23/4 - 3.5 2:05 Example: 11.7 - 3/4 3:39 Subtracting Rational Numbers in Word Problems 4:37 Puppy's Weight 4:41 Extra Example 1: Subtracting with Unlike Denominators 6:48 Extra Example 2: Subtracting Rational Numbers 7:27 Extra Example 3: Rainfall 10:32 Extra Example 4: Decorating Your House 12:06 Multiplying Rational Numbers 11m 2s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:17 Rational Number 0:18 Exponent 0:30 Power 0:35 Multiplying Decimals 0:47 Example 1: Multiplying Decimals 0:50 Example 2: Multiplying Decimals 1:18 Multiplying Rational Numbers in Word Problems 1:51 Example: Length of Pipes 1:56 Raising a Fraction to a Power 2:58 Examples: Raising Fractions to Power 2:59 Extra Example 1: Multiplying Fractions 4:45 Extra Example 2: Compare Fractions 5:34 Extra Example 3: Flour 7:28 Extra Example 4: Income 8:50 Dividing Rational Numbers 12m 8s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:22 Reciprocal 0:26 Finding the Reciprocal 0:56 Example: Reciprocal of 2/3 1:00 Example: Reciprocal of 8 1:04 Example: Reciprocal of -1/2 1:10 Dividing Rational Numbers 1:28 Example 1: Dividing Rational Numbers 1:35 Example 2: Dividing Rational Numbers 2:09 Example 3: Dividing Rational Numbers 2:35 Dividing Rational Numbers in Word Problems 3:56 Example: Chocolate Peanuts 4:00 Extra Example 1: Dividing Decimals 5:17 Extra Example 2: Dividing Fractions 7:09 Extra Example 3: Search Committee 8:34 Extra Example 4: Stake 10:18 Solving Equations by Adding or Subtracting 13m 43s Intro 0:00 What You'll Learn and Why 0:06 Topics Overview 0:07 What You'll Learn and Why 0:23 Isolate 0:24 Solving Equations with Fractions 0:41 Example: n + 1/2 = 11/12 0:44 Example: 3/5 - a = 13/20 1:34 Writing Equations with Fractions 3:08 Example: Thickness 3:11 Extra Example 1: Solving Equations 6:01 Extra Example 2: Solving Equations 6:58 Extra Example 3: School Lunches 8:23 Extra Example 4: Fashion Designer 10:44 Solving Equations by Multiplying 11m 10s Intro 0:00 What You'll Learn and Why 0:06 Topics Overview 0:07 Vocabulary 0:20 Multiplicative Inverse 0:21 Solve Each Equation 0:51 Example 1: Solve the Equation 0:57 Example 2: Solve the Equation 2:39 Writing Multiplication Equations 3:30 Example: Water Level 3:34 Extra Example 1: Solve the Equation 5:15 Extra Example 2: Solve the Equation 6:28 Extra Example 3:Money 7:26 Extra Example 4: Solve the Equation 9:45 Zero and Negative Exponents 6m 41s Intro 0:00 What You'll Learn and Why 0:06 Topics Overview 0:07 Vocabulary 0:22 Zero Exponents: Arithmetic Definition 0:27 Zero Exponents: Algebra Definition 0:39 Negative Exponents: Arithmetic Definition 0:46 Negative Exponents: Algebra Definition 1:03 Simplifying Exponents 1:18 Examples: Simplifying Exponents 1:21 Fractions and Negative Exponents 2:41 Examples: Fractions and Negative Exponents 2:45 Extra Example 1: Negative Exponent 3:58 Extra Example 2: Zero Exponent 4:30 Extra Example 3: Fraction and Negative Exponent 4:40 Extra Example 4: Subtracting Numbers with Exponents 5:00 Section 6: Rates and Proportions Ratios 7m 5s Intro 0:00 What You'll Learn and Why 0:06 Topics Overview 0:07 Vocabulary 0:22 Ratio 0:23 Equivalent Ratios 0:40 Writing a Ratio 1:18 Write the Ratio in Three Ways 1:21 Writing an Equivalent Ratio 2:07 Different Ratios Equivalent to 10:12 2:08 Writing a Ratio in Simplest Form 2:47 Write the Ratio in Simplest Form 2:50 Extra Example 1: Write the Ratio 3:32 Extra Example 2: Ratio in Simplest Form 4:38 Extra Example 3: Write the Ratio 5:08 Extra Example 4: Ratio in Simplest Form 5:48 Rates 12m 45s Intro 0:00 What You'll Learn and Why 0:04 Topics Overview 0:05 Vocabulary 0:18 Rate 0:22 Unit Rate 0:40 Finding a Unit Rate 1:00 Example: Delivery Rate 1:03 Using a Unit Rate 1:46 Example: Miles and Gallon of Gas 1:49 Comparing Unit Rates 2:52 Example: Which is the Better Buy 2:58 Extra Example 1: Calories 6:30 Extra Example 2: Typing Speed 7:22 Extra Example 3: Which is the Better Buy 8:23 Extra Example 4: Wages 10:48 Dimensional Analysis 16m 19s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:15 Conversion Factor 0:16 Dimensional Analysis 0:23 Conversion Chart: Length 0:32 Conversion Chart: in, ft, yd, and m 0:33 Conversion Chart: Weight 0:49 Conversion Chart: oz, lb, and t 0:50 Conversion Chart: Capacity 0:59 Conversion Chart: fl oz, cup, pt, qt, and gal 1:00 Converting Units 1:17 Example: Convert 1.3 Miles Into Feet 1:18 Converting Units 3:14 Example: Convert Pounds to Ounces 3:15 Example: Convert Cups to Fluid Ounces 3:52 Converting Units in a Rate 4:30 Unit Rate: 2,200 m in 17.2 min 4:31 Using Dimensional Analysis 8:06 Example: Planning Project 8:07 Extra Example 1: Converting Units 9:15 Extra Example 2: Unit Rate 10:31 Extra Example 3: Planning Project 12:15 Extra Example 4: Converting Units 13:45 Application of Rates 17m 42s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:18 Rate 0:22 Unit Rate 0:27 Finding Total Distance 0:32 Example: Total Distance 0:33 Finding Average Speed 2:49 Example: Car's Average Speed 2:53 Using a Unit Rate 6:31 Example: Weight and Spring 6:32 Extra Example 1: Total Distance 8:08 Extra Example 2: Bird's Average Speed 10:33 Extra Example 3: Cost of Shirts 13:37 Extra Example 4: Cost of Bottles 15:22 Proportions 14m 36s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:16 Proportion 0:17 Cross Products 0:45 Identifying a Proportion 11:28 Example: Do the Ratios Form a Proportion? 11:29 Solving Proportions Using Two Methods 2:47 Example: x/4 = 12/21 3:03 Example: 20/y = 15/9 4:43 Solving Proportions in Word Problems 5:56 Example: Find the Unit Rate of Exchange 6:00 Extra Example 1: Does the Ratio Form a Proportion? 9:55 Extra Example 2: Solving Proportions 10:53 Extra Example 3: Find the Length of the Photo 11:33 Extra Example 4: Distance 12:59 Section 7: Applications of Percent Fractions, Decimals, and Percents 10m 13s Intro 0:00 What You'll Learn and Why 0:06 Topics Overview 0:07 Vocabulary 0:24 Percent 0:25 Compare a Fraction to a Decimal to a Percent 0:35 Example: 31 out of 100 0:39 Write a Percent as a Fraction 1:11 Example: Write 47% as a Fraction 1:12 Example: Write 25% as a Fraction 1:59 Write a Fraction as a Percent 2:38 Example: Write 2/10 as a Percent 2:39 Example: Write 57/100 as a Percent 3:22 Example: Write 3/25 as a Percent 3:57 Write a Decimal as a Percent 4:48 Example: Write 0.85 as a Percent 4:49 Example: Write 0.3 as a Percent 5:05 Example: Write 0.04 as a Percent 5:17 Extra Example 1: Write Percent as Decimal and Fraction 5:34 Extra Example 2: Write Fraction as Decimal and Percent 6:47 Extra Example 3: Write Percent as Fraction and Decimal 7:41 Extra Example 4: Fraction, Percent, and Decimal of Students 8:36 Finding a Percent of a Number 8m 39s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:20 Percent 0:21 Finding a Percent 0:26 Example: Find 15% of 100 0:27 Example: Find 10% of 135 1:08 Finding More Percents 2:23 Example: Find 8% of 65 2:25 Example: Find 120% of 50 2:45 Estimating a Percent 3:25 Example: Estimate 10% of 31.05 3:38 Example: Estimate 15% of 31.05 3:54 Example: Estimate 20% of 31.05 4:55 Extra Example 1: Find 7% of 120 5:35 Extra Example 2: Find 125% of 75 6:01 Extra Example 3: Estimate 15% Tip 7:02 Extra Example 4: Estimate 20% Tip 8:01 Percents and Proportions 15m 38s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:23 Proportion 0:33 Cross Products 0:38 Finding Part of a Whole 1:07 Method 1: Find 50% of 36 1:11 Method 2: Find 50% of 36 2:23 Finding Part of a Whole, cont. 3:06 Example: Find 65% of 143 3:07 Finding the Whole Amount 3:44 Example: How Many Students are in 8th Grade? 3:45 Finding the Whole Amount, cont. 5:18 Example: How Many Students are in the School? 5:19 Finding a Percent 6:38 Method 1: What Percent of 175 is 105 7:21 Method 2: What Percent of 175 is 105 8:49 Finding a Percent, cont. 9:57 What Percent of 115 is 46? 9:58 Extra Example 1: Find 8% of 48 11:09 Extra Example 2: 9 is 25% of What Number? 11:34 Extra Example 3: How Many Students are in the Class? 12:39 Extra Example 4: 66 is What Percent of 55? 14:25 Percent of Change 11m 27s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:17 Percent of Change 0:18 Amount of Change/ Original Amount 0:26 Finding a Percent of Increase 1:04 Example: Find the Percent of Increase 1:06 Converting Units 2:56 Example: Converting Units and Percent Increase 3:00 Finding a Percent of Decrease 5:08 Example: Find the Percent of Decrease 5:09 Extra Example 1: Find the Percent of Increase 6:32 Extra Example 2: Find the Percent of Decrease 7:30 Extra Example 3: Find the Percent of Increase 8:28 Extra Example 4: Find the Percent of Decrease 10:23 Applications of a Percent 12m 55s Intro 0:00 What You'll Learn and Why 0:06 Topics Overview 0:07 Vocabulary 0:22 Markup 0:23 Selling Price 0:28 Formula of Markup 0:32 Formula for Discount 1:06 Finding a Percent of Markup 1:16 Example: Find the Percent of Markup 1:19 Finding a Percent of Discount 3:19 Example: Find the Percent of Discount 3:21 Finding a Sale Price 5:03 Example: Find the Sale Price of the Stereo System 5:10 Extra Example 1: Find the Percent of Markup 6:39 Extra Example 2: Find the Percent of Discount 8:57 Extra Example 3: Find the Percent of Discount 9:52 Extra Example 4: Find the Percent of Discount 10:44 Section 8: Linear Functions and Graphing Graphing in the Coordinate Plane 15m 13s Intro 0:00 What You'll Learn and Why 0:06 Topics Overview 0:07 Vocabulary 0:17 Coordinate Plane 0:23 y-axis 0:29 x-axis 0:33 Quadrants 0:37 More Vocabulary 0:49 Origin 0:50 Ordered Pair 0:53 x-coordinate 1:00 y-coordinate 1:08 Labeling Vocabulary 1:24 Example: Label the Vocabulary 1:25 Graphing Points on a Coordinate Plane 3:08 Example: Graph and Label the Locations 3:12 Quadrants 6:18 Example: Name the Quadrants of Each Ordered Pair 6:23 Extra Example 1: Draw and Label 10:18 Extra Example 2: Graph the Ordered Pair 11:20 Extra Example 3: Graph the Ordered Pair 12:42 Extra Example 4: Name the Quadrants of Each Ordered Pair 13:32 Length in the Coordinate Plane 18m 36s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:18 x-coordinate 0:21 y-coordinate 0:28 Pythagorean Theorem 0:34 Finding Lengths of Line Segments 1:02 Example: Find the Length of the Horizontal Line Segment 1:05 Finding Lengths of Line Segments 3:50 Example: Find the Length of the Vertical Line Segment 3:54 Finding Distance in the Coordinate Plane 5:59 Example: Find the Length of the Hypotenuse 6:02 Extra Example 1: Find the Distance Between Two Points 7:36 Extra Example 2: Find the Length of the Line Segment 10:13 Extra Example 3: Find the Length of the Line Segment 14:28 Extra Example 4: How Far is Your School from the Arcade? 16:02 Functions 14m 29s Intro 0:00 What You'll Learn and Why 0:04 Topics Overview 0:05 Vocabulary 0:24 Function 0:25 Function Rule 0:51 Evaluating a Function Rule 0:59 Example: Table of Input and Output 1:00 Using Function Notation 2:56 Example: Write the Equation and Evaluate the Cost 2:59 Writing Functions 4:40 Example: Writing Function 4:41 Extra Example 1: Complete the Table 6:02 Extra Example 2: Complete the Table and Find the Function 7:38 Extra Example 3: Function Notation 9:39 Extra Example 4: Write a Function and Find the Total Cost 11:49 Graphing Linear Functions 16m 2s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:21 Solution 0:22 Linear Equation 0:29 Linear Function 0:44 Making a Graph from a Table 1:05 Example: Total Savings in Dollars 1:08 Graphing a Linear Function 3:03 Example: Graph the Linear Function 3:07 Extra Example 1: How Much Cereal is Left? 5:42 Extra Example 2: Graph the Value 7:45 Extra Example 3: Graph the Linear Function 10:17 Extra Example 4: Graph the Linear Function 12:28 Slope 17m 53s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:15 Slope Definition 1 0:18 Slope Definition 2 0:40 Slope 1:20 Positive Slope 1:32 Negative Slope 1:39 Slope of 0 2:10 Undefined Slope 2:25 Finding the Slope of a Line 3:57 Example: Using Rise/Run to Find Slope 3:58 Finding the Slope of a Line 6:01 Example: Using Coordinates to Find Slope 6:02 Finding Slope of a Line With Given Coordinates 9:00 Example: Slope of (4,1) and (3, -2) 9:01 Extra Example 1: Find the Slope of the Line 10:17 Extra Example 2: Find the Slope of the Line 12:19 Extra Example 3: Find the Slope of the Line 14:54 Extra Example 4: Find the Slope of the Line 16:35 Slope and Direct Variation 13m 50s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:33 Direct Variation 0:34 Identifying a Direct Variation 0:47 Steps in Identifying a Direct Variation 0:56 Slope and Direct Variation 3:01 Example: Slope and Direct Variation 3:05 Extra Example 1: Direct Variation 6:16 Extra Example 2: Direct Variation 7:14 Extra Example 3: Graphing Direct Variation 8:10 Extra Example 4: Slope and Direct Variation 11:23 Section 9: Inequalities Writing Inequalities 17m 32s Intro 0:00 What You'll Learn and Why 0:05 Topics Overview 0:06 Vocabulary 0:15 Inequality 0:18 System of Inequalities 1:31 Solution of an Inequality 1:50 Writing an Inequality 2:10 Example: Price p is More than$6
2:20
Writing an Inequality
3:53
Example: Wage w is at Least \$8.25
3:54
Writing a System of Inequalities
5:24
Example: System of Inequalities for Wind Speed
5:32
Writing a System of Inequalities
9:03
Example: Price of a Room in Las Vegas
9:04
Identifying Solutions of an Inequality
10:33
Example: Driver's Permit
10:37
Extra Example 1: Writing Inequalities
12:03
Extra Example 2: Writing a System of Inequalities
13:24
Extra Example 3: Writing Inequalities
14:51
Extra Example 4: Using Inequality to Solve Word Problem
15:31
Solving Inequalities by Adding or Subtracting
9m 16s
Intro
0:00
What You'll Learn and Why
0:07
Topics Overview
0:08
Vocabulary
0:20
0:31
Subtraction Property of Equality
0:43
Example: x > 7
0:57
1:33
Example: Solve x - 8 = 10
1:37
Example: Solve x - 8 < 10
2:05
2:21
Example: 2 ≤ t - 5
2:22
Example: a - 8 > 15
2:59
Solving an Inequality by Subtracting
3:14
Example: How Many Students can Board the Bus?
3:22
Solving an Inequality by Subtracting
4:13
Example: How Many More Tickets can be Sold?
4:14
Extra Example 1: Solve the Inequality
5:16
Extra Example 2: Solve 8 ≥ 3 + m
5:52
Extra Example 3: MP3 Player
6:29
Extra Example 4: Write and Solve an Inequality
7:34
Solving Inequalities by Dividing
12m 15s
Intro
0:00
What You'll Learn and Why
0:06
Topics Overview
0:07
Vocabulary
0:22
Division Property of Inequality
0:23
Example: 3x > 6
0:37
Dividing by a Positive Number
1:08
Example: Hotel Elevator
1:11
Dividing by a Negative Number
2:29
Example: Solve -6x ≥ -18
2:32
Example: Suppose x = 2
3:13
Dividing by a Negative Number
4:58
Example: Solve -3t ≥ 51
5:05
Example: Solve -8m < -56
5:24
Extra Example 1: Photo Album
5:49
Extra Example 2: Banquet
8:05
Extra Example 3: Solve -0.5x > 18
9:24
Extra Example 4: How Many Crates can the Crane Lift?
10:30
Solving Inequalities by Multiplying
14m 33s
Intro
0:00
What You'll Learn and Why
0:07
Topics Overview
0:08
Vocabulary
0:17
Multiplication Property of Inequality
0:18
Multiplying by a Positive Number
1:25
Example: Write and Solve an Inequality
1:28
Multiplying by a Positive Number
3:38
Example: Write and Solve an Inequality
3:39
Multiplying by a Negative Number
5:42
Example: Solve x/-4 > 28
5:45
Example: Solve (-1/2)y < -8
6:10
Example: t/-7 < 5
6:42
Extra Example 1: Bowling League
7:12
Extra Example 2: Street Performers
8:27
Extra Example 3: Write and Solve the Inequality
9:52
Extra Example 4: Solve and Graph the System of Inequalities
11:26
Solving Two-Step Inequalities
14m 15s
Intro
0:00
What You'll Learn and Why
0:05
Topics Overview
0:06
Vocabulary
0:15
Inequality
0:16
Properties of Inequality
0:27
Solving Two-Step Inequalities
0:37
Example: Solve -2x - 8 > -14
0:41
Example: Solve (x/4) - 7 > 25
1:40
Example: Solve -5y + 9 ≤ 54
2:12
Writing Two-Step Inequalities
3:16
Example: How Many Pairs of Socks?
3:21
Writing Two-Step Inequalities
5:49
Example: How Many Folders?
5:53
Extra Example 1: Solve 15 < -3 ( x + 1 )
7:32
Extra Example 2: Solve the Inequalities
8:43
Extra Example 3: Muffin
10:37
Extra Example 4: Birthday Party
11:51
Section 10: Exponents and Equations
Properties of Exponents
12m 7s
Intro
0:00
What You'll Learn and Why
0:05
Topics Overview
0:06
Vocabulary
0:26
Exponent
0:29
Power
0:44
Multiplying Powers with the Same Base
1:04
Example: Multiplying Powers with the Same Base
1:07
Multiplying Expressions with Exponents
2:25
Examples
2:26
Dividing Powers with the Same Base
3:24
Example: Dividing Powers with the Same Base
3:25
Dividing Expression with Exponents
3:59
Example: How Long Sunlight Takes to Reach the Comet
4:02
Dividing Expression with Exponents
6:44
Example: How Long Sunlight Takes to Reach Earth
6:45
Extra Example 1: Multiplying Expressions with Exponents
8:22
Extra Example 2: Dividing Expression with Exponents
8:46
Extra Example 3: How Long Sunlight Takes to Reach Saturn
9:12
Extra Example 4: Sun's Diameter and Earth's Diameter
10:34
Power Rules
11m 58s
Intro
0:00
What You'll Learn and Why
0:04
Topics Overview
0:05
Vocabulary
0:15
Exponent
0:18
Power
0:34
Raising a Power to a Power
0:44
Example: Raising a Power to a Power
0:47
Raising a Power to a Power
2:38
More Examples
2:42
Raising a Product to a Power
3:00
Example: Raising a Product to a Power
3:01
Raising a Product to a Power
4:00
Example: Surface Area of a Plant Cell
4:12
Example: Surface Area of the Moon
6:15
Extra Example 1: Raising Power to a Power
8:08
Extra Example 2: Complete the Inequality Statement
8:22
Extra Example 3: Find the Area of a Square
8:51
Extra Example 4: Find the Area of a Circle
10:28
Exploring Roots
9m 39s
Intro
0:00
What You'll Learn and Why
0:05
Topics Overview
0:06
Vocabulary
0:17
Square Root
0:21
Roots
1:13
The Root Symbol
2:10
Root Symbols
2:11
Finding Roots of a Number
2:41
Examples
2:42
Simplifying Expressions with Roots
4:41
Example: Simplify the Expression
4:42
Simplifying Expressions with Roots
5:42
Example: Simplify the Expression
5:43
Extra Example 1: Finding Roots of a Number
6:36
Extra Example 2: Simplifying Expressions with Roots
7:11
Extra Example 3: Simplifying Expressions with Roots
7:36
Extra Example 4: Simplifying Expressions with Roots
8:34
Simplifying Algebraic Expressions
12m 43s
Intro
0:00
What You'll Learn and Why
0:07
Topics Overview
0:08
Vocabulary
0:23
Term
0:28
Like Terms
0:35
Combining Like Terms
1:18
Example: 2y + y - 15y
1:20
Example: -x - 5x
2:16
Writing and Simplifying Expressions
2:57
Example: Total Cost of Drinks
2:58
Writing and Simplifying Expressions
4:48
Example: Total Cost of Apples
4:49
Distributing and Simplifying
5:42
Simplify: 4x - 2( x + 6 )
5:46
Simplify: 3( 2y + 2 ) - 4y
6:57
Extra Example 1: Simplify the Expression
7:52
Extra Example 2: Distributing and Simplifying
8:18
Extra Example 3: Writing and Simplifying Expressions
9:35
Extra Example 4: Distributing and Simplifying
10:50
Solving Multi-Step Equations
18m 35s
Intro
0:00
What You'll Learn and Why
0:06
Topics Overview
0:07
Vocabulary
0:17
Like Terms
0:21
Distributive Property
0:49
Simplifying Before Solving an Equation
1:37
Example: 8x + 45 - 12x = 9
1:40
Example: -15 = 6b + 12 - 3b + 6
3:25
Using the Distributive Property
4:50
Example: Haiti Relief Efforts
4:51
Using the Distributive Property
7:50
Example: Amusement Park
7:51
Extra Example 1: Simplify and Solve
11:34
Extra Example 2: Simplify and Solve
12:21
Extra Example 3: Simplify and Solve
13:18
Extra Example 4: Mailing Letters
14:53
Solving Equations with Variables on Both Sides
18m 50s
Intro
0:00
What You'll Learn and Why
0:07
Topics Overview
0:08
Vocabulary
0:26
Term
0:30
Like Terms
0:44
Variables on Both Sides
1:10
Example: 3x + 24 = 9x
1:11
Example: -7 - 3x = 1 + 5x
2:16
Using the Distributive Property
4:01
Example: Height of Two Plants
4:02
Using the Distributive Property
9:01
Example: Running Laps
9:02
Extra Example 1: Solving Equations with Variables on Both Sides
11:59
Extra Example 2: Solving Equations with Variables on Both Sides
12:46
Extra Example 3: Solving Equations with Variables on Both Sides
14:18
Extra Example 4: Cost of Renting Video
15:24
Section 11: Test-Taking Strategies
Test Taking Strategy Part 1
12m 55s
Intro
0:00
Finding Needed Information
1:02
1:07
Example: Finding Needed Information
1:25
Extra Example 1: Finding Needed Information
2:36
Extra Example 2: Finding Needed Information
4:10
Using Mental Math
6:10
Use Mental Math to Eliminate Unreasonable Answers
6:16
Example: Using Mental Math
6:47
Extra Example 1: Simplify Using Mental Math
8:54
Extra Example 2: Mental Math and Total Cost
9:36
Extra Example 3: Account Balance
11:15
Test Taking Strategy Part 2
11m 24s
Intro
0:00
Working Backward
0:06
0:08
Example: Working Backward
0:38
Extra Example 1: Equivalent Expression
2:54
Extra Example 2: Miles Per Gallon
4:13
Extra Example 3: Miles Per Hour
5:14
Choosing the Process
6:12
Strategies for Choosing the Process
6:19
Example: Dimensions of the Fenced Area
6:48
Extra Example 1: Choosing the Process
8:28
Extra Example 2: Choosing the Process
10:01
Test Taking Strategy Part 3
11m 36s
Intro
0:00
0:06
0:07
Example: Height of a Plant
0:49
Extra Example 1: Distance
2:36
Extra Example 2: Decimal
4:11
Using a Variable
4:50
Variables Represent the Unknown
4:51
Example: Company Logo
5:20
6:31
Extra Example 2: Proportion
9:03
Test Taking Strategy Part 4
20m 13s
Intro
0:00
0:06
0:07
Example: Election
0:28
Extra Example 1: Football Kicker
2:42
Extra Example 2: Percent
4:09
6:19
6:20
Example: Inequality
6:43
Extra Example 1: Cost of Cheese
8:12
Extra Example 2: Inequality
9:40
Drawing a Diagram
11:35
Diagrams
11:36
Example: Drawing a Diagram to Show Distance
12:03
Extra Example 1: Drawing a Diagram to Show Distance
15:22
Extra Example 2: Height
17:48
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0 answersPost by Shirley Wang on July 26, 2018Hi your videos are great 0 answersPost by sherman boey on August 22, 2014better check the answers for this topic and the previous alot of wrong answers this makes new students confuse..
Solving Equations with Variables on Both Sides
• To solve equations with variables on both sides, bring all the variable terms to one side of the equation. Use the Distributive Property, if necessary, collect like terms, and simplify.
• To solve equations with variables on both sides, simplify as much as you can on both sides before you apply inverse operations to isolate the variable.
Solving Equations with Variables on Both Sides
5x + 15 = 8x. Solve for x.
• 5x − 8x = − 15
• − 3x = − 15
x = 3
− 8 − 4x = 10 + 3x. Solve for x.
• − 8 − 10 = 3x + 7x
• − 18 = 10x
• x = − [18/10]
x = − [9/5]
4x + 5 = 3(2x + 1). Solve for x.
• 4x + 5 = 6x + 3
• 4x − 6x = 3 − 5
• − 2x = − 2
x = 1
11 − 5x + 3 = − 3x − 10. Solve for x.
• − 5x + 3x = − 10 − 11 − 3
• − 2x = − 24
x = 12
3.6x = 4(x − 2). Solve for x.
• 3.6x = 4x − 8
• 3.6x − 4x = − 84
• − 0.4x = − 8
x = 20
35 + 7x = 24 + x + 1. Solve for x.
• 7x − x = 24 + 1 − 35
• 6x = 25 − 35
• 6x = − 104
x = − [5/3]
14.5 + 5x = − 3.3x − 10.4. Solve for x.
• 5x + 3.3x = − 10.4 − 14.5
• 8.3x = − 24.9
x = − 3
− 3(5x − 2) = − 18x + 10x. Solve for x.
• − 15x + 6 = − 8x
• − 15x + 8x = − 6
• − 7x = − 6
x = [6/7]
Two boats are taking the same path to a destination. Ship A travels at 25 mi/hr, while ship B travels at 30 mi/hr and leaves 30 min after ship A does. How long after ship A leaves will ship B catch up to ship A?
• Let t = time that ship A travels before ship B catches up.
• distanceship A = distanceship B
distance = rate × time
• 25 mi/hr ×t = 30 mi/hr × (t - 0.5)
• 25t = 30t − 15
• 25t − 30t = − 15
• − 5t = − 15
t = 3 hours
Lisa runs [1/8] mi/min. George runs [1/10] mi/min. Lisa starts running 5 min after George starts. How long after George starts running have they run the same distance?
• Let t = number of minutes
• distanceLisa = distanceGeorge
distance = rate × time
• [1/8](t − 5) = [1/10]t
• [1/8]t − [5/8] = [1/10]t
• [1/8]t − [1/10]t = [5/8]
• [5/40]t − [4/40]t = [25/40]
• [1/40]t = [25/40]
t = 25 min
*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.
Solving Equations with Variables on Both Sides
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
• Intro 0:00
• What You'll Learn and Why 0:07
• Topics Overview
• Vocabulary 0:26
• Term
• Like Terms
• Variables on Both Sides 1:10
• Example: 3x + 24 = 9x
• Example: -7 - 3x = 1 + 5x
• Using the Distributive Property 4:01
• Example: Height of Two Plants
• Using the Distributive Property 9:01
• Example: Running Laps
• Extra Example 1: Solving Equations with Variables on Both Sides 11:59
• Extra Example 2: Solving Equations with Variables on Both Sides 12:46
• Extra Example 3: Solving Equations with Variables on Both Sides 14:18
• Extra Example 4: Cost of Renting Video 15:24
OR
Start Learning Now
Our free lessons will get you started (Adobe Flash® required). | 2021-06-14 02:36:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48379987478256226, "perplexity": 12025.232551349938}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487611320.18/warc/CC-MAIN-20210614013350-20210614043350-00013.warc.gz"} |
http://qudt.org/vocab/quantitykind/Pressure | quantitykind:Pressure
Type
Description
Properties
Pressure is an effect which occurs when a force is applied on a surface. Pressure is the amount of force acting on a unit area. Pressure is distinct from stress, as the former is the ratio of the component of force normal to a surface to the surface area. Stress is a tensor that relates the vector force to the vector area.
$$p = \frac{dF}{dA}$$, where $$dF$$ is the force component perpendicular to the surface element of area $$dA$$.
Annotations
Pressure(en)
Generated 2021-06-03T11:44:47.320-07:00 by lmdoc version 1.1 with TopBraid SPARQL Web Pages (SWP) | 2021-06-14 09:47:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.770036518573761, "perplexity": 426.824878123858}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487611641.26/warc/CC-MAIN-20210614074543-20210614104543-00061.warc.gz"} |
https://www.circuitlab.com/blog/2020/08/10/ideal-diodes-in-circuitlab/ | Ideal Diodes in CircuitLab
Aug 10 2020, 9:30 AM PDT · 0 comments »
We’re introducing a new component to the CircuitLab toolbox: the ideal diode.
We’ve had semiconductor PN junction diodes since we’ve launched, which show the exponential current-voltage relationship and accurately model real-world diodes.
In contrast, the ideal diode is more like a simulated on-off switch: the I-V curve would be piecewise linear. It acts like it’s open-circuit when reverse biased, and short-circuit when forward biased. You can simply drag the ideal diode from the toolbox into your circuit, and optionally double-click to configure its parameters.
Here’s an example using four ideal diodes to build a full-wave rectifier:
Click to open and simulate the circuit above. Observe how the 4 diodes turn on and off at different times in the AC cycle.
Here’s an simulation comparing the regular PN Junction Diode with the ideal diode:
Click to open and simulate the circuit above.
Note that D2 (a PN junction diode) gives us a gentle knee as it transitions from off to on, as real diodes do. D2’s curve is smooth in the calculus sense: its derivative is continuous.
In contrast, ideal diodes D1 and D3 show an abrupt (piecewise-linear) knee when they change from off to on. These are not smooth in the calculus sense: their derivatives are discontinuous.
Use the PN junction diode when:
• Intending to accurately simulate real-world devices
• Simulation runtime is not a constraint
Use the ideal diode when:
• Learning about signal rectification (where PN junction complexity is unnecessary)
• Simulating signal clamping behaviors (where hard clamping is acceptable)
• Simulating switching power supplies: buck, boost converters, etc. (where simulation performance is a constraint)
• Other cases where much faster-running simulations are preferred to accurate diode modeling
Both diode models, as well as Zener Diodes, photodiodes, and LEDs, are nows available in the CircuitLab toolbox: | 2023-01-31 23:13:22 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8374963402748108, "perplexity": 5463.920870343009}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499891.42/warc/CC-MAIN-20230131222253-20230201012253-00288.warc.gz"} |
http://mathhelpforum.com/algebra/88510-solve-following-equation-print.html | # Solve the following equation
• May 10th 2009, 08:50 PM
Hapa
Solve the following equation
Having trouble getting started on a few of these.
1. 5^(x-10) = 125^x
2. (3/4) log x ^(3/4) + 9 = 0
thanks again!
• May 11th 2009, 09:03 AM
inzaghina
Hello..
1) 125^x=5^(3x) => x-10=3x => x=-5
2) 3/4 * log x ^(3/4) + 9 = 0 => 3/4*3/4 * log x = -9 => 1/16 * log x = -1 => log x = -16
if log = ln => x=e^(-16)
• May 11th 2009, 09:13 AM
Soroban
Hello, Hapa!
Quote:
$1)\;\;5^{x-10} \:=\: 125^x$
We have: . $5^{x-10} \;=\;\left(5^3\right)^x \quad\Rightarrow\quad 5^{x-10} \:=\:5^{3x}$
Equate exponents: . $x-10 \:=\:3x\quad\hdots\;\text{ etc.}$
Quote:
$2)\;\;\tfrac{3}{4}\log\left(x ^{\frac{3}{4}}\right) + 9 \:=\: 0$
We have: . $\tfrac{3}{4}\log\left(x^{\frac{3}{4}}\right) \:=\:-9 \quad\Rightarrow\quad \tfrac{3}{4}\cdot\tfrac{3}{4}\log(x) \:=\:-9 \quad\Rightarrow\quad \tfrac{9}{16}\log(x) \:=\:-9$
. . . . . . $\log(x) \:=\:-16 \quad\Rightarrow\quad x \:=\:10^{-16}$
Edit: Too slow ... again!
• May 11th 2009, 09:43 AM
inzaghina
soroban, how do you do to write the equations so beautiful ? :)
• May 11th 2009, 04:11 PM
lmasud
he used mathtype to write it like that
and here is another way to solve:
5^(x-10)= 125^x
5^(x-10)= 5^(3x)
Take "log" of both sides:
(x-10)log5= (3x)log5
dividing both sides by log5:
x-10= 3x
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https://se.mathworks.com/help/optim/ug/fmincon.html | # fmincon
Find minimum of constrained nonlinear multivariable function
## Syntax
x = fmincon(fun,x0,A,b)
x = fmincon(fun,x0,A,b,Aeq,beq)
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub)
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)
x = fmincon(problem)
[x,fval] = fmincon(___)
[x,fval,exitflag,output] = fmincon(___)
[x,fval,exitflag,output,lambda,grad,hessian] = fmincon(___)
## Description
Nonlinear programming solver.
Finds the minimum of a problem specified by
b and beq are vectors, A and Aeq are matrices, c(x) and ceq(x) are functions that return vectors, and f(x) is a function that returns a scalar. f(x), c(x), and ceq(x) can be nonlinear functions.
x, lb, and ub can be passed as vectors or matrices; see Matrix Arguments.
example
x = fmincon(fun,x0,A,b) starts at x0 and attempts to find a minimizer x of the function described in fun subject to the linear inequalities A*x ≤ b. x0 can be a scalar, vector, or matrix. NotePassing Extra Parameters explains how to pass extra parameters to the objective function and nonlinear constraint functions, if necessary.
example
x = fmincon(fun,x0,A,b,Aeq,beq) minimizes fun subject to the linear equalities Aeq*x = beq and A*x ≤ b. If no inequalities exist, set A = [] and b = [].
example
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables in x, so that the solution is always in the range lb ≤ x ≤ ub. If no equalities exist, set Aeq = [] and beq = []. If x(i) is unbounded below, set lb(i) = -Inf, and if x(i) is unbounded above, set ub(i) = Inf. NoteIf the specified input bounds for a problem are inconsistent, fmincon throws an error. In this case, output x is x0 and fval is [].For the default 'interior-point' algorithm, fmincon sets components of x0 that violate the bounds lb ≤ x ≤ ub, or are equal to a bound, to the interior of the bound region. For the 'trust-region-reflective' algorithm, fmincon sets violating components to the interior of the bound region. For other algorithms, fmincon sets violating components to the closest bound. Components that respect the bounds are not changed. See Iterations Can Violate Constraints.
example
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon) subjects the minimization to the nonlinear inequalities c(x) or equalities ceq(x) defined in nonlcon. fmincon optimizes such that c(x) ≤ 0 and ceq(x) = 0. If no bounds exist, set lb = [] and/or ub = [].
example
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options) minimizes with the optimization options specified in options. Use optimoptions to set these options. If there are no nonlinear inequality or equality constraints, set nonlcon = [].
example
x = fmincon(problem) finds the minimum for problem, a structure described in problem.
example
[x,fval] = fmincon(___), for any syntax, returns the value of the objective function fun at the solution x.
example
[x,fval,exitflag,output] = fmincon(___) additionally returns a value exitflag that describes the exit condition of fmincon, and a structure output with information about the optimization process.
example
[x,fval,exitflag,output,lambda,grad,hessian] = fmincon(___) additionally returns:lambda — Structure with fields containing the Lagrange multipliers at the solution x.grad — Gradient of fun at the solution x.hessian — Hessian of fun at the solution x. See fmincon Hessian.
## Examples
collapse all
Find the minimum value of Rosenbrock's function when there is a linear inequality constraint.
Set the objective function fun to be Rosenbrock's function. Rosenbrock's function is well-known to be difficult to minimize. It has its minimum objective value of 0 at the point (1,1). For more information, see Solve a Constrained Nonlinear Problem, Solver-Based.
fun = @(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2;
Find the minimum value starting from the point [-1,2], constrained to have $x\left(1\right)+2x\left(2\right)\le 1$. Express this constraint in the form Ax <= b by taking A = [1,2] and b = 1. Notice that this constraint means that the solution will not be at the unconstrained solution (1,1), because at that point $x\left(1\right)+2x\left(2\right)=3>1$.
x0 = [-1,2]; A = [1,2]; b = 1; x = fmincon(fun,x0,A,b)
Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2 0.5022 0.2489
Find the minimum value of Rosenbrock's function when there are both a linear inequality constraint and a linear equality constraint.
Set the objective function fun to be Rosenbrock's function.
fun = @(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2;
Find the minimum value starting from the point [0.5,0], constrained to have $x\left(1\right)+2x\left(2\right)\le 1$ and $2x\left(1\right)+x\left(2\right)=1$.
• Express the linear inequality constraint in the form A*x <= b by taking A = [1,2] and b = 1.
• Express the linear equality constraint in the form Aeq*x = beq by taking Aeq = [2,1] and beq = 1.
x0 = [0.5,0]; A = [1,2]; b = 1; Aeq = [2,1]; beq = 1; x = fmincon(fun,x0,A,b,Aeq,beq)
Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2 0.4149 0.1701
Find the minimum of an objective function in the presence of bound constraints.
The objective function is a simple algebraic function of two variables.
fun = @(x)1+x(1)/(1+x(2)) - 3*x(1)*x(2) + x(2)*(1+x(1));
Look in the region where $x$ has positive values, , and .
lb = [0,0]; ub = [1,2];
There are no linear constraints, so set those arguments to [].
A = []; b = []; Aeq = []; beq = [];
Try an initial point in the middle of the region.
x0 = (lb + ub)/2;
Solve the problem.
x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub)
Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2 1.0000 2.0000
A different initial point can lead to a different solution.
x0 = x0/5; x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub)
Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2 10-6 × 0.4000 0.4000
To determine which solution is better, see Obtain the Objective Function Value.
Find the minimum of a function subject to nonlinear constraints
Find the point where Rosenbrock's function is minimized within a circle, also subject to bound constraints.
fun = @(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2;
Look within the region , .
lb = [0,0.2]; ub = [0.5,0.8];
Also look within the circle centered at [1/3,1/3] with radius 1/3. Copy the following code to a file on your MATLAB® path named circlecon.m.
% Copyright 2015 The MathWorks, Inc. function [c,ceq] = circlecon(x) c = (x(1)-1/3)^2 + (x(2)-1/3)^2 - (1/3)^2; ceq = [];
There are no linear constraints, so set those arguments to [].
A = []; b = []; Aeq = []; beq = [];
Choose an initial point satisfying all the constraints.
x0 = [1/4,1/4];
Solve the problem.
nonlcon = @circlecon; x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)
Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance. x = 0.5000 0.2500
Set options to view iterations as they occur and to use a different algorithm.
To observe the fmincon solution process, set the Display option to 'iter'. Also, try the 'sqp' algorithm, which is sometimes faster or more accurate than the default 'interior-point' algorithm.
options = optimoptions('fmincon','Display','iter','Algorithm','sqp');
Find the minimum of Rosenbrock's function on the unit disk, . First create a function that represents the nonlinear constraint. Save this as a file named unitdisk.m on your MATLAB® path.
function [c,ceq] = unitdisk(x) c = x(1)^2 + x(2)^2 - 1; ceq = [];
Create the remaining problem specifications. Then run fmincon.
fun = @(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2; A = []; b = []; Aeq = []; beq = []; lb = []; ub = []; nonlcon = @unitdisk; x0 = [0,0]; x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)
Iter Func-count Fval Feasibility Step Length Norm of First-order step optimality 0 3 1.000000e+00 0.000e+00 1.000e+00 0.000e+00 2.000e+00 1 12 8.913011e-01 0.000e+00 1.176e-01 2.353e-01 1.107e+01 2 22 8.047847e-01 0.000e+00 8.235e-02 1.900e-01 1.330e+01 3 28 4.197517e-01 0.000e+00 3.430e-01 1.217e-01 6.172e+00 4 31 2.733703e-01 0.000e+00 1.000e+00 5.254e-02 5.705e-01 5 34 2.397111e-01 0.000e+00 1.000e+00 7.498e-02 3.164e+00 6 37 2.036002e-01 0.000e+00 1.000e+00 5.960e-02 3.106e+00 7 40 1.164353e-01 0.000e+00 1.000e+00 1.459e-01 1.059e+00 8 43 1.161753e-01 0.000e+00 1.000e+00 1.754e-01 7.383e+00 9 46 5.901601e-02 0.000e+00 1.000e+00 1.547e-02 7.278e-01 10 49 4.533081e-02 2.898e-03 1.000e+00 5.393e-02 1.252e-01 11 52 4.567454e-02 2.225e-06 1.000e+00 1.492e-03 1.679e-03 12 55 4.567481e-02 4.406e-12 1.000e+00 2.095e-06 1.501e-05 13 58 4.567481e-02 0.000e+00 1.000e+00 2.159e-09 1.511e-05 Local minimum possible. Constraints satisfied. fmincon stopped because the size of the current step is less than the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance. x = 0.7864 0.6177
Include gradient evaluation in the objective function for faster or more reliable computations.
Include the gradient evaluation as a conditionalized output in the objective function file. For details, see Including Gradients and Hessians. The objective function is Rosenbrock's function,
function [f,g] = rosenbrockwithgrad(x) % Calculate objective f f = 100*(x(2) - x(1)^2)^2 + (1-x(1))^2; if nargout > 1 % gradient required g = [-400*(x(2)-x(1)^2)*x(1)-2*(1-x(1)); 200*(x(2)-x(1)^2)]; end
Save this code as a file named rosenbrockwithgrad.m on your MATLAB® path.
Create options to use the objective function gradient.
options = optimoptions('fmincon','SpecifyObjectiveGradient',true);
Create the other inputs for the problem. Then call fmincon.
fun = @rosenbrockwithgrad; x0 = [-1,2]; A = []; b = []; Aeq = []; beq = []; lb = [-2,-2]; ub = [2,2]; nonlcon = []; x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)
Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance. x = 1.0000 1.0000
Solve the same problem as in Nondefault Options using a problem structure instead of separate arguments.
Create the options and a problem structure. See problem for the field names and required fields.
options = optimoptions('fmincon','Display','iter','Algorithm','sqp'); problem.options = options; problem.solver = 'fmincon'; problem.objective = @(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2; problem.x0 = [0,0];
The nonlinear constraint function unitdisk appears at the end of this example. Include the nonlinear constraint function in problem.
problem.nonlcon = @unitdisk;
Solve the problem.
x = fmincon(problem)
Iter Func-count Fval Feasibility Step Length Norm of First-order step optimality 0 3 1.000000e+00 0.000e+00 1.000e+00 0.000e+00 2.000e+00 1 12 8.913011e-01 0.000e+00 1.176e-01 2.353e-01 1.107e+01 2 22 8.047847e-01 0.000e+00 8.235e-02 1.900e-01 1.330e+01 3 28 4.197517e-01 0.000e+00 3.430e-01 1.217e-01 6.172e+00 4 31 2.733703e-01 0.000e+00 1.000e+00 5.254e-02 5.705e-01 5 34 2.397111e-01 0.000e+00 1.000e+00 7.498e-02 3.164e+00 6 37 2.036002e-01 0.000e+00 1.000e+00 5.960e-02 3.106e+00 7 40 1.164353e-01 0.000e+00 1.000e+00 1.459e-01 1.059e+00 8 43 1.161753e-01 0.000e+00 1.000e+00 1.754e-01 7.383e+00 9 46 5.901601e-02 0.000e+00 1.000e+00 1.547e-02 7.278e-01 10 49 4.533081e-02 2.898e-03 1.000e+00 5.393e-02 1.252e-01 11 52 4.567454e-02 2.225e-06 1.000e+00 1.492e-03 1.679e-03 12 55 4.567481e-02 4.406e-12 1.000e+00 2.095e-06 1.501e-05 13 58 4.567481e-02 0.000e+00 1.000e+00 2.159e-09 1.511e-05 Local minimum possible. Constraints satisfied. fmincon stopped because the size of the current step is less than the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2 0.7864 0.6177
The iterative display and solution are the same as in Nondefault Options.
The following code creates the unitdisk function.
function [c,ceq] = unitdisk(x) c = x(1)^2 + x(2)^2 - 1; ceq = []; end
Call fmincon with the fval output to obtain the value of the objective function at the solution.
The Minimize with Bound Constraints example shows two solutions. Which is better? Run the example requesting the fval output as well as the solution.
fun = @(x)1+x(1)./(1+x(2)) - 3*x(1).*x(2) + x(2).*(1+x(1)); lb = [0,0]; ub = [1,2]; A = []; b = []; Aeq = []; beq = []; x0 = (lb + ub)/2; [x,fval] = fmincon(fun,x0,A,b,Aeq,beq,lb,ub)
Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance.
x = 1×2 1.0000 2.0000
fval = -0.6667
Run the problem using a different starting point x0.
x0 = x0/5; [x2,fval2] = fmincon(fun,x0,A,b,Aeq,beq,lb,ub)
Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance.
x2 = 1×2 10-6 × 0.4000 0.4000
fval2 = 1.0000
This solution has an objective function value fval2 = 1, which is higher than the first value fval = –0.6667. The first solution x has a lower local minimum objective function value.
To easily examine the quality of a solution, request the exitflag and output outputs.
Set up the problem of minimizing Rosenbrock's function on the unit disk, . First create a function that represents the nonlinear constraint. Save this as a file named unitdisk.m on your MATLAB® path.
function [c,ceq] = unitdisk(x) c = x(1)^2 + x(2)^2 - 1; ceq = [];
Create the remaining problem specifications.
fun = @(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2; nonlcon = @unitdisk; A = []; b = []; Aeq = []; beq = []; lb = []; ub = []; x0 = [0,0];
Call fmincon using the fval, exitflag, and output outputs.
[x,fval,exitflag,output] = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)
Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance. x = 0.7864 0.6177 fval = 0.0457 exitflag = 1 output = struct with fields: iterations: 24 funcCount: 84 constrviolation: 0 stepsize: 6.9162e-06 algorithm: 'interior-point' firstorderopt: 2.4373e-08 cgiterations: 4 message: '...' bestfeasible: [1x1 struct]
• The exitflag value 1 indicates that the solution is a local minimum.
• The output structure reports several statistics about the solution process. In particular, it gives the number of iterations in output.iterations, number of function evaluations in output.funcCount, and the feasibility in output.constrviolation.
fmincon optionally returns several outputs that you can use for analyzing the reported solution.
Set up the problem of minimizing Rosenbrock's function on the unit disk. First create a function that represents the nonlinear constraint. Save this as a file named unitdisk.m on your MATLAB® path.
function [c,ceq] = unitdisk(x) c = x(1)^2 + x(2)^2 - 1; ceq = [];
Create the remaining problem specifications.
fun = @(x)100*(x(2)-x(1)^2)^2 + (1-x(1))^2; nonlcon = @unitdisk; A = []; b = []; Aeq = []; beq = []; lb = []; ub = []; x0 = [0,0];
Request all fmincon outputs.
[x,fval,exitflag,output,lambda,grad,hessian] = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)
Local minimum found that satisfies the constraints. Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance. x = 0.7864 0.6177 fval = 0.0457 exitflag = 1 output = struct with fields: iterations: 24 funcCount: 84 constrviolation: 0 stepsize: 6.9162e-06 algorithm: 'interior-point' firstorderopt: 2.4373e-08 cgiterations: 4 message: '...' bestfeasible: [1x1 struct] lambda = struct with fields: eqlin: [0x1 double] eqnonlin: [0x1 double] ineqlin: [0x1 double] lower: [2x1 double] upper: [2x1 double] ineqnonlin: 0.1215 grad = -0.1911 -0.1501 hessian = 497.2903 -314.5589 -314.5589 200.2392
• The lambda.ineqnonlin output shows that the nonlinear constraint is active at the solution, and gives the value of the associated Lagrange multiplier.
• The grad output gives the value of the gradient of the objective function at the solution x.
• The hessian output is described in fmincon Hessian.
## Input Arguments
collapse all
Function to minimize, specified as a function handle or function name. fun is a function that accepts a vector or array x and returns a real scalar f, the objective function evaluated at x.
Specify fun as a function handle for a file:
x = fmincon(@myfun,x0,A,b)
where myfun is a MATLAB® function such as
function f = myfun(x) f = ... % Compute function value at x
You can also specify fun as a function handle for an anonymous function:
x = fmincon(@(x)norm(x)^2,x0,A,b);
If you can compute the gradient of fun and the SpecifyObjectiveGradient option is set to true, as set by
options = optimoptions('fmincon','SpecifyObjectiveGradient',true)
then fun must return the gradient vector g(x) in the second output argument.
If you can also compute the Hessian matrix and the HessianFcn option is set to 'objective' via optimoptions and the Algorithm option is 'trust-region-reflective', fun must return the Hessian value H(x), a symmetric matrix, in a third output argument. fun can give a sparse Hessian. See Hessian for fminunc trust-region or fmincon trust-region-reflective algorithms for details.
If you can also compute the Hessian matrix and the Algorithm option is set to 'interior-point', there is a different way to pass the Hessian to fmincon. For more information, see Hessian for fmincon interior-point algorithm. For an example using Symbolic Math Toolbox™ to compute the gradient and Hessian, see Calculate Gradients and Hessians Using Symbolic Math Toolbox™.
The interior-point and trust-region-reflective algorithms allow you to supply a Hessian multiply function. This function gives the result of a Hessian-times-vector product without computing the Hessian directly. This can save memory. See Hessian Multiply Function.
Example: fun = @(x)sin(x(1))*cos(x(2))
Data Types: char | function_handle | string
Initial point, specified as a real vector or real array. Solvers use the number of elements in, and size of, x0 to determine the number and size of variables that fun accepts.
• 'interior-point' algorithm — If the HonorBounds option is true (default), fmincon resets x0 components that are on or outside bounds lb or ub to values strictly between the bounds.
• 'trust-region-reflective' algorithm — fmincon resets infeasible x0 components to be feasible with respect to bounds or linear equalities.
• 'sqp', 'sqp-legacy', or 'active-set' algorithm — fmincon resets x0 components that are outside bounds to the values of the corresponding bounds.
Example: x0 = [1,2,3,4]
Data Types: double
Linear inequality constraints, specified as a real matrix. A is an M-by-N matrix, where M is the number of inequalities, and N is the number of variables (number of elements in x0). For large problems, pass A as a sparse matrix.
A encodes the M linear inequalities
A*x <= b,
where x is the column vector of N variables x(:), and b is a column vector with M elements.
For example, to specify
x1 + 2x2 ≤ 10
3x1 + 4x2 ≤ 20
5x1 + 6x2 ≤ 30,
enter these constraints:
A = [1,2;3,4;5,6]; b = [10;20;30];
Example: To specify that the x components sum to 1 or less, use A = ones(1,N) and b = 1.
Data Types: double
Linear inequality constraints, specified as a real vector. b is an M-element vector related to the A matrix. If you pass b as a row vector, solvers internally convert b to the column vector b(:). For large problems, pass b as a sparse vector.
b encodes the M linear inequalities
A*x <= b,
where x is the column vector of N variables x(:), and A is a matrix of size M-by-N.
For example, consider these inequalities:
x1 + 2x2 ≤ 10
3x1 + 4x2 ≤ 20
5x1 + 6x2 ≤ 30.
Specify the inequalities by entering the following constraints.
A = [1,2;3,4;5,6]; b = [10;20;30];
Example: To specify that the x components sum to 1 or less, use A = ones(1,N) and b = 1.
Data Types: double
Linear equality constraints, specified as a real matrix. Aeq is an Me-by-N matrix, where Me is the number of equalities, and N is the number of variables (number of elements in x0). For large problems, pass Aeq as a sparse matrix.
Aeq encodes the Me linear equalities
Aeq*x = beq,
where x is the column vector of N variables x(:), and beq is a column vector with Me elements.
For example, to specify
x1 + 2x2 + 3x3 = 10
2x1 + 4x2 + x3 = 20,
enter these constraints:
Aeq = [1,2,3;2,4,1]; beq = [10;20];
Example: To specify that the x components sum to 1, use Aeq = ones(1,N) and beq = 1.
Data Types: double
Linear equality constraints, specified as a real vector. beq is an Me-element vector related to the Aeq matrix. If you pass beq as a row vector, solvers internally convert beq to the column vector beq(:). For large problems, pass beq as a sparse vector.
beq encodes the Me linear equalities
Aeq*x = beq,
where x is the column vector of N variables x(:), and Aeq is a matrix of size Me-by-N.
For example, consider these equalities:
x1 + 2x2 + 3x3 = 10
2x1 + 4x2 + x3 = 20.
Specify the equalities by entering the following constraints.
Aeq = [1,2,3;2,4,1]; beq = [10;20];
Example: To specify that the x components sum to 1, use Aeq = ones(1,N) and beq = 1.
Data Types: double
Lower bounds, specified as a real vector or real array. If the number of elements in x0 is equal to the number of elements in lb, then lb specifies that
x(i) >= lb(i) for all i.
If numel(lb) < numel(x0), then lb specifies that
x(i) >= lb(i) for 1 <= i <= numel(lb).
If there are fewer elements in lb than in x0, solvers issue a warning.
Example: To specify that all x components are positive, use lb = zeros(size(x0)).
Data Types: double
Upper bounds, specified as a real vector or real array. If the number of elements in x0 is equal to the number of elements in ub, then ub specifies that
x(i) <= ub(i) for all i.
If numel(ub) < numel(x0), then ub specifies that
x(i) <= ub(i) for 1 <= i <= numel(ub).
If there are fewer elements in ub than in x0, solvers issue a warning.
Example: To specify that all x components are less than 1, use ub = ones(size(x0)).
Data Types: double
Nonlinear constraints, specified as a function handle or function name. nonlcon is a function that accepts a vector or array x and returns two arrays, c(x) and ceq(x).
• c(x) is the array of nonlinear inequality constraints at x. fmincon attempts to satisfy
c(x) <= 0 for all entries of c.
• ceq(x) is the array of nonlinear equality constraints at x. fmincon attempts to satisfy
ceq(x) = 0 for all entries of ceq.
For example,
x = fmincon(@myfun,x0,A,b,Aeq,beq,lb,ub,@mycon)
where mycon is a MATLAB function such as
function [c,ceq] = mycon(x) c = ... % Compute nonlinear inequalities at x. ceq = ... % Compute nonlinear equalities at x.
If the gradients of the constraints can also be computed and the SpecifyConstraintGradient option is true, as set by
options = optimoptions('fmincon','SpecifyConstraintGradient',true)
then nonlcon must also return, in the third and fourth output arguments, GC, the gradient of c(x), and GCeq, the gradient of ceq(x). GC and GCeq can be sparse or dense. If GC or GCeq is large, with relatively few nonzero entries, save running time and memory in the interior-point algorithm by representing them as sparse matrices. For more information, see Nonlinear Constraints.
Data Types: char | function_handle | string
Optimization options, specified as the output of optimoptions or a structure such as optimset returns.
Some options apply to all algorithms, and others are relevant for particular algorithms. See Optimization Options Reference for detailed information.
Some options are absent from the optimoptions display. These options appear in italics in the following table. For details, see View Options.
All Algorithms Algorithm Choose the optimization algorithm:'interior-point' (default)'trust-region-reflective''sqp''sqp-legacy' (optimoptions only)'active-set'For information on choosing the algorithm, see Choosing the Algorithm.The trust-region-reflective algorithm requires: A gradient to be supplied in the objective functionSpecifyObjectiveGradient to be set to trueEither bound constraints or linear equality constraints, but not both If you select the 'trust-region-reflective' algorithm and these conditions are not all satisfied, fmincon throws an error.The 'active-set', 'sqp-legacy', and 'sqp' algorithms are not large-scale. See Large-Scale vs. Medium-Scale Algorithms. CheckGradients Compare user-supplied derivatives (gradients of objective or constraints) to finite-differencing derivatives. Choices are false (default) or true. For optimset, the name is DerivativeCheck and the values are 'on' or 'off'. See Current and Legacy Option Names. ConstraintTolerance Tolerance on the constraint violation, a positive scalar. The default is 1e-6. See Tolerances and Stopping Criteria.For optimset, the name is TolCon. See Current and Legacy Option Names. Diagnostics Display diagnostic information about the function to be minimized or solved. Choices are 'off' (default) or 'on'. DiffMaxChange Maximum change in variables for finite-difference gradients (a positive scalar). The default is Inf. DiffMinChange Minimum change in variables for finite-difference gradients (a positive scalar). The default is 0. Display Level of display (see Iterative Display): 'off' or 'none' displays no output.'iter' displays output at each iteration, and gives the default exit message.'iter-detailed' displays output at each iteration, and gives the technical exit message.'notify' displays output only if the function does not converge, and gives the default exit message.'notify-detailed' displays output only if the function does not converge, and gives the technical exit message.'final' (default) displays only the final output, and gives the default exit message.'final-detailed' displays only the final output, and gives the technical exit message. FiniteDifferenceStepSize Scalar or vector step size factor for finite differences. When you set FiniteDifferenceStepSize to a vector v, the forward finite differences delta aredelta = v.*sign′(x).*max(abs(x),TypicalX);where sign′(x) = sign(x) except sign′(0) = 1. Central finite differences aredelta = v.*max(abs(x),TypicalX);Scalar FiniteDifferenceStepSize expands to a vector. The default is sqrt(eps) for forward finite differences, and eps^(1/3) for central finite differences. For optimset, the name is FinDiffRelStep. See Current and Legacy Option Names. FiniteDifferenceType Finite differences, used to estimate gradients, are either 'forward' (default), or 'central' (centered). 'central' takes twice as many function evaluations but should be more accurate. The trust-region-reflective algorithm uses FiniteDifferenceType only when CheckGradients is set to true.fmincon is careful to obey bounds when estimating both types of finite differences. So, for example, it could take a backward, rather than a forward, difference to avoid evaluating at a point outside bounds. However, for the interior-point algorithm, 'central' differences might violate bounds during their evaluation if the HonorBounds option is set to false. For optimset, the name is FinDiffType. See Current and Legacy Option Names. FunValCheck Check whether objective function values are valid. The default setting, 'off', does not perform a check. The 'on' setting displays an error when the objective function returns a value that is complex, Inf, or NaN. MaxFunctionEvaluations Maximum number of function evaluations allowed, a positive integer. The default value for all algorithms except interior-point is 100*numberOfVariables; for the interior-point algorithm the default is 3000. See Tolerances and Stopping Criteria and Iterations and Function Counts. For optimset, the name is MaxFunEvals. See Current and Legacy Option Names. MaxIterations Maximum number of iterations allowed, a positive integer. The default value for all algorithms except interior-point is 400; for the interior-point algorithm the default is 1000. See Tolerances and Stopping Criteria and Iterations and Function Counts. For optimset, the name is MaxIter. See Current and Legacy Option Names. OptimalityTolerance Termination tolerance on the first-order optimality (a positive scalar). The default is 1e-6. See First-Order Optimality Measure. For optimset, the name is TolFun. See Current and Legacy Option Names. OutputFcn Specify one or more user-defined functions that an optimization function calls at each iteration. Pass a function handle or a cell array of function handles. The default is none ([]). See Output Function and Plot Function Syntax. PlotFcn Plots various measures of progress while the algorithm executes; select from predefined plots or write your own. Pass a built-in plot function name, a function handle, or a cell array of built-in plot function names or function handles. For custom plot functions, pass function handles. The default is none ([]): 'optimplotx' plots the current point'optimplotfunccount' plots the function count'optimplotfval' plots the function value'optimplotfvalconstr' plots the best feasible objective function value found as a line plot. The plot shows infeasible points as red, and feasible points as blue, using a feasibility tolerance of 1e-6.'optimplotconstrviolation' plots the maximum constraint violation'optimplotstepsize' plots the step size'optimplotfirstorderopt' plots the first-order optimality measure Custom plot functions use the same syntax as output functions. See Output Functions for Optimization Toolbox™ and Output Function and Plot Function Syntax.For optimset, the name is PlotFcns. See Current and Legacy Option Names. SpecifyConstraintGradient Gradient for nonlinear constraint functions defined by the user. When set to the default, false, fmincon estimates gradients of the nonlinear constraints by finite differences. When set to true, fmincon expects the constraint function to have four outputs, as described in nonlcon. The trust-region-reflective algorithm does not accept nonlinear constraints. For optimset, the name is GradConstr and the values are 'on' or 'off'. See Current and Legacy Option Names. SpecifyObjectiveGradient Gradient for the objective function defined by the user. See the description of fun to see how to define the gradient in fun. The default, false, causes fmincon to estimate gradients using finite differences. Set to true to have fmincon use a user-defined gradient of the objective function. To use the 'trust-region-reflective' algorithm, you must provide the gradient, and set SpecifyObjectiveGradient to true. For optimset, the name is GradObj and the values are 'on' or 'off'. See Current and Legacy Option Names. StepTolerance Termination tolerance on x, a positive scalar. The default value for all algorithms except 'interior-point' is 1e-6; for the 'interior-point' algorithm, the default is 1e-10. See Tolerances and Stopping Criteria. For optimset, the name is TolX. See Current and Legacy Option Names. TypicalX Typical x values. The number of elements in TypicalX is equal to the number of elements in x0, the starting point. The default value is ones(numberofvariables,1). fmincon uses TypicalX for scaling finite differences for gradient estimation.The 'trust-region-reflective' algorithm uses TypicalX only for the CheckGradients option. UseParallel When true, fmincon estimates gradients in parallel. Disable by setting to the default, false. trust-region-reflective requires a gradient in the objective, so UseParallel does not apply. See Parallel Computing. Trust-Region-Reflective Algorithm FunctionTolerance Termination tolerance on the function value, a positive scalar. The default is 1e-6. See Tolerances and Stopping Criteria. For optimset, the name is TolFun. See Current and Legacy Option Names. HessianFcn If [] (default), fmincon approximates the Hessian using finite differences, or uses a Hessian multiply function (with option HessianMultiplyFcn). If 'objective', fmincon uses a user-defined Hessian (defined in fun). See Hessian as an Input. For optimset, the name is HessFcn. See Current and Legacy Option Names. HessianMultiplyFcn Hessian multiply function, specified as a function handle. For large-scale structured problems, this function computes the Hessian matrix product H*Y without actually forming H. The function is of the formW = hmfun(Hinfo,Y)where Hinfo contains a matrix used to compute H*Y. The first argument is the same as the third argument returned by the objective function fun, for example[f,g,Hinfo] = fun(x)Y is a matrix that has the same number of rows as there are dimensions in the problem. The matrix W = H*Y, although H is not formed explicitly. fmincon uses Hinfo to compute the preconditioner. For information on how to supply values for any additional parameters hmfun needs, see Passing Extra Parameters.NoteTo use the HessianMultiplyFcn option, HessianFcn must be set to [], and SubproblemAlgorithm must be 'cg' (default).See Hessian Multiply Function. See Minimization with Dense Structured Hessian, Linear Equalities for an example.For optimset, the name is HessMult. See Current and Legacy Option Names. HessPattern Sparsity pattern of the Hessian for finite differencing. Set HessPattern(i,j) = 1 when you can have ∂2fun/∂x(i)∂x(j) ≠ 0. Otherwise, set HessPattern(i,j) = 0.Use HessPattern when it is inconvenient to compute the Hessian matrix H in fun, but you can determine (say, by inspection) when the ith component of the gradient of fun depends on x(j). fmincon can approximate H via sparse finite differences (of the gradient) if you provide the sparsity structure of H as the value for HessPattern. In other words, provide the locations of the nonzeros.When the structure is unknown, do not set HessPattern. The default behavior is as if HessPattern is a dense matrix of ones. Then fmincon computes a full finite-difference approximation in each iteration. This computation can be very expensive for large problems, so it is usually better to determine the sparsity structure. MaxPCGIter Maximum number of preconditioned conjugate gradient (PCG) iterations, a positive scalar. The default is max(1,floor(numberOfVariables/2)) for bound-constrained problems, and is numberOfVariables for equality-constrained problems. For more information, see Preconditioned Conjugate Gradient Method. PrecondBandWidth Upper bandwidth of preconditioner for PCG, a nonnegative integer. By default, diagonal preconditioning is used (upper bandwidth of 0). For some problems, increasing the bandwidth reduces the number of PCG iterations. Setting PrecondBandWidth to Inf uses a direct factorization (Cholesky) rather than the conjugate gradients (CG). The direct factorization is computationally more expensive than CG, but produces a better quality step towards the solution. SubproblemAlgorithm Determines how the iteration step is calculated. The default, 'cg', takes a faster but less accurate step than 'factorization'. See fmincon Trust Region Reflective Algorithm. TolPCG Termination tolerance on the PCG iteration, a positive scalar. The default is 0.1. Active-Set Algorithm FunctionTolerance Termination tolerance on the function value, a positive scalar. The default is 1e-6. See Tolerances and Stopping Criteria. For optimset, the name is TolFun. See Current and Legacy Option Names. MaxSQPIter Maximum number of SQP iterations allowed, a positive integer. The default is 10*max(numberOfVariables, numberOfInequalities + numberOfBounds). RelLineSrchBnd Relative bound (a real nonnegative scalar value) on the line search step length. The total displacement in x satisfies |Δx(i)| ≤ relLineSrchBnd· max(|x(i)|,|typicalx(i)|). This option provides control over the magnitude of the displacements in x for cases in which the solver takes steps that are considered too large. The default is no bounds ([]). RelLineSrchBndDuration Number of iterations for which the bound specified in RelLineSrchBnd should be active (default is 1). TolConSQP Termination tolerance on inner iteration SQP constraint violation, a positive scalar. The default is 1e-6. Interior-Point Algorithm HessianApproximation Chooses how fmincon calculates the Hessian (see Hessian as an Input). The choices are:'bfgs' (default)'finite-difference''lbfgs'{'lbfgs',Positive Integer}NoteTo use HessianApproximation, both HessianFcn and HessianMultiplyFcn must be empty entries ([]). For optimset, the name is Hessian and the values are 'user-supplied', 'bfgs', 'lbfgs', 'fin-diff-grads', 'on', or 'off'. See Current and Legacy Option Names. HessianFcn If [] (default), fmincon approximates the Hessian using finite differences, or uses a supplied HessianMultiplyFcn. If a function handle, fmincon uses HessianFcn to calculate the Hessian. See Hessian as an Input.For optimset, the name is HessFcn. See Current and Legacy Option Names. HessianMultiplyFcn User-supplied function that gives a Hessian-times-vector product (see Hessian Multiply Function). Pass a function handle.NoteTo use the HessianMultiplyFcn option, HessianFcn must be set to [], and SubproblemAlgorithm must be 'cg'.For optimset, the name is HessMult. See Current and Legacy Option Names. HonorBounds The default true ensures that bound constraints are satisfied at every iteration. Disable by setting to false. For optimset, the name is AlwaysHonorConstraints and the values are 'bounds' or 'none'. See Current and Legacy Option Names. InitBarrierParam Initial barrier value, a positive scalar. Sometimes it might help to try a value above the default 0.1, especially if the objective or constraint functions are large. InitTrustRegionRadius Initial radius of the trust region, a positive scalar. On badly scaled problems it might help to choose a value smaller than the default $\sqrt{n}$, where n is the number of variables. MaxProjCGIter A tolerance (stopping criterion) for the number of projected conjugate gradient iterations; this is an inner iteration, not the number of iterations of the algorithm. This positive integer has a default value of 2*(numberOfVariables - numberOfEqualities). ObjectiveLimit A tolerance (stopping criterion) that is a scalar. If the objective function value goes below ObjectiveLimit and the iterate is feasible, the iterations halt, because the problem is presumably unbounded. The default value is -1e20. ScaleProblem true causes the algorithm to normalize all constraints and the objective function. Disable by setting to the default false. For optimset, the values are 'obj-and-constr' or 'none'. See Current and Legacy Option Names. SubproblemAlgorithm Determines how the iteration step is calculated. The default, 'factorization', is usually faster than 'cg' (conjugate gradient), though 'cg' might be faster for large problems with dense Hessians. See fmincon Interior Point Algorithm. TolProjCG A relative tolerance (stopping criterion) for projected conjugate gradient algorithm; this is for an inner iteration, not the algorithm iteration. This positive scalar has a default of 0.01. TolProjCGAbs Absolute tolerance (stopping criterion) for projected conjugate gradient algorithm; this is for an inner iteration, not the algorithm iteration. This positive scalar has a default of 1e-10. SQP and SQP Legacy Algorithms ObjectiveLimit A tolerance (stopping criterion) that is a scalar. If the objective function value goes below ObjectiveLimit and the iterate is feasible, the iterations halt, because the problem is presumably unbounded. The default value is -1e20. ScaleProblem true causes the algorithm to normalize all constraints and the objective function. Disable by setting to the default false. For optimset, the values are 'obj-and-constr' or 'none'. See Current and Legacy Option Names.
Example: options = optimoptions('fmincon','SpecifyObjectiveGradient',true,'SpecifyConstraintGradient',true)
Problem structure, specified as a structure with the following fields:
Field NameEntry
objective
Objective function
x0
Initial point for x
Aineq
Matrix for linear inequality constraints
bineq
Vector for linear inequality constraints
Aeq
Matrix for linear equality constraints
beq
Vector for linear equality constraints
lbVector of lower bounds
ubVector of upper bounds
nonlcon
Nonlinear constraint function
solver
'fmincon'
options
Options created with optimoptions
You must supply at least the objective, x0, solver, and options fields in the problem structure.
Data Types: struct
## Output Arguments
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Solution, returned as a real vector or real array. The size of x is the same as the size of x0. Typically, x is a local solution to the problem when exitflag is positive. For information on the quality of the solution, see When the Solver Succeeds.
Objective function value at the solution, returned as a real number. Generally, fval = fun(x).
Reason fmincon stopped, returned as an integer.
All Algorithms: 1 First-order optimality measure was less than options.OptimalityTolerance, and maximum constraint violation was less than options.ConstraintTolerance. 0 Number of iterations exceeded options.MaxIterations or number of function evaluations exceeded options.MaxFunctionEvaluations. -1 Stopped by an output function or plot function. -2 No feasible point was found. All algorithms except active-set: 2 Change in x was less than options.StepTolerance and maximum constraint violation was less than options.ConstraintTolerance. trust-region-reflective algorithm only: 3 Change in the objective function value was less than options.FunctionTolerance and maximum constraint violation was less than options.ConstraintTolerance. active-set algorithm only: 4 Magnitude of the search direction was less than 2*options.StepTolerance and maximum constraint violation was less than options.ConstraintTolerance. 5 Magnitude of directional derivative in search direction was less than 2*options.OptimalityTolerance and maximum constraint violation was less than options.ConstraintTolerance. interior-point, sqp-legacy, and sqp algorithms: -3 Objective function at current iteration went below options.ObjectiveLimit and maximum constraint violation was less than options.ConstraintTolerance.
Information about the optimization process, returned as a structure with fields:
iterations Number of iterations taken funcCount Number of function evaluations lssteplength Size of line search step relative to search direction (active-set and sqp algorithms only) constrviolation Maximum of constraint functions stepsize Length of last displacement in x (not in active-set algorithm) algorithm Optimization algorithm used cgiterations Total number of PCG iterations (trust-region-reflective and interior-point algorithms) firstorderopt Measure of first-order optimality bestfeasible Best (lowest objective function) feasible point encountered. A structure with these fields:xfvalfirstorderoptconstrviolationIf no feasible point is found, the bestfeasible field is empty. For this purpose, a point is feasible when the maximum of the constraint functions does not exceed options.ConstraintTolerance.The bestfeasible point can differ from the returned solution point x for a variety of reasons. For an example, see Obtain Best Feasible Point. message Exit message
Lagrange multipliers at the solution, returned as a structure with fields:
lower Lower bounds corresponding to lb upper Upper bounds corresponding to ub ineqlin Linear inequalities corresponding to A and b eqlin Linear equalities corresponding to Aeq and beq ineqnonlin Nonlinear inequalities corresponding to the c in nonlcon eqnonlin Nonlinear equalities corresponding to the ceq in nonlcon
Gradient at the solution, returned as a real vector. grad gives the gradient of fun at the point x(:).
Approximate Hessian, returned as a real matrix. For the meaning of hessian, see Hessian Output.
## Limitations
• fmincon is a gradient-based method that is designed to work on problems where the objective and constraint functions are both continuous and have continuous first derivatives.
• For the 'trust-region-reflective' algorithm, you must provide the gradient in fun and set the 'SpecifyObjectiveGradient' option to true.
• The 'trust-region-reflective' algorithm does not allow equal upper and lower bounds. For example, if lb(2)==ub(2), fmincon gives this error:
Equal upper and lower bounds not permitted in trust-region-reflective algorithm. Use either interior-point or SQP algorithms instead.
• There are two different syntaxes for passing a Hessian, and there are two different syntaxes for passing a HessianMultiplyFcn function; one for trust-region-reflective, and another for interior-point. See Including Hessians.
• For trust-region-reflective, the Hessian of the Lagrangian is the same as the Hessian of the objective function. You pass that Hessian as the third output of the objective function.
• For interior-point, the Hessian of the Lagrangian involves the Lagrange multipliers and the Hessians of the nonlinear constraint functions. You pass the Hessian as a separate function that takes into account both the current point x and the Lagrange multiplier structure lambda.
• When the problem is infeasible, fmincon attempts to minimize the maximum constraint value.
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### Hessian as an Input
fmincon uses a Hessian as an optional input. This Hessian is the matrix of second derivatives of the Lagrangian (see Equation 1), namely,
${\nabla }_{xx}^{2}L\left(x,\lambda \right)={\nabla }^{2}f\left(x\right)+\sum {\lambda }_{i}{\nabla }^{2}{c}_{i}\left(x\right)+\sum {\lambda }_{i}{\nabla }^{2}ce{q}_{i}\left(x\right).$ (1)
For details of how to supply a Hessian to the trust-region-reflective or interior-point algorithms, see Including Hessians.
The active-set and sqp algorithms do not accept an input Hessian. They compute a quasi-Newton approximation to the Hessian of the Lagrangian.
The interior-point algorithm has several choices for the 'HessianApproximation' option; see Choose Input Hessian Approximation for interior-point fmincon:
• 'bfgs'fmincon calculates the Hessian by a dense quasi-Newton approximation. This is the default Hessian approximation.
• 'lbfgs'fmincon calculates the Hessian by a limited-memory, large-scale quasi-Newton approximation. The default memory, 10 iterations, is used.
• {'lbfgs',positive integer}fmincon calculates the Hessian by a limited-memory, large-scale quasi-Newton approximation. The positive integer specifies how many past iterations should be remembered.
• 'finite-difference'fmincon calculates a Hessian-times-vector product by finite differences of the gradient(s). You must supply the gradient of the objective function, and also gradients of nonlinear constraints (if they exist). Set the 'SpecifyObjectiveGradient' option to true and, if applicable, the 'SpecifyConstraintGradient' option to true. You must set the 'SubproblemAlgorithm' to 'cg'.
### Hessian Multiply Function
The interior-point and trust-region-reflective algorithms allow you to supply a Hessian multiply function. This function gives the result of a Hessian-times-vector product, without computing the Hessian directly. This can save memory. For details, see Hessian Multiply Function.
## Algorithms
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### Choosing the Algorithm
For help choosing the algorithm, see fmincon Algorithms. To set the algorithm, use optimoptions to create options, and use the 'Algorithm' name-value pair.
The rest of this section gives brief summaries or pointers to information about each algorithm.
### Interior-Point Optimization
This algorithm is described in fmincon Interior Point Algorithm. There is more extensive description in [1], [41], and [9].
### SQP and SQP-Legacy Optimization
The fmincon 'sqp' and 'sqp-legacy' algorithms are similar to the 'active-set' algorithm described in Active-Set Optimization. fmincon SQP Algorithm describes the main differences. In summary, these differences are:
### Active-Set Optimization
fmincon uses a sequential quadratic programming (SQP) method. In this method, the function solves a quadratic programming (QP) subproblem at each iteration. fmincon updates an estimate of the Hessian of the Lagrangian at each iteration using the BFGS formula (see fminunc and references [7] and [8]).
fmincon performs a line search using a merit function similar to that proposed by [6], [7], and [8]. The QP subproblem is solved using an active set strategy similar to that described in [5]. fmincon Active Set Algorithm describes this algorithm in detail.
### Trust-Region-Reflective Optimization
The 'trust-region-reflective' algorithm is a subspace trust-region method and is based on the interior-reflective Newton method described in [3] and [4]. Each iteration involves the approximate solution of a large linear system using the method of preconditioned conjugate gradients (PCG). See the trust-region and preconditioned conjugate gradient method descriptions in fmincon Trust Region Reflective Algorithm.
## Alternative Functionality
### App
The Optimize Live Editor task provides a visual interface for fmincon.
## References
[1] Byrd, R. H., J. C. Gilbert, and J. Nocedal. “A Trust Region Method Based on Interior Point Techniques for Nonlinear Programming.” Mathematical Programming, Vol 89, No. 1, 2000, pp. 149–185.
[2] Byrd, R. H., Mary E. Hribar, and Jorge Nocedal. “An Interior Point Algorithm for Large-Scale Nonlinear Programming.” SIAM Journal on Optimization, Vol 9, No. 4, 1999, pp. 877–900.
[3] Coleman, T. F. and Y. Li. “An Interior, Trust Region Approach for Nonlinear Minimization Subject to Bounds.” SIAM Journal on Optimization, Vol. 6, 1996, pp. 418–445.
[4] Coleman, T. F. and Y. Li. “On the Convergence of Reflective Newton Methods for Large-Scale Nonlinear Minimization Subject to Bounds.” Mathematical Programming, Vol. 67, Number 2, 1994, pp. 189–224.
[5] Gill, P. E., W. Murray, and M. H. Wright. Practical Optimization, London, Academic Press, 1981.
[6] Han, S. P. “A Globally Convergent Method for Nonlinear Programming.” Journal of Optimization Theory and Applications, Vol. 22, 1977, pp. 297.
[7] Powell, M. J. D. “A Fast Algorithm for Nonlinearly Constrained Optimization Calculations.” Numerical Analysis, ed. G. A. Watson, Lecture Notes in Mathematics, Springer-Verlag, Vol. 630, 1978.
[8] Powell, M. J. D. “The Convergence of Variable Metric Methods For Nonlinearly Constrained Optimization Calculations.” Nonlinear Programming 3 (O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, eds.), Academic Press, 1978.
[9] Waltz, R. A., J. L. Morales, J. Nocedal, and D. Orban. “An interior algorithm for nonlinear optimization that combines line search and trust region steps.” Mathematical Programming, Vol 107, No. 3, 2006, pp. 391–408. | 2020-12-03 23:44:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7391195297241211, "perplexity": 2632.851215557565}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141732835.81/warc/CC-MAIN-20201203220448-20201204010448-00648.warc.gz"} |
https://www.overleaf.com/latex/examples/grunnur-ii-hlutaprof-1/qpytmmrkpdhx | #### Learn by Example
Our searchable collection of LaTeX examples is a great place to look when you're using a LaTeX package for the first time (or the first time in a while!). | 2018-01-16 21:01:42 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8115718364715576, "perplexity": 1059.408068676138}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084886739.5/warc/CC-MAIN-20180116204303-20180116224303-00588.warc.gz"} |
https://socratic.org/questions/how-to-show-that-s-sn-1-3-n-given-s-9-4-and-sn-9-4-1-1-3-n | # How to show that S_oo- S_n= (-1/3)^n ,given, S_oo=9/4 and S_n=9/4(1-(-1/3)^n)?
Jun 21, 2018
I think there's a $\frac{9}{4}$ factor missing
#### Explanation:
Since you're given both ${S}_{\setminus} \infty$ and ${S}_{n}$, in order to compute ${S}_{\setminus} \infty - {S}_{n}$ you can simply subtract the two expressions:
color(red)(S_\infty)-color(blue)(S_n) = color(red)(9/4) - color(blue)(9/4(1-(-1/3)^n)
We can expand ${S}_{n}$ as follows:
$\frac{9}{4} \left(1 - {\left(- \frac{1}{3}\right)}^{n}\right) = \frac{9}{4} - \frac{9}{4} \setminus \times {\left(- \frac{1}{3}\right)}^{n}$
So, we have
${S}_{\setminus} \infty - {S}_{n}$
$= \frac{9}{4} - \left(\frac{9}{4} - \frac{9}{4} \setminus \times {\left(- \frac{1}{3}\right)}^{n}\right)$
$= \cancel{\frac{9}{4}} - \cancel{\frac{9}{4}} + \frac{9}{4} \setminus \times {\left(- \frac{1}{3}\right)}^{n}$
So,
${S}_{\setminus} \infty - {S}_{n} = \frac{9}{4} \setminus \times {\left(- \frac{1}{3}\right)}^{n}$ | 2022-09-24 22:57:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 11, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7272918224334717, "perplexity": 3405.9180321820922}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030333541.98/warc/CC-MAIN-20220924213650-20220925003650-00063.warc.gz"} |
http://www.r-bloggers.com/the-tf-idf-statistic-for-keyword-extraction/ | # The tf-idf-Statistic For Keyword Extraction
February 27, 2014
By
(This article was first published on joy of data » R, and kindly contributed to R-bloggers)
The tf-idf-statistic (“term frequency – inverse document frequency”) is a common tool for the purpose of extracting keywords from a document by not just considering a single document but all documents from the corpus. In terms of tf-idf a word is important for a specific document if it shows up relatively often within that document and rarely in other documents of the corpus. I used tf-idf for extracting keywords from protocols of sessions of the German Bundestag and am quite happy with the results. Given that I was dealing with (so far) 18 documents, together containing more than one million words which would have to be aggregated for the term frequency, then outer joined and then fed to the formula I was first a bit worried about how R would perform. To my surprise the whole processing from reading the files from disk to the final table of tf-idf-values took about 8 seconds. That’s not bad at all.
# The tf part
The tf-formula is a ratio of a term’s occurences in a document and the number of occurences of the most frequent word within the same document. Why this makes sense is pretty self explanatory. But obviously we would end up with stop words yielding high scores – and even if those would have been discarded before, a lot of words naturally show up often in a long text but aren’t relevant to the specific document.
# The idf part
And this is exactly where the idf-factor comes into play as it represents the inverse of the share of the documents in which the regarded term can be found. The lower the number of containing documents relative to the size of the corpus, the higher the factor. The reason why this ratio is not used directly but instead its logarithm is because otherwise the effective scoring penalty of showing up in two documents would be too extreme. As you can see in the plot – the idf for a term found in just one document is twice the idf for a term found in two. This would heavily bias the ranking in favor of super-rare words even if the tf-factor indicates a high relevance. It is very unlikely that a word is of high relevance in one document but never used anywhere else.
# R code for above chart
library(ggplot2)
df <- data.frame(
x = c(1:20,1:20),
y = c(1/1:20,log(20/1:20)/log(20)),
idf = c(rep("1/n",20),rep("log",20))
ggplot(data = df), aes(x,y)) +
geom_line(aes(color = idf)) +
geom_point(aes(color = idf)) +
scale_x_discrete() +
labs(title = "comparison of relative impact of idf-formula (scaled to 1) if term occurs in more or less documents") +
xlab("number of documents a term is contained in") +
ylab("") +
theme(axis.title.x = element_text(color="grey20"))
# tf and idf together
Both factors react positively to higher relevance – one using local information, the other taking a more global perspective. So we can simply take the product and we have the “traditional” tf-idf-formula. But this formula is not set in stone and depending on whether you want to put emphasis on the tf- or rather on the idf-part it might make sense to get a feeling for the ranking behaviour in your use case and apply adjustments to it until you are d’accord with the result.
For example in the case of the Bundestag protocols I came to the conclusion that the final rankings are more useful if I put some more weight on the idf-part – which leads to effectively penalizing a word’s ranking if the word shows up in more than one document. This seems to make sense right now as my corpus only keeps 18 documents. So what I did is I added the idf-part and divided it with the size of the corpus – see tfidf’ in the top image. That has the effect that the penalization of non-exclusivity is decreasing with time as the corpus grows in size (more protocols being published).
# calculates tf-idf for different parameters and using
# different tf-idf-versions
tab_tfidf <- function(ncorpus=20) {
# I assume a maximum word frequency of 4000
max_ft <- 4000
# tf-idf without log
tfidf0 <- function(ft,max_ft,ndocs,ncorpus) (ft/max_ft) * (ncorpus/ndocs)
tfidf1 <- function(ft,max_ft,ndocs,ncorpus) (ft/max_ft) * log(ncorpus/ndocs)
# tf-idf with added idf/N
tfidf2 <- function(ft,max_ft,ndocs,ncorpus) (1/ncorpus + ft/max_ft) * log(ncorpus/ndocs)
# ft = frequency of term / ndocs = how often it showed up in other documents
df <- expand.grid(ft=c(5,10,20,30),ndocs=c(1,2,3,5,10))
res0 <- apply(df,1,function(r) tfidf0(r["ft"],max_ft,r["ndocs"],ncorpus))
ranks0 <- order(order(-res0))
res1 <- apply(df,1,function(r) tfidf1(r["ft"],max_ft,r["ndocs"],ncorpus))
ranks1 <- order(order(-res1))
res2 <- apply(df,1,function(r) tfidf2(r["ft"],max_ft,r["ndocs"],ncorpus))
ranks2 <- order(order(-res2))
result <- cbind(df,res0,res1,res2,ranks0,ranks1,ranks2)
result <- result[order(result$ft),] return(list("ncorpus" = ncorpus, "max_ft" = max_ft, result)) } # tf-idf for combinations of term frequency in {10,20,30} and # occurences in {1,2,3} relative to (20, 2) get_change_matrix <- function(res, colname) { m <- matrix(res[res$ft %in% c(10,20,30) & res\$ndocs %in% 1:3,colname], ncol=3)
# num of documents where word is assumed to be present
rownames(m) <- as.character(1:3)
# num of occurences within the hypothetical document
colnames(m) <- as.character(1:3*10)
# (A-B)/B
m <- round((m - m[2,2])/m[2,2],2)
return(m)
}
On that note – let’s see how the two versions of tf-idf compare – assuming a corpus containing 20 documents and the most frequent word to show up 4000 times.
> res <- tab_tfidf()
> get_change_matrix(res[[3]],"res1")
10 20 30
1 -0.35 0.30 0.95
2 -0.50 0.00 0.50
3 -0.59 -0.18 0.24
# tfidf' (my flavor)
> get_change_matrix(res[[3]],"res2")
10 20 30
1 0.24 0.30 0.36
2 -0.05 0.00 0.05
3 -0.21 -0.18 -0.14
Let’s for example compare a word A occuring 20 times in the document d and in 2 documents of the corpus to another word B occuring 10 times in the document d and in only 1 document (d, of course) of the corpus. In case of the traditional formula the tf-idf-statistic of B would be 35% lower compared to the tf-idf-statistic of A. In case of the alternated tf-idf-formula on the other hand we would experience an increase of B’s tf-idf-statistic by 24% compared to A’s! So it makes sense to think a bit about how exactly one would like the score to behave / rank words.
I didn’t want to spend too much time on this sub-project, so I kept it at my rule of thumb optimization. But when you start thinking about the tf-idf idea a lot of possible tweaks come to mind. To give an examle, true to the motto “once won’t do any harm” we could restrict the denominator of idf to taking only documents into account where the term shows up at least twice.
# Constraints
Obviously the tf-idf approach to quantification of a keyword’s relevance has its limits. For example it is easily conceivable that a word will show up in a lot of documents of the corpus and yet play a central role in of it. Or a subject is covered in several documents because it is very important – but tf-idf would penalize terms typical for this subject exactly because of that reason. This is why tf-idf is most certainly not the answer to everything. I came across another idea described in a paper from 2009 where the density of a term is used to infer its relevance. The basic idea is that a very relevant word will show relatively strong local densities compared to a common word with a more uniform density. Below you see the a density approximation for three stop words (“and”,”the” (male) and “the” (female)) and the densities for three terms that scored highest with respect to tf-idf in protocol #11.
# vector of words from protocol #11 - length: 127375 words
# wv <- c("Inhaltsverzeichnis", "Plenarprotokoll", "Deutscher", ...)
plot(density(which(wv=="und"), bw="SJ", n=1000, adjust=.5, from=0,
to=length(wv)), main="black: 'und', blue: 'die', green: 'der'",
xlab="index of word")
lines(density(which(wv=="die"), bw="SJ", n=1000, adjust=.5, from=0,
to=length(wv)),col="blue")
lines(density(which(wv=="der"), bw="SJ", n=1000, adjust=.5, from=0,
to=length(wv)),col="green")
plot(density(which(wv=="Tierhaltung"), bw="SJ", n=1000, adjust=.5, from=0,
to=length(wv)), main="black: 'Tierhaltung', blue: 'Verbraucherpolitik',
green: 'Rechtspolitik'", xlab="index of word")
lines(density(which(wv=="Verbraucherpolitik"), bw="SJ", n=1000, adjust=.5,
from=0, to=length(wv)),col="blue")
lines(density(which(wv=="Rechtspolitik"), bw="SJ", n=1000, adjust=.5,
from=0, to=length(wv)),col="green")
If it is possible to take the distribution into account then we can even determine relevancy for insulated single documents.
The post The tf-idf-Statistic For Keyword Extraction appeared first on joy of data.
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If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook... | 2014-10-31 17:11:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.45635995268821716, "perplexity": 2595.6728936685936}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414637900031.50/warc/CC-MAIN-20141030025820-00160-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://www.i2m.univ-amu.fr/events/entangleability-of-cones/ | # Entangleability of cones
Guillaume AUBRUN
ICJ, Université Claude Bernard Lyon 1
http://math.univ-lyon1.fr/~aubrun/
Date(s) : 11/02/2022 iCal
14 h 30 min - 15 h 30 min
We solve a long-standing conjecture by Barker, proving that the minimal and maximal tensor products of two finite-dimensional proper cones coincide if and only if one of the two cones is generated by a linearly independent set. Here, given two proper cones C1, C2, their minimal tensor product is the cone generated by products of the form x1 \otimes x2, where x1 \in C1 and x2 \in C2, while their maximal tensor product is the set of tensors that are positive under all product functionals f1 \in f2, where f1 is positive on C1 and f2 is positive on C2. Our proof techniques involve a mix of convex geometry, elementary algebraic topology, and computations inspired by quantum information theory. Our motivation comes from the foundations of physics: as an application, we show that any two non-classical systems modelled by general probabilistic theories can be entangled.
(arXiv:1911.09663, joint with Ludovico Lami, Carlos Palazuelos, Martin Plavala)
Catégories | 2023-03-20 22:43:09 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9731758832931519, "perplexity": 1091.8097609710383}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943562.70/warc/CC-MAIN-20230320211022-20230321001022-00124.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/36-maximize-utility-consumer-allocate-money-income-goods-services-consumed-elasticity-dema-q3012356 | 36. To maximize utility a consumer should allocate money income between goods and services consumed so that the:
A) elasticity of demand on all products purchased is the same.
B) marginal utility obtained from the last dollar spent on each product is the same.
C) total utility derived from each product consumed is the same.
D) marginal utility of the last unit of each product consumed is the same. | 2016-08-30 09:24:04 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8219192624092102, "perplexity": 1797.9822127998061}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982974985.86/warc/CC-MAIN-20160823200934-00139-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://love2d.org/forums/viewtopic.php?f=4&t=84496 | ## math.random > love.math.random
Questions about the LÖVE API, installing LÖVE and other support related questions go here.
Forum rules
Before you make a thread asking for help, read this.
KayleMaster
Party member
Posts: 212
Joined: Mon Aug 29, 2016 8:51 am
### math.random > love.math.random
I did some test while optimizing a problematic loop and these are my findings:
2621440 tests
math.random ~25ms
love.math.random ~35 ms
Is this because of the extra access table (love.) ?
Just something interesting I found.
Nixola
Inner party member
Posts: 1940
Joined: Tue Dec 06, 2011 7:11 pm
Location: Italy
### Re: math.random > love.math.random
It may be, although a 40% increase in time looks excessive. Can you try making both functions local when testing?
lf = love.filesystem
ls = love.sound
la = love.audio
lp = love.physics
li = love.image
lg = love.graphics
KayleMaster
Party member
Posts: 212
Joined: Mon Aug 29, 2016 8:51 am
### Re: math.random > love.math.random
They are both local, in fact, it's just one function, I just change math.random to love.math.random in the function and compile again. Did this 10 times and got consistent results.
Anyways, it's not that noticeable, and no, It wasn't the problem of my loop ( I wish it was ).
0x25a0
Prole
Posts: 36
Joined: Mon Mar 20, 2017 10:08 pm
Contact:
### Re: math.random > love.math.random
As far as I know the pseudo-random number generator in LOVE is not based on Lua's PRNG. So, the performance difference is not because of an extra table access, it's because they are inherently different PRNGs
Nixola
Inner party member
Posts: 1940
Joined: Tue Dec 06, 2011 7:11 pm
Location: Italy
### Re: math.random > love.math.random
Yeah, LÖVE's is indeed slower. I just thought it was faster for some reason.
lf = love.filesystem
ls = love.sound
la = love.audio
lp = love.physics
li = love.image
lg = love.graphics
zorg
Party member
Posts: 2732
Joined: Thu Dec 13, 2012 2:55 pm
Location: Absurdistan, Hungary
Contact:
### Re: math.random > love.math.random
It's better though, Löve's implementation, i mean.
Me and my stuff True Neutral Aspirant. Why, yes, i do indeed enjoy sarcastically correcting others when they make the most blatant of spelling mistakes. No bullying or trolling the innocent tho.
Nixola
Inner party member
Posts: 1940
Joined: Tue Dec 06, 2011 7:11 pm
Location: Italy
### Re: math.random > love.math.random
Yeah, that's a given; it's also consistent across all platforms and Lua interpreters (while LuaJIT packs a consistent PRNG, LÖVE can also use standard Lua).
lf = love.filesystem
ls = love.sound
la = love.audio
lp = love.physics
li = love.image
lg = love.graphics
erasio
Party member
Posts: 118
Joined: Wed Mar 15, 2017 8:52 am
Location: Germany
### Re: math.random > love.math.random
Plus a larger period and actual randomness without staying by setting a seed and calling random 3 times.
I mean math.random is just weird. Functional when you follow the docs. But I wouldn't recommend using it.
Plus love has random generator objects allowing you to have random generators assigned to parts of your game for actually consistent seed based randomness.
Instead of just having one which may play into combat, map generation and more meaning the order of random calls will affect the result and make debugging a pain.
KayleMaster
Party member
Posts: 212
Joined: Mon Aug 29, 2016 8:51 am
### Re: math.random > love.math.random
Yeah but I just need random tile gen (different shades) so it'd do fine for me.
vrld
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Posts: 917
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Location: Germany
Contact:
### Re: math.random > love.math.random
I was curious, so I did my own test. I also computed the variation in run time to get a sense if the difference is (statistically) significant. This is the code I used:
Code: Select all
-- compute mean and standard deviation from sample
local function mean_std(t)
assert(#t >= 2)
local mean, scatter = 0, 0
for n,x in ipairs(t) do
local d = x - mean
mean = mean + d/n
scatter = scatter + d * (x - mean)
end
return mean, math.sqrt(scatter / (#t - 1))
end
-- take N samples of K runs of f().
local function profile(f, N, K)
local t = {}
for n = 1,N or 10000 do
local t0 = love.timer.getTime()
for k = 1,K or 100 do
f()
end
t[#t+1] = (love.timer.getTime() - t0) / (K or 100)
end
return t
end
-- first run to warm up the JIT compiler
mean_std(profile(math.random))
mean_std(profile(love.math.random))
-- actual measurement run
m1,s1 = mean_std(profile(math.random), 1000000, 10000)
m2,s2 = mean_std(profile(love.math.random), 1000000, 10000)
local ns = 1e9 -- one second is 1000000000 nanoseconds
print("libc:", ("%.3f ns ± %.3f ns"):format(m1*ns, s1*ns))
print("love:", ("%.3f ns ± %.3f ns"):format(m2*ns, s2*ns))
love.event.quit()
And this is one result (you may get other numbers, even across runs):
Code: Select all
libc: 8.958 ns ± 2.154 ns
love: 14.195 ns ± 2.300 ns
What does that tell us? Well, your numbers are most probably correct: 2621440 runs take, on average, 25ms for math.random() or 35ms for love.math.random(). The difference also seems to be significant.
But: One evaluation of either function takes less than 20 nanoseconds. Nanoseconds! Given the benefits of love's PRNG (that is: consistency over OSes and better statistical properties), it seems as if this is not the right place to start optimizing your code.
I have come here to chew bubblegum and kick ass... and I'm all out of bubblegum.
hump | HC | SUIT | moonshine
### Who is online
Users browsing this forum: Bing [Bot], Google [Bot] and 7 guests | 2019-12-10 11:03:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.354235976934433, "perplexity": 13868.14837664244}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540527205.81/warc/CC-MAIN-20191210095118-20191210123118-00032.warc.gz"} |
https://animalshelterz.com/how-tall-is-the-pitbull-hulk/ | # How tall is the pitbull Hulk?
Table of Contents
## How tall is the pitbull Hulk?
Hulk The Pitbull Weighs 174 Pounds And Is 6 Feet Tall.
## How is the Hulk Pitbull so big?
Hulk, the giant dog, is believed by some to be a cross between an American bulldog and a American pit bull terrier. Not even fully grown yet at 22 months, Hulk clocked in at 175 pounds. Hulk fathers puppies who will grow into enormous dogs so his owner can rake in the cash.
## How Big Is Hulk The pitbull 2020?
The public debut of Hulk, whose owners claim he is the world’s largest Pit Bull, has caused a great deal of controversy. The 174-pound dog sparked debate over backyard breeding, the safety of Pit Bulls as a breed, and the ethics of breeding dogs (in any capacity) for size and aggression.
## How old is Hulk the pitbull?
He is 8 years old as of this year, and unknown is his real name.
Biography: Who is Hulk Dog?
First and last name: Hulk Dog
Date of birth: September 7, 2013
Birthplace: Florida
Location: Florida
Current Age: 8
## What are the tallest pitbulls?
One of the tallest pit bull breeds is the American bulldog. Males of this breed range in height from 22 to 27 inches, with females ranging from 20 to 25 inches in height. The second tallest is the bull terrier.
## Is Kong bigger than Hulk?
Is Kong Bigger than Hulk? Kong isn’t bigger than the Hulk yet, but he looks as though he will be in the future. Currently, Kong weighs 160 lbs which is still 15 lbs less than what Hulk weighs.
## Did Hulk The pitbull die 2020?
When did Hulk the Dog Die? Contrary to what many people believe, Hulk is still alive and healthy as of May 2020. He still trains at the DDK9’s kennel and still fathers a lot of pups. Ace died when two other DDK9 dogs attacked him in a contest for a female.
## How did Hulk the pitbull died?
He was attacked and fatally wounded by two much larger, younger dogs on June 26, 2018, at the company’s New Hampshire HQ after accidentally being let out at the same time as both of them and, crucially, a lone female.
## Can pit bulls weigh 100 pounds?
Pit Bull Size
In addition, there are people who claim to have a 100-pound pit bull. The two breeds of Terrier and Bull Dog have never come in weighing 100 pounds at all. A Pit is a medium size dog that isn’t in the large dog club.
## How much is the pitbull Hulk worth?
Hulk the famous pitbull guard dog is now a dad. The 175-pound dog, who is worth a whopping $500,00 thanks to his size and guarding skills, recently welcomed a litter of eight puppies, which is estimated at another$500,000 combined.
## Does Hulk ever die?
Although Marvel recently revealed that the Hulk is, in a way, immortal, he is still able to and has died multiple times throughout the comics. The Hulk is arguably one of the strongest characters in the Marvel universe. Regardless of his stipulated immortality, he can (and has) in fact still die.
## What does Hulk Pitbull eat?
Probably the most important question regarding Hulk is “What do you feed a 175 pound pit bull?” The answer: 4 pounds of ground beef. Daily. That’s right. Hulk takes in 4 pounds of raw ground beef every single day.
## How Much Does Hulk The pitbull weigh 2021?
The massive dog looks scary, but he’s sweet with his owners’ 3-year-old son. His enormous size made him go viral, but his owners say Hulk is a gentle giant with their 3-year-old son. — — This dog just may be the world’s largest Pit Bull. Only 18-months-old, Hulk weighs a hefty 175 pounds.
## Is there a pitbull bigger than Hulk?
Another dog that is bigger than the hulk is my good friend Gina from Rhode Island. Azore one of the biggest bully pitbulls in the breeding world has been videoed with being weighed on a scale at around 195 pounds.
## Can a pitbull kill a lion?
Personally I think no dog can kill a lion alone. But yes we have few dog breeds like rotwiller,pit bull and bhutia dog.. they can fight with lion but alone they can’t win.
## How heavy can a pitbull get?
Pit bull/Mass
Related posts On Pitbull : | 2023-02-08 00:49:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19004833698272705, "perplexity": 9080.145811888355}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500664.85/warc/CC-MAIN-20230207233330-20230208023330-00733.warc.gz"} |
https://gamedev.stackexchange.com/questions/98873/turn-the-front-wheels-of-a-car-according-to-the-cars-bezier-path | # Turn the front wheels of a car according to the car's bezier path
I have a 3D car which follows a predefined 3D Bezier path. I want the car's front wheels' rotation to match the car's changing direction.
I had the idea to match the wheel's orientation to the derivative of the path's direction (3D vector), aka the 2nd degree derivative of the Bezier path.
For some reason, this barely works. At some point it seems to work fine, while at others the wheel spins like hell. I noted that the 2nd degree derivative changes even when the Bezier path is a straight line: AFAIK in this case it should be 0.
So, my 1st question is if my idea to match the wheel's rotation to the 2nd degree is the way to go. If yes, my 2nd question is what on earth is going wrong?
Here is my Bezier 3D curve code:
package fanlib.math {
import flash.geom.Vector3D;
public class BezierCubic3D
{
public const anchor1:Vector3D = new Vector3D();
public const anchor2:Vector3D = new Vector3D();
public const control1:Vector3D = new Vector3D();
public const control2:Vector3D = new Vector3D();
/**
* Gets values from both 'getPointAt' and 'getDirectionAt'
*/
public const result:Vector3D = new Vector3D();
private const previous:Vector3D = new Vector3D(); // temporary (optimization)
// normalization aka arc-parameterization
public var arcLengths:Vector.<Number> = new Vector.<Number>;
public var steps:Number = 100;
private var _length:Number;
public function BezierCubic3D()
{
}
/**
* To get a point between anchor1 and anchor2, pass value [0...1]
* @param t
*/
public function getPointAt(t:Number):Vector3D {
const t2:Number = t*t;
const t3:Number = t*t2;
const threeT:Number = 3*t;
const threeT2:Number = 3*t2;
result.x = getPointAxisAt(anchor1.x, anchor2.x, control1.x, control2.x, t3, threeT, threeT2);
result.y = getPointAxisAt(anchor1.y, anchor2.y, control1.y, control2.y, t3, threeT, threeT2);
result.z = getPointAxisAt(anchor1.z, anchor2.z, control1.z, control2.z, t3, threeT, threeT2);
return result;
}
public function getPointAxisAt(a1:Number,a2:Number,c1:Number,c2:Number, t3:Number, threeT:Number, threeT2:Number):Number {
return t3 * (a2+3*(c1-c2)-a1) +
threeT2 * (a1-2*c1+c2) +
threeT * (c1-a1) +
a1;
}
/**
* @param t
* @return Un-normalized Vector3D!
*/
public function getDirectionAt(t:Number):Vector3D {
const threeT2:Number = 3 * t * t;
const sixT:Number = 6 * t;
result.x = getDirAxisAt(anchor1.x, anchor2.x, control1.x, control2.x, threeT2, sixT);
result.y = getDirAxisAt(anchor1.y, anchor2.y, control1.y, control2.y, threeT2, sixT);
result.z = getDirAxisAt(anchor1.z, anchor2.z, control1.z, control2.z, threeT2, sixT);
return result;
}
public function getDirAxisAt(a1:Number,a2:Number,c1:Number,c2:Number, threeT2:Number, sixT:Number):Number {
return threeT2 * (a2+3*(c1-c2)-a1) +
sixT * (a1-2*c1+c2) +
3 * (c1-a1);
}
public function getDirectionDerivativeAt(t:Number):Vector3D {
const sixT:Number = 6 * t;
result.x = getDirDerAxisAt(anchor1.x, anchor2.x, control1.x, control2.x, sixT);
result.y = getDirDerAxisAt(anchor1.y, anchor2.y, control1.y, control2.y, sixT);
result.z = getDirDerAxisAt(anchor1.z, anchor2.z, control1.z, control2.z, sixT);
return result;
}
public function getDirDerAxisAt(a1:Number,a2:Number,c1:Number,c2:Number, sixT:Number):Number {
return sixT * (a2+3*(c1-c2)-a1) +
6 * (a1-2*c1+c2);
}
/**
* Call this after any change to defining points and before accessing normalized points of curve.
*/
public function recalc():void {
arcLengths.length = steps + 1;
arcLengths[0] = 0;
const step:Number = 1 / steps;
previous.copyFrom(getPointAt(0));
_length = 0;
for (var i:int = 1; i <= steps; ++i) {
_length += Vector3D.distance(getPointAt(i * step), previous);
arcLengths[i] = _length;
previous.copyFrom(result);
}
}
/**
* 'recalc' must have already been called if any changes were made to any of the defining points
* @param u
* @return u normalized/converted to t
*/
public function normalizeT(u:Number):Number {
var targetLength:Number = u * arcLengths[steps];
var low:int = 0,
high:int = steps,
index:int; // TODO : have a look-up table of starting low/high indices for each step!
while (low < high) {
index = low + ((high - low) >>> 1);
if (arcLengths[index] < targetLength) {
low = index + 1;
} else {
high = index;
}
}
if (this.arcLengths[index] > targetLength) {
--index;
}
var lengthBefore:Number = arcLengths[index];
if (lengthBefore === targetLength) {
return index / steps;
} else {
return (index + (targetLength - lengthBefore) / (arcLengths[index + 1] - lengthBefore)) / steps;
}
}
public function getNormalizedPointAt(u:Number):Vector3D {
return getPointAt(normalizeT(u));
}
/**
* "Normalized" goes for t, not the return Vector3D!!!
* @param u
* @return Un-normalized Vector3D!
*/
public function getNormalizedDirectionAt(u:Number):Vector3D {
return getDirectionAt(normalizeT(u));
}
public function getNormalizedDirectionDerivativeAt(u:Number):Vector3D {
return getDirectionDerivativeAt(normalizeT(u));
}
public function get length():Number
{
return _length;
}
}
}
And here is the code that applies the 2nd degree derivative orientation to the car's wheels:
const dirDer:Vector3D = bezier.getDirectionDerivativeAt(time);
dirDer.negate(); // negate vector's values; for some reason, this gives better results
for each (wheel in dirWheels) {
wheel.setRotation(0,0,0); // must nullify before below line
const localDirDer:Vector3D = wheel.globalToLocalVector(dirDer); // convert dirDer vector to wheel's local axis; wheel translation does NOT affect conversion
wheel.setOrientation(localDirDer); // orients the object in a specific direction; Up-vector's default value = (0,1,0)
}
I even tried (to no avail):
for each (wheel in dirWheels) {
const localDirDer:Vector3D = wheel.parent.globalToLocalVector(dirDer); // convert dirDer vector to wheel's local axis; wheel translation does NOT affect conversion
wheel.setOrientation(localDirDer); // orients the object in a specific direction; Up-vector's default value = (0,1,0)
}
One clear example that something is wrong: even when the car is on a straight line, the wheel originally is non-rotated (as it should), but after the car passes the center point of the line, the wheel rotates 180 degrees!
I haven't digested all of your code. Consider the effect of using the center of the rear axle as the vehicle's origin and have that point ride the actual Bezier. As such, the car's forward direction is coincident with the tangent. Both of these relate to the path of a real car. The front wheels just need to always point directly toward their next-frame-location (t+0.0001).
If the Bezier is too "tight", the back wheels "break lose" from the pavement and the same effect occurs at the center of the front axle, instead of the back.
Edit: Added control points/lines in red, as well as, first- and second-order derivatives for t=0.5. The car/wheel locations for t=0.5 are in blue.
When interpolating the car's location between magenta and cyan, also lerp the wheel rotation between magenta and cyan. The positive difference in 't' between magenta and cyan is arbitrary, since it will be updated every frame.
Edit2: I use an AutoCAD addon called AutoTurn to design driveways that can accommodate fire-truck and/or semi-truck turn radii; it makes similar paths to the one shown below, in yellow. It is an extremely specific path based on the vehicle's lock-angle, wheel-base, etc., etc. It is not even remotely parametric.
Interpolating using the direction between (normalizedT) and (normalizedT + 0.0001) should look great.
• Yes, that's what I thought. I thought of using 2nd degree derivative for more accurate results (instead of approximating with next-frame-location (t+0.0001)). Maybe I have a bug there with my 2nd degree der function, but I can't see it. Or maybe I need to combine 1st and 2nd degree derivatives..? Dunno – Bill Kotsias Apr 24 '15 at 6:13
• @BillKotsias, aren't you interpolating movement along the Bezier anyway, and by deltaTime (probably ~0.01666)? – Jon Apr 24 '15 at 6:42
• Interpolating, no. I get the Bezier's exact position over time with getPointAt and assign that directly to the car's position. I don't use deltaTime but exact time values. – Bill Kotsias Apr 24 '15 at 6:47
• @BillKotsias, do you suspect that the error is somewhere within the basis and/or derivative calculation? – Jon Apr 24 '15 at 7:15
• After thinking about it, no, there can't be an error in the derivative calculation. I think my concept is wrong: the 2nd degree der shows how the direction changes, NOT where it goes. I think I should add the 2nd degree to the 1st degree to get the tires' direction. In the end, if I can't find out how to fix it, I'll just do it exactly as you describe! :-) – Bill Kotsias Apr 24 '15 at 8:11 | 2019-07-20 15:51:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4441238045692444, "perplexity": 9411.384901517627}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526536.46/warc/CC-MAIN-20190720153215-20190720175215-00506.warc.gz"} |
http://cms.math.ca/cjm/kw/complex%20manifolds | Effective Actions of the Unitary Group on Complex Manifolds We classify all connected $n$-dimensional complex manifolds admitting effective actions of the unitary group $U_n$ by biholomorphic transformations. One consequence of this classification is a characterization of $\CC^n$ by its automorphism group. Keywords:complex manifolds, group actionsCategories:32Q57, 32M17 | 2015-05-27 05:55:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8512712121009827, "perplexity": 412.13153334049855}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207928907.65/warc/CC-MAIN-20150521113208-00139-ip-10-180-206-219.ec2.internal.warc.gz"} |
http://experiment-ufa.ru/Prime-factorization-of-128 | # Prime factorization of 128
If it's not what You are looking for type in the field below your own integer, and You will get the solution.
Prime factorization of 128:
By prime factorization of 128 we follow 5 simple steps:
1. We write number 128 above a 2-column table
2. We divide 128 by the smallest possible prime factor
3. We write down on the left side of the table the prime factor and next number to factorize on the ride side
4. We continue to factor in this fashion (we deal with odd numbers by trying small prime factors)
5. We continue until we reach 1 on the ride side of the table
128 prime factors number to factorize 2 64 2 32 2 16 2 8 2 4 2 2 2 1
Prime factorization of 128 = 1×2×2×2×2×2×2×2= $1 × 2^7$
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http://en.wikipedia.org/wiki/Newman-Penrose_formalism | # Newman–Penrose formalism
(Redirected from Newman-Penrose formalism)
The Newman–Penrose (NP) formalism[1][2] is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism,[3] where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the space-time, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The most often-used variables in the formalism are the Weyl scalars, derived from the Weyl tensor. In particular, it can be shown that one of these scalars--$\Psi_4$ in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.[4]
Newman and Penrose introduced the following functions as primary quantities using this tetrad:[1][2]
• Twelve complex spin coefficients (in three groups) which describe the change in the tetrad from point to point: $\kappa, \rho, \sigma, \tau\,; \lambda, \mu, \nu, \pi\,; \epsilon, \gamma, \beta, \alpha.$.
• Five complex functions encoding Weyl tensors in the tetrad basis: $\Psi_0, \ldots, \Psi_4$.
• Ten functions encoding Ricci tensors in the tetrad basis: $\Phi_{00}, \Phi_{11}, \Phi_{22}, \Lambda$ (real); $\Phi_{01}, \Phi_{10}, \Phi_{02}, \Phi_{20}, \Phi_{12}, \Phi_{21}$ (complex).
In many situations—especially algebraically special spacetimes or vacuum spacetimes—the Newman–Penrose formalism simplifies dramatically, as many of the functions go to zero. This simplification allows for various theorems to be proven more easily than using the standard form of Einstein's equations.
In this article, we will only employ the tensorial rather than spinorial version of NP formalism, because the former is easier to understand and more popular in relevant papers. One can refer to ref.[5] for a unified formulation of these two versions.
## Null tetrad and sign convention
The formalism is developed for four-dimensional spacetime, with a Lorentzian-signature metric. At each point, a tetrad (set of four vectors) is introduced. The first two vectors, $l^\mu$ and $n^\mu$ are just a pair of standard (real) null vectors such that $l^a n_a = -1$. For example, we can think in terms of spherical coordinates, and take $l^a$ to be the outgoing null vector, and $n^a$ to be the ingoing null vector. A complex null vector is then constructed by combining a pair of real, orthogonal unit space-like vectors. In the case of spherical coordinates, the standard choice is
$m^\mu = \frac{1}{\sqrt{2}}\left( \hat{\theta} + i \hat{\phi} \right)^\mu\ .$
The complex conjugate of this vector then forms the fourth element of the tetrad.
Two sets of signature and normalization conventions are in use for NP formalism: $\{(+,-,-,-); l^a n_a=1\,,m^a \bar{m}_a=-1\}$ and $\{(-,+,+,+); l^a n_a=-1\,,m^a \bar{m}_a=1\}$. The former is the original one that was adopted when NP formalism was developed[1][2] and has been widely used[6][7] in black-hole physics, gravitational waves and various other areas in general relativity. However, it is the latter convention that is usually employed in contemporary study of black holes from quasilocal perspectives[8] (such as isolated horizons[9] and dynamical horizons[10][11]). In this article, we will utilize $\{(-,+,+,+); l^a n_a=-1\,,m^a \bar{m}_a=1\}$ for a systematic review of the NP formalism (see also refs.[12][13][14]).
It's important to note that, when switching from $\{(+,-,-,-)\,,l^a n_a=1\,,m^a \bar{m}_a=-1\}$ to $\{(-,+,+,+)\,,l^a n_a=-1\,,m^a \bar{m}_a=1\}$, definitions of the spin coefficients, Weyl-NP scalars $\Psi_{i}$ and Ricci-NP scalars $\Phi_{ij}$ need to change their signs; this way, the Einstein-Maxwell equations can be left unchanged.
In NP formalism, the complex null tetrad contains two real null (co)vectors $\{\ell\,,n\}$ and two complex null (co)vectors $\{m\,, \bar m\}$. Being null (co)vectors, self-normalization of $\{\ell\,,n\}$ are naturally vanishes,
$l_a l^a=n_a n^a=m_a m^a=\bar{m}_a \bar{m}^a=0$,
so the following two pairs of cross-normalization are adopted
$l_a n^a=-1=l^a n_a\,,\quad m_a \bar{m}^a=1=m^a \bar{m}_a\,,$
while contractions between the two pairs are also vanishing,
$l_a m^a=l_a \bar{m}^a=n_a m^a=n_a \bar{m}^a=0$.
Here the indices can be raised and lowered by the global metric $g_{ab}$ which in turn can be obtained via
$g_{ab}=-l_a n_b - n_a l_b +m_a \bar{m}_b +\bar{m}_a m_b\,, \quad g^{ab}=-l^a n^b - n^a l^b +m^a \bar{m}^b +\bar{m}^a m^b\,.$
## NP Quantities and Tetrad Equations
### Four directional derivatives
First of all, there are four directional covariant derivatives along with each tetrad vector,
$D:= \nabla_\ell=l^a\nabla_a\,,\; \Delta:= \nabla_\mathbf{n}=n^a\nabla_a\,, \;\delta := \nabla_\mathbf{m}=m^a\nabla_a\,, \;\bar{\delta} := \nabla_\mathbf{\bar{m}}=\bar{m}^a\nabla_a\,,$
which are reduced to $\{D=l^a\partial_a\,, \Delta=n^a\partial_a\,,\delta=m^a\partial_a\,,\bar{\delta}=\bar{m}^a\partial_a \}$ when acting on scalar functions.
### Twelve spin coefficients
In NP formalism, instead of using index notations as in orthogonal tetrads, each Ricci rotation coefficient $\gamma_{ijk}$ in the null tetrad is assigned a lower-case Greek letter, which constitute the 12 complex spin coefficients (in three groups),
$\kappa:= -m^aDl_a=-m^a l^b \nabla_b l_a\,,\quad \tau:= -m^a\Delta l_a=-m^a n^b \nabla_b l_a\,,$
$\sigma:= -m^a\delta l_a=-m^a m^b\nabla_b l_a\,, \quad \rho := -m^a\bar{\delta} l_a=-m^a \bar{m}^b \nabla_b l_a\,;$
$\pi:= \bar{m}^aDn_a=\bar{m}^al^b\nabla_b n_a\,, \quad \nu:= \bar{m}^a\Delta n_a=\bar{m}^a n^b\nabla_b n_a\,,$
$\mu:= \bar{m}^a\delta n_a=\bar{m}^a m^b\nabla_b n_a\,, \quad \lambda:= \bar{m}^a\bar{\delta} n_a=\bar{m}^a \bar{m}^b \nabla_b n_a\,;$
$\varepsilon:= -\frac{1}{2}\big(n^aDl_a-\bar{m}^aDm_a \big)=-\frac{1}{2}\big(n^al^b\nabla_b l_a-\bar{m}^al^b\nabla_b m_a \big)\,,$
$\gamma:= -\frac{1}{2}\big(n^a\Delta l_a-\bar{m}^a\Delta m_a \big)= -\frac{1}{2}\big(n^a n^b\nabla_b l_a-\bar{m}^a n^b\nabla_b m_a \big)\,,$
$\beta:= -\frac{1}{2}\big(n^a\delta l_a-\bar{m}^a\delta m_a \big)=-\frac{1}{2}\big(n^a m^b\nabla_b l_a-\bar{m}^am^b\nabla_b m_a \big)\,,$
$\alpha:= -\frac{1}{2}\big(n^a\bar{\delta} l_a-\bar{m}^a\bar{\delta}m_a \big)=-\frac{1}{2}\big(n^a\bar{m}^b\nabla_b l_a-\bar{m}^a\bar{m}^b\nabla_b m_a \big)\,.$
Spin coefficients are the primary quantities in NP formalism, with which all other NP quantities (as defined below) could be calculated indirectly using the NP field equations. Thus, NP formalism is sometimes referred to as spin-coefficient formalism as well.
### Transportation equations
Apply of the directional derivative operators to tetrad vectors and one could obtain the transportation/propagation equations:[5][13]
$D l^a=(\varepsilon+\bar{\varepsilon})l^a-\bar{\kappa}m^a-\kappa\bar{m}^a\,,$
$\Delta l^a=(\gamma+\bar{\gamma})l^a-\bar{\tau}m^a-\tau\bar{m}^a\,,$
$\delta l^a =(\bar{\alpha}+\beta)l^a-\bar{\rho}m^a-\sigma\bar{m}^a\,,$
$\bar{\delta} l^a=(\alpha+\bar{\beta})l^a-\bar{\sigma}m^a-\rho\bar{m}^a\,;$
$D n^a=\pi m^a+\bar{\pi}\bar{m}^a-(\varepsilon+\bar{\varepsilon})n^a\,,$
$\Delta n^a=\nu m^a+\bar{\nu}\bar{m}^a-(\gamma+\bar{\gamma})n^a\,,$
$\delta n^a=\mu m^a+\bar{\lambda}\bar{m}^a-(\bar{\alpha}+\beta)n^a\,,$
$\bar{\delta} n^a=\lambda m^a+\bar{\mu}\bar{m}^a-(\alpha+\bar{\beta})n^a\,;$
$D m^a=(\varepsilon-\bar{\varepsilon})m^a+\bar{\pi}l^a-\kappa n^a\,,$
$\Delta m^a=(\gamma-\bar{\gamma})m^a+\bar{\nu}l^a-\tau n^a\,,$
$\delta m^a=(\beta-\bar{\alpha})m^a+\bar{\lambda}l^a-\sigma n^a\,,$
$\bar{\delta} m^a=(\alpha-\bar{\beta})m^a+\bar{\mu}l^a-\rho n^a\,;$
$D m^a=(\bar{\varepsilon}-\varepsilon)m^a+\pi l^a-\bar{\kappa} n^a\,,$
$\Delta m^a=(\gamma-\bar{\gamma})m^a+\nu l^a-\bar{\tau} n^a\,,$
$\delta m^a=(\beta-\bar{\alpha})m^a+\mu l^a-\bar{\rho} n^a\,,$
$\bar{\delta} m^a=(\alpha-\bar{\beta})m^a+\lambda l^a-\bar{\sigma} n^a\,.$
### Commutators
The metric-compatibility or torsion-freeness of the covariant derivative is recast into the commutators of the directional derivatives,
$\Delta D-D\Delta=(\gamma+\bar{\gamma})D+(\varepsilon+\bar{\varepsilon})\Delta-(\bar{\tau}+\pi)\delta-(\tau+\bar{\pi})\bar{\delta}\,,$
$\delta D-D\delta=(\bar{\alpha}+\beta-\bar{\pi})D+\kappa\Delta-(\bar{\rho}+\varepsilon-\bar{\varepsilon})\delta-\sigma\bar{\delta}\,,$
$\delta\Delta-\Delta\delta=-\bar{\nu}D+(\tau-\bar{\alpha}-\beta)\Delta+(\mu-\gamma+\bar{\gamma})\delta+\bar{\lambda}\bar{\delta}\,,$
$\bar{\delta}\delta-\delta\bar{\delta}=(\bar{\mu}-\mu)D+(\bar{\rho}-\rho)\Delta+(\alpha-\bar{\beta})\delta-(\bar{\alpha}-\beta)\bar{\delta}\,,$
which imply that
$\Delta l^a-D n^a=(\gamma+\bar{\gamma})l^a+(\varepsilon+\bar{\varepsilon})n^a-(\bar{\tau}+\pi)m^a-(\tau+\bar{\pi})\bar{m}^a\,,$
$\delta l^a-D m^a=(\bar{\alpha}+\beta-\bar{\pi})l^a+\kappa n^a-(\bar{\rho}+\varepsilon-\bar{\varepsilon}) m^a-\sigma\bar{m}^a\,,$
$\delta n^a-\Delta m^a=-\bar{\nu}l^a+(\tau-\bar{\alpha}-\beta)n^a+(\mu-\gamma+\bar{\gamma})m^a+\bar{\lambda}\bar{m}^a\,,$
$\bar{\delta}m^a-\delta\bar{m}^a=(\bar{\mu}-\mu)l^a+(\bar{\rho}-\rho)n^a+(\alpha-\bar{\beta})m^a-(\bar{\alpha}-\beta)\bar{m}^a\,.$
Note: (i) The above equations can be regarded either as implications of the commutators or combinations of the transportation equations; (ii) In these implied equations, the vectors $\{l^a,n^a,m^a,\bar{m}^a\}$ can be replaced by the covectors and the equations still hold.
### Weyl-NP and Ricci-NP scalars
The 10 independent components of Weyl's tensor can be encoded into 5 complex Weyl-NP scalars,
$\Psi_0:= C_{abcd} l^a m^b l^c m^d\,,\quad \Psi_1:= C_{abcd} l^a n^b l^c m^d\,,\quad \Psi_2:= C_{abcd} l^a m^b\bar{m}^c n^d\,,\quad \Psi_3:= C_{abcd} l^a n^b\bar{m}^c n^d\,,\quad \Psi_4:= C_{abcd} n^a \bar{m}^b n^c \bar{m}^d\,.$
The 10 independent components of the Ricci tensor are encoded into 4 real scalars $\{\Phi_{00}$, $\Phi_{11}$, $\Phi_{22}$, $\Lambda\}$ and 3 complex scalars $\{\Phi_{10},\Phi_{20},\Phi_{21} \}$ (with their complex conjugates),
$\Phi_{00}:=\frac{1}{2}R_{ab}l^a l^b\,, \quad \Phi_{11}:=\frac{1}{4}R_{ab}(\,l^a n^b+m^a\bar{m}^b)\,, \quad\Phi_{22}:=\frac{1}{2}R_{ab}n^a n^b\,, \quad\Lambda:=\frac{R}{24}\,;$
$\Phi_{01}:=\frac{1}{2}R_{ab}l^a m^b\,, \quad\; \Phi_{10}:=\frac{1}{2}R_{ab}l^a \bar{m}^b=\overline{\Phi_{01}}\,,$
$\Phi_{02}:=\frac{1}{2}R_{ab}m^a m^b\,, \quad \Phi_{20}:=\frac{1}{2}R_{ab}\bar{m}^a \bar{m}^b=\overline{\Phi_{02}}\,,$
$\Phi_{12}:=\frac{1}{2}R_{ab} m^a n^b\,, \quad\; \Phi_{21}:=\frac{1}{2}R_{ab} \bar{m}^a n^b=\overline{\Phi_{12}}\,.$
In these definitions, $R_{ab}$ could be replaced by its trace-free part $\displaystyle Q_{ab}=R_{ab}-\frac{1}{4}g_{ab}R$[13] or by the Einstein tensor $\displaystyle G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R$ because of the normalization relations. Also, $\Phi_{11}$ is reduced to $\Phi_{11}=\frac{1}{2}R_{ab}l^a n^b=\frac{1}{2}R_{ab}m^a\bar{m}^a$ for electrovacuum ($\Lambda=0$).
## Einstein-Maxwell-NP Equations
### NP field equations
In a complex null tetrad, Ricci identities give rise to the following NP field equations connecting spin coefficients, Weyl-NP and Ricci-NP scalars (recall that in an orthogonal tetrad, Ricci rotation coefficients would respect Cartan's first and second structure equations),[5][13]
$D\rho -\bar{\delta}\kappa=(\rho^2+\sigma\bar{\sigma})+(\varepsilon+\bar{\varepsilon})\rho-\bar{\kappa}\tau-\kappa(3\alpha+\bar{\beta}-\pi)+\Phi_{00}\,,$
$D\sigma-\delta\kappa=(\rho+\bar{\rho})\sigma+(3\varepsilon-\bar{\varepsilon})\sigma-(\tau-\bar{\pi}+\bar{\alpha}+3\beta)\kappa+\Psi_0\,,$
$D\tau-\Delta\kappa=(\tau+\bar{\pi})\rho+(\bar{\tau}+\pi)\sigma+(\varepsilon-\bar{\varepsilon})\tau-(3\gamma+\bar{\gamma})\kappa+\Psi_1+\Phi_{01}\,,$
$D\alpha-\bar{\delta}\varepsilon=(\rho+\bar{\varepsilon}-2\varepsilon)\alpha+\beta\bar{\sigma}-\bar{\beta}\varepsilon-\kappa\lambda-\bar{\kappa}\gamma+(\varepsilon+\rho)\pi+\Phi_{10}\,,$
$D\beta-\delta\varepsilon=(\alpha+\pi)\sigma+(\bar{\rho}-\bar{\varepsilon})\beta-(\mu+\gamma)\kappa-(\bar{\alpha}-\bar{\pi})\varepsilon+\Psi_1\,,$
$D\gamma-\Delta\varepsilon=(\tau+\bar{\pi})\alpha+(\bar{\tau}+\pi)\beta-(\varepsilon+\bar{\varepsilon})\gamma-(\gamma+\bar{\gamma})\varepsilon+\tau\pi-\nu\kappa+\Psi_2+\Phi_{11}-\Lambda\,,$
$D\lambda-\bar{\delta}\pi=(\rho\lambda+\bar{\sigma}\mu)+\pi^2+(\alpha-\bar{\beta})\pi-\nu\bar{\kappa}-(3\varepsilon-\bar{\varepsilon})\lambda+\Phi_{20}\,,$
$D\mu-\delta\pi=(\bar{\rho}\mu+\sigma\lambda)+\pi\bar{\pi}-(\varepsilon+\bar{\varepsilon})\mu-(\bar{\alpha}-\beta)\pi-\nu\kappa+\Psi_2+2\Lambda\,,$
$D\nu-\Delta\pi=(\pi+\bar{\tau})\mu+(\bar{\pi}+\tau)\lambda+(\gamma-\bar{\gamma})\pi-(3\varepsilon+\bar{\varepsilon})\nu+\Psi_3+\Phi_{21}\,,$
$\Delta\lambda-\bar{\delta}\nu=-(\mu+\bar{\mu})\lambda-(3\gamma-\bar{\gamma})\lambda+(3\alpha+\bar{\beta}+\pi-\bar{\tau})\nu-\Psi_4\,,$
$\delta\rho-\bar{\delta}\sigma=\rho(\bar{\alpha}+\beta)-\sigma(3\alpha-\bar{\beta})+(\rho-\bar{\rho})\tau+(\mu-\bar{\mu})\kappa-\Psi_1+\Phi_{01}\,,$
$\delta\alpha-\bar{\delta}\beta=(\mu\rho-\lambda\sigma)+\alpha\bar{\alpha}+\beta\bar{\beta}-2\alpha\beta+\gamma(\rho-\bar{\rho})+\varepsilon(\mu-\bar{\mu})-\Psi_2+\Phi_{11}+\Lambda\,,$
$\delta\lambda-\bar{\delta}\mu=(\rho-\bar{\rho})\nu+(\mu-\bar{\mu})\pi+(\alpha+\bar{\beta})\mu+(\bar\alpha-3\beta)\lambda-\Psi_3+\Phi_{21}\,,$
$\delta\nu-\Delta\mu=(\mu^2+\lambda\bar{\lambda})+(\gamma+\bar{\gamma})\mu-\bar{\nu}\pi+(\tau-3\beta-\bar{\alpha})\nu+\Phi_{22}\,,$
$\delta\gamma-\Delta\beta=(\tau-\bar{\alpha}-\beta)\gamma+\mu\tau-\sigma\nu-\varepsilon\bar{\nu}-(\gamma-\bar{\gamma}-\mu)\beta+\alpha\bar{\lambda}+\Phi_{12}\,,$
$\delta\tau-\Delta\sigma=(\mu\sigma+\bar{\lambda}\rho)+(\tau+\beta-\bar{\alpha})\tau-(3\gamma-\bar{\gamma})\sigma-\kappa\bar{\nu}+\Phi_{02}\,,$
$\Delta\rho-\bar{\delta}\tau=-(\rho\bar{\mu}+\sigma\lambda)+(\bar{\beta}-\alpha-\bar{\tau})\tau+(\gamma+\bar{\gamma})\rho+\nu\kappa-\Psi_2-2\Lambda\,,$
$\Delta\alpha-\bar{\delta}\gamma=(\rho+\varepsilon)\nu-(\tau+\beta)\lambda+(\bar{\gamma}-\bar{\mu})\alpha+(\bar{\beta}-\bar{\tau})\gamma-\Psi_3\,.$
Also, the Weyl-NP scalars $\Psi_{i}$ and the Ricci-NP scalars $\Phi_{ij}$ can be calculated indirectly from the above NP field equations after obtaining the spin coefficients rather than directly using their definitions.
### Maxwell-NP scalars, Maxwell equations in NP formalism
The six independent components of the Faraday-Maxwell 2-form (i.e. the electromagnetic field strength tensor) $F_{ab}$ can be encoded into three complex Maxwell-NP scalars[12]
$\phi_0:= -F_{ab}l^a m^b \,,\quad \phi_1:= -\frac{1}{2} F_{ab}\big(l^an^a-m^a\bar{m}^b \big)\,, \quad \phi_2 := F_{ab} n^a \bar{m}^b\,,$
and therefore the eight real Maxwell equations $d\mathbf{F}=0$ and $d^{\star}\mathbf{F}=0$ (as $\mathbf{F}=dA$) can be transformed into four complex equations,
$D\phi_1 -\bar{\delta}\phi_0=(\pi-2\alpha)\phi_0+2\rho\phi_1-\kappa\phi_2\,,$
$D\phi_2 -\bar{\delta}\phi_1=-\lambda\phi_0+2\pi\phi_1+(\rho-2\varepsilon)\phi_2\,,$
$\Delta\phi_0-\delta\phi_1=(2\gamma-\mu)\phi_0-2\tau\phi_1+\sigma\phi_2\,,$
$\Delta\phi_1-\delta\phi_2=\nu\phi_0-2\mu\phi_1+(2\beta-\tau)\phi_2\,,$
with the Ricci-NP scalars $\Phi_{ij}$ related to Maxwell scalars by[12]
$\Phi_{ij}=\,2\,\phi_i\,\overline{\phi_j}\,,\quad (i,j\in\{0,1,2\})\,.$
It is worthwhile to point out that, the supplementary equation $\Phi_{ij}=2\,\phi_i\, \overline{\phi_j}$ is only valid for electromagnetic fields; for example, in the case of Yang-Mills fields there will be $\Phi_{ij}=\,\text{Tr}\,(\digamma_i \,\bar{\digamma}_j)$ where $\digamma_i (i\in\{0,1,2 \})$ are Yang-Mills-NP scalars.[15]
To sum up, the aforementioned transportation equations, NP field equations and Maxwell-NP equations together constitute the Einstein-Maxwell equations in Newman–Penrose formalism.
## Applications of NP formalism to gravitational radiation field
The Weyl scalar $\Psi_4$ was defined by Newman & Penrose as
$\Psi_4 = -C_{\alpha\beta\gamma\delta} n^\alpha \bar{m}^\beta n^\gamma \bar{m}^\delta\$
(note, however, that the overall sign is arbitrary, and that Newman & Penrose worked with a "timelike" metric signature of $(+,-,-,-)$). In empty space, the Einstein Field Equations reduce to $R_{\alpha\beta}=0$. From the definition of the Weyl tensor, we see that this means that it equals the Riemann tensor, $C_{\alpha\beta\gamma\delta} = R_{\alpha\beta\gamma\delta}$. We can make the standard choice for the tetrad at infinity:
$l^{\mu} = \frac{1}{\sqrt{2}} \left( \hat{t} + \hat{r} \right)\ ,$
$n^{\mu} = \frac{1}{\sqrt{2}} \left( \hat{t} - \hat{r} \right)\ ,$
$m^{\mu} = \frac{1}{\sqrt{2}} \left( \hat{\theta} + i\hat{\phi} \right)\ .$
In transverse-traceless gauge, a simple calculation shows that linearized gravitational waves are related to components of the Riemann tensor as
$\frac{1}{4}\left( \ddot{h}_{\hat{\theta}\hat{\theta}} - \ddot{h}_{\hat{\phi}\hat{\phi}} \right) = -R_{\hat{t}\hat{\theta}\hat{t}\hat{\theta}} = -R_{\hat{t}\hat{\phi}\hat{r}\hat{\phi}} = -R_{\hat{r}\hat{\theta}\hat{r}\hat{\theta}} = R_{\hat{t}\hat{\phi}\hat{t}\hat{\phi}} = R_{\hat{t}\hat{\theta}\hat{r}\hat{\theta}} = R_{\hat{r}\hat{\phi}\hat{r}\hat{\phi}}\ ,$
$\frac{1}{2} \ddot{h}_{\hat{\theta}\hat{\phi}} = -R_{\hat{t}\hat{\theta}\hat{t}\hat{\phi}} = -R_{\hat{r}\hat{\theta}\hat{r}\hat{\phi}} = R_{\hat{t}\hat{\theta}\hat{r}\hat{\phi}} = R_{\hat{r}\hat{\theta}\hat{t}\hat{\phi}}\ ,$
assuming propagation in the $\hat{r}$ direction. Combining these, and using the definition of $\Psi_4$ above, we can write
$\Psi_4 = \frac{1}{2}\left( \ddot{h}_{\hat{\theta} \hat{\theta}} - \ddot{h}_{\hat{\phi} \hat{\phi}} \right) + i \ddot{h}_{\hat{\theta}\hat{\phi}} = -\ddot{h}_+ + i \ddot{h}_\times\ .$
Far from a source, in nearly flat space, the fields $h_+$ and $h_\times$ encode everything about gravitational radiation propagating in a given direction. Thus, we see that $\Psi_4$ encodes in a single complex field everything about (outgoing) gravitational waves.
### Radiation from a finite source
Using the wave-generation formalism summarised by Thorne,[16] we can write the radiation field quite compactly in terms of the mass multipole, current multipole, and spin-weighted spherical harmonics:
$\Psi_4(t,r,\theta,\phi) = - \frac{1}{r\sqrt{2}} \sum_{l=2}^{\infty} \sum_{m=-l}^l \left[ {}^{(l+2)}I^{lm}(t-r) -i\ {}^{(l+2)}S^{lm}(t-r) \right] {}_{-2}Y_{lm}(\theta,\phi)\ .$
Here, prefixed superscripts indicate time derivatives. That is, we define
${}^{(l)}G(t) = \left( \frac{d}{dt} \right)^l G(t)\ .$
The components $I^{lm}$ and $S^{lm}$ are the mass and current multipoles, respectively. ${}_{-2}Y_{lm}$ is the spin-weight -2 spherical harmonic.
## References
1. ^ a b c Ezra T. Newman and Roger Penrose (1962). "An Approach to Gravitational Radiation by a Method of Spin Coefficients". Journal of Mathematical Physics 3 (3): 566–768. Bibcode:1962JMP.....3..566N. doi:10.1063/1.1724257. The original paper by Newman and Penrose, which introduces the formalism, and uses it to derive example results.
2. ^ a b c Ezra T Newman, Roger Penrose. Errata: An Approach to Gravitational Radiation by a Method of Spin Coefficients. Journal of Mathematical Physics, 1963, 4(7): 998.
3. ^ Chandrasekhar, S. (1998). The Mathematical Theory of Black Holes (Reprinted ed.). Oxford University Press. p. 40. ISBN 0-19850370-9. Retrieved 13 May 2013. The Newman-Penrose formalism is a tetrad formalism with a special choice of the basis vectors.
4. ^ Saul Teukolsky (1973). "Perturbations of a rotating black hole". Astrophysical Journal 185: 635–647. Bibcode:1973ApJ...185..635T. doi:10.1086/152444.
5. ^ a b c Peter O'Donnell. Introduction to 2-Spinors in General Relativity. Singapore: World Scientific, 2003.
6. ^ Subrahmanyan Chandrasekhar. The Mathematical Theory of Black Holes. Chicago: University of Chikago Press, 1983.
7. ^ J B Griffiths. Colliding Plane Waves in General Relativity. Oxford: Oxford University Press, 1991.
8. ^ Ivan Booth. Black hole boundaries. Canadian Journal of Physics, 2005, 83(11): 1073-1099. [arxiv.org/abs/gr-qc/0508107 arXiv:gr-qc/0508107v2]
9. ^ Abhay Ashtekar, Christopher Beetle, Jerzy Lewandowski. Geometry of generic isolated horizons. Classical and Quantum Gravity, 2002, 19(6): 1195-1225. arXiv:gr-qc/0111067v2
10. ^ Abhay Ashtekar, Badri Krishnan. Dynamical horizons: energy, angular momentum, fluxes and balance laws. Physical Review Letters, 2002, 89(26): 261101. [arxiv.org/abs/gr-qc/0207080 arXiv:gr-qc/0207080v3]
11. ^ Abhay Ashtekar, Badri Krishnan. Dynamical horizons and their properties. Physical Review D, 2003, 68(10): 104030. [arxiv.org/abs/gr-qc/0308033 arXiv:gr-qc/0308033v4]
12. ^ a b c Jeremy Bransom Griffiths, Jiri Podolsky. Exact Space-Times in Einstein's General Relativity. Cambridge: Cambridge University Press, 2009. Chapter 2.
13. ^ a b c d Valeri P Frolov, Igor D Novikov. Black Hole Physics: Basic Concepts and New Developments. Berlin: Springer, 1998. Appendix E.
14. ^ Abhay Ashtekar, Stephen Fairhurst, Badri Krishnan. Isolated horizons: Hamiltonian evolution and the first law. Physical Review D, 2000, 62(10): 104025. Appendix B. gr-qc/0005083
15. ^ E T Newman, K P Tod. Asymptotically Flat Spacetimes, Appendix A.2. In A Held (Editor): General Relativity and Gravitation: One Hundred Years After the Birth of Albert Einstein. Vol(2), page 27. New York and London: Plenum Press, 1980.
16. ^ Thorne, Kip S. (April 1980). "Multipole expansions of gravitational radiation". Rev. Mod. Phys. 52 (2): 299–339. Bibcode:1980RvMP...52..299T. doi:10.1103/RevModPhys.52.299. A broad summary of the mathematical formalism used in the literature on gravitational radiation. | 2014-12-26 08:01:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 133, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8960230946540833, "perplexity": 1329.9173054389241}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1419447548655.55/warc/CC-MAIN-20141224185908-00087-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/1476159/how-to-find-the-inverse-of-an-upper-triangular-matrix | # How to find the inverse of an upper triangular matrix
I want to find the inverse of an upper triangular matrix in an efficient way. I googled a lot, but all the articles discussed about a lower triangular matrix.
Is it possible to edit the matlab code in this answer so that its suitable for an upper triangular matrix? https://stackoverflow.com/a/12240951/919177
• The transpose of an upper triangular matrix is lower triangular. This should help you. – J.R. Oct 12 '15 at 9:45
• I think matlab's backslash operator will automatically make use of the upper triangular structure. Why not just use backslash? But note that if you want to write your own solver, you can use back substitution to solve an upper triangular system. You know the last component of $x$ immediately, and that's a good start. – littleO Oct 12 '15 at 9:51
• I guess I will just use the transpose on the forward substitution code given in the link. Thanks everyone. – vipin Oct 12 '15 at 10:32
If you really want to find the inverse $M$ of an invertible upper triangular matrix $U$, note that $U M = I \implies M^T U^T = I$, which shows that $M^T$ is the inverse of the lower triangular matrix $U^T$.
So, you can find $M^T$ using the code you already have to invert a lower triangular matrix. This gives you $M$.
However, a rule of thumb is that you rarely want to compute the inverse of a matrix explicitly. If you ever need to solve $Ux = b$, you can just use back substitution. | 2019-10-18 02:46:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7648051381111145, "perplexity": 117.8641150238551}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986677412.35/warc/CC-MAIN-20191018005539-20191018033039-00469.warc.gz"} |
https://physics.stackexchange.com/questions/694070/thermodynamics-and-daltons-partial-pressure-law | # Thermodynamics and dalton's partial pressure law
By kinetic equation of ideal gas, $$PV=\frac{1}{3}mNu_{rms}^{2}$$ Where, $$P$$ is pressure $$m$$ is mass of molecule $$N$$ is number of moles considered and $$u_{rms}$$ is root mean speed. In my book to prove the Dalton's law of partial pressure, he considers the expression that $$p_{total}V=\text{sum of kinetic energies of all molecules}$$ What is the intuition behind it?
Similarly, see what I did but I get the wrong answer,
Let's take two equal volume($$V$$) containers with two different gases. Let's suppose that pressure,volume, temperature of both the gases are different. Then by kinetic equation for ideal gas we have for 1st container $$P_1V=\frac{1}{3}m_1N_1u_1^{2}=\frac{2N_1\beta T_1}{3}$$ For some constant $$\beta$$. Similarly for 2nd container we have, $$P_2V=\frac{1}{3}m_2N_2u_2^{2}=\frac{2N_2\beta T_2}{3}$$
Consider another container with the same volume and slowly mix these both gases from those containers into this new one and let's ay it's pressure now is $$P$$
Now when both the gases are mixed then according to law of thermodynamics we have that gases exchange their temperatures until equilibrium is reached. Let's calculate it as follows, $$\text{temperature of mixture of gases}=T_1-\Delta T_1=T_2+\Delta T_1=T$$ Where $$\Delta T_1$$ is the change in temperature due to equilibrium. Solving for final temperature via $$T_1$$ and $$T_2$$ gives, $$T=\frac{T_1+T_2}{2}$$
Is this all true?
(a) $$\sum \text{translational KEs of molecules} =N \times \tfrac12 m c_{rms}^2=\tfrac12 N m c_{rms}^2$$ But $$pV=\tfrac13 N m c_{rms}^2$$ So the correct relationship for an ideal gas is $$\text{Total translational KE} =\tfrac32pV$$
(b) Dalton's law states that the pressure exerted by a mixture of gases (in a container) is the sum of the pressures that would be exerted by each gas if it were the only gas in the container. So the law doesn't say anything about what happens when you mix gases at different temperatures. The gases that the law deals with are already mixed and therefore at the same temperature!
(c) When you talk of your textbook proving Dalton's law, you mean, I think, deriving it from the kinetic theory of ideal gases. You can do the derivation without considering the sum of molecules' kinetic energies. You simply extend the assumption of additivity of momentum changes due to molecules' impacts with the wall to include molecules of different species but with the same $$m c_{rms}^2$$.
(d) At the end of your question you assume that when you mix two gases at different temperatures the temperature rise of one gas is equal to the temperature fall of the other. This will not, in general, be true. Just think of what would happen if you mix 10 mole of a hot gas with 1 mole of a colder gas. It is, of course, the case that both gases will end up at the same temperature.
(e) How should one find an expression for the final pressure of equal volumes, $$V$$, of two gases at different temperatures and pressures when mixed together and confined to a volume, $$V$$?
You know that the number of moles is conserved. Hence, using the ideal gas equation, you can find a relationship involving just pressures and temperatures.
But the confinement won't take place spontaneously, as you suggest: "Consider another container with the same volume and slowly mix these both gases from those containers into this new one." You'll have to do work to squash 2 volumes $$V$$ of gas into volume $$V$$. This stops internal energy being conserved even if the mixing is adiabatic, so a nice easy equation is denied you.
But internal energy would be conserved if the final volume of the mixture were allowed to be $$2V$$, provided that the mixing were adiabatic. If you further assume that the molar heat capacity of the two gases is the same, you get a very simple result for the final pressure in terms of the initial pressures. | 2022-08-14 19:50:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 25, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8006290197372437, "perplexity": 197.60457259052944}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572063.65/warc/CC-MAIN-20220814173832-20220814203832-00163.warc.gz"} |