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https://stats.stackexchange.com/questions/244666/interpreting-estimates-of-a-bivariate-regression-model-with-a-categotical-and-a/244670 | # Interpreting estimates of a bivariate regression model with a categotical and a numeric variable
How to interpret the intercept in a bivariate regression with one numeric and one categorical variable?
The following model has a numeric variable (log10(N_Total_e)), namely the log transformed amount of nitrogen combined with a categorical variable divided in three levels of fertilizer class (NH4 only, NO3 only, NH4+NO3 combined). Dependent variable is species richness.
Multivariate Meta-Analysis Model (k = 211; method: REML)
logLik Deviance AIC BIC AICc
-33.0673 66.1345 76.1345 92.7981 76.4331
Variance Components:
estim sqrt nlvls fixed factor
sigma^2 0.0125 0.1116 73 no Primary_Study
Test for Residual Heterogeneity:
QE(df = 207) = 132.2270, p-val = 1.0000
Test of Moderators (coefficient(s) 2,3,4):
QM(df = 3) = 13.9606, p-val = 0.0030
Model Results:
estimate se zval pval ci.lb ci.ub
intrcpt 1.2338 0.1358 9.0846 <.0001 0.9676 1.5000 ***
log10(N_Total_e) -0.1938 0.0667 -2.9071 0.0036 -0.3245 -0.0632 **
F_typeNH4+NO3 -0.1328 0.0579 -2.2952 0.0217 -0.2463 -0.0194 *
F_typeNO3 -0.1221 0.1368 -0.8927 0.3720 -0.3903 0.1460
Is it right to say that the amount of nitrogen (log10(N_Total_e)) has in general a negative effect species richness (estimate = -0.19)? Should I interpret F_typeNH4+NO3 and F_typeNO3 estimates equals to 1.10 and 1.11 (namely the sum of the estimate with the intercept)? So specifying the type of fertilizer will always increase species richness when one is making predictions according to this model. Is this interpretation correct?
• Note that you can get an omnibus test of Ftype by using the btt parameter in rma.mv – mdewey Nov 7 '16 at 14:23
• You might also benefit from reading this Q&A stats.stackexchange.com/questions/242956/… especially @Wolfgang's suggestion about a three level model. – mdewey Nov 8 '16 at 14:26
Since you are using dummy coding to handle the categorical variable, your intercept and the regression coefficients for the dummy codes will reference the group you left out of the dummy coding (NH4 only). The intercept is the species richness with NH4 only and a nitrogen log value of $0.0$, and its p-value describes if this value is different from zero. The regression coefficient for each dummy code is the difference between that level of the categorical variable and the reference group (NH4 only).
You are correct that you always need to add in the intercept. So species richness with NH4 only and a nitrogen log value of $0.0$ is $1.2338$. And when the fertilizer type is N03 with a nitrogen log value of $0.0$, species richness equals $1.2338-0.1228=1.111$. For every log unit increase in nitrogen, species richness decreases by another $0.1938$. | 2019-10-20 07:22:55 | {"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": 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": 0.29705294966697693, "perplexity": 2834.978330455406}, "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/1570986703625.46/warc/CC-MAIN-20191020053545-20191020081045-00548.warc.gz"} |
https://learn.careers360.com/engineering/question-please-help-the-resistive-network-shown-below-is-connected-to-a-dc-source-of-16-v-the-power-consumed-by-the-network-is-4-watt-the-value-of-r-is/ | # The resistive network shown below is connected to a D.C. source of 16 V. The power consumed by the network is 4 Watt. The value of R is : Option 1) $8\Omega$ Option 2) $16\Omega$ Option 3) $1\Omega$ Option 4) $6\Omega$
Heat developed in a resistor -
When a steady current flows through a resistance R for time t , the loss in electric potential energy appears as increased thermal energy(Heat H) of resistor and $H=i^2Rt$
The power devoleped = $\frac{energy}{time}=i^2R=iR=\frac{V^2}{R}$ (from Ohm's law)
- wherein
Unit of heat is joule (J)
Unit of power is watt (W)
$P=\frac{v^{2}}{Req}$
$\Rightarrow Req=\frac{v^{2}}{P}=\frac{16^{2}}{4}=64\Omega$
$\Rightarrow 64=2R+R+4R+R$
$\Rightarrow 64=8R$
$\Rightarrow 8=R$
$\Rightarrow R=8\Omega$
Option 1)
$8\Omega$
Option 2)
$16\Omega$
Option 3)
$1\Omega$
Option 4)
$6\Omega$
Exams
Articles
Questions | 2020-05-25 02:56:23 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 16, "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": 0.9157089591026306, "perplexity": 3073.0141416657157}, "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-2020-24/segments/1590347387155.10/warc/CC-MAIN-20200525001747-20200525031747-00307.warc.gz"} |
https://www.electro-tech-online.com/threads/power-supply-for-our-jacobs-ladder-project.151068/ | # Power supply for our Jacob's Ladder project.
Status
Not open for further replies.
#### tcmtech
##### Banned
Standard neon sign transformer or fuel oil ignition transformer will work for that.
So will an ignition coil with the correct driver circuit.
#### dr pepper
##### Well-Known Member
Zvs flyback drivers are not hard to make if you can find an old Crt Tv as a donor.
#### DerStrom8
There is absolutely no reason to buy a Jacob's Ladder kit, especially not for $56.99! Jacob's Ladders are one of the easiest projects to make and you can generally make it at very little cost, if not free. I have made Jacob's Ladders with neon sign transformers, oil burner ignition transformers, automotive ignition coils, TV line output transformers, etc. Any of these will work great, though the last two requite additional driver circuitry in order to generate the high voltage. The easiest to use are neon sign transformers (NSTs) or oil burner ignition transformers (OBITs). They plug directly into an outlet and just work, no added circuitry needed. You can often find NSTs and OBITs in scrap yards or dumpsters, plumbing/HVAC shops, repair shops, thrift stores, etc. In order to make a reliable Jacob's Ladder I would recommend shooting for a transformer that can supply at least 6000 volts at 10mA. #### shortbus= ##### Well-Known Member Most Helpful Member There is absolutely no reason to buy a Jacob's Ladder kit, especially not for$56.99! Jacob's Ladders are one of the easiest projects to make and you can generally make it at very little cost, if not free.
I agree, usually the most expensive thing is the tube to surround the electrodes. Fist one I made with a tube the tube was some type of plastic, didn't last long. Think it was the ozone that made the plastic go bad. A better one was made from glass tubing that was for a milking system, but expensive at the time. Don't know the cost today though. The surround tube lets the electrodes used be a longer length, more like in the old Frankenstein movies, a lot of fun as a kid.
#### DerStrom8
##### Super Moderator
I agree, usually the most expensive thing is the tube to surround the electrodes. Fist one I made with a tube the tube was some type of plastic, didn't last long. Think it was the ozone that made the plastic go bad. A better one was made from glass tubing that was for a milking system, but expensive at the time. Don't know the cost today though. The surround tube lets the electrodes used be a longer length, more like in the old Frankenstein movies, a lot of fun as a kid.
True, though I never bothered with the tube. Stiff enough wire (coathanger wire, for example) is usually strong enough to hold itself up, and even if it couldn't I could always put some sort of non-conductive (often plastic or dry wood) cap at the top to hold the wires the correct distance apart.
#### shortbus=
##### Well-Known Member
True, though I never bothered with the tube. Stiff enough wire (coathanger wire, for example) is usually strong enough to hold itself up, and even if it couldn't I could always put some sort of non-conductive (often plastic or dry wood) cap at the top to hold the wires the correct distance apart.
That's how I started. The tube does make a big difference. Something about the heat from the arc is contained and it can then jump a bigger gap, if I remember correctly. Or maybe it is the ionized air? Been many years since messing with one.
#### DerStrom8
##### Super Moderator
That's how I started. The tube does make a big difference. Something about the heat from the arc is contained and it can then jump a bigger gap, if I remember correctly. Or maybe it is the ionized air? Been many years since messing with one.
The one thing I can think of is that it blocks any external air currents form extinguishing the arc easily. Perhaps that's it?
I always used higher voltage anyway (10-18kV) at 30-60mA so jumping the gap wasn't ever really a problem for me. | 2021-08-05 17:53:33 | {"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": 0.42399945855140686, "perplexity": 1702.4584613208872}, "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/1627046156141.29/warc/CC-MAIN-20210805161906-20210805191906-00168.warc.gz"} |
https://socratic.org/questions/a-student-has-earned-scores-of-87-81-and-88-on-the-first-3-of-4-tests-if-the-stu | # A student has earned scores of 87, 81, and 88 on the first 3 of 4 tests. If the student wants an average (arithmetic mean) of exactly 87, what score must she earn on the fourth test?
Jul 6, 2016
She must earn a score of $82$ on her fourth test.
#### Explanation:
To have an average of $87$,
if we call the 4th test results $f$
then
$\textcolor{w h i t e}{\text{XXX}} \frac{87 + 81 + 88 + f}{4} = 87$
$\textcolor{w h i t e}{\text{XXX}} 256 + f = 87 \times 4 = 348$
$\textcolor{w h i t e}{\text{XXX}} f = 82$ | 2022-08-14 18:20:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 6, "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": 0.3115905821323395, "perplexity": 938.3547561731583}, "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-20220814173832-20220814203832-00791.warc.gz"} |
https://relativitydigest.com/category/uncategorized/ | ## Optimal Strategies for Winning The Democratic Primaries
Election season is upon us again, and a number of people from political analysts to campaign advisors are making a huge deal about winning the Iowa caucuses. This seems to be the standard “wisdom”. I decided to run some analysis on the data to see if it was true.
I looked at every Democratic primary since 1976 and tried to find which states are absolutely “must-win” for a candidate to be the Democratic presidential nominee. Because the data from a data science perspective is scarce, I had to run Monte Carlo bootstrap sampling on the dataset to come up with the results.
Interestingly, irrespective of the number of bootstrap samples, three classification tree results kept coming up, which I now present:
Very interestingly, from the classification tree above, one sees that actually the most important state for a candidate to win to ensure the highest probability of being the Democratic nominee is Illinois.
The other result from bootstrap sampling was as follows:
Here we see that winning Texas is of paramount importance. In fact, all subsequent paths to the nomination stem from winning Texas.
There is also a third result that came from the bootstrap simulation:
We see that in this simulation, once again Illinois is of prime importance. However, even if a candidate does lose Illinois, evidently a path to the nomination is still possible if that candidate wins Maryland and Arizona.
Conclusion: We see that from analyzing the data that Iowa and New Hampshire are actually not very important in becoming the Democratic party nomination. Rather, Illinois and Texas are much more important to ensure a candidate of a high probability of being the Democratic nominee.
## NBA Analytics Dashboard
Here is an embedded dashboard that shows a number of statistical insights for NBA teams, their opponents, and individual players as well. You can compare multiple teams and players. Navigate through the different pages by clicking through the scrolling arrow below. (The data is based on the most recent season “per-game” numbers.)
(If you cannot see the dashboard embedded below for whatever reason, click here to be taken directly to the dashboard in a separate page.)
## So, What’s Wrong with the Knicks?
As I write this post, the Knicks are currently 12th in the Eastern conference with a record of 22-32. A plethora of people are offering the opinions on what is wrong with the Knicks, and of course, most of it being from ESPN and the New York media, most of it is incorrect/useless, here are some examples:
A while ago, I wrote this paper based on statistical learning that shows the common characteristics for NBA playoff teams. Basically, I obtained the following important result:
This classification tree shows along with arguments in the paper, that while the most important factor in teams making the playoffs tends to be the opponent number of assists per game, there are paths to the playoffs where teams are not necessarily strong in this area. Specifically, for the Knicks, as of today, we see that:
opp. Assists / game : 22.4 > 20. 75, STL / game: 7. 2 < 8.0061, TOV / game : 14.1 < 14.1585, DRB / game: 33.8 > 29.9024, opp. TOV / game: 13.0 < 13.1585.
So, one sees that what is keeping the Knicks out of the playoffs is specifically pressure defense, in that, they are not forcing enough turnovers per game. Ironically, they are very close to the threshold, but, it is not enough.
A probability density approximation of the Knicks’ Opp. TOV/G is as follows:
This PDF has the approximate functional form:
P(oTOV) =
Therefore, by computing:
$\int_{A}^{\infty} P(oTOV) d(oTOV)$,
=
,
where Erfc is the complementary error function, and is given by:
$erfc(z) = \frac{2}{\sqrt{\pi}} \int_{z}^{\infty} e^{-t^2} dt$
Given that the threshold for playoff-bound teams is more than 13.1585 opp. TOV/game, setting A = 13 above, we obtain: 0.435. This means that the Knicks have roughly a 43.5% chance of forcing more than 13 TOV in any single game. Similarly, setting A = 14, one obtains: 0.3177. This means that the Knicks have roughly a 31.77% chance of forcing more than 14 TOV in any single game, and so forth.
Therefore, one concludes that while the Knicks problems are defensive-oriented, it is specifically related to pressure defense and forcing turnovers.
By: Dr. Ikjyot Singh Kohli, About the Author
## Some Thoughts on The US GDP
Here are some thoughts on the US GDP based on some data I’ve been looking at recently, mostly motivated by some Donald Trump supporters that have been criticizing President Obama’s record on the GDP and the economy.
First, analyzing the real GDP’s average growth per year, we obtain that (based on a least squares regression analysis)
According to these calculations, President Clinton’s economic policies led to the best average GDP growth rate at $436 Billion / year. President Reagan and President Obama have almost identical average GDP growth rates in the neighbourhood of$320 Billion / year. However, an obvious caveat is that President Obama’s GDP record is still missing two years of data, so I will need to revisit these calculations in two years! Also, it should be noted that, historically, the US GDP has grown at an average of about \$184 Billion / year.
The second point I wanted to address is several Trump supporters who keep comparing the average real GDP annual percentage change between President Reagan and President Obama. Although they are citing the averages, they are not mentioning the standard deviations! Computing these we find that:
Looking at these calculations, we find that Presidents Clinton and Obama had the most stable growth in year-to-year real GDP %. Presidents Bush and Reagan had highly unstable GDP growth, with President Bush’s being far worse than President Reagan’s. Further, Trump supporters and most Republicans seem quick to point out the mean of 3.637% figure associated with President Reagan, but the point is this is +/- 2.55%, which indicates high volatility in the GDP under President Reagan, which has not been the case under President Obama.
Another observation I would like to point out is that very few people have been mentioning the fact that the annual real US GDP % is in fact correlated to that of other countries. Based on data from the World Bank, one can compute the following correlations:
One sees that the correlation between the annual growth % of the US real GDP and Canada is 0.826, while for Estonia and The UK is roughly close to 0.7. Therefore, evidently, any President that claims that his policies will increase the GDP, is not being truthful, since, it is quite likely that these numbers also depend on those for other countries, which, I am not entirely convinced a US President has complete control over!
My final observation is with respect to the quarterly GDP numbers. There are some articles that I have seen in recent days in addition to several television segments in which Trump supporters are continuously citing how better Reagan’s quarterly GDP numbers were compared to Obama’s. We now show that in actuality this is not the case.
The problem is that most of the “analysts” are just looking at the raw data, which on its face value actually doesn’t tell you much, since, as expected, fluctuates. Below, we analyze the quarterly GDP% data during the tenure of both Presidents Reagan and Obama, from 1982-1988 and 2010-2016 respectively, comparing data from the same length of time.
For Reagan, we obtain:
For Obama, we obtain:
The only way to reasonably compare these two data sets is to analyze the rate at which the GDP % has increased in time. Since the data is nonlinear in time, this means we must calculate the derivatives at instants of time / each quarter. We first performed cubic spline interpolation to fit curves to these data sets, which gave extremely good results:
We then numerically computed the derivative of these curves at each quarter and obtained:
The dashed curves in the above plot are plots of the derivatives of each curve at each quarter. In terms of numbers, these were found to be:
Summarizing the table above in graphical format, we obtain:
As can be calculated easily, Obama has higher GDP quarterly growth numbers for 15/26 (57.69%) quarters. Therefore, even looking at the quarterly real GDP numbers, overall, President Obama outperforms President Reagan.
Thanks to Hargun Singh Kohli, B.A. Honours, LL.B. for the data collection and processing part of this analysis.
## Stephen Curry and Mahmoud Abdul-Rauf?
As usual, Phil Jackson made another interesting tweet today:
And, as usual received many criticisms from “Experts”, who just looked at the raw numbers from each players, and saw that there is just no way such a statement is justified, but it is not that simple!
When you compare two players (or two objects) who have very different data feature values, it is not that they can’t be compared, you must effectively normalize the data somehow to make the sets comparable.
In this case, I used the data from Basketball-Reference.com to compare Chris Jackson’s 6 seasons in Denver to Stephen Curry’s last 6 seasons (including this one) and took into account 45 different statistical measures, and came up with the following correlation matrix/similarity matrix plot:
Dark blue circles indicate a strong correlation, while dark red circles indicate a weak correlation between two sets of features.
What would be of interest in an analysis like this is to examine the diagonal of this matrix, which offers a direct comparison between the two players:
One can see that there are many features that have strong correlation coefficients.
Therefore, it is true that Stephen Curry and Chris Jackson do in fact share many strong similarities!
## New Paper on Stochastic Eternal Inflation
Our new paper was accepted for publication in Physical Review D. The goal of the paper was to calculate the probability that a multiverse could emerge from a more general background spacetime, in this case, Bianchi Type I coupled to a chaotic inflaton potential. Basically, we found that a multiverse being generated from such a scenario has a small probability of occurring. Further, the fine-tuning problem that the multiverse / eternal inflation is supposed to solve doesn’t actually occur, because fine-tuning is still required of the geometry of the background spacetime, the initial conditions, and most importantly, the amount of anisotropy.
The preprint can be read on the arXiv here. | 2020-04-09 07:11:29 | {"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": 2, "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": 0.5263352990150452, "perplexity": 1415.3588813303697}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371830894.88/warc/CC-MAIN-20200409055849-20200409090349-00347.warc.gz"} |
https://cstwiki.wtb.tue.nl/index.php?title=Group2_19-1_Week4&oldid=80936 | # Group2 19-1 Week4
Hallo Tijn, ik weet niet of je al begonnen bent met lezen, maar we zijn nog even bezig met het updaten van de Wiki. Ik zal regelmatig opslaan tijdens het updaten, zodat je simpelweg de pagina af en toe kan refreshen om de wijzigingen te laden.
## Notulen tutor meeting
• Make requirements measurable
• Subdivide requirements
• Requirements, preferences and constraints
• Wie gaat het maken? Andere groep stakeholders/gebruikers
• Expand explanation for translation to exoplanets
• Nut van de achievements van de missie
• Wiki hoeft pas klaar te zijn na de presentatie, dus week 8
• Keuze voor experimenten moet echt gemaakt worden, deze week eigenlijk nog'
• Of we in een beschaving mogen/kunnen komen is een ethisch vraagstuk waar we wat meer op in moeten gaan
• Conclusies gaan trekken over constraints en keuzes
• Hoe zeker is er een oceaan op Europa, onze afweging moet duidelijk zijn voordat we ingaan op een cryobot sturen
• Planning weer updaten (week 5 report -> andere tijdsbesteding)
• Modelleren/berekenen
• Anthropologische deontologie
• Zekerheid oceaan
#### To do voor donderdag:
• Requirements updaten (zie notes Tijn)(Wouter)
• Modelletje temperatuur (Wouter)
• Hoe omgaan met beschavingen (Wouter)
• Experimenten updaten (Marco)
• Planning updaten (Marco)
• Aanwezigheid oceaan bevestigen (Kasper)
• Fiber optic cable at 100 K (Marco)
• Communication options via fiber or alternative options (Kasper)
## Week 4 Logbook
### Presence of an ocean
The first theories that Europa has a sub-surface ocean came after the fly-by mission of Voyager 1. This spacecraft was, in march 1979, the first that made images in significant detail of Europa’s surface, with a resolution of about 2 kilometers per pixel. These images revealed a surprisingly smooth surface, brighter than that of earth’s moon, crisscrossed with numerous bands and ridges. Researchers noted that some of the dark bands had opposite sides that matched extremely well, comparable to pieces of a jigsaw puzzle. These cracks had separated, and dark, icy material appeared to have flowed into the opened gaps, suggesting that the surface had been active at some time in the past. The images also showed only a handful of big craters, which are expected to build up over billions of years as the planetary surface is bombarded by meteorite, until the surface is covered in craters. Thus, a lack of much craters suggested that Europa’s surface was relatively young and implied that something erased the craters, such as icy, volcanic flows. Next to that, scientist found patterns of some of the longest linear features in the images that did not match the predicted patterns of the features, created by tides as Europa orbits Jupiter. They determined that the found patterns would fit very well if Europa’s surface could move independently and was not locked to the rest of the interior.
These interesting findings led to the next mission to Europa, Galileo. This spacecraft was launched in 1989 and entered orbit around Jupiter in 1995. Galileo eventually made 12 close flybys of the icy moon, including images of Europa at a range of scales, revealing new details about the surface and providing context for how those details were related to the moon as a whole.
One important measurement made by the Galileo mission showed how Jupiter’s magnetic field was disrupted in the space around Europa, implying that a special type of magnetic field is being created within Europa by a deep layer of some electrically conductive fluid beneath the surface. Scientists belief, based on Europa’s icy composition, that the most likely material to create this magnetic signature is a global ocean of salty water. Above described are four strong indications of a sub-surface ocean on Europa, which is why the common belief under scientists is that the ocean really exists. (https://europa.nasa.gov/about-europa/ocean/)
### Physical considerations about the presence of water on Europa
Phase diagram of water
One of the reasons to assume that water won’t be present in its liquid form on Europa is in the phase diagram of water, shown to the right. As can be seen in the image, the lowest pressure at which water can still exist in liquid form is its triple point at 611.73 Pa (0.0061 atm), at the common freezing temperature of 273.15 K (0 degrees C). Below that pressure, water has no liquid form. Since the pressure at Europa’s surface is about 10^-10 atm, this means that liquid water can not stably exist on Europa. It may surface for a brief moment, but will almost instantaneously either freeze or boil, leaving no water remaining.
It should be noted that indeed this diagram does not extend below 10^-5 atm, and that based on this image it is thus technically not possible to say that water does not have a liquid form at such ultra-low pressures. However, it is first of all unlikely that such an out-of-place phase change exists based on this and other phase diagrams. Secondly, this ‘liquid’ may not be liquid as we know it and still be unable to support life. Much like solid water has different crystalline structures at different temperatures and pressures, so can this liquid water have very different properties based on the environment it is in. Hence based purely on physical grounds it is unlikely that liquid water in a familiar form exists on the surface of Europa.
### Modelling temperature of the digger
A 10x1 digger in near vacuum air with a 1 kW power source
The 10x1 digger in ice with a 100 kW power source
The 10x1 digger in ice with a 200 kW power source
The following data was used for making the model, all derived from Engineeringtoolbox
Thermal conductivities (W/m K): Stainless steel: 14.35 Ice: 2.18 Air: 0.06
Specific heat: (J/kg K) Air: 700 Steel: 500
Densities: (kg m^-3) Steel: 7850
(Matlab’s PDE solver didn’t seem to be able to work with convection, and was not able to make 3D models. WebNutils proved unable to work with solids and multiple domains, and modelling steel as a super-dense fluid with an ultra-high viscosity doesn’t increase trust in the model. Instead, I used Quickfield Student Edition to make the following pictures. This did have the disbenefit of modelling over only 255 data points.)
In the following pictures, the digger will be modeled as being somewhat pill-shaped: a 9 m long cylinder with a 1 m radius, capped off with a hemisphere of the same radius. Both are modelled as being solid steel. The models are axially symmetric, with the bottom border of the picture always forming the axis of symmetry. The other outside borders are always set to be -175 oC (~100 K) to simulate the environment. The models are steady-state, as it is assumed that the digger will be digging through the ice long enough for the temperature to equilibrate. The bit of air around the digger was incorporated in the model to account for the fact that the slow progression of the digger may heat the ice or air around it a little, thereby insulating the digger somewhat. The power source was always adjusted such that it homogeneously produced the desired power.
The upper left image is modelled in air with a 1 kW power source. In this case, it is visible that the digger will have trouble maintaining a sufficient temperature gradient around the power source, as the steel around it effectively takes away the thermal energy but is unable to give it to the environment. This implies that an RTG, as well as other devices that require temperature gradients may not work efficiently in these conditions.
A smaller digger with a 10 kW power source
The two pictures to the right show the digger in ice with a power source of 100 (above) and 200 kW (below). In this case the gradient forms, as the digger can lose its heat, but is not sufficiently large. Even at 200 kW the total temperature difference is only 50 K, and the difference between the in- and outside of the power source is only 15 K.
The solution may be found in the last model, where the digger has been scaled down by a factor of 3.28 (m have been set to ft), yet the power per unit volume was still scaled to produce similar powers. The result (seen on the bottom left) is a much more localised power source capable of producing much stronger temperature gradients. In this case, the total power was 10 kW (10 times less than the other models), yet the total difference is twice as high. Taking into account that this difference occurs over a 3.28 times shorter difference, means that the gradient can already become up to 6.5 times larger.
### Communication options via fiber
Fiber-optic communication
Is a method of transmitting information from one place to another by sending pulses of light through an optical fiber.
Fiber-optic use in missions:
In the ARTEMIS mission in 2015 a fiber optic link was used. The function of this fiber was to monitor ARTEMIS under water and allow the scientists to display the real time outputs of the sensors on a screen. There were high noise levels on some sensors. The fiber was 15 km long, but ARTEMIS was at its maximum 10 km from the base. Due to the ocean currents the fiber couldn’t be re-used. https://caves.org/section/commelect/drupal/files/Presentations/NSS2016%20Under%20Ice%20Nav%20And%20Com.pdf
https://onlinelibrary.wiley.com/doi/full/10.1002/rob.21740?sid=worldcat.org
Glass fiber:
• Tensile strength in touch up to 300 degrees of Celsius. Can’t find any information about the tensile strength in freezing temperatures
• Very high tensile strength (3445 MPa for E-glass)
• 100% insensitive for interferential signals, for example from electromagnetic fields.
• Density E-glass: 2.58 g/cm^3. With a typical diameter between 3.8 and 20 µm.
With the smallest diameter, an glass fiber of 10 km would weight only 0,29 gram. With a diameter of 20 µm, the fiber would weight 8,1 gram https://www.sciencedirect.com/topics/engineering/glass-fibre
“The cryobot uses the laser energy to heat water with which to melt the ice in front of it, while the water re-freezes behind it around the fiber, allowing communications and power flow to be maintained.” (Link used before) https://www.space.com/29644-cryobot-tunneling-robot-explore-icy-moons.html | 2023-02-02 21:21:23 | {"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": 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": 0.5518507361412048, "perplexity": 3099.3079020982004}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500041.18/warc/CC-MAIN-20230202200542-20230202230542-00672.warc.gz"} |
https://forum.allaboutcircuits.com/threads/project-resistor-parts-storage.12341/ | # Project: Resistor Parts Storage
Discussion in 'The Completed Projects Collection' started by Wendy, Jun 14, 2008.
1. ### Wendy Thread Starter Moderator
Mar 24, 2008
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A couple of years ago I decided I was tired of buying resistors piecemeal, and started a resistor kit. This leads to the question, when you have over 150 different resistors, how do you store them? Shelfs really do take too much space, and can be hard to maintain. Lord help you if you drop one and scatter the contents to the 4 winds.
My answer at the time was custom made bags made with an impulse sealer. If you've never seen an impulse sealer, think plastic welder. They work on static bags and just about anything plastic, by using a nichrome strip and Teflon coated tape to melt and weld two pieces of plastic together. Here is my impulse sealer, which I bought from Harbor Freight for around \$20. I think they've discontinued this particular make/model, but they are available elsewhere. Mine is an 8" sealer, if I had it to do over again I would probably get longer (maybe not).
Any bags will work, but I use either sandwich bags or snack bags, each of which makes 2" x 3¼". I am aware of a certain French invented system out there, but I'll be using inches throughout. Here is a before and after shot in the same picture.
And I ended up with a kit that looked something like this...
There were about 7 big bags full of small bags. This was awkward and made for hard to find parts, but IMO still better than shelves. My boy Jim had a better idea, which I tried out and am sharing here.
Instead of a big bag of randomly placed parts, how about a big bag with organized parts? The end result looked something like this, with plenty of room for later changes. Notice how easy it is to take inventory with the new layout.
Last edited: Jun 16, 2008
2. ### Wendy Thread Starter Moderator
Mar 24, 2008
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Continued from previous page...
This in turn folded pretty neatly into the previous bag.
I made this bag starting from a large 1 gallon storage bag, and dividing it up into 6 parts 1¾" wide with a indelible marker. I cut a piece of cardboard from a coke box 1½" X 10½" for an insert later in the process. I slide this cardboard into the first slot I was making, and seam sealed it in. None of the dimensions are critical, in fact the whole layout can be pretty sloppy.
I then slide the cardboard as far up as it would go and used it as a backing to allow cutting on only one plastic side, leaving the other intact.
I repeated this until I had all the panels sealed and cut, then removed the zip lock seal from the top of the bag.
Dang belly, keeps getting in the way!
Last edited: Jun 14, 2008
3. ### Wendy Thread Starter Moderator
Mar 24, 2008
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The only thing left for that panel at this point was to divide it into 1/3 sections. I marked the plastic and seam sealed accordingly, to form the individual pockets.
There was still one problem, there weren't enough pockets for all the resistor bags. Remember though, a seam sealer is a welder, so I made another one and attached it. Wallah! One down, 7 to go.
I take all the bags and store them in one large storage bag. To bad I can't do this with capacitors, but the variations in size are a killer.
Last edited: Jun 15, 2008
4. ### Wendy Thread Starter Moderator
Mar 24, 2008
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Finishing Up
Materials needed:
Impulse Sealer
Box of 1 Gallon Storage Bags
Sharpie (indelible marker)
Ruler with metal edge for cutting
Good pair of scissors
Sharp knife
Cardboard (I used a Coke box)
Scotch Tape (it will not seam weld, so is useful holding two panels together for welding)
Notes:
I took 20 minutes to make a panel, 45 for a whole sheet using 2 panels. This is not an evening project, and it takes longer if you get tired due to stupid mistakes. When I say this isn't precision, don't be surprised at ¼" variation from the manufacturer themselves. I went through 3 sharpies, mostly because they were used to begin with, and dried out during use.
I started trimming the cardboard inserts edge at the beginning of each panel, since it will get sliced up. The cardboard insert was pretty handy for handling, since the plastic just flops around otherwise. I gave up using the boxes shown in the pictures to line things up, they helped but were more in the way overall.
My kit has 8 ranges, starting at $10^{-1}$ (1 - 9.9 Ω) and ending with $10^{6}$ (10-99 MΩ). Not all used 2 panels, but the usual number of resistor bags was 21 (each panel has 18 slots).
*****************
One last thought, after using this a little while, I wish I had made the pockets deeper. The little bags are about right, but if I had it to do over I would have made the pockets 3½" X 2" instead of 3½" X 1¾". Not worth redoing, but 5 X 3 pockets per panel with the new dimensions would still be 10½" X 10" and would still give enough spaces to use, when figured between 2 panels.
2" X 5 = 10", 1¾" X 6 = 10½"
A longer impulse sealer would have speed things up a little too, since I could have done 1 seal per line instead of the 2 seals per line I was doing.
Last edited: Jun 16, 2008
5. ### beenthere Retired Moderator
Apr 20, 2004
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Sure beats the coffee can and ohmmeter method.
6. ### thingmaker3 Retired Moderator
May 16, 2005
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Bill, do you think the bag-matrix could be glued to a card and put in a binder? I have visions of a "library" of parts...
7. ### Wendy Thread Starter Moderator
Mar 24, 2008
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Don't see why not, though the deeper pockets would almost be required. It would be easy to dump too, so maybe some foam to press everything in place when the binder is closed. Maybe a large flap seam welded when you make the initial panels?
Actually, small envelopes made of paper combined with the small bags might even be better, as envelopes have flaps. The thin plastic used in bags is way too flexable, plastic static stickies perhaps? If you are ever looking for high quality double sided tape think carpet tape, it ages well, simply will not let go (don't slip when using it), and cuts well with scissors. I've used it to make home made decals along with my ink jet printer with excellent results.
There are a ton of solutions thinking about it, I like the book idea.
Small capacitors would be a logical canidate, as are zener diodes (and some other small type diodes).
Resistors were my bugaboo though, you can compensate for almost anything design wise with a nice selection of resistors.
Last edited: Jun 17, 2008
8. ### SgtWookie Expert
Jul 17, 2007
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Interesting method, Bill
I started using 3"x 4.5" coin envelopes about 25 years ago. I write the resistance value and wattage along the flap hinge on the opposite side in pencil; and have them all standing up in a couple of boxes that are 3 1/4" wide, 3" tall and a foot long. If I take an envelope out, I just lift up the one right behind it and raise the flap so I can easily spot the place where the envelope I have out goes. Taking a ruler and drawing a diagonal line across the top edges of the envelopes with a colored Sharpie makes it a snap to spot an envelope that was errantly filed (this is a trick I learned when using IBM punched cards for ensuring the card decks were in sequence.)
I use pencil for the markings because ink would run if water or isopropyl were splashed/spilled on it.
About a year ago, I bought a 1/4W E24 8,400 piece resistor assortment on Ebay for around \$19. It came with the resistors in 24 plastic bags, labeled "1R0 - 10M", "1R1 - 1M1",... "9R1 - 9M1". In each bag, there are 7 strips of 50 resistors that came off a tape reel. They store fairly compactly in a 7"x7"x6" box, and I can locate the value I need in just a few moments by reading the multiplier band.
9. ### Wendy Thread Starter Moderator
Mar 24, 2008
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I just got through buying around 801 resistors, in groups of 20. Tanner's had 1 20MΩ resistor (DANGIT). My current spread is 10, 11, 12, 13, 15, 16, 18, 20, 22, 24, 27, 30, 33, 36, 39, 43, 47, 51, 56, 62, 68, 75, 82, and 91, or 24 per range. It starts at 1Ω and goes to 10MΩ. Looks like I'll just have to live without a full pack on the 10-99MΩ range. I'm not sure what I'll use 1.1Ω or 1.3Ω for, but I'll think of something. I wound up buying 11, 13, and 16 for every range, I hadn't originally added those to my basic range set.
I like plastic bags, other than their ESD liablilities. Cheap, easy to replace if they wear out, and extremely portable. Add a seam welder and it gets interesting.
Any ideas on relatively cheap sources for the high end megohm scale? I figure 20 of everything is enough, other than a few basics (I like large quantities of 1KΩ and 10KΩ, they are my personal default values).
Isopropal lets me reuse the bags. In the big bag their pretty safe.
10. ### SgtWookie Expert
Jul 17, 2007
22,201
1,806
Well, seems like my suppliers for the really large values have dried up. The fellow I bought the E24 assortment from hasn't had them since not long after I got mine - and the largest values in that assortment was 10M.
But using really high resistance values isn't typical, unless you're building HV probes or working with HV tube sets or something like that. They were easier to get about 10 years ago.
High value resistors are handy to use as forms for winding small inductors on, BTW.
11. ### Wendy Thread Starter Moderator
Mar 24, 2008
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This kit is 5% ¼W, so their pretty much the same size, pretty irregular cylinders. I've used ½W and 1W to make inductors, especially the nice cylindrical units. They're getting less common from what I've seen.
Last edited: Jun 17, 2008
12. ### SgtWookie Expert
Jul 17, 2007
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The "nice cylindrical ones" are things of the past now, come to think of it. Good reason for that, too - all of the NOS of that kind that I have, even mil-spec 1% resistors, have shifted wildly in value over the past 20+ years. The newer type that look more like coated miniature dumbells seem to be steady as a rock.
If you have some particular values in mind, E-mail me a list of what you need. I'll check my favorite haunts next time I head that way.
13. ### Wendy Thread Starter Moderator
Mar 24, 2008
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OK, finished kitting up my latest aquisitions, and had to make another panel (probably the last). I wound up making the pockets 2 1/8" X 3 1/2", it helped a lot but still isn't perfect. If anyone else trys this out you might think about flaps. I drew the panels up for reference.
My $10^{7}$ kit looks like it will never truely by finished, the only values I have for it are 10MΩ, 15MΩ, 22MΩ, and 82MΩ. I'll keep my eyes open, but it's not a biggie. Values this large are used to tweak test selects IMO.
Last edited: Jun 21, 2008
Jul 30, 2008
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15. ### studiot AAC Fanatic!
Nov 9, 2007
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Here is a storage method, courtesy a now defunct magazine Electronics Today International (Jan 1973).
Their version was all sheet metal work but mine was made from thin plywood.
The tubes were plastic pill tubes, obtainable at the local chemist for £1 per hundred. Film canisters are also good.
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16. ### Wendy Thread Starter Moderator
Mar 24, 2008
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Lets see, slightly over 8 ranges (X10E-1 to X10E6) X 24 resistors = 200+ values. (Yes, I actually bought them over time).
I count 75 slots in that unit, need more shelves....
17. ### linchiek Active Member
Jul 23, 2008
110
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Wow...!
Bill, u r creative man...!
18. ### vetterick Active Member
Aug 11, 2008
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I liked this so much I joined the forum just to tell you about it
I used the 3 ring sheet protectors (Avery #PV119XL-10), made 2 vertical seams at 3" and 6" (9" wide), marked the horizontals every 2 1/4" (sound familiar?) and cut slits 1/4" down from the lines before welding on the lines, I also welded across the top for a little more support.
This is very easy to layout if you use one of those sliding paper cutters.
It took about 1 1/2 hours to do and will take much longer to fill the 150 spaces, and its easily expandable.
Thanks very much for the idea, keep em commin.
Rick
19. ### thingmaker3 Retired Moderator
May 16, 2005
5,072
8
Hmmm... my wife buys bulk beads in tubes... I've even used them as parts storage in the past...
20. ### iONic AAC Fanatic!
Nov 16, 2007
1,640
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Seems a bit time consuming, but very handy. Could save the effort sorting and looking for the right resistor in a hurry. If you could find Horizontally orientated zip-loc bags the right size you could save a lot of time in construction. I will be on the lookout for some now! | 2019-04-26 02:05:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 3, "/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": 0.36088529229164124, "perplexity": 3767.8886050408337}, "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-2019-18/segments/1555578747424.89/warc/CC-MAIN-20190426013652-20190426035652-00525.warc.gz"} |
http://fallout.wikia.com/wiki/Talk:Damage_Threshold | # Talk:Damage Threshold
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## Reconsidering DR versus DTEdit
Playing Fallout 3 for two years now, I question the whole point of having the SPECIAL version of DR. I mostly agree with Bethesda streamlining things to a single value. But It seems to me that DT is more practical for modeling armor than DR. I always hated how easily Paladin Hoss is killed by those puny little .32 pistol caliber rifles, or how I can blow away a top-level Sentry bot (Fallout 3) with Paulson's Revovler. To make things even more annoying, Bethesda allowed all Energy weapons to bypass DR. No Telsa resistence that I can see.
I welcome the reintroduction of DT to New Vegas, but I think that Obsidian should drop DR alltogether. It's redundant in the face of the threshold value, it isn't as intuitive or believable for gameplay purposes, it is easily abused by exploiting VATS and unique weapons, and it limits the options for making challenging NPCs. Instead of only inflating a Ghoul Reaver to 1100 HP, developers can add DT for it's metal armor, making it more resilient to lower-level weapons than DR allows. I think this also allows developers to stick by the game rules without borking the challenge; 635 HP is the max allowed at level 30, which makes for a pretty tough NPC as it is. Add a realistic DT to the armor and that Reaver will be potent without having to break rules or abuse DR.http://brianranzoni.com 11:02, August 10, 2010 (UTC)
## The difference between DR and DT?Edit
So it seems to me that the main difference between DR and DT is that DR reduces damaged received by a percentage of damage taken while DT reduces damage received by a set number. Correct?
As far as I know, that is specifically and exactly accurate. --Kris mailbox 01:25, August 27, 2010 (UTC)
In the damage formula, damage is first reduced by any applicable resistances (including DR) as a percentage of the damage. These resistances stack additively, so if you had a 20% weakness to energy and 40 DR, you'd end up with 20% resistance against lasers and plasma. After resistance is done, the result is then reduced by a set value equal to DT. The game uses fMinDamMultiplier to set a minimum percentage. By default the value is 0.2, so regardless of resistance and DT, you cannot take less than 20% damage from a single attack.
Resistances (including DR) are valuable up to the point where you'd be taking the minimum damage. By default, resistance caps at 85%, even though fMinDamMultiplier is 0.2 (so 85 DR would be effectively equal to 80 DR). DT doesn't seem to have any cap, but like DR, it's only valuable up to the point where you'd be taking 20% damage anyway. As one final note: creatures no longer have doubled DT like they did when the game first shipped, but they still do get doubled DR for some reason. Any DR effect you assign to a creature will be twice as powerful as if an NPC had it.--SushiSquid 13:09, August 17, 2011 (UTC)
And I now realize that it's likely you were talking about the first games. What I posted was related to New Vegas.--SushiSquid 14:32, August 17, 2011 (UTC)
So basically... Damage Resistance: Damage Percentage reduction (If you have 25% Damage Resistance, any damage taken will be reduced by that amount, as in, 25 DR=25% damage reduction) Damage threshold: If your bullets DT is at 10, and the targets DT is at 5, You'll penetrate their armor with no damage reduction, However, if your bullet has 5 DT and your target has 20 DT, then the number between the DT and your bullets DT will be reduced by that amount in percent. For example: 5 Bullet DT vs. 20 Bullet DT. (20DT(Armor)-5DT(Bullet)=15DT=15% damage reduction. Amirite?
This ^^^ is completely wrong for Damage Threshold (DT) and slightly wrong for Damage Resistance (DR). In the example given where the bullets do 10 damage and the victims damage threshold is 5, what would happen is that after the bullets go through the armour, their damage is reduced by 5. 10-5=5. What is left is that the bullets do 5 damage to the victim. This system has some caveats though. Damage resistance must be calculated first and can not reduce damage to below 25% of the original damage. If it tries to do that then the original damage is just multiplied by 25%. Damage threshold can not reduce damage to below 20% of the DR damage. How DR and DT work together is as follows.
1) Damage Resistance is calculated. The original damage is multiplied by (100%-DR%) If that value is less than 25% of the original damage, then the equation (original damage * 25%) is used.
2) Damage Threshold is calculated. The value calculated from step #1 is used with Damage Threshold. The Damage Threshold is subtracted from The value calculated in step #1. (#1_value - DT). If this value is less than 20% of of the #1 value, then the equation (#1_value * 20%). is used.
3) The players health drops down by the amount of #2 value.
Here is one example of how this works. A shot of damage 40 is fired against a character with 95 DR and 10 DT. Damage Resistance is first calculated and the damage is reduced to 6 (95 DR is capped down to 85% reduction). Next Damage Threshold is calculated. The value from the previous step (6), has 10 subtracted from it. The result (-4) is less than 20% of the pre-adjusted original (6), so the final damage is instead 1.2 (6 * 20% = 1.2).
I Think it just might be the percentage of damage reduction, and if anything penetrates/is higher then that number, then it removes the protection of the armor.
## DT is better? Edit
So with a DT of 20, if you got with an attack of 14 you'd take no damage, whereas an attack of 21 you'd recive one damage, correct?
Sorry forgot to sign- 70.72.160.21 00:07, October 19, 2010 (UTC)
No. With 20 DT, you'd take some damage from both examples in New Vegas. You will always do a very small amount damage, regardless of DT. In the old games, you would only take damage from the 21 point attack. Nitty Tok. 00:09, October 19, 2010 (UTC)
Makes things a lot more realistic... Now armor can actually be armor...--KnightNapier 13:11, October 21, 2010 (UTC)
This mod brings back the original DT http://www.newvegasnexus.com/downloads/file.php?id=34872. So, back to your example, 14 DAM vs 20 DT = no damage taken. 21 DAM vs 20 DT = 1 damage taken. No bleedthrough if your DAM is below the DT (or at least, so says the mod author). Servius 16:50, October 25, 2010 (UTC)
Oh, nice mod. I'd like to see pool cue wielding fiends damage my power armour now (so long as I'm not taking on radscorps with hollow points)! Out of curiosity, what is the damage that occurs when you attac kor are attacked by something that doesn't meet the DT? --DragonJTSLeave me a message 18:39, October 29, 2010 (UTC)
## DT vs DR: which is applied first? Edit
Which is applied first, the DT or the DR? Some people might not be as math-minded as me, so I'll explain why this would matter. Let's say you have 8 DT and 50% DR of whatever damage type you'll be receiving. Let's say your attacker hits you with 20 damage before your DT/R is applied. If DT is applied first, the damage goes down to 12, then your DR cuts it down to 6 damage. If DR is applied first, it cuts it down to 10 damage, then your DT subtracts 8, bringing it down to a mere 2 damage. Anyone know which one is the case? NightChime 22:40, November 3, 2010 (UTC)
The Fallout2 manual states that DT is applied first, then DR. I assume it's the same for Fallout 1 and Tactics 84.246.5.26 10:46, November 9, 2010 (UTC)dwaz
## Player DT in NV? Edit
Does anybody know what determines the Player Character's DT in NV? My base DT when wearing nothing (new character) is -2 and I'm wondering if that's the default DT. I looked around the Wiki and the BethSoft forums and didn't see anything specific.--Sdpens 22:37, November 5, 2010 (UTC)
0 by default. You had -2 because you took Kamikaze trait. 24.155.188.162 17:39, November 8, 2010 (UTC)
Thanks, totally forgot about that trait.--Sdpens 23:11, November 8, 2010 (UTC)
## When is DT applied? Edit
In New Vegas, is DT applied before or after other damage multipliers? Say I have a hunting rifle (45 damage, 45 crit) with .308 JSP (x1.5 damage) and I do a critical hit against an opponent with 20 DT.
Would it be calculated as (45 + 45) * 1.5 - 20 = 125 or (45 - 20 + 45) * 1.5 = 105?
And how would sneak attack criticals factor in?
84.246.5.26 10:46, November 9, 2010 (UTC)dwaz
I believe DT/R reduction is applied after all damage multiplayers. —17:19, August 14, 2011 (UTC)
## DTEdit
I have rewritten the whole DT article for FNV section, hopefully it will help explain how DT works and what impact it has on the calculations and answering most of the questions already here. GhostAvatar 20:50, November 13, 2010 (UTC)
## DT and DR Stats Forthcoming? Edit
Is there any chance someone can post up the DT & DR% values for the creatures of FONV? Not to sound cynical, but what is the point of the Damage Inflicted formula, without the necessary stats to input into it? Some DT values have been added to some of the creature info boxes, but without the DR% this formula is moot.32.97.110.55 00:01, November 18, 2010 (UTC)Pulsar
Thats because very few things have any thing other than a 0 assigned to the DR value. In fact other than the Rebreather I am finding it hard to find anything which has a DR value, it is pretty much a defunct value in New Vegas. Which leads me to believe that the Rebreather was simply a oversight by the devs and DR was supposed to be removed all together by setting everything to 0 in favor of DT.
As for the NPC's, this is something I have not looked into with much detail, but from what I can gather the values are set to 0 with most of the DT coming from any armour. The formula is of more use for calc your own protection more than anything else. Avatar 01:16, November 21, 2010 (UTC)
Med-X and Slasher add 25% DR each--Christian Broach (talk) 02:12, August 4, 2012 (UTC)
## Shotguns and DT Edit
Using a Sawed-Off Shotgun has not been too effective for me against people with armor, even at close range. Does the DT affect each pellet, i.e. is 7.1x14 damage ineffective vs DT 8 or higher? 198.95.226.224 23:33, November 22, 2010 (UTC)Thanks, Mike
Yes, the DT is applied vs. each pellet. This is the reason for the Shotgun Surgeon perk. --Kris 02:33, November 23, 2010 (UTC)
Slugs work well against armor, too! Nitty Tok. 02:39, November 23, 2010 (UTC)
Yeah, that's 'cause slugs are a single projectile instead of 7 :P --Kris 04:34, November 23, 2010 (UTC)
## Yep and Nope Edit
Now I love Bethesda to death, but I must say DT is a more accurate model of armor than DR is. Beth put it in FO3 because it already existed in TES:IV (and the cap was 85% believe it or not). The only problem I have is this. Hollow points are just about useless against most everything. It bugs me. Now I understand how insects and arachnids like scorpions and ants and shit have that whole "exoskeleton" thing going on, so they are somewhat hardened against puny rounds... but deathclaws? Come on. Deathclaws are big and strong and evil but they're soft. Any soft target, i.e. a human/ghoul wearing civilian clothes or a big muscly deathclaw should have a very low DT and so I should be able to use hollow points against meaty targets without fear. I don't give a shit about "thick hides." Grumble. --Jackarunda 20:48, November 29, 2010 (UTC)
Personally I hate DT because it renders automatic weapons like the Gatling laser weak as shit, but ant's have no DT so hollow-points should work fine. As for the Deathclaws their DT is what makes them dangerous and they aren't natural (not even radioactive natural) the U.S Military created by genetic engineering a bunch of different species for combat situations so it makes sense that they would have some kind of hardened shell to protect them in dangerous situations. --1stRecon 01:23, November 30, 2010 (UTC)
Apart from that, ever seen rhino skin? Even in our world, some big creatures have skin thick enough to turn bullets, at least the lighter variety. Wunengzi 13:15, August 17, 2011 (UTC)
## DT Cap? Edit
I know DR in Fallout 3 capped out at 85%, like in Oblivion. Does DT in New Vegas cap somewhere? I'm going for a maxed DT character (Toughness X 2, Sub-Dermal Armor, Remnant Armor, etc.), and wondering if I'm getting much mileage out of going past 40 or so. Super Duper Mutant 21:13, December 2, 2010 (UTC)
If you read the page you will see that a minimum of 20% damage bleeds through regardless of DT. So if you are hit with a shot that does 10 damage you will always take a minimum of 2 damage unless you DT is lower than 10 regardless of how extremely high your DT is. ☣Avatar☣ 21:29, December 2, 2010 (UTC)
Yeah, I got that, but is there a number where DT just totally caps out? As in, have DT of X with no helmet equipped, equip Power Helment and DT doesn't go up because it's already at the max. 98.236.179.112 04:18, December 5, 2010 (UTC)
No it's not capped; but again it is if you fight opponents that have a low DMG/attack weapon :) --94.212.252.40 17:05, December 7, 2010 (UTC)
## DT bleedthrough and difficulty Edit
Is the increased damage at higher difficulties calculated before, or after DT?
Say I have a DT of 20 (and 0 DR) and I'm hit for 15 damage, normally I'd take 3 damage thanks to the 20% bleedthrough. When playing at very hard, damage is doubled, but would that mean that 15 damage is doubled to 30 and then DT is applied, so I'd takeh 10 damage, or that the damage after DT is doubled, leaving me with 6 damage taken? Dwaz 10:11, December 8, 2010 (UTC)
## Wrong Formula Picture? Edit
The text say that the minimum damage would 20% of the weapon damage, regardless of damage threshold and damage reduction. So the formula "max((Shot Damage-DT)x(100%-DR%),(Shot Damage x 20%)x(100%-DR%))= Damage inflicted" should be wrong. With 80% DR for example, the second part of the formula would produce only 4% of the Shot Damage and not the minimum of 20%.
Corrected Formula should look like this: max((Shot Damage-DT)x(100%-DR%),(Shot Damage x 20%))= Damage inflicted --130.83.73.236 11:22, December 8, 2010 (UTC)
Are you fucking kidding me? Where the hell are we supposed to get a screenshot of a concept?! You ask too much of us, Internet.
I think the internet just owned you <.<
## I started playing Fallout 1Edit
And I wonder where is DT shown? I dont know what my DT is ;(
## Deathclaws... Edit
If I recall right I heard that deathclaws go straight through DT or DR (can't remember which.) or is it both? Basically I want to know if one cannot be reduced or is less effective than the other. So far I've just taken that perk that gives me faster run speed in light armor + I have Kamikaze most of the time, so I haven't worried about it to much.... Roflmaomgz 08:12, June 25, 2011 (UTC)
## Faded DT? Edit
Alright so i'm playing Fallout: New Vegas and i notice i've been more susceptible to damage then usual, so i check my pip-boy. After checking the EFFECTS to no avail, i switched to Items to check my armor. My armor is in perfect condition, but i looked to where it says DT and instead of being the normal color it is darker, and faded, but only my DT is like this. I've looked everywhere, but i can't find anything about this, and i fear it may mean that my DT is just being ignored completely, by EVERYTHING. Anyone know what might be wrong? --67.241.105.0 05:30, August 1, 2011 (UTC)
I've noticed the samething in my game, even when I put it on very easy, my health was dropping like a rock and that was against even weak oponents (I have a DT 21). Macilnar 12:28, August 2, 2011 (UTC)
## Inconsistency Edit
DR article says DR applies before DT reduction. DT article presents formula where DR is applied after DT reduction. —16:40, August 14, 2011 (UTC)
## Head shots? Edit
I see the game has head gear, but does the game differentiate DT for different body parts? Or it's just one summary DT for the whole body? If latter then the most important benefit of head shots is practically out of the window. —17:44, August 14, 2011 (UTC)
Oh, headshots do 2x damage.
## It seems that DT (armor) is only useful at low levels... Edit
Because, for example, against Deathclaw's attacks a standard DT of 20 oftentimes won't change anything, you would still die from same number of hits. Am I right? —Preceding unsigned comment added by 46.138.158.3 (talkcontribs). Please sign your posts with ~~~~!
It would be more accurate to say that DT is useless against deathclaws, because deathclaws' attacks ignore DT. High DT is still useful against most other high-end Mojave dangers, and I think a DR is still effective against deathclaws, what little there is to be had in FNV that is (Med-X and the rebreather are all that come to mind). --Kris 12:38, August 17, 2011 (UTC)
What on earth is the point of this feature, anyway? Just to make sure deathclaws can hurt you no matter what? It would seem far more logical to allow them to do what they would in fact do in real life, knock you over and give you concussion before they managed to make it through your armor. Wunengzi 13:19, August 17, 2011 (UTC)
## Grammar issue Edit
Stop changing the section under bugs it's grammatically correct the way the had it--Your friend, Austinsnoop 03:53, September 12, 2011 (UTC)
You are wrong. Do not keep reposting grammatically incorrect updates to the wiki. The definite article is not capitalized when part of a proper noun structure, so stop doing it.--SushiSquid 21:38, September 12, 2011 (UTC)
## Missing DR/DT value Edit
My pip boy no longer shows DR/DT, the article mentions something about this but not how to recover if the problem persists. Any ideas?
update: Adding Sub-Dermal Armor seemed to solve the problem, not sure what to do if it turns up again though.
Update 2: Equipping the Rebreather (with has DR rather than DT) seemed to fix the problem, just make sure the pip boy is displaying DT, not DR or in between.
## Other ModifiersEdit
Here's something I don't quite understand: the formula given here claims that 'other modifiers' occurs before DT reduction, which is fair enough. However, the 20% bleed through effect uses 20% of the ADJUSTED damage instead of BASE damage. What I'm asking is does the ammunition types and their effects come under 'other modifiers'? If so then this system is a little weird.
Here's an example: if I use a weapon with 11.8 DAM with normal rounds and hollow point rounds on a target, as you increase the target's DT, hollow point will obviously hit the 20% barrier before normal rounds. In this case, hollow point would hit the barrier at around 6 DT while normal rounds will hit the barrier at 10 DT.
BUT, since the target is taking 20% of the 'adjusted dam', then if its DT>10, where both ammo types hit their barriers, overall damage from hollow type rounds is still higher than standard. This makes very little sense, because what we're effectively saying is to maximise damage, switch to hollow point again when you're normal rounds isn't sufficient: you won't pierce the DT but you'll still be doing more damage. --101Phase 23:00, May 4, 2012 (UTC)
That's a good point. The adjusted damage may only include DR. Or, the mechanics may be such that adjusted damage is actually base damage and the interaction between DR/other multipliers and DT is actually more complicated than listed. I need to do some experiments... I'm going to update the page with the more sensical formula (other multipliers not a part of the "adjusted") since obviously HP isn't always better than normal. I'm going to do some testing later.
It may also be that really just the BASE is used for the 20%, and if that's the case, that means there's a slight bug in the game where effectively DR higher than 80 is useless. Again, bears testing. (Thelee (talk) 20:29, July 19, 2012 (UTC))
Upon reflection, the damage equation is most likely something like:
$adj = base \times ammo\ multiplier$
$postDT = max(adj - max(DT \times ammo\ multiplier + ammo\ mod, 0), base \times .2)$
$final = postDT \times DR \times others...$
So DR would come after DT, like in other fallout games. I say this is the most likely equation mainly because it requires the least amount of mental hurdles to try and reason through any possible interaction between all of the effects, and also because in Oblivion, straight up DR came last after any other effects (like damage reflection). (Thelee (talk) 20:54, July 19, 2012 (UTC))
I think given the above, it makes sense to go ahead and adjust all combat equations appropriately. The burden of proof is to demonstrate that DR is handled specially (like an ammo multiplier, but somehow with a special exception to allow for 85 DR). I plan on testing at some point when I have time, but if others test before me and find that the equation is not as i hypothesize, then please re-fix the equations. (Thelee (talk) 20:56, July 19, 2012 (UTC))
Wow, way to *ing go - pushing edits onto articles basing solely on hypothesis. Until you get down to testing your ideas, please refrain to adding changes to math formulas. I did extensive research & testings on the damage formula, and kept testing until I was sure it is correct before releasing the changes. --Slider2k (talk) 20:49, February 8, 2013 (UTC)
## DAMAGE THRESHOLD INCREASE ON SUITS?Edit
I was browsing through the wiki searching for the best armor and i noticed another guy on the internet asking for looks or armor. The point is that he showed his Assassins suit dt being a 28. When the wiki shows it being only 14. What is the explanation to this. Is there a way to increase a suits dt? Please answer soon! Also tell me what you think are the 2 best armors in the light, medium, and heavy category and do me a favor and give an explanation to why.
## DT not showing Edit
In New Vegas i just noticed that my DT stopped displaying. My armor is repaired to normal condition, what could be causing this? | 2015-08-31 00:34:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 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": 0.5394097566604614, "perplexity": 2687.8281923662307}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644065464.19/warc/CC-MAIN-20150827025425-00292-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://matheducators.stackexchange.com/tags/definitions/hot | # Tag Info
54
"Because we said so" is a bit of a conversation closer, I agree. But "Because some people agreed a long time ago to define it that way so we could have conversations where we all understood each other. Does that seem like it would be a good idea?" is both more inclusive and more correct. I don't even think it's that hairy to talk through the FTA with ...
38
I'm confused. Are you really going to try to make this sort of distinction when teaching geometric figures to students "around 9-13 years old"? Students that age (and engineers my age -- much, much older) think that a triangle is a triangle. It's a polygon formed by three non-colinear points. A triangle has many ways you can think about it. ...
37
In the U.S. Common Core standards, functions are supposed to be introduced in the 8th grade, i.e., around age 13-14. So arguably age 15 is a year or two behind where they ought to be. The standard for the 8th grade says: Understand that a function is a rule that assigns to each input exactly one output. So honestly that really doesn't seem like a hugely ...
34
I cannot answer the OP's question about cross-cultural/international perspectives, but here is a historical perspective that may be helpful. The issue here (whether the category "rectangles" includes or excludes the category "squares") is one aspect of a larger question having to do with whether the classification of quadrilaterals ...
34
28
Maybe, your students have a belief problem. They will rarely (maybe never) have encountered problems where something was not well-defined. If you have never been in trouble since everything you were shown was well-defined, then you don't even understand the problem! (Even harder: after proving that something is well-defined, the world looks right like it ...
26
In my experience, students are often predisposed to "learn" by memorizing facts; that's how much of their early education worked, so that's what they're used to. When you give them a definition and then a bunch of consequences of the definition, they don't think "okay, I have a definition and then 87 examples of using the definition", ...
24
One encounters exactly the same issue teaching multivariable calculus when one treats integrals over three-dimensional regions and integrals over the surfaces that are their boundaries. In particular the word sphere is particularly confusing in this context. (Mathematicians use sphere to mean the the two-dimensional surface; colloquial speech and some ...
21
I think the distinction you are raising is not natural to students at this age. I teach undergraduates and graduate students, not elementary schoolers, but I find that it is not natural for undergraduates who have not had a theoretical math course. In my experience, students do not naturally think of geometric figures as sets of points. If $P = (-1,-1)$, $Q =... 19 I think it is a good thing to talk about how there are some concepts where there are choices for where you start when definining them. It happens in linear algebra too, with the definition of linear dependence. You need to talk about how there is this web of connected properties, and it depends on what you're trying to achieve as to where you start. I do ... 18 I think there are serious pedagogical problems with such an approach. Here is a good general rule for explaining any kind of math: Skipping over the motivation doesn't make something easier to understand. As math educators, our instinct is always to simplify the math that we are presenting as much as possible. We always want students to understand the ... 18 This is a case where you might be looking for a distinction that's pretty subtle. By definition, the y-intercept occurs at x=0. In one notation, it's literally f(0), where the x is explicitly offered. I'd be ok with a student's answer to "What is the y-intercept?" to be simply the y value, or the$(0,y_0)$point. If a teacher prefers one, you can ask ... 16 It's funny you should ask this now, because I just taught a Math Ed graduate class on this topic the other day, so a lot of these thoughts are fresh in my mind. The pedagogical sequence we conventionally follow with respect to complex numbers -- first introducing them in a "just pretend" way, and only later (for some students) providing a geometric ... 16 Not formal research, but some decades of experience teaching both undergrad and graduate level courses, and "editing" PhD theses and such: It appears that even many serious professional mathematicians do not understand the difference between a "definitional" iff and an "assertive" iff. This is entirely parallel to an assignment equality versus an assertive ... 16$2x$is an expression, a phrase. Compare it to "two ducks". This is neither true nor false. It doesn't have a 'truth value'.$2x = 4$is an equation, a statement. Compare it to "two ducks have four legs". This is true (edit: for the ducks, but not necessarily for the$x$). The meaning of the word "ducks" has not changed. The grammar of what is with that ... 16 Functions are far broader and more applicable than you give them credit for. Consider the following: Country or state Capital Elevation (in meters) Bolivia Sucre 2783 Ecuador Quito 2763 Colombia Bogata 2619 Eritrea Asmara 2363 Ethiopia Addis Ababa 2362 Mexico Ciudad de Mexico 2216 New Mexico Santa Fe 2152 Wyoming Cheyenne 1856 Colorado Denver 1613 ... 16 First of all, you should test them on remembering the definitions. Second, there are probably a significant number of your students who do not understand the definitions. Suppose you gave them an example of a grammar that was not right-linear and the definition of a right-linear grammar, and asked them why, according to the definition, the example grammar ... 15 Many high school geometry textbooks define an angle as simply the union of two rays with a common endpoint The advantage of this definition is its simplicity. Among its disadvantages: It does not serve well for capturing the idea of a "direction": That is, there is no way to distinguish between a clockwise and a counterclockwise rotation. It more or ... 15 A linear function is not necessarily a first degree polynomial function: zero function is also linear. In France the terminology is more appropriate than the traditional English one: a linear function is a function of the form$f(x) = ax$, while a function of the form$f(x) = ax + b$is called an affine function. So, strictly speaking, constant functions ... 14 I think a lot depends on context -- what course you are teaching and what the characteristics of that course are. If by "Freshman college calculus" you mean what I think you mean -- namely, a non-rigorous course that treats a lot of things using intuitive and heuristic arguments -- then I wonder if "define" is even the right verb to use. Students come into ... 14 I've never tried this in a classroom, but I suspect a lot of the trouble with functions is that students haven't been taught the vocabulary to deal with things that are weaker than functions. For example, they are trying to define a function$f: A \to B$. What they have written down probably defines SOMETHING: Perhaps a subset$R$of$A \times B'$for some$...
14
This is a very difficult question to answer; I recommend as a first place to look: Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. The ideas of algebra, K-12, 8, 19. Link (no paywall). As Usiskin writes (emphasis in original): My thesis is that the purposes we have for teaching algebra, the conceptions we have of the subject, and ...
13
I'm going to rewrite this answer to clarify what I think the issue is. I think the OP is imagining a different definition of the ring $k[x]$ than most answerers are. Here are two reasonable definitions: $k[x]$ is the ring of formal expressions of the form $\sum_{j=0}^{\infty} p_j x^j$ with $p_j \in k$ and we require that $p_j$ is $0$ for $j$ sufficiently ...
12
The situation does indeed change your interpretation of the terminology. Volume is a general concept of the amount of 3D space something takes up. For a solid object like a brick, or for a liquid there is no ambiguity and you can simply do a calculation (like $V = \pi r^2 h$ for a solid cylinder), or possibly compare its mass to the mass of a known volume ...
12
While I haven't done a systematic survey, my impression is that the overwhelming majority of pre-calculus and calculus texts define a linear function to be one of the form $f(x) = mx + b$ with no stipulation that $m \neq 0$. Thus, if you define linear to be a polynomial of degree 1 you are likely to be contradicting whatever textbook you are using. From a ...
11
I think that you hit the nail on the head, when you said some are not even aware of what well-defined means. As Anschewski suggests the problem may be that students have not encountered enough non-well-defined operations to fully appreciate the problem. This Spring I was teaching freshman algebra, and while explaining equivalence relations (prior to getting ...
11
Turning my comment into an answer as per request: I think the graphical approach gets the idea across. You could represent $3\times 1$ by a row of three dots; likewise, $3\times 2$ is two rows of three dots; etc. If you asked them to guess what $3\times 0$ is, some would probably tell you you'd have no rows so no dots, so the answer is zero. Humans are very ...
11
Have a look at the paper written by Nunez et all: EMBODIED COGNITION AS GROUNDING FOR SITUATEDNESS AND CONTEXT IN MATHEMATICS EDUCATION. In essence, they argue that it is better to be causious if you want to "motivate the formal definition of continuity starting from the intuition" you have suggested in your question. In the following passage, "natural ...
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https://www.smogon.com/forums/threads/ultra-sun-ultra-moon-battle-mechanics-research-read-post-2.3620030/page-10 | # Ultra Sun & Ultra Moon Battle Mechanics Research (Read Post #2)
#### VCrakeV
When does a Pokemon get burnt if attacking a Pokemon using Beak Blast? Is it before or after damage calculation?
It checks and deals damage first, then causes burn.
#### sumwun
Can fling activate magician? Bulbapedia says it can't, but I think it did in a Showdown! battle.
what types exactly are impostored arceus formes without plates? this keeps getting changed back and forth on showdown.
http://replay.pokemonshowdown.com/gen7balancedhackmons-870175890
right now it is always the underlying type that gets copied ie. normal unless having a plate, but just a few weeks earlier it was the superficial type
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#### Wheaties
Do Photon Geyser and Light-That-Burns-The-Sky take into account the attacker's abilities? For example, I would expect skill swapping Pure Power onto Ultra Necrozma would affect the power and physical/special designation of those moves. (not sure if that situation is possible in game, but the question still stands)
Photon Geyser determines which stat is higher after applying stat boosts, but nothing else. Huge Power and Choice Band will both be ignored.
Imprison itself was bugged on sim earlier today. It's a self-targeting move, and should never have an opportunity to run into Magic Coat or Magic Bounce.
Weight mechanics:
• Wailord (398.0 kg) holding a Float Stone, and with its ability swapped to Light Metal, is hit for 80 power by Low Kick. This proves that both Float Stone and Light Metal are cumulative, reducing the weight to 99.5 in this case.
• Raichu (30.0 kg) gets 80 power from using Heavy Slam on Reuniclus (20.1 kg) @ Float Stone. This proves that weight-tracking has a granularity of 0.1 kg, and halving effects round down (so 20.1 becomes 10.0, not 10.1 or 10.05).
• Azelf (0.3 kg) gets 80 power from using Heavy Slam on Kartana (0.1 kg) @ Float Stone. This proves that weight has a minimum of 0.1, and cannot be reduced below that. In fact, the same is true if Azelf is also holding a Float Stone and has its ability swapped to Heavy Metal, which proves that Heavy Metal must apply before Float Stone (otherwise Azelf would round down from 0.15 to 0.1, and only double back up to 0.2).
• Celesteela (999.9 kg) with its ability swapped to Heavy Metal gets 80 power from using Heavy Slam on Avalugg (505.0 kg). This proves that unlike the minimum, there is no maximum at 999.9, and Celesteela must be counting as 1999.8 to reach that power level.
• If the Wailord from before uses Autotomize, Low Kick still hits it for 80 power. This remains true whether abilities and items are swapped around before the use of Autotomize, or afterwards. What this proves is that regardless of their chronological order of acquisition, Autotomize always counts before either Light Metal or Float Stone, so 398 -> 298 -> 149 -> 74.5. If Autotomize came after one of them, the sequence would be 398 -> 199 -> 99 -> 49.5 (which is only 60 power); if it came after both, it would be 398 -> 199 -> 99.5 -> 0.1 (only 20 power).
• If a Lairon (120.0 kg) with Heavy Metal and Eviolite uses Autotomize, Low Kick hits it for 60 power. This proves that Autotomize also comes before Heavy Metal (120 -> 20 -> 40, not 120 -> 240 -> 140 which would be 100 power).
• Autotomize displays a "<Pokemon> became nimble!" message upon use, and this message can appear separately on multiple uses of the move. However, it will not appear if Autotomize failed to change the stat (likely by already being at +6), or if the user's weight is already bottomed out at 0.1 kg. Each use of the move that causes the message to appear creates a separate, cumulative weight-reduction effect, and using moves like Haze, Heart Swap, or Topsy-Turvy may change the stat levels but they will not alter or remove the existing weight reductions. To back this up, a Wailord with a normal item and ability can use Autotomize 3 times in a row and Low Kick will proceed to hit it for 80 power (three separate 100kg reductions knock it down to 98.0). After a fourth Autotomize (which can't raise the speed any higher), no message appears about becoming nimble, and Low Kick still hits for 80. After a Haze and a fifth Autotomize, Low Kick now hits for 20, showing that the weight reductions are still intact and the new one drops it all the way to 0.1.
In summary: however many Autotomize effects exist, apply them first, then Heavy Metal, then Light Metal and Float Stone (rounding down to a granularity of 0.1 kg, and also a minimum of 0.1kg).
Note that while Haze does not remove Autotomize's weight-loss effect, any form change does. This most commonly happens with Aegislash, but it could also feasibly be a factor for, say, an Aggron that holds off on going mega for a turn in order to get a speed boost with Sturdy insurance. Both Rock Polish and Autotomize exist as options for this, but Autotomize might be used for a "psych-out" play, as it can then go mega the next turn and be back to its full 395kg, not 295, to give it that bit of extra power on Heavy Slam that the opponent might not have expected it to use based on its willingness to reduce its own weight.
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#### urkerab
weight-tracking has a granularity of 0.1 kg, and halving effects round down (so 20.1 becomes 10.0, not 10.1 or 10.05).
To avoid floating-point rounding errors, do you think the sim should store the weight in hectograms, only dividing by 10 for the /details command?
As for the sim mechanics, since Float Stone and Light Metal don't truncate to 0.1, it currently does Autotomize first and then the other three, but apart from that, all the rest of the mechanics appear to be correct.
Do you know whether Transform copies the effect of Autotomize on weight?
Do you know whether Transform copies the effect of Autotomize on weight?
Yes. If a Smeargle uses Autotomize and then transforms into something that hasn't, it will lose the weight reduction effect (this much is consistent with treating the act of transformation as a form change), but if it hasn't used Autotomize and transforms into something that has, it will gain the weight reduction from its transformed body's weight (and if they used the move multiple times, it will copy all such reductions).
On a rather unrelated note, I believe I've sorted out the full order of steps that current games make when they check for a move to be successful. If a move could fail for multiple different reasons, this chart shows which one will take precedence, and where along the way you get effects like PP subtraction or Protean triggering if the move hasn't failed yet. It's a very long list, so...
1. check for sleep/freeze (unless using a move that can execute despite that status)
2. check for disobedience
3. check for Truant loafing turn
4. if move is Focus Punch, check for prior damage this turn
5. check for flinch
6. check for confusion self-damage
7. check for full paralysis
8. check for attraction
9. check if there was no PP left for the move
10. if move is Z powered, display Z dance, and if move is status, APPLY Z effect
11. if not Z powered, check if move is Disabled
12. if not Z powered, check if move is blocked by Heal Block
13. check if move is blocked by Gravity (applies even to Z moves)
14. if not Z powered, check if sound move is blocked by Throat Chop
15. (G7) check for move failure due to being Choice-locked into a different move
16. if not Z powered, check if status move is blocked by Taunt
17. if not Z powered, check if move is sealed by Imprison
18. if user has a Choice item, LOCK into the chosen move
19. if move can call another move (Metronome/Assist/etc.), REPLACE the running move and repeat steps 12-14
20. if ability is Stance Change, CHANGE form if appropriate (in G6, move this before step 1)
21. check for move type changing due to Aerilate/Pixilate/Refrigerate/Normalize/Galvanize/Liquid Voice
22. set move type for Hidden Power/Judgment/Multi-Attack/Natural Gift/Revelation Dance/Techno Blast/Weather Ball/Z-Weather Ball
23. check for move type changing due to Electrify/Ion Deluge/Plasma Fists
24. if move is Curse, user is a ghost, and target is self (which shouldn't be valid for that version of the move), change target to a random opponent
25. check for target redirection by Lightning Rod/Storm Drain
26. check for target redirection by Follow Me/Rage Powder/Spotlight/Z-Destiny Bond/Z-Grudge
27. SUBTRACT the appropriate amount of PP from the move (if the move called another move at step 19, use the original move for PP subtraction)
28. check for move condition failures, part 1
Sucker Punch/Me First: target doesn't have an eligible move pending
Burn Up: user is not Fire type
Fake Out/First Impression/Mat Block: user has already performed an action
protection moves: automatically fail if there are no other pending moves this turn, otherwise check for failure chance if repeated
(G7) Destiny Bond: move is repeated and previous use was successful
Fling/Natural Gift: Embargo or Magic Room are in effect, or ineligible held item
Stockpile: existing Stockpile count is 3
Swallow/Spit Up: existing Stockpile count is 0
Bide: no energy to unleash
Counter/Mirror Coat/Metal Burst: target hasn't hit with an eligible damaging move this turn
Encore: target hasn't used a move, or has no PP left on the move, or move is exempt from the effect
Hyperspace Fury/(G7) Dark Void: ineligible user
Rest: user has full HP, or is already asleep (through Sleep Talk, or user's Comatose)
Rest: user has Insomnia/Vital Spirit (this check happens after the previous one)
Snore/Sleep Talk: user isn't asleep
Future Sight/Doom Desire: target slot already has the future attack effect
29. if move allows self-defrost, UNTHAW the user
30. check if move is blocked by Primordial Sea/Desolate Land
31. check for Fire move failure and backfire due to Powder
32. if move is Future Sight/Doom Desire, CREATE the effect and EXIT (does not count as failing the move)
33. if move is part 1 of an eligible Pledge combo, fail the current move and TRIGGER teammate to pick up the conclusion (they start over from step 1)
34. check if move is blocked by Damp
35. if ability is Protean and user can change type with it, CHANGE type to match with move
36. if move requires charging, APPLY Defense boost if Skull Bash, and unless it's Solar Beam/Solar Blade in sun, or Power Herb can be consumed, SUSPEND the move until next turn (does not count as failing)
37. check for move failure due to lack of target (includes Helping Hand/Ally Switch in singles)
38. check if move is STOLEN by a pending Snatch (go back to step 12 for the Pokemon who snatched it)
39. if move is Explosion/Self-Destruct/Mind Blown, LOSE 100/100/50% of own max HP respectively, but don't fail the move from user fainting
40. check for move condition failures, part 2
Rest: Uproar is in effect
recovery moves other than Rest: user has full HP
Substitute/Belly Drum: not enough HP to make one
Conversion/Camouflage: user is already the type it would change to
Conversion 2: no prior move, or all resistances to the move's type (if any) are already in user's own types
Sky Drop: target is a teammate
41. check for move failure due to target in Fly/Dig/Dive/Bounce/Sky Drop/Phantom Force/Shadow Force
42. check if priority move is blocked by Dazzling/Queenly Majesty
43. check if priority move is blocked by Psychic Terrain
44. if move is Sky Drop, check for failure due to target too heavy (200kg)
45. check if move is blocked by Quick Guard/Wide Guard/Crafty Shield (if Z powered attack, only 75% reduction)
46. check if move is blocked by Protect/Detect/King's Shield/Spiky Shield/Baneful Bunker (if Z powered attack, only 75% reduction)
47. check if move is reflected by Magic Coat
48. if move is Telekinesis, check for failure vs. Diglett/Dugtrio/Sandygast/Palossand/Gengar-Mega, Smacked Down, or Ingrained target
49. check if move is reflected by Magic Bounce
50. check if move is blocked by Mat Block (if Z powered attack, only 75% reduction)
51. check for ability-based immunities, part 1
Sap Sipper: Grass type moves
Flash Fire: Fire type moves
Dry Skin, Storm Drain, Water Absorb: Water type moves
Lightning Rod, Motor Drive, Volt Absorb: Electric type moves
Soundproof: sound moves
Telepathy: damaging moves from a teammate
Wonder Guard: move is a type that isn't super effective
52. if non-status move or Thunder Wave, and target does not hold Ring Target, check for type chart immunity (G6: move this check to step 42)
53. check for Levitate immunity (G6: move this check to step 43)
54. check for Magnet Rise/Telekinesis/Air Balloon immunity
55. check for Overcoat immunity
56. check for Safety Goggles immunity
57. check for type-based move condition immunities
powder moves: Grass type
burning moves: Fire type
poisoning moves: Poison/Steel type (unless user has Corrosion)
paralysis moves: Electric type
(G7) Sheer Cold: Ice type
Mean Look/Spider Web/Block: Ghost type
Sky Drop on the way down: Flying type
58. check for other move condition immunities
Synchronoise: non-matching type
Dream Eater/Nightmare: target is awake
Attract: gender failure
Endeavor: target has less HP than user
Fissure/Horn Drill/Guillotine/Sheer Cold: target has higher level than user
59. check for ability-based immunities, part 2
Sturdy: OHKO moves
Sticky Hold: Trick/Switcheroo
Bulletproof: bullet/bomb moves
Oblivious: Attract/Captivate/Taunt
60. if move is Captivate, check for gender immunity
61. if move is Curse, user isn't a ghost, and target isn't itself, change target to self
62. check for generic move failure due to redundancy
status affliction when target already has one
attempted creation of weather/field effect/pseudo-status that already exists
stat changing moves that can't go any higher or lower
Trick/Switcheroo/Bestow with no item, or unmovable item
Instruct/Disable/Sketch vs. no move, or ineligible move (note that Encore is checked earlier)
Healing Wish/Lunar Dance with no one to switch to
various other moves whose effects would naturally do nothing
etc.
63. check if move is blocked by Safeguard
64. check if move is blocked by Electric Terrain/Misty Terrain
65. check if move is blocked by Sweet Veil or Comatose/Shields Down/Insomnia/Vital Spirit/Leaf Guard
66. check for move accuracy
67. check if move is blocked by target's Substitute
68. check if move is blocked by Mist
69. check for ability-based immunities, part 3
Clear Body/White Smoke/Full Metal Body: all stat drop moves
Hyper Cutter: attack drop moves
Big Pecks: defense drop moves
Keen Eye: accuracy drop moves
Aroma Veil: Attract, Disable, Encore, Heal Block, Taunt, Torment
Flower Veil: all stat drop moves, all status inflicting moves
Water Veil, Water Bubble: burn moves
Immunity: poisoning moves
Limber: paralysis moves
Own Tempo: confusion moves
Suction Cups: Roar, Whirlwind
70. if move is Roar or Whirlwind, check for failure due to Ingrain
71. if move's priority was elevated due to Prankster, check for Dark type immunity
72. if move is Brick Break/Psychic Fangs/Spectral Thief/Pay Day, APPLY its side effect now
73. perform move
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#### urkerab
Yes. If a Smeargle uses Autotomize and then transforms into something that hasn't, it will lose the weight reduction effect (this much is consistent with treating the act of transformation as a form change), but if it hasn't used Autotomize and transforms into something that has, it will gain the weight reduction from its transformed body's weight (and if they used the move multiple times, it will copy all such reductions).
Looking at the sim, it suggests this is a change from Gen 6, where it used to keep its own Autotomize weight loss rather than copying its target; can you confirm this? Edit: Also, the forme change resetting weight mechanic is new in Gen 7?
#### sumwun
Can sleep talk call currently disabled moves? Also,
Can fling activate magician? Bulbapedia says it can't, but I think it did in a Showdown! battle.
#### viol and bass
Is Future Sight boosted by Psychic Terrain? If so, does it check for terrain on the turn it hits or the turn it’s selected?
#### DaWoblefet
##### Demonstrably so
Can sleep talk call currently disabled moves?
Yes, Sleep Talk can call Disabled moves.
Can fling activate magician? Bulbapedia says it can't, but I think it did in a Showdown! battle.
Using Fling with a Magician Pokemon into another target that has a stealable held item will not steal that held item on cartridge. You were correct that the interaction is currently bugged on Showdown.
Is Future Sight boosted by Psychic Terrain? If so, does it check for terrain on the turn it hits or the turn it’s selected?
From Marty's profile:
Marty: Psychic Terrain has to be active when Future Sight hits, and will only boost it if the user is grounded at that time, even if it wasn't grounded when the move was used. If the user is currently switched out, it's always considered grounded unless it's a Flying type (a Pokemon's Ability and item aren't checked while it's inactive).
Aurella: Okay, so to describe to what I think you are saying: Levitate Bronzong uses Future Sight > Tapu Lele switches in on the turn that future sight will hit > since Levitate Bronzong (future sight user) isn't active on the field (since Tapu Lele is in currently) then terrain will boost the future sight attack since the original user of the move is not in field and therefore it's ungrounding ability isn't checked during terrain/damage calculation.
Marty: That's correct, yeah.
Turns out there's still more bitwise rollovers to be found!
So I had this ordinary level 1 Alola Geodude, 6 points in Attack, with Galvanize, Muscle Band, and the moves Defense Curl/Rollout/Charge/Explosion. Selecting each of those moves once and in order, Rollout had the obligatory Mimikyu hit to store a 32x multiplier. Then immediately after Charge, while Electric Terrain had been summoned, the rest of the field was: L100 Mantine with 209 Def/373 HP, L80 Wishiwashi with 237 Def/186 HP, and L50 Oranguru with 118 Def/165 HP. Mantine gave Geodude a helping hand, instructed once by Mantine, to cheer on the incoming explosion.
Those stats were calibrated so that, based on existing knowledge of Rollout storage, the Explosion would take everyone into the red, without any chance of a KO except on a critical. But this setup was also chosen for another reason: Helping Hand, Muscle Band, Galvanize, and Charge are all multipliers that work specifically on the move power, before we get to any other part of the formula. When we tally everything up, this Explosion should have a nominal power of 71,273, which happens to exceed 16 bits--so what I was curious about was whether the move really did count as the full 71273 power, or if they put the same 16-bit limitation on power that they also put on stats, in which case I could expect it to wrap around to 5737 power, and in that case the Explosion would do a bit less than 10% damage to everyone.
The end result was a bit surprising: Mantine took 28 damage, Wishiwashi took 6, and Oranguru took 4. The 28 figure is consistent with what a 5737-power move should do, implying a rollover there, but the other two figures appear too low. Even at the lower, rolled-over power, both opponents should have taken at least 12 damage.
Turns out if you use the lingering multiplier from Rollout storage on a spread move, it considers each target one at a time. First is your own teammate (if the move includes them in its targeting scope, as Explosion does), then the opponent's left slot (the one on your right), then the opponent's right slot. With the teammate coming first, it uses up the stored 32x power multiplier, then there's no multiplier left to see when it comes to checking damage against the opponents. So instead of 71273 or even 5737 power, Explosion was counting for just 2227 against them, as you would get from putting Galvanize, Muscle Band, two Helping Hands, Charge, and Electric Terrain onto a starting point of 250 rather than 8000. Presumably in the further-desynced version where Rollout breaks two or more disguises, it proceeds the same way--one at a time, consider each target in order and play "take a number" until there are no more numbers left in the dispenser to take. It's not a mechanic that's likely to have serious competitive impact in any case, but now I know this is how it works.
On a related note, since this proves that move power is capped at 16 bits, and I already knew there were only two bytes devoted to each stat in the Pokemon data structure, it seemed like a done deal that in-battle calculated stats for the damage formula behaved similarly. It's not possible to explore this without hacking: the highest you can ever legally get the stats in question, no matter how contrived, is 14712 in Attack and 9808 in Defense, neither of which is even a quarter of the way there (and in Custom Game, where it is possible to get stats that high, Showdown removes all bitwise rollovers as a matter of policy). So this is purely a curiosity.
If you're going to hack stats to unnatural levels, you might as well make them clean unnatural levels, so for example you could set a Pikachu's attack stat to 16384 then give it Huge Power and a Light Ball to go directly to 65536, wrapping around to a clean 0. 0-point attack stats can be realized even without hacking, by using the effects of Slow Start or Defeatist, and as expected, a 65536-point Attack stat behaves just like a 0. You do very little damage, even against weak opponents, not much to say about that.
The analogous case for the defense stat is a bit more interesting. First off, there's nothing like Slow Start that works on defensive stats to allow those to realize a 0-point value naturally, so we don't have any hackless precedent in the current games to give us an idea of the behavior we should expect there. And in older games where we do have such prior precedent (the first two gens where you could have 256 Attack vs. 3 Defense, and it would scale down both stats one time to try and make them fit within 8 bits), the result is a game crash, and that outcome wouldn't be helpful to anyone.
The good news is games are more robust now than they were back then, and this doesn't cause a crash for attempting to divide by 0. Actually, the tester made a typo the first time and put in the starting defense stat as 16834 instead of 16384, on a Ditto with Fur Coat and Metal Powder. Doubling the wrong value twice gives 67336, which wraps around to 1800, and if anything that gave us an additional useful data point. 67336 did appear to act just like a stat of 1800, taking little damage from physical attacks but not being so completely incomprehensible on the scale of opponents' stats, at least when they have reasonably high levels.
After they corrected to the right value, the defense hit a value that should be wrapping around to 0. But what does that even mean? Well, throwing several attacks at the Ditto to see what sticks, against low- and high-level opponents alike, they were all ending up around 1-8 damage depending on factors such as STAB, criticals, and the spread move penalty. Some of them should have been plenty strong enough, such as a Machamp Dynamicpunch, to prey on an ostensible 0 defense, but only ended up dealing 2 damage anyway.
With a variety of permutations at hand, I believe I have an explanation for all of them. As we learned from Gyro Ball, the game really doesn't like trying to divide by 0, but instead of outright crashing, now they just make an emergency exit from the routine. In Gyro Ball's case, it sets the move power to the minimum 1, and here they seem to be breaking out of an attempted X / 0 calculation by returning 0. This means you can effectively think of it as a 0-point defensive stat (albeit after rollover in this case) gets treated as an infinitely high stat. Yes, this is the opposite of what you'd expect from a low stat, just like how being slow is normally good for powering up your Gyro Ball, but reaching the ultimate in slowness ends up reducing the move to minimum power.
After the 0 return, there are still a few things left to put in the formula. The immediate following step is the static "+2", which takes you from 0 to 2, and then the usual suspects: spread move penalty, weather, critical, random variation (which at this stage is really just "-1 damage, 15/16 of the time"), STAB...but none of them have anything to do with either side's stats at this point, and the end result will usually be in single digits from what little building can be done from that base of 2.
[UPDATE] As you may recall, the multipliers for stat stages/ranks go like this: -6 is 2/8, -5 is 2/7, ... +0 is 2/2, ... +6 is 8/2. The denominator in every case is at least 2, and they don't try to simplify the fractions...so of course, the stat is still capped at 16 bits even during the middle of this calculation. It can never happen in a real match with legal stats, but this means that during this step, stats effectively roll over at half the cap, or 32768. If you somehow had a raw stat of 8193 and went to +6, that would be 8/2 so you'd take 8193 * 8 (65544), wrap around to 8, and divide by 2 to get the effective stat value: 4 (although you could go on to boost this value with things like Choice Band).
I have a low-level damage calculation spreadsheet that you can make a copy of, and explore the nitty-gritty details of every possible rollover condition thus far.
The more you know?
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#### DaWoblefet
##### Demonstrably so
If you're going to hack stats to unnatural levels, you might as well make them clean unnatural levels, so for example you could set a Pikachu's attack stat to 16384 then give it Huge Power and a Light Ball to go directly to 65536, wrapping around to a clean 0. 0-point attack stats can be realized even without hacking, by using the effects of Slow Start or Defeatist, and as expected, a 65536-point Attack stat behaves just like a 0. You do very little damage, even against weak opponents, not much to say about that.
I do have one thing to add! With SadisticMystic's research on 16-bit truncation for damage, it was demonstrated that this 65535 maximum is enforced after the check to make sure damage is at least 1. This allows us to deal 0 damage with an otherwise super-powered attack, as I have also showed with an independent setup. Typically, if a Base Power, Attack, or Defense would be 0, the game makes sure it becomes 1 prior to the start of the base damage calculation (seen below). However, a similar occurrence happens for the Attack and Defense should the effective Attack or Defense reach a multiple of 65536; when it's wrapped around back to 0, it stays at 0, and does not get bumped back up to one, and so 0 gets used in the initial base damage calculation.
Suppose you have a case like a level 100 Smeargle Transformed into a level 1 -6 Attack Archen that has 6 or 7 Attack, with Smeargle's HP being low enough for Defeatist to be active. Smeargle-Archen's Attack would first be quartered by the -6 stage in attack, and the result floored. Floor(6*0.25) = Floor(1.5) = 1. Defeatist further cuts this in half, and this time is pokeRounded down from a mere 0.5 to a lonely 0. However, this is bumped back up to 1 prior to the base damage because of an explicit check by the game to make sure an attack stat can't actually be 0 (little did they know we can bypass this with enormously high Attack stats!).
In my test with my Level 100 16384 Attack Huge Power Light Ball Pikachu, I dealt exactly 1 damage with a Double-Edge into a level 2 Buneary with 7 Defense. If you took this 16-bit cutoff to occur prior to the 1 Attack check, then our starting value for Attack should have been 1, not 0. But this doesn't line up with the math! Let's see what happens if we assume the Attack was 1:
$\dpi{150}&space;Base&space;Damage&space;=&space;Floor\left&space;\lfloor&space;\frac{Floor\left&space;\lfloor&space;\frac{Floor\left&space;\lfloor&space;\frac{2&space;\times&space;Level}{5}&space;+&space;2&space;\right&space;\rfloor&space;\times&space;Base&space;Power&space;\times&space;Attack}{Defense}&space;\right&space;\rfloor}{50}\right&space;\rfloor&space;+&space;2$
Plugging in our values, 100 for Level, 120 for Base Power, 1 for Attack and 7 for Defense, our base damage should have been 16. The random factor will allow this attack to have a range of damage from 13-16, but it cannot mathematically hit 1 damage.
However, if you had an initial value of 0 Attack, then the entire calculation gets essentially "zeroed out", leaving a lone 2 in its place. With the effect of the random factor, it should be possible to deal 1-2 damage with this attack - and that's exactly what Double-Edge did. Given that Defense enters the base damage formula as 0 as well (which is why we had to explain why the game didn't crash in an attempt to divide by 0), it seems consistent to say the following:
• For Attack, Defense, Base Power modifiers, and the final damage done at the end of damage calculation: after doing checks to make sure its value is at least 1, apply a 16-bit cutoff, so that the maximum amount of Attack, Defense, BP, or actual damage can at most be 65535.
EDIT: Speed also behaves in this way. A hacked Ditto with 32768 Speed and Quick Powder moved after a level 3 Caterpie. This notably implies the 16-bit check is done prior to the check for 10000 Speed.
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#### Mijzelffan
Zoom Lens' description says "If the holder moves after the foe, its accuracy will be boosted.", does this include switches? I can't find research on this anywhere, and this mechanic has been known to differ between for instance analytic and payback.
#### Marty
##### Always more to find
Zoom Lens' description says "If the holder moves after the foe, its accuracy will be boosted.", does this include switches? I can't find research on this anywhere, and this mechanic has been known to differ between for instance analytic and payback.
Analytic and Payback behave differently because they're different mechanics. Payback checks if the target had its place in turn order before the user, and Analytic checks if the user moved last in turn order.
The difference between Payback in Gen 4 and Gen 5+ is that the action of switching in changed from counting as your position in turn order to counting as nothing for turn order, which is why it's boosted against switches in Gen 4 but not since then.
Since Analytic checks the user's own position in turn order, in Singles if your opponent switches out then the switch-in doesn't have a place in turn order, so even though technically you moved first you also moved last and you get the boost.
So to answer your question, Zoom Lens behaves the same way as Payback. Your target has to have been on the field and taken its turn before you.
#### VCrakeV
If a Pokemon holding a colbur berry is hit with knock off, can the berry be brought back with recycle or harvest?
Focus Sash, Colbur Berry, Enigma Berry, and Weakness Policy (assuming the target is weak to dark in the latter three cases) all get used up early enough that Knock Off will not print a message about knocking them off. Those items are then still eligible to be recovered with Recycle.
#### Feria-Aeris
I'm going to ask a weird question and use a weird replay for it, but I was told y'all might have an answer for me...
https://replay.pokemonshowdown.com/gen7metronomebattle-993197978
How was Kakuna's Synchronoise able to hit my Gallade and Gardevoir? Isn't that not supposed to happen? The current hypothesis is that since the other Kakuna was hit, the other guys were hit too... Any insight on this? | 2019-11-15 10:06:28 | {"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": 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": 0.29715442657470703, "perplexity": 4068.1411543621825}, "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-2019-47/segments/1573496668618.8/warc/CC-MAIN-20191115093159-20191115121159-00543.warc.gz"} |
https://www.aa.quae.nl/cgi-bin/glossary.cgi?l=en&o=Cyllene | Astronomy Answers: From the Astronomical Dictionary
Astronomy AnswersFrom the Astronomical Dictionary
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The description of the word you requested from the astronomical dictionary is given below.
Cyllene
A moon of 2 km diameter at about 24,349,000 km from the planet Jupiter. The gravity at its surface is about 0.0001 times as strong as on Earth. The moon goes once around its planet in about 776.0 days. The moon was discovered in 2003. Also called JXLVIII (Jupiter forty-eight). Its provisional designation was S/2003 J13. | 2018-11-21 03:04:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.4458802342414856, "perplexity": 3092.8021710913713}, "config": {"markdown_headings": false, "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-2018-47/segments/1542039746926.93/warc/CC-MAIN-20181121011923-20181121033923-00484.warc.gz"} |
https://ktln2.org/2017/08/31/rhme3-exploitation-writeup/ | # Writeup CTF RHME3: exploitation
Also this year there will be a CTF from Riscure mainly targeted for hardware security people, but before that, from the 8th of August until the 28th there was the qualification phase: three challenges to solve in order to qualify and to receive a physical board with the real challenges.
In this post I’ll write about the only challenge that I was able to solve named exploitation; the other two weren’t too complicated but more about side channel attacks and very specific tools that I didn’t know about so I preferred to do not waste too much of my time and wait to learn from other people writeup (by the way, I was very near the solution of the tracing challenge, only matter of changing the algorithm in deadpool but this is a story for another time).
The challenge was a remote one, but the binary (main.elf) was provided together with its system library (libc).
The protections (all but PIE are enabled)
$checksec --file main.elf [*] 'main.elf' Arch: amd64-64-little RELRO: Partial RELRO Stack: Canary found NX: NX enabled PIE: No PIE (0x400000) hints us that probably a ropchain would be necessary. ## Preliminaries First of all was necessary to find the port number to which connect to the server: using radare2 I reversed the main() function [0x00400ec0]> s main [0x004021a1]> pd 20 ;-- main: / (fcn) main 205 | main (); | ; var int local_9h @ rbp-0x9 | ; var int local_8h @ rbp-0x8 | ; var int local_4h @ rbp-0x4 | ; DATA XREF from 0x00400edd (entry0) | 0x004021a1 55 push rbp | 0x004021a2 4889e5 mov rbp, rsp | 0x004021a5 4883ec10 sub rsp, 0x10 | 0x004021a9 c645f700 mov byte [local_9h], 0 | 0x004021ad bf18264000 mov edi, 0x402618 | 0x004021b2 e87ceeffff call sym.background_process | 0x004021b7 bf39050000 mov edi, 0x539 | 0x004021bc e85eefffff call sym.serve_forever I found at address 0x004021b7 that a value of 0x539 is passed as argument to serve_forever(); using the numerical conversion functionality inside my reversing tool I obtained [0x004021a1]> ? 0x539 1337 0x539 02471 1.3K 0000:0539 1337 "9\x05" 0000010100111001 1337.0 1337.000000f 1337.000000 trying that value as port number now I can connect to the challenge:$ nc pwn.rhme.riscure.com 1337 Welcome to your TeamManager (TM)! 0.- Exit 1.- Add player 2.- Remove player 3.- Select player 4.- Edit player 5.- Show player 6.- Show team Your choice:
The application setup routine checks for a specific path and user existence, if they don’t exist it exits. Otherwise it forks in order to daemonize() itself and forks again when a connection is made. This makes attach gdb to the process a pain in the ass.
To make easier to debug the executable I preferred to nop the binary in the daemonize part using again radare2
$cp main.elf main_modified.elf$ r2 -w -A main_modified.elf [x] Analyze all flags starting with sym. and entry0 (aa) [x] Analyze len bytes of instructions for references (aar) [x] Analyze function calls (aac) [x] Use -AA or aaaa to perform additional experimental analysis. [x] Constructing a function name for fcn.* and sym.func.* functions (aan) -- Change the size of the file with the 'r' (resize) command [0x00400ec0]> s 0x004021ad [0x004021ad]> wx 9090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090 [0x004021ad]> o 3 * main_modified.elf : -rw- size=0x4c68 +0x0 0x0 - 0x4c68 : -rw- : [0x004021ad]> wc
Now it was possible to have a not forking executable that uses its own stdin and stdout for interact with the user, much more easy to deal with.
Otherwise you can this Dockerfile
FROM ubuntu:17.04 RUN apt-get update RUN apt-get upgrade -y #RUN apt-get install -y gdb RUN adduser pwn --home /opt/riscure/pwn COPY main.elf /opt/riscure/pwn COPY libc.so.6 /opt/riscure/pwn EXPOSE 1337 ENV LD_LIBRARY_PATH=/opt/riscure/pwn CMD "/opt/riscure/pwn/main.elf"
with which you can create a container and launch it with the followind command
$docker build -t rhme3/exploitation .$ docker run -p 1337:1337 rhme3/exploitation
Now you have the challenge running at localhost.
## Analysis
It seems a kind of players list application; after a little bit of usage a strange pattern appears: in order to visualize and edit, you need to select before a player, instead to remove one you enter the index after, this weird UI is probably source of some bugs.
Indeed if you select a player, remove it and then visualize it you obtain a strange looking output. we have here an user after free vulnerability! Below an example
0.- Exit 1.- Add player 2.- Remove player 3.- Select player 4.- Edit player 5.- Show player 6.- Show team Your choice: 5 Name: A/D/S/P: 39363632,0,1,1
With a well-crafted input this should become an info leak usable to defeat ASLR, so studying the data structure and input retrieving is the next step.
Analyzing the way in which the player data is accessed I suppose that the original data structure could be the following (in C):
struct _player { uint32_t attack; uint32_t defense; uint32_t speed; uint32_t precision; char* name; }
When a player is created there are two allocation in the heap, the first one is of 0x20 bytes and it’s for this data structure.
The name is allocated with enough space to contain it, but take in mind that there is a limit of 256 bytes in the character name (including the NULL terminating byte) when processed by the function readline(). When a name is modified and extended then the buffer’s address is passed to realloc(). All the data passing is done via strcpy(), so we cannot insert a NULL byte in the middle of a payload.
## Setup the weird machine dancing with the heap
How to exploit this UAE? first of all we need to understand how the heap usually works: here is used the glibc’s allocator but the internals are not necessary (if you are interested this is a must read), the only thing that matters is that, for performance reason, when a block of a given size is deallocated is put in a LIFO where will be available for future allocation of the same size. Technically speaking this is true only for allocation with size lower than 0x80 bytes and the list of unallocated objects is called fastbin.
Our target is to create a player having the name allocated where we previously had a selected player: this is possible just playing with the allocations.
The rules to take in mind for the allocations/deallocations of a player are
1. the last unallocated chunk with the same size is used for a new allocation with the same size
2. the player’s name is the last allocated when a player is added and the first deallocated when a player is removed
3. the player’s data structure has always the same size of 0x20 bytes
4. the player’s name can have a size up to 255 bytes.
Below I used some diagrams created with villoc to better explain the process.
First of all we create two players, one with a name that can fit in a 0x20 bytes:
Then I add a second player with a name that doesn’t fit into 0x20 bytes
At this point I select the first player and I’m ready to alter the heap configuration with my dance: I remove the first player
and then remove the second:
now we have a fastbin with 3 elements with size 0x20 and one with size 0x30.
Adding a new player with the name fitting in 0x20 I obtain a name allocated where the first player’s data structure was!
## R/W primitives vs the GOT
With this configuration we have now a read/write primitive to an user controlled address: using a GOT entry inside the executable as part of the name of the player we can now read what’s the address resolved by the dynamic loader so to bypass ASLR; then using a write I can overwrite that address to the one pointing to a function more valuable: if I overwrite the GOT entry for the free() function with the system(), it’s possible to execute a shell simply removing a player with name /bin/sh;)
This approach has two drawbacks:
• you cannot write the bytes \n and \0, so if your address has a digit with these values you need to change the overwritten function if possible or use a more generic primitive.
• it’s one-shot: you cannot change the address without redoing the dance
## Exploit
For completeness below you can read the exploit used to solve the challenge, it needs the executable and its system library in the same directory where this file lives. Remember to export LD_LIBRARY_PATH if you want to try it and to install pwntools.
The important parts are the functions setup() and __main__.
$python exploit.py [*] trying to overwrite got @0x603018 [*] libc function address 0x7f676dc944f0 [*] libc base address 0x7f676dc10000 [*] one gadget at 0x7f676dc55390 [*] Switching to interactive mode$ ls flag main.elf \$ cat flag RHME3{h3ap_0f_tr0uble?}
It was a very interesting challenge: didn’t use the very common buffer overflow but a simple logic bug; in particular shows that also with a number of modern protection the exploiting it’s very possible. Probably enabling PIE should have make more troublesome the creation of a weird machine. | 2017-12-16 12:37:29 | {"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": 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": 0.19084933400154114, "perplexity": 3140.8661247710347}, "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/1512948588072.75/warc/CC-MAIN-20171216123525-20171216145525-00765.warc.gz"} |
https://www.physicsforums.com/threads/position-function-of-time.334567/ | # Position Function of Time
1. Sep 3, 2009
### Zhalfirin88
New to PF, so here it goes. Oh just a question, did this site use to have a dark skin? I can't remember if that was this or something else.
1. The problem statement, all variables and given/known data
The position of a particle as a function of time (in s) is given by C1 + C2t + C3t2. Let C1 = 11.0 m, C2 = 9.5 m/s and C3 = -0.49 m/s2. What is the velocity of the particle at time t = 15.0 s? And what is the particle's acceleration at time t = 15.0 s?
2. Relevant equations
Velocity is the derivative of position.(How do you make it look all nice?)
Acceleration is the 2nd derivative of position.
3. The attempt at a solution
Where do I start? I don't even understand the equation that's given. Judging from the question (it's online) it looks like those t are variables and not subscripts.
Last edited: Sep 3, 2009
2. Sep 3, 2009
### Staff: Mentor
Welcome to the PF. The skins have changed several times, so it's possible that a previous skin was dark.
The "t" in the equation is time, so you would just plug in the time to get the position as a funtion of time. You are correct about the velocity being the derivative of the position, and then the acceleration is the derivative of the velocity. Using LaTex, it looks like this:
$$x(t) = C_1 + C_2 t + C_3 t^2$$
$$v(t) = \frac{dx(t)}{dt} = \frac{d(C_1 + C_2 t + C_3 t^2)}{dt}$$
$$a(t) = \frac{dv(t)}{dt}$$
So just do the differentiations, and plug in the respective times to get the answers.
3. Sep 3, 2009
### Zhalfirin88
Just to check my derivatives since I haven't done them in months, I'm assuming that the C's aren't constants, right?
I don't know the syntax for LaTeX so, the derivative of postion would be:
C2 + 2C3t
2nd derivative would be:
2C3
4. Sep 3, 2009
### Staff: Mentor
Correct. To see how the LaTex is formed, you can just QUOTE my post, to see the tags that are embedded in it. Also, there is a stickie thread with a LaTex tutorial at the top of the Learning Materials forum:
https://www.physicsforums.com/forumdisplay.php?f=151
.
5. Sep 3, 2009
### Zhalfirin88
I don't understand why it's saying my answer for the acceleration is wrong. My velocity answer was correct though.
2C3 = 2(-0.49) = -0.98 m/s2
Thanks for the sticky, I'll check it out when I finish my homework :P
Ah never mind, forgot to type in the negative sign -.- | 2018-02-25 10:37:07 | {"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": 0.7139046788215637, "perplexity": 1084.7067299409675}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891816351.97/warc/CC-MAIN-20180225090753-20180225110753-00577.warc.gz"} |
https://gemseo.readthedocs.io/en/3.1.0/tutorials/benchmark_problems.html | # Benchmark problems¶
In this section, we describe the GEMSEO’s benchmark MDO problems:
## Sellar’s problem¶
The Sellar’s problem is considered in different tutorials:
### Description of the problem¶
The Sellar problem is defined by analytical functions:
\begin{split}\begin{aligned} \text{minimize the objective function }&obj=x_{local}^2 + x_{shared,2} +y_1^2+e^{-y_2} \\ \text{with respect to the design variables }&x_{shared},\,x_{local} \\ \text{subject to the general constraints } & c_1 \leq 0\\ & c_2 \leq 0\\ \text{subject to the bound constraints } & -10 \leq x_{shared,1} \leq 10\\ & 0 \leq x_{shared,2} \leq 10\\ & 0 \leq x_{local} \leq 10. \end{aligned}\end{split}
where the coupling variables are
$\text{Discipline 1: } y_1 = \sqrt{x_{shared,1}^2 + x_{shared,2} + x_{local} - 0.2\,y_2},$
and
$\text{Discipline 2: }y_2 = |y_1| + x_{shared,1} + x_{shared,2}.$
and where the general constraints are
\begin{align}\begin{aligned}c_1 = 3.16 - y_1^2\\c_2 = y_2 - 24.\end{aligned}\end{align}
The Sellar disciplines are also available with analytic derivatives in GEMSEO, as well as a DesignSpace:
### Creation of the disciplines¶
To create the Sellar disciplines, use the function create_discipline:
from gemseo.api import create_discipline
disciplines = create_discipline(["Sellar1", "Sellar2", "SellarSystem"])
### Importation of the design space¶
To import the Sellar design space, use the class create_discipline:
from gemseo.problems.sellar.sellar_design_space import SellarDesignSpace
design_space = SellarDesignSpace()
Then, you can visualize it with print(design_space):
+----------+-------------+--------+-------------+-------+
| name | lower_bound | value | upper_bound | type |
+----------+-------------+--------+-------------+-------+
| x_local | 0 | (1+0j) | 10 | float |
+ + + + + +
| x_shared | -10 | (4+0j) | 10 | float |
+ + + + + +
| x_shared | 0 | (3+0j) | 10 | float |
+ + + + + +
| y_1 | -100 | (1+0j) | 100 | float |
+ + + + + +
| y_2 | -100 | (1+0j) | 100 | float |
+----------+-------------+--------+-------------+-------+
See Tutorial: How to solve a MDO problem to create the Sellar problem from scratch
## Aerostructure problem¶
The Sobieski’s SSBJ test case is considered in the different tutorials:
### Description of the problem¶
The Aerostructure problem is defined by analytical functions:
\begin{split}\text{OVERALL AIRCRAFT DESIGN} = \left\{ \begin{aligned} &\text{minimize }\text{range}(\text{thick\_airfoils}, \text{thick\_panels}, \text{sweep}) = 8\times10^{11}\times\text{lift}\times\text{mass}/\text{drag} \\ &\text{with respect to }\text{thick\_airfoils},\,\text{thick\_panels},\,\text{sweep} \\ &\text{subject to }\\ & \text{rf}-0.5 = 0\\ & \text{lift}-0.5 \leq 0 \end{aligned}\right.\end{split}
where
\begin{split}\text{AERODYNAMICS} = \left\{ \begin{aligned} &\text{drag}=0.1\times((\text{sweep}/360)^2 + 200 + \text{thick\_airfoils}^2 - \text{thick\_airfoils} - 4\times\text{displ})\\ &\text{forces}=10\times\text{sweep} + 0.2\times\text{thick\_airfoils}-0.2\times\text{displ}\\ &\text{lift}=(\text{sweep} + 0.2\times\text{thick\_airfoils}-2\times\text{displ})/3000 \end{aligned} \right.\end{split}
and
\begin{split}\text{STRUCTURE} = \left\{ \begin{aligned} &\text{mass}=4000\times(\text{sweep}/360)^3 + 200000 + 100\times\text{thick\_panels} + 200\times\text{forces}\\ &\text{rf}=3\times\text{sweep} - 6\times\text{thick\_panels} + 0.1\times\text{forces} + 55\\ &\text{displ}=2\times\text{sweep} + 3\times\text{thick\_panels} - 2\times\text{forces} \end{aligned} \right.\end{split}
\begin{split}\text{OVERALL AIRCRAFT DESIGN} = \left\{ \begin{aligned} &\text{minimize }\text{range}(\text{thick\_airfoils}, \text{thick\_panels}, \text{sweep}) = 8\times10^{11}\times\text{lift}\times\text{mass}/\text{drag} \\ &\text{with respect to }\text{thick\_airfoils},\,\text{thick\_panels},\,\text{sweep} \\ &\text{subject to }\\ & \text{rf}-0.5 = 0\\ & \text{lift}-0.5 \leq 0 \end{aligned}\right.\end{split}
where
\begin{split}\text{AERODYNAMICS} = \left\{ \begin{aligned} &\text{drag}=0.1\times((\text{sweep}/360)^2 + 200 + \text{thick\_airfoils}^2 - \text{thick\_airfoils} - 4\times\text{displ})\\ &\text{forces}=10\times\text{sweep} + 0.2\times\text{thick\_airfoils}-0.2\times\text{displ}\\ &\text{lift}=(\text{sweep} + 0.2\times\text{thick\_airfoils}-2\times\text{displ})/3000 \end{aligned} \right.\end{split}
and
\begin{split}\text{STRUCTURE} = \left\{ \begin{aligned} &\text{mass}=4000\times(\text{sweep}/360)^3 + 200000 + 100\times\text{thick\_panels} + 200\times\text{forces}\\ &\text{rf}=3\times\text{sweep} - 6\times\text{thick\_panels} + 0.1\times\text{forces} + 55\\ &\text{displ}=2\times\text{sweep} + 3\times\text{thick\_panels} - 2\times\text{forces} \end{aligned} \right.\end{split}
The Aerostructure disciplines are also available with analytic derivatives in the classes Mission, Aerodynamics and Structure, as well as a AerostructureDesignSpace:
### Creation of the disciplines¶
To create the aerostructure disciplines, use the function create_discipline():
from gemseo.api import create_discipline
disciplines = create_discipline(["Aerodynamics",
"Structure",
"Mission"])
### Importation of the design space¶
To import the aerostructure design space, use the class create_discipline:
from gemseo.problems.aerostructure.aerostructure_design_space import AerostructureDesignSpace
design_space = AerostructureDesignSpace()
Then, you can visualize it with print(design_space):
+----------------+-------------+-------------+-------------+-------+
| name | lower_bound | value | upper_bound | type |
+----------------+-------------+-------------+-------------+-------+
| thick_airfoils | 5 | (15+0j) | 25 | float |
| thick_panels | 1 | (3+0j) | 20 | float |
| sweep | 10 | (25+0j) | 35 | float |
| drag | 100 | (340+0j) | 1000 | float |
| forces | -1000 | (400+0j) | 1000 | float |
| lift | 0.1 | (0.5+0j) | 1 | float |
| mass | 100000 | (100000+0j) | 500000 | float |
| displ | -1000 | (-700+0j) | 1000 | float |
| rf | -1000 | 0j | 1000 | float |
+----------------+-------------+-------------+-------------+-------+
See MDO formulations and scalable models for a toy example in aerostructure to see an application of this problem.
## Sobieski’s SSBJ test case¶
The Sobieski’s SSBJ test case is considered in the different tutorials:
### Origin of the test case¶
This test was taken from the reference article by Sobieski, the first publication on the BLISS formulation. It is based on a 1996 AIAA student competition organized by the AIAA/United Technologies/Pratt & Whitney Individual Undergraduate Design Competition. This competition initially focused on both the technical and economical challenges of a development.
The formulas used for each are based on semi-empirical and/or analytical models. Depending on the , some examples can be found in the following references [Niu88][And36][Ray06]. As specified by Sobieski et al., additional design and state variables (not present in the original problem) were introduced in the disciplinary analyses for testing purposes.
### The MDO problem¶
The aim of the problem is to maximize the range of a Super-Sonic Business under various constraints.
The problem is built from three disciplines: structure, aerodynamics and propulsion.
A fourth discipline, not coupled to the other ones, is required to compute the range.
### Input and output variables¶
The next tables display the input variables required by each of the four disciplines.
As they are shared by several disciplines, global design variables, implemented as $$x\_shared$$, are provided to all disciplines.
Coupling variables are implemented as $$y\_ij$$, which is an output of discipline $$i$$ and an input of discipline $$j$$.
Disciplines are listed as follows
1. Structure,
2. Aerodynamics,
3. Propulsion,
4. Range.
Variable
Description
Bounds
Symbol
$$t/c$$
Thickness to chord ratio
$$0.01\leq t/c\leq 0.09$$
$$x\_shared[0]$$
$$h$$
Altitude ($$ft$$)
$$30000\leq h \leq 60000$$
$$x\_shared[1]$$
$$M$$
Mach number
$$1.4\leq M\leq 1.8$$
$$x\_shared[2]$$
$$AR=b^2/S_W$$
Aspect ratio
$$2.5\leq AR\leq 8.5$$
$$x\_shared[3]$$
$$\Lambda$$
Wing sweep ($$\deg$$)
$$40\leq\Lambda\leq70$$
$$x\_shared[4]$$
$$S_W$$
Wing surface area ($$ft^2$$)
$$500\leq S\leq 1500$$
$$x\_shared[5]$$
$$\lambda = {c_{tip}}/{c_{root}}$$
Wing taper ratio
$$0.1\leq\lambda\leq0.4$$
$$x\_1[0]$$
$$x$$
Wingbox x-sectional area
$$0.75\leq x \leq 1.25$$
$$x\_1[1]$$
$$L$$
Lift from aerodynamics ($$N$$)
$$y\_21[0]$$
$$W_{E}$$
Engine mass from propulsion ($$lb$$)
$$y\_31[0]$$
$$C_f$$
Skin friction coefficient
$$0.75\leq Cf\leq 1.25$$
$$x\_2[0]$$
$$W_T$$
Total aircraft mass from structure
$$y\_12[0]$$
$$\Delta\alpha_v$$
Wing twist from structure
$$y\_12[1]$$
$$ESF$$
Engine scale factor from propulsion
$$y\_32[0]$$
$$Th$$
Throttle setting (engine mass flow)
$$0.1\leq T\leq 1.25$$
$$x\_3[0]$$
$$D$$
Drag from aerodynamics ($$N$$)
$$y\_23[0]$$
$$L/D$$
Lift-over-drag ratio from aerodynamics
$$y\_24[0]$$
$$W_T$$
Total aircraft mass from structure
$$y\_14[0]$$
$$W_F$$
Fuel mass from structure
$$y\_14[1]$$
$$SFC$$
Specific Fuel Consumption from propulsion
$$y\_34[0]$$
Table: Input variables of Sobieski’s problem
Variable
Description
Constraint
Symbol
$$\sigma_1$$
Stress constraints on wing section 1
$$\sigma_1<1.09$$
$$g\_1[0]$$
$$\sigma_2$$
Stress constraints on wing section 2
$$\sigma_2<1.09$$
$$g\_1[1]$$
$$\sigma_3$$
Stress constraints on wing section 3
$$\sigma_3<1.09$$
$$g\_1[2]$$
$$\sigma_4$$
Stress constraints on wing section 4
$$\sigma_4<1.09$$
$$g\_1[3]$$
$$\sigma_5$$
Stress constraints on wing section 5
$$\sigma_5<1.09$$
$$g\_1[4]$$
$$W_T$$
Total aircraft mass ($$lb$$)
$$y\_1[0]$$
$$W_F$$
Fuel mass ($$lb$$)
$$y\_1[1]$$
$$\Delta\alpha_{v}$$
Wing twist ($$\deg$$)
$$0.96<\Delta\alpha_{v}<1.04$$
$$y\_1[2],g_1[5]$$
$$L$$
Lift ($$N$$)
$$y\_2[0]$$
$$D$$
Drag ($$N$$)
$$y\_2[1]$$
$$L/D$$
Lift-over-drag ratio
$$y\_2[2]$$
$$dp/dx$$
$$dp/dx<1.04$$
$$g\_2[0]$$
$$SFC$$
Specific Fuel Consumption
$$y\_3[0]$$
$$W_E$$
Engine mass ($$lb$$)
$$y\_3[1]$$
$$ESF$$
Engine Scale Factor
$$0.5\leq ESF \leq 1.5$$
$$y\_3[2],g_3[0]$$
$$T_E$$
Engine temperature
$$T_E\leq 1.02$$
$$g\_3[1]$$
$$Th$$
Throttle setting constraint
$$Th\leq Th_{uA}$$
$$g\_3[2]$$
$$R$$
Range ($$nm$$)
$$y\_4[0]$$
Table: Output variables of Sobieski’s problem
### Creation of the disciplines¶
To create the SSBJ disciplines :
from gemseo.api import create_discipline
disciplines = create_discipline(["SobieskiStructure",
"SobieskiPropulsion",
"SobieskiAerodynamics",
"SobieskiMission"])
### Reference results¶
This problem was implemented by Sobieski et al. in Matlab and Isight. Both implementations led to the same results.
As all gradients can be computed, we resort to gradient-based optimization methods. All Jacobian matrices are coded analytically in GEMSEO.
Reference results using the MDF formulation are presented in the next table.
Variable
Initial
Optimum
Range (nm)
535.79
3963.88
$$\lambda$$
0.25
0.38757
$$x$$
1
0.75
$$C_f$$
1
0.75
$$Th$$
0.5
0.15624
$$t/c$$
0.05
0.06
$$h$$ $$(ft)$$)
45000
60000
$$M$$
1.6
1.4
$$AR$$
5.5
2.5
$$\Lambda$$ $$(\deg)$$
55
70
$$S_W$$ $$(ft^2)$$
1000
1500
## The Propane combustion problem¶
The Propane MDO problem can be found in [PAG96] and [TM06]. It represents the chemical equilibrium reached during the combustion of propane in air. Variables are assigned to represent each of the ten combustion products as well as the sum of the products.
The optimization problem is as follows:
\begin{split}\begin{aligned} \text{minimize the objective function }& f_2 + f_6 + f_7 + f_9 \\ \text{with respect to the design variables }&x_{1},\,x_{3},\,x_{6},\,x_{7} \\ \text{subject to the general constraints } & f_2(x) \geq 0\\ & f_6(x) \geq 0\\ & f_7(x) \geq 0\\ & f_9(x) \geq 0\\ \text{subject to the bound constraints } & x_{1} \geq 0\\ & x_{3} \geq 0\\ & x_{6} \geq 0\\ & x_{7} \geq 0\\ \end{aligned}\end{split}
where the System Discipline consists of computing the following expressions:
\begin{split}\begin{aligned} f_2(x) & = & 2x_1 + x_2 + x_4 + x_7 + x_8 + x_9 + 2x_{10} - R, \\ f_6(x) & = & K_6x_2^{1/2}x_4^{1/2} - x_1^{1/2}x_6(p/x_{11})^{1/2}, \\ f_7(x) & = & K_7x_1^{1/2}x_2^{1/2} - x_4^{1/2}x_7(p/x_{11})^{1/2}, \\ f_9(x) & = & K_9x_1x_3^{1/2} - x_4x_9(p/x_{11})^{1/2}. \\ \end{aligned}\end{split}
Discipline 1 computes $$(x_{2}, x_{4})$$ by satisfying the following equations:
\begin{split}\begin{aligned} x_1 + x_4 - 3 &=& 0,\\ K_5x_2x_4 - x_1x_5 &=& 0.\\ \end{aligned}\end{split}
Discipline 2 computes $$(x_2, x_4)$$ such that:
\begin{split}\begin{aligned} K_8x_1 + x_4x_8(p/x_{11}) &=& 0,\\ K_{10}x_{1}^{2} - x_4^2x_{10}(p/x_{11}) &=& 0.\\ \end{aligned}\end{split}
and Discipline 3 computes $$(x_5, x_9, x_{11})$$ by solving:
\begin{split}\begin{aligned} 2x_2 + 2x_5 + x_6 + x_7 - 8&=& 0,\\ 2x_3 + x_9 - 4R &=& 0, \\ x_{11} - \sum_{j=1}^{10} x_j &=& 0. \\ \end{aligned}\end{split}
### Creation of the disciplines¶
The Propane combustion disciplines are available in GEMSEO and can be imported with the following code:
from gemseo.api import create_discipline
disciplines = create_discipline(["PropaneComb1",
"PropaneComb2",
"PropaneComb3",
"PropaneReaction"])
A gemseo.algos.design_space.DesignSpace file propane_design_space.txt is also available in the same folder, which can be read using the gemseo.api.read_design_space() method.
### Problem results¶
The optimum is $$(x1,x3,x6,x7) = (1.378887, 18.426810, 1.094798, 0.931214)$$. The minimum objective value is $$0$$. At this point, all the system-level inequality constraints are active. | 2022-10-07 19:35:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9917589426040649, "perplexity": 7515.044047985686}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030338244.64/warc/CC-MAIN-20221007175237-20221007205237-00013.warc.gz"} |
https://www.physicsforums.com/threads/minimum-distance-between-two-curves.775960/ | # Minimum Distance between two curves
1. Oct 13, 2014
### ManInTheSuit97
Minimum Distance between y^2=4x and x^2+y^2-12x+31=0.
Attempt:I got that the parabola has vertex at(0,0) and focus at(1,0).The Circle is centred at (6,0)and its radius is sqrt 5.I figured that the double ordinate that passed through (6,0) would be bisected at the point.So I found out the chord of contact and it turned out to be x=6.I substituted the value of x in the parabola and found out y= sqrt24.I thought maybe the minimum distance is the difference in vertical distance and so my answer was sqrt 24-sqrt5.But it is not the answer and so I probably have messed up somewhere.I want to know the approach I should take
2. Oct 13, 2014
### willem2
For any point P outside a circle, the closest point to P on the circle will lie on the line from P to the center of the circle.
The distance From P to the circle will be the distance from P to the center of the circle minus the radius of the circle.
Compute the distance from a point on the parabola to the center of the circle and find a minimum by differentiating this distance.
(it will be easier to find a minimum for the square of the distance)
3. Oct 13, 2014
### RUber
If you are looking to solve this without differentiating, you could first solve the parabola I terms of x, ie $(x,2\sqrt{x})$.
Then write the equation for distance to (6,0).
$d=\sqrt{(x-6)^2+(2\sqrt{x})^2}$
You could use the quadratic equation to find the zero for distance. Then, note that the closest real point to an imaginary number a+bi is just the real part a. | 2017-11-25 01:22:02 | {"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": 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": 0.5853067636489868, "perplexity": 371.34308813146146}, "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-2017-47/segments/1510934809160.77/warc/CC-MAIN-20171124234011-20171125014011-00470.warc.gz"} |
https://apple.stackexchange.com/questions/291149/permanently-change-mount-point-of-volume/291154 | # Permanently change mount point of volume?
I'd like to set another user's home folder to use a certain volume on my disk. To that end, I'd like to be able to mount said volume at a specific location, say, at Users/foobar
I can do this using the command line by running sudo mount -t hfs /dev/disk0s4 /Users/, however this is only a temporary solution. If I unmount and remount the disk, it will mount to its usual location in Volumes/
Is there some way to permanently specify the mount point of a disk?
• Set an entry in: /etc/fstab – user3439894 Jul 17 '17 at 15:40
• @user3439894 I already tried, as per the instructions here. However when I use the File System UUID which DiskUtility gives, an error occurs when opening the disk with sudo mount -a, namely GetMasterBlock: Error 2 opening UUID=5E75BA88-7C74-34A9-8CE6-266C752CE2CA GetMasterBlock: Error 2 opening UUID=5E75BA88-7C74-34A9-8CE6-266C752CE2CA mount_hfs: error on mount(): error = -1. mount_hfs: No such file or directory – bingbomboom Jul 17 '17 at 15:56
• Never mind, it turns out that mount -a just doesn't function as expected, and that setting the entry in /etc/fstab works fine if you mount the disk using DiskUtility – bingbomboom Jul 17 '17 at 15:59
Set an entry in /etc/fstab as directed here. To summarize.
1. Open Disk Utility, unmount the relevant volume.
2. Click on the volume you're trying to mount, and click the "info" button. Note down the drive's File System UUID, which should look something like 5E85BA88-7C74-34A9-8CE6-267C752CE2BA. I'm just gonna use 123abc as shorthand for it.
3. Open up /etc/fstab using your text editor of choice (run it as root!) Edit: as per klanomath's comment below, run sudo vifs to safely edit etc/fstab, and add the following line.
UUID=123abc /desired/mount/path hfs rw 0 2
4. Mount the volume, the mount point should show the path you specified.
Some notes:
• I have absolutely no clue what the "1 2" does at the end of that line. It was there in the link, so I kept it. Thanks to fd0 for helping me out here!
• The page I linked instructs you to reload /etc/fstab using some niload command. I couldn't figure out how to install whatever package that required, but the good news is that I think DiskUtility will automatically reload the file, so don't worry about that.
• Some sources will tell you to reload etc/fstab using the command mount -a. It seems like this should be a perfectly sensible way to do so, but it does not work. Don't worry if it tells you it can't mount a volume.
• I suggest that you read the man 5 fstab manual. The fifth field in the fstab entry is the instruction on whether you allow the filesystem to be dumped (crippled in macOS, 0 would be a better choice). The sixth field is used to determine the order in which fsck is run. – fd0 Jul 17 '17 at 16:29
• Editing of fstab is preferably done with sudo vifs! – klanomath Jul 17 '17 at 16:35
• Thank you very much for the advice, I'll edit my answer to include it tomorrow – bingbomboom Jul 17 '17 at 21:28
• Had no joy with this but what did work for me was to write an Applescript where I pass the path using the hdiutil command. I then save as an Application and run on startup from my User setting in System Preferences. Syntax is: hdiutil mount -mountpoint /path/to/mountpoint <named.dmg> – Tony Barganski Oct 4 '18 at 13:13
• Thanks! I was so worried about this, in my case my hard drive is APFS so I edited vifs with UUID=123abc /desired/mount/path apfs rw 1 2 and it worked, I have no idea what I am doing but it worked. The main reason I did this was to have that folder be case sensitive for runing docker with volumes... problem solved. – santiago arizti Jul 26 at 21:34
In Disk Utility right tap on the APFS volume and click "Rename"
Now opening (mounting) the volume via Finder will mount it to /Volumes/home or whatever new name you give it. | 2019-10-16 14:16:13 | {"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": 0.2579561173915863, "perplexity": 2585.4436382629488}, "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-2019-43/segments/1570986668994.39/warc/CC-MAIN-20191016135759-20191016163259-00213.warc.gz"} |
http://icpc.njust.edu.cn/Problem/Local/1706/ | Version Controlled IDE
Time Limit: 8000ms
Memory Limit: 524287 kb
Description
Programmers use version control systems to manage files in their projects, but in these systems,versions are saved only when you manually submit.
Can you implement an IDE that automatically saves a new version whenever you insert or delete a string?
Positions in the buffer are numbered from 1 from left to right. Initially, the buffer is empty and in version 0. Then you can execute 3 commands (vnow means the version before executing the command, and L[v] means the length of buffer at version v):
1 p s: insert string s after position p(0<=p<=L[vnow], p=0 means insert before the start of the buffer). s contains at most 1 and at most 100 letters.
2 p c: remove c characters starting at position p(p>=1, p+c<=L[vnow]+1). The remaining charactesr (if any) will be shifted left, filling the blank
3 v p c: print c characters starting at position p(p>=1, p+c<=L[v]+1), in version v(1<=v<=vnow).
The first command is guaranteed to be command 1(insert). After executing each command 1 or 2, version is incremented by 1.
Input
There is only one test case. It begins with a single integer n (1<=n<=50,000), the number of commands.
Each of the following n lines contains a command. The total length of all inserted string will not exceed 1,000,000.
Output
Print the results of command 3, in order. The total length of all printed strings will not exceed 200,000.
Sample Input
6
1 0 abcdefgh
2 4 3
3 1 2 5
3 2 2 3
1 2 xy
3 3 2 4
Sample Output
bcdef
bcg
bxyc
Hint
In order to prevent you from preprocessing the command, we adopt the following obfuscation scheme:
Each type-1 command becomes 1 p+d s
Each type-2 command becomes 2 p+d c+d
Each type-3 command becomes 3 v+d p+d c+d
Where d is the number of lowercase letter 'c' you printed, before processing this command. After the obfuscation, the sample input would be:
6
1 0 abcdefgh
2 4 3
3 1 2 5
3 3 3 4
1 4 xy
3 5 4 6 | 2019-09-15 12:34:14 | {"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": 0.3554763197898865, "perplexity": 3128.7709340389497}, "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-2019-39/segments/1568514571360.41/warc/CC-MAIN-20190915114318-20190915140318-00167.warc.gz"} |
https://scicomp.stackexchange.com/questions/35143/comparison-of-integrals-with-a-function/35209 | # Comparison of integrals with a function:
Consider the following integral:
$$S(q)=\int_{x=2}^q\sin^2\left(\frac{π\Gamma(x)}{2x}\right)dx$$
And consider the functions :
$$R(q)=\frac{q}{\log(q)}$$
$$T(q)=\int_2^q\frac{1}{\log(x)}dx$$
I want to compare them with each other (at least numerically for a large interval of values)
If graph for very large intervals (up to at least $$10^4$$) possible please add (please add three graphs in one axis system, so that I can compare)
(Does numerics suggest $$S(q) \sim R(q)$$ or $$T(q)$$? )
Note: Can't calculate the first integral on Mathematica for large values.
See this MSE post and this MSE post for more details.
There is no need for numerical computation here.
First, $$T(q)$$ is a well-known function, the logarithmic integral. Repeated integration by parts gives an asymptotic expansion
$$\mathrm{Li}(q) = \frac{q}{\log q}\sum_{k=0}^{K-1} \frac{k!}{\log^k q} + O\left(\frac{q}{\log^{K+1}q}\right).$$
There's also a fairly rapidly convergent representation due to Ramanujan which you can find on Wikipedia
Second, regarding the main integral, it has a different asymptotic. First, $$\sin^2\theta = \frac{1-\cos(2\theta)}{2}$$, we remove the $$\frac{q-2}{2}$$ coming from the constant and concentrate on getting cancellation in the oscillatory part. Second, recall the digamma function (the logarithmetic derivative of the gammafunction) $$\digamma(x) = \frac{\Gamma'(x)}{\Gamma(x)}$$ which satisfies $$\log x - \frac{1}{x}\leq\digamma(x) \leq \log x - \frac{1}{2x}$$ and $$\digamma'(x) = \frac1x + \frac{1}{2x^2} +O(x^{-3})$$
Letting $$u = \frac{\Gamma(x)}x$$ we have $$\frac{du}{u} = d(\log u) = (\digamma(x)-\frac1x)dx$$ so that
$$\begin{split}\frac{q-2}{2} - S(q) &= \int_{x=2}^{x=q} \cos\left(\pi\frac{\Gamma(x)}{2x}\right)dx \\ &= \int_{x=2}^{x=q} \frac{\cos(\pi u)}{u}\frac{du}{\digamma(x)-\frac1x} \end{split}$$
We now integrate by parts and get
$$\begin{split} &= \left[-\frac{\sin(\pi u)}{\pi u}\frac{1}{\digamma(x)-\frac1x}\right]_{x=2}^{x=q} - \int_{x=2}^{x=q} \frac{\sin(\pi u)}{\pi u^2}\frac{du}{\digamma(x)-\frac1x} \\ &- \int_{x=2}^{x=q} \frac{\sin(\pi u)}{\pi u}\frac{\digamma'(x)+\frac1{x^2}}{\left(\digamma(x)-\frac1x\right)^2} \frac{dx}{du}du \\ & = \left[-\frac{\sin(\pi u)}{\pi u}\frac{1}{\digamma(x)-\frac1x}\right]_{x=2}^{x=q} - \int_{x=2}^{x=q} \frac{\sin(\pi u)}{\pi u^2}\frac{du}{\digamma(x)-\frac1x} \\ &- \int_{x=2}^{x=q} \frac{\sin(\pi u)}{\pi u}\frac{\digamma'(x)+\frac1{x^2}}{\left(\digamma(x)-\frac1x\right)^3} du \end{split}$$
The first term is $$O(1) + O\left(\frac{q}{\Gamma(q)}\right)$$ and in particular is bounded. The second is similarly $$O\left(\int_{x=2}^{x=q} \frac{du}{u^2}\right) = O(1)+ O\left(\frac{q}{\Gamma(q)}\right)$$. For the last term divide the interval into two parts: up to $$2\leq x\leq q^\delta$$ and $$q^\delta \leq x \leq q$$ for some $$\delta < 1$$. On the first interval we use that $$\digamma'(x)+\frac1{x^2} = O(\frac{1}{x}) = O(1)$$ to bound the integral as $$O(1)+O(\log u(q^\delta) = O(\log(\Gamma(q^\delta)) = O(\delta q^\delta \log q)$$. On the second interval we have $$\digamma'(x)+\frac1{x^2} = O(q^{-\delta})$$ so the whole integral is $$O(q^{1-\delta}\log q)$$. Taking $$\delta = \frac12$$ we conclude that
$$S(q) = \frac{q-2}{2} + O(q^{1/2}\log q)$$
And in particular has a different asymptotic.
Finally, a more careful analysis using the $$\log^3x$$ in the denominator of the third term would give the error term $$O\left(\frac{q^{1/2}}{\log^2 q}\right)$$.
• I think multiple integrations by parts will give a $S(q) = \frac12 q + O_\epsilon(q^\epsilon)$ for any $\epsilon>0$. – Lior Silberman May 23 at 21:34
• The problem is that the connection to prints only occurs at integral values of $x$, and the function is so oscillatory that the sum over the integers is a terrible approximation to the integral. – Lior Silberman May 28 at 20:25
• thank you for the reply . But you can see the post on growth condition on the functional, which seems optimistic. – Bambi May 29 at 20:56
You write $$S(q)$$ and $$T(q)$$ as integrals, but it is easier to think of them as solutions of ODEs: $$S'(q) = \sin^2\left(\frac{π\Gamma(q)}{2q}\right)$$ with initial conditions $$S(2) = 0,$$ and similarly for $$T(q)$$. You can then use any of the common ODE integrators in matlab, mathematica, maple, ..., to solve and plot the solutions so that you can compare them.
• This is a great idea. Unfortunately, I don't think it solves the problem. See my edit above. – Spencer Bryngelson May 15 at 22:19
• @SpencerBryngelson Why not? Maybe I don't see what your edit is, but I see no reason why you couldn't use this approach to compare the three functions. Can you explain what precisely doesn't work for you? – Wolfgang Bangerth May 16 at 3:45
• I think the RHS has something like 10^5 oscillations by $q=50$. If you hope to resolve those, you’d need at least 10x more time steps than that. For larger $q$, the number of oscillations grows more rapidly still. I just don’t think it’s possible for the $q$ values OP mentioned. – Spencer Bryngelson May 16 at 7:02
• @SpencerBryngelson -- well, but then no other integration method will be able to do it either unless you use particular properties of the integrand. For example, you could see whether you can approximating the integrand over one period of the sine and think about whether that helps you to approximate the integral by counting how many periods you cover from $q=2$ to wherever you need to go. – Wolfgang Bangerth May 16 at 15:34
• agreed. That was the spirit of my comment about oscillatory integration methods, which do just that. – Spencer Bryngelson May 17 at 0:21
The only chance you stand to deal with this problem from a numerical perspective is oscillatory integration methods. Filon/Levin-type methods can sometimes handle problems like this, particularly when they are of $$\sin$$ or $$\cos$$ type, though the $$\Gamma$$ function and its run-away growth may be prohibitive for the $$q$$ you are hoping for. In any case, I was able to accurately evaluate the integral with Mathematica using Levin's method up to around $$q = 50$$ via
NIntegrate[Sin[Pi Gamma[x]/(2 x)]^2, {x, 2, 50}, MinRecursion -> 9, Method -> "LevinRule"]
• If you want to analyze this from an analytical perspective, Math.SE is likely more appropriate
• Mathematica can likely get you further if you can derive the amplitude/oscillation matrices and specify them in the function call. See, e.g. here. I'm not sure if this is possible in your case or not, but reading the classic papers by Levin from 1996/7 might be helpful.
• Further questions regarding Mathematica implementation should probably be directed to Mathematica.SE
Edit:
Wolfgang had the clever idea to just solve this in ODE form. However, I don't think solves the problem in the end. The integrand (or RHS in derivative form) is just too highly oscillatory for even modest $$q$$. For example, using standard methods:
NDSolve[{s'[x] == Sin[Pi Gamma[x]/(2 x)]^2, s[2] == 0}, s[x], {x, 2, q}, MaxSteps -> 10^7]
I am only able to reach $$q=17$$ in these $$10^7$$ time steps. The problem gets much worse for still larger $$q$$. A quick plot of number of oscillations per $$q$$ reveals this. I think somehow re-casting the problem in a non-oscillatory form (e.g. Levin's method) in the only real way forward, though I'm happy to learn otherwise.
• I already posted the question on MSE but no answers – Bambi May 15 at 7:53 | 2020-09-29 09:03:02 | {"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": 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": 45, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7319864630699158, "perplexity": 420.9554080631578}, "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-2020-40/segments/1600401632671.79/warc/CC-MAIN-20200929060555-20200929090555-00019.warc.gz"} |
http://michaelnielsen.org/polymath1/index.php?title=Szemer%C3%A9di%27s_theorem&oldid=534 | # Szemerédi's theorem
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Finite version. For every $\delta\gt0$ and every positive integer k there exists n such that every subset $A\subset [n]$ of size at least $\delta n$ contains an arithmetic progression of length k.
It is known that for fixed $\delta$ and n sufficiently large one can take k to be $\log\log\log\log\log\log n.$ (This is not quite the best known bound but is easier to state.) | 2018-12-11 05:51:54 | {"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": 0.9557494521141052, "perplexity": 362.4340997968657}, "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/1544376823565.27/warc/CC-MAIN-20181211040413-20181211061913-00371.warc.gz"} |
http://math.stackexchange.com/questions/191807/convergence-of-the-real-analysis | # Convergence of the Real analysis
The question is find the Fourier series of "|cost| for all t". I already found the fourier series
But now the question asks " At which values of $x$, does the series fail to converge to ? To what values does it converge at those points? "
So what to do ?
-
That will converge for all $x$; the function is Lipschitz continuous, and so its Fourier series converges everywhere. (see, for instance, the answer to this question) | 2014-08-29 12:10:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9757103323936462, "perplexity": 259.43168589158853}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500832155.37/warc/CC-MAIN-20140820021352-00342-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://academic.naver.com/article.naver?doc_id=14887240 | Search
### Journal Articles
#### VSI ${\overline{X}}$-CRL 합성관리도의 경제적 설계
Economic design of VSI ${\overline{X}}$-CRL Synthetic Control Chart
Author
송서일, 박현규, 정혜진
Journal
Publisher
한국산업경영시스템학회 in 2005
Cited Count
0
Partners
KISTI
Category
Keywords
synthetic control, economic design of control chart, VSI control chart
##### Abstract
This paper is designed a VSI ${\overline{X}}$-CRL synthetic control chart in aspect of economy. We found the optimal sampling interval and various control limit factors under various cost parameters using cost function, proposed Lorenzen and Vance. Optimal design parameters include the sample size, control limit width, sampling interval, CRL/S chart control limit; L. Comparison and analysis of cost parameters are applied between synthetic VSI ${\overline{X}}$-CRL chart and FSI ${\overline{X}}$-CRL chart. The result of this paper shows that VSI ${\overline{X}}$-CRL chart brings cost-cutting effect of 3.04% control expense less than FSI control chart. It may not be difficult to establish the optimal economic control parameters to apply the practical cost parameters in the field.
##### References
No relevant information is available
If you register references through the customer center, the reference information will be registered as soon as possible.
view options | 2018-09-19 04:18:17 | {"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": 0.2992398142814636, "perplexity": 13568.90135334985}, "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-2018-39/segments/1537267155817.29/warc/CC-MAIN-20180919024323-20180919044323-00171.warc.gz"} |
https://ftp.aimsciences.org/article/doi/10.3934/jimo.2008.4.271 | American Institute of Mathematical Sciences
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A nonsmooth Newton's method for discretized optimal control problems with state and control constraints
April 2008, 4(2): 271-285. doi: 10.3934/jimo.2008.4.271
New approach to global minimization of normal multivariate polynomial based on tensor
1 School of Mathematics Sciences and Computing Technology, Central South University, Hunan Changsha, 410083, China 2 School of Information Science and Engineering, Central South University, Changsha, 410083, China
Received January 2007 Revised September 2007 Published April 2008
In this paper, we first present a concise representation of multivariate polynomial, based on which we deduce the calculation formulae of its derivatives using tensor. Then, we propose a solution method to determine a global descent direction for the minimization of general normal polynomial. At a local and non-global maximizer or saddle point, we could use this method to get a global descent direction of the objective function. By using the global descent direction, we can transform an $n$-dimensional optimization problem into a one-dimensional one. Based on some efficient algorithms for one dimensional global optimization, we develop an algorithm to compute the global minimizer of normal multivariate polynomial. Numerical examples show that the proposed algorithm is promising.
Citation: Zhong Wan, Chunhua Yang. New approach to global minimization of normal multivariate polynomial based on tensor. Journal of Industrial & Management Optimization, 2008, 4 (2) : 271-285. doi: 10.3934/jimo.2008.4.271
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2019 Impact Factor: 1.366 | 2021-01-19 16:03:04 | {"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": 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": 0.44117236137390137, "perplexity": 6519.913024223792}, "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-2021-04/segments/1610703519395.23/warc/CC-MAIN-20210119135001-20210119165001-00691.warc.gz"} |
https://mathoverflow.net/questions/338359/is-binary-integer-linear-programming-solvable-in-polynomial-time/338360 | # Is Binary Integer Linear Programming solvable in polynomial time?
The paper Solving the Binary Linear Programming Model in Polynomial Time claims that Binary Integer Linear Programming is in P. However, it seems that no subsequent literature in the mainstream has done any further study on this. I am a bit doubtful regarding the correctness of the claim made in that paper. Therefore, I have put this question here.
You can also find the claim that BIP $$\in$$ NPC in many class notes, e.g., this set. | 2020-02-20 07:06:37 | {"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5464824438095093, "perplexity": 196.70904597831884}, "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/1581875144708.87/warc/CC-MAIN-20200220070221-20200220100221-00357.warc.gz"} |
https://owenduffy.net/blog/?paged=9&cat=11 | ## Measure transmission line Zo – nanoVNA – CCS RG6
There are many ways to get a good estimate of the characteristic impedance Zo of a transmission line.
One method is to measure the input impedances of a section of line with both a short circuit and open circuit termination. From Zsc and Zoc we can calculate the Zo, and the complex propagation constant $$\gamma=\alpha + \jmath \beta$$, and from that, MLL.
Calculation of Zo is quite straightforward.
The solution for γ involves the log of a complex number $$r \angle \theta$$ which is one of the many possible values $$ln(r) + j \left(\theta + 2 \pi k \right)$$ for +ve integer k. Conveniently, the real part α is simply $$ln(r)$$. The real part of γ is the attenuation in Np/m which can be scaled to dB/m, and the imaginary part is the phase velocity in c/m. The challenge is finding k.
Let’s take an example from recent measurements of 35m of CCS RG6 coax, and extract the s11 values recorded in saved .s1p files @ 1.87MHz. The saved data in MA format, magnitude and angle (in degrees).
Calculate Zo and gamma is flexible and can accept the MA format data directly.
Above, the results. Zo is 74.73-j1.156Ω, and matched line loss MLL is 0.03281dB/m. This MLL is quite a deal higher than you might find in many line loss calculators, they often fail on CCS cables. Continue reading Measure transmission line Zo – nanoVNA – CCS RG6
## Nichols: The Two Bird Experiment
With the following introduction, (Nichols nd) tries to demonstrate some important principles. He says…
This really tests your understanding of transmission line theory.
Above is Nichols’ test setup, simple enough.
With the transmitter keyed, the transmatch is adjusted to show zero reflected power on Bird Wattmeter #1. Transmitter is then adjusted to generate exactly 100 watts of forward power indicated on Bird Wattmeter #1. Bird directional Wattmeter #2 indicates about 36 watts of REFLECTED power. (Charts are readily available to show that a 4:1 mismatch gives about 36% reflected power).
## nanoVNA – measure Transmission Loss – example 5
This article is demonstration of measurement of Transmission Loss in a section of two wire transmission line embedded in a common mode choke. The scenario is based on an online article MEASURING DM ATTENUATION of YOUR CMC USING THE NANOVNA AND NANOVNA SAVER.
The reference article publishes measured attenuation or loss being -1.45dB @ 28.4MHz. Of course, the -ve value hints that the author is lost in hamdom where all losses MUST be -ve dB..
The meaning of loss in a generic sense (ie without further qualification) is $$loss=\frac{Power_{in}}{Power_{out}}$$ and can be expressed in dB as $$loss_{dB}=10 log_{10}(loss)$$.
Some might interpret the result to imply that $$(1-10^{\frac{-loss}{10}})*100=28 \%$$ of input power is converted to heat in the choke.
The result given (and corrected) as 1.45dB was taken simply from the nanoVNA $$|s21|$$ result, and so it is actually InsertionLoss, not simply Loss.
What is the difference? Continue reading nanoVNA – measure Transmission Loss – example 5
## Disturbing the thing being measured – coax line
An issue that often arises in online discussions inability to reconcile the VSWR indicated by a transceiver (or possibly an inline VSWR meter) and an antenna analyser.
Is this Segal’s law at play?
There are several common contributors including:
• faulty, dirty, or not properly mated connectors and cables;
• VSWR meters that are not accurate at low power levels; and
• influence of the common mode current path on VSWR.
I see online discussions struggling to try to work out if a receiving system is sufficiently good for a certain application.
Let’s work an example using Simsmith to do some of the calculations.
Scenario:
• 20m ground mounted vertical base fed against a 2.4m driven earth electrode @ 0.5MHz;
• 10m RG58A/U coax; and
• Receiver with 500+j0Ω ohms input impedance and Noise Figure 20dB.
An NEC-4.2 model of the antenna gives a feed point impedance of 146-j4714Ω and radiation efficiency of 0.043%, so radiation resistance $$Rr=146 \cdot 0.00043=0.0063$$.
Above, the NEC antenna model summary. Continue reading Quantifying performance of a simple broadcast receive system on MF
## nanoVNA – measure Transmission Loss – example 4
This article is demonstration of measurement of Transmission Loss in a section of coaxial transmission line. The scenario is chosen to expose the experiment to some of the things that complicate such measurements.
The very popular nanoVNA-H will be used to make the measurements.
The scenario:
• nanoVNA fully calibrated from 1.5-1.8MHz using a 200mm length coax lead on Port 2 (nanoVNA CH1);
• 10m of RG58C/U; and
• f=1.65MHz.
Above is a block diagram of the test configuration. nanoVNA measurements are wrt 50Ω, so $$P=\frac{V^2}{50}$$ and $$V=\sqrt{50P}$$. Continue reading nanoVNA – measure Transmission Loss – example 4
## KL7AJ on the Conjugate Match Theorem – analytical solution – Simsmith
KL7AJ on the Conjugate Match Theorem asked the question Should we have expected this outcome?
Let us solve a very similar problem analytically where measurement errors do not contribute to the outcome.
Taking the load impedance to be the same 10.1+j0.2Ω, and calculating for a T match similar to the MFJ-949E (assuming L=26µH, QL=200, and ideal capacitors) with Simsmith we can find a near perfect match.
The capacitors are 177.2 and 92.9pF for the match. Continue reading KL7AJ on the Conjugate Match Theorem – analytical solution – Simsmith
## nanoVNA – measure Transmission Loss – example 3
This article is demonstration of measurement of Transmission Loss in a section of coaxial transmission line. The scenario is chosen to expose the experiment to some of the things that complicate such measurements.
The very popular nanoVNA-H will be used to make the measurements.
The scenario:
• nanoVNA fully calibrated from 1-5MHz using a 200mm length coax lead on Port 2 (nanoVNA CH1);
• 35m of CCS RG6/U (close to an electrical quarter wavelength);
• 75-50Ω Minimum Loss Pad (5.72dB); and
• f=1.65MHz (close to a quarter wavelength.
Above is a block diagram of the test configuration. nanoVNA measurements are wrt 50Ω, so $$P=\frac{V^2}{50}$$ and $$V=\sqrt{50P}$$. Continue reading nanoVNA – measure Transmission Loss – example 3
## nanoVNA – measure Transmission Loss – example 2
This article is demonstration of measurement of Transmission Loss in a section of coaxial transmission line. The scenario is chosen to expose the experiment to some of the things that complicate such measurements.
The very popular nanoVNA-H will be used to make the measurements.
The scenario:
• nanoVNA fully calibrated from 1-5MHz using a 200mm length coax lead on Port 2 (nanoVNA CH1);
• 35m of CCS RG6/U (close to an electrical quarter wavelength);
• three 50Ω terminations in shunt with VNA Port 2; and
• f=1.65MHz (close to a quarter wavelength.
The transmission line load is four 50Ω loads in parallel, one of them being VNA Port 2. Only one quarter of the output power is captured by the VNA, so there is effectively a loss of 6.02dB in that configuration. It also delivers a 12.5+j0Ω load the the transmission lines, VSWR is about 6. Note this power division is based on the assumption that Zin of Port 2 is 50+j0Ω, and error in Zin flows into the result. A 10dB attenuator is fitted to Port 2 prior to calibration to improve accuracy of Zin.
Above is a block diagram of the test configuration. nanoVNA measurements are wrt 50Ω, so $$P=\frac{V^2}{50}$$ and $$V=\sqrt{50P}$$. Continue reading nanoVNA – measure Transmission Loss – example 2
## nanoVNA – measure Transmission Loss – example 1
This article is demonstration of measurement of Transmission Loss in a section of coaxial transmission line. The scenario is chosen to expose the experiment to some of the things that complicate such measurements.
The very popular nanoVNA-H will be used to make the measurements.
The scenario:
• nanoVNA fully calibrated from 1-5MHz using a 200mm length coax lead on Port 2 (nanoVNA CH1);
• 35m of CCS RG6/U (close to an electrical quarter wavelength); and
• f=1.65MHz (close to a quarter wavelength.
Above is a block diagram of the test configuration. nanoVNA measurements are wrt 50Ω, so $$P=\frac{V^2}{50}$$ and $$V=\sqrt{50P}$$. Continue reading nanoVNA – measure Transmission Loss – example 1 | 2022-06-27 03:04:18 | {"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": 0.47534260153770447, "perplexity": 4042.4281059686905}, "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/1656103324665.17/warc/CC-MAIN-20220627012807-20220627042807-00711.warc.gz"} |
https://www.learncram.com/ml-aggarwal/ml-aggarwal-class-7-solutions-for-icse-maths-chapter-8-objective-type-questions/ | # ML Aggarwal Class 7 Solutions for ICSE Maths Chapter 8 Algebraic Expressions Objective Type Questions
## ML Aggarwal Class 7 Solutions for ICSE Maths Chapter 8 Algebraic Expressions Objective Type Questions
Mental Maths
Question 1.
Fill in the blanks:
(i) The terms with different algebraic factors are called ……….
(ii) The number of terms in a monomial is ………
(iii) An algebraic expression having two unlike terms is called a ……..
(iv) 3a2b and -7ba2 are ……….. terms.
(v) -6a2b and -6ab2 are ……… terms.
(vi) The number of unlike terms in the algebraic expression 3x2 – 2xy + 5x2 is ………
(vii) The factors of the term -3p2q2 are ……..
(viii) The perimeter of a triangle whose sides measure 2a, b and a + b is ………
(ix) The value of the expression 2x3 – 7x2 + 5x – 3 when x = 1 is ………..
(x) In the term -7a2bc, the coefficient of a is ……..
(xi) The degree of the polynomial 3 – 5x2 + 7x3 – x4 is ……….
(xii) The degree of the polynomial 3x2 – 2xy2 + 5 is ………
Solution:
Question 2.
State whether the following statements are true (T) or false (F).
(i) The expression 5x + 7 – 2x is a trinomial.
(ii) (7x – 10) – (3x – 5) = 4x – 15.
(iii) The coefficient of 3x in -3x3y is -xy.
(iv) The constant term in the expression 2x2 – 3xy – 7 is 7.
(v) If x = 3 and y = $$\frac { 1 }{ 3 }$$ then the value of xy (x2 + y2) is 9$$\frac { 1 }{ 9 }$$.
(vi) (3x – y + 5) – (x + y) is a binomial.
(vii) Sum of 2 and p is 2p.
(viii) Sum of x2 + x and y2 + y is 2x2 + 2y2.
(ix) In like terms, variables and their powers are the same.
(x) Every polynomial is a monomial.
(xi) If we add a monomial and a binomial, then answer can never be a monomial.
(xii) If we subtract a monomial from a binomial, then the answer is at least a binomial.
(xiii) if we add a monomial and a trinomial, then the answer can be a monomial.
(xiv) If we add a monomial and a binomial, then the answer can be a trinomial.
Solution:
Multiple Choice Questions
Choose the correct answer from the given four options (3 to 16):
Question 3.
The algebraic expression for the statement ‘Thrice square of a number x subtracted from five times the sum of y and 2’ is
(a) 5y + 2 – 3x2
(b) 3x2 – (5y + 2)
(c) 5(y + 2) – 3x2
(d) 5(y + 2) – (3x)2
Solution:
Question 4.
The expression 7x – 5(x2 + y2) is a
(a) monomial
(b) binomial
(c) trinomial
(d) none of these
Solution:
Question 5.
The coefficient of 5a2 in -5a3bc is
(a) -bc
(b) a2bc
(c) -a2bc
(d) -abc
Solution:
Question 6.
Which of the following is a pair of like terms?
(a) -5xy, 5x
(b) -5xy, 3yz
(c) -5xy, -5y
(d) -5xy, 7yx
Solution:
Question 7.
The like terms is the expressions 3x(3 – 2y) and 2(xy + x2) are
(a) 9x and 2x2
(b) -6xy and 2xy
(c) 9x and 2xy
(d) -6xy and 2x2
Solution:
Question 8.
Identify the binomial out of the following:
(a) 3xy2 + 5y – x2y
(b) 2x2y – 5y – 2x2y
(c) 3xy2 + 5y – xy2
(d) xy + yz + zx
Solution:
Question 9.
The number of (unlike) terms in the expression 3xy2 + 2y2z – y2x + y(xz + yz) – 5
(a) 3
(b) 4
(c) 5
(d) 6
Solution:
Question 10.
The value of the expression x3 + y3 when x = 2 and y = -2 is
(a) 0
(b) 8
(c) 16
(d) -16
Solution:
Question 11.
-xy – (-5xy) is equal to
(a) -6xy
(b) 6xy
(c) -4xy
(d) 4xy
Solution:
Question 12.
On subtracting 7x + 5y – 3 from 5y – 3x – 9, we get
(a) 10x + 6
(b) -10x – 6
(c) 10x + 10y – 12
(d) -10x – 12
Solution:
Question 13.
The value of the expression $$\frac { 5 }{ 3 }$$ x2 + 1 when x = -2 is
(a) $$\frac { -17 }{ 3 }$$
(b) $$\frac { -7 }{ 3 }$$
(c) $$\frac { 21 }{ 3 }$$
(d) $$\frac { 23 }{ 3 }$$
Solution:
Question 14.
The number of sides in a pattern having 3 hexagons arranged in a row as shown in the given figure is
(a) 18
(b) 17
(c) 16
(d) 15
Solution:
Question 15.
The degree of the polynomial 3x3y – 5xy4 – 2x + 1 is
(a) 5
(b) 4
(c) 3
(d) 2
Solution:
Higher Order Thinking Skills (HOTS)
Question 1.
The length of a rectangle is 3x – 4y + 6z and the perimeter is 7x + 8y + 17z, find the breadth of the rectangle.
Solution:
Question 2.
Solution:
Question 3.
If a = 3, b = -1, then find the value of each of the following:
Solution: | 2021-03-09 03:52:27 | {"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": 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": 0.6644139885902405, "perplexity": 2294.890596993572}, "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-20210309030723-20210309060723-00591.warc.gz"} |
https://stacks.math.columbia.edu/tag/00IP | # The Stacks Project
## Tag 00IP
Lemma 10.50.4 (Krull's intersection theorem). Let $R$ be a Noetherian local ring. Let $I \subset R$ be a proper ideal. Let $M$ be a finite $R$-module. Then $\bigcap_{n \geq 0} I^nM = 0$.
Proof. Let $N = \bigcap_{n \geq 0} I^nM$. Then $N = I^nM \cap N$ for all $n \geq 0$. By the Artin-Rees Lemma 10.50.2 we see that $N = I^nM \cap N \subset IN$ for some suitably large $n$. By Nakayama's Lemma 10.19.1 we see that $N = 0$. $\square$
The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 11634–11639 (see updates for more information).
\begin{lemma}[Krull's intersection theorem]
\label{lemma-intersect-powers-ideal-module-zero}
Let $R$ be a Noetherian local ring. Let $I \subset R$ be
a proper ideal. Let $M$ be a finite $R$-module.
Then $\bigcap_{n \geq 0} I^nM = 0$.
\end{lemma}
\begin{proof}
Let $N = \bigcap_{n \geq 0} I^nM$.
Then $N = I^nM \cap N$ for all $n \geq 0$.
By the Artin-Rees Lemma \ref{lemma-Artin-Rees}
we see that $N = I^nM \cap N \subset IN$ for
some suitably large $n$. By Nakayama's Lemma \ref{lemma-NAK}
we see that $N = 0$.
\end{proof}
There are no comments yet for this tag.
## Add a comment on tag 00IP
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner). | 2017-10-21 23:07:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 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": 0.9598196744918823, "perplexity": 480.12441836160633}, "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-2017-43/segments/1508187824899.75/warc/CC-MAIN-20171021224608-20171022004608-00794.warc.gz"} |
http://www.gradesaver.com/of-mice-and-men/q-and-a/what-is-the-particular-importance-of-crooks-initial-commitment-to-the-idea-of-buying-a-ranch-and-then-his-later-rejection-of-the-idea-303572 | # What is the particular importance of Crooks' initial commitment to the idea of buying a ranch, and then his later rejection of the idea?
what is the particular importance of crooks' initial commitment to the idea of buying a ranch, and then his later rejetion of the idea?
(notes on section four question 5) | 2017-06-29 09:32:22 | {"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": 0.8346701264381409, "perplexity": 2781.6356114142204}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128323895.99/warc/CC-MAIN-20170629084615-20170629104615-00662.warc.gz"} |
https://tex.stackexchange.com/questions/559079/partially-expanding-macro-argument | # Partially expanding macro argument
I have a library of tikz pictures, eg.
\newcommand*{\mypic}[1]{
\begin{tikzpicture}
%...
\end{tikzpicture}
}
which I would like to print together with their names (including arguments). However,
\renewcommand{\show}[1]{\verb|#1|\quad #1}
\show{\mypic{a}}
produces
Package tikz Error: Sorry, some package has redefined the meaning of the math-mode dollar sign. This is incompatible with tikz and its calc library and might cause unrecoverable errors.
I guess \show is trying to expand its argument inside \verb. I experimented with \expandafter but it didn't help. When I replaced \verb| | with \texttt{}, \show didn't print the macro name at all.
• \verb cannot appear inside another macro. Aug 18 '20 at 5:03
• Try instead \renewcommand{\show}[1]{\detokenize{#1}\quad #1} Aug 18 '20 at 5:03
• Also \show only takes a single token. If you want to print multiple, try \message{...} or \errmessage{...}. Aug 18 '20 at 5:04
Firstly, it's better to not redefine primitive, for example \show. Just choose another command name. Then, \Verb from fvextra package is more robust than \verb, when used inside another macro.
\documentclass{article}
\usepackage{fvextra}
\usepackage{tikz}
\newcommand*{\mypic}[1]{
\begin{tikzpicture}
\node[draw, circle] {text};
\end{tikzpicture}
} | 2021-10-25 01:01:41 | {"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": 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": 0.8949640989303589, "perplexity": 10918.50293429791}, "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/1634323587608.86/warc/CC-MAIN-20211024235512-20211025025512-00654.warc.gz"} |
https://nrich.maths.org/public/topic.php?code=47&cl=3&cldcmpid=303 | # Resources tagged with: Creating and manipulating expressions and formulae
Filter by: Content type:
Age range:
Challenge level:
### There are 131 results
Broad Topics > Algebraic expressions, equations and formulae > Creating and manipulating expressions and formulae
### The Medieval Octagon
##### Age 14 to 16Challenge Level
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
### Pareq Calc
##### Age 14 to 16Challenge Level
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
### Pythagoras Proofs
##### Age 14 to 16Challenge Level
Can you make sense of these three proofs of Pythagoras' Theorem?
### Semi-square
##### Age 14 to 16Challenge Level
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
### Triangles Within Triangles
##### Age 14 to 16Challenge Level
Can you find a rule which connects consecutive triangular numbers?
### Triangles Within Pentagons
##### Age 14 to 16Challenge Level
Show that all pentagonal numbers are one third of a triangular number.
### Gutter
##### Age 14 to 16Challenge Level
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
### Christmas Chocolates
##### Age 11 to 14Challenge Level
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
### Nicely Similar
##### Age 14 to 16Challenge Level
If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?
### Partitioning Revisited
##### Age 11 to 14Challenge Level
We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4
### Beach Huts
##### Age 11 to 14Challenge Level
Can you figure out how sequences of beach huts are generated?
### Regular Hexagon Loops
##### Age 11 to 14Challenge Level
Make some loops out of regular hexagons. What rules can you discover?
### A Tilted Square
##### Age 14 to 16Challenge Level
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
### Magic W
##### Age 14 to 16Challenge Level
Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.
### Interactive Number Patterns
##### Age 14 to 16Challenge Level
How good are you at finding the formula for a number pattern ?
### Cubes Within Cubes Revisited
##### Age 11 to 14Challenge Level
Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
##### Age 14 to 16Challenge Level
Kyle and his teacher disagree about his test score - who is right?
### The Simple Life
##### Age 11 to 14Challenge Level
The answer is $5x+8y$... What was the question?
### Triangles Within Squares
##### Age 14 to 16Challenge Level
Can you find a rule which relates triangular numbers to square numbers?
### Attractive Tablecloths
##### Age 14 to 16Challenge Level
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
### Multiplication Square
##### Age 14 to 16Challenge Level
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
### Salinon
##### Age 14 to 16Challenge Level
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
### The Pillar of Chios
##### Age 14 to 16Challenge Level
Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.
### Sitting Pretty
##### Age 14 to 16Challenge Level
A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?
### Three Four Five
##### Age 14 to 16Challenge Level
Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
### Pair Products
##### Age 14 to 16Challenge Level
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
### Number Pyramids
##### Age 11 to 14Challenge Level
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
### Partly Painted Cube
##### Age 14 to 16Challenge Level
Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?
### More Number Pyramids
##### Age 11 to 14Challenge Level
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
### Simplifying Doughnut
##### Age 14 to 18Challenge Level
An algebra task which depends on members of the group noticing the needs of others and responding.
### Matchless
##### Age 14 to 16Challenge Level
There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?
### Janine's Conjecture
##### Age 14 to 16Challenge Level
Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .
### Marbles in a Box
##### Age 11 to 16Challenge Level
How many winning lines can you make in a three-dimensional version of noughts and crosses?
### What's Possible?
##### Age 14 to 16Challenge Level
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
### Generating Triples
##### Age 14 to 16Challenge Level
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
### Always a Multiple?
##### Age 11 to 14Challenge Level
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
### Magic Squares for Special Occasions
##### Age 11 to 16
This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.
##### Age 11 to 14Challenge Level
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
### Always the Same
##### Age 11 to 14Challenge Level
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?
### Steel Cables
##### Age 14 to 16Challenge Level
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
### Seven Squares
##### Age 11 to 14Challenge Level
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
### Hallway Borders
##### Age 11 to 14Challenge Level
What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?
### Odd Differences
##### Age 14 to 16Challenge Level
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
### Terminology
##### Age 14 to 16Challenge Level
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
### AMGM
##### Age 14 to 16Challenge Level
Can you use the diagram to prove the AM-GM inequality?
### Pick's Theorem
##### Age 14 to 16Challenge Level
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
### Series Sums
##### Age 14 to 16Challenge Level
Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.
### Chocolate Maths
##### Age 11 to 14Challenge Level
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
### DOTS Division
##### Age 14 to 16Challenge Level
Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.
### One and Three
##### Age 14 to 16Challenge Level
Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . . | 2021-03-01 17:40:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.5294557213783264, "perplexity": 2007.3606869960734}, "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/1614178362741.28/warc/CC-MAIN-20210301151825-20210301181825-00553.warc.gz"} |
https://paucasedmatura.alvb.in/en/blog/haskell/gloss-haskell.html | # Playing with drawings and animations in Haskell
@ 2012-01-04 by João Paulo Pizani Flor
It’s an old adage of the computing profession that “Computer programming is the most fun you can have with your clothes on”, and certainly the most fun you can have with programming is when your program draws something on the screen, or does some animation. It’s one of those activities that can make you spend hours late at night…
On December 31, 2011, I spent my morning and afternoon writing a program that would help me check whether I had won the New Year’s lottery. Of course, being such an important program, it HAD to have a nice, beautiful graphical interface. And of course the language I was going to use was Haskell; after all, I wanted to have some FUN writing it :)
I remembered that some months ago I came across the fantastic gloss library for Haskell, but didn’t have much time to play with it… so I decided to give “gloss” a try this time. Gloss is a Haskell library that allows the user to create and run 2D static images, animations and GAMES! It runs on top of OpenGL, so - if you have a reasonable graphics card - your game can even be fast :)
What I needed to draw in my lottery program was actually too simple and boring (just some colorful numbers), so in this post we’ll create a nicer animation with gloss: we will design a clock! Yes, a classic analog clock with three dials, sort of…
First of all, a huge part of the stuff that gloss manipulates has the type Picture. And there are several convenient functions for creating fundamental pictures and transforming one picture into another. In fact, there is a “rectangleSolid” function, which is exactly what we need to create the dials of our clock. Here’s a black 50px by 200px rectangle:
rectangleSolid 50 200
We need three of these rectangles for our “clock”, one for each dial (seconds, minutes, hours). And each of them has a differente length. So this is how we defined our dials:
baseLength w = 0.75 * w
secondsDialLength x = baseLength x
minutesDialLength x = secondsDialLength x * 0.75
hoursDialLength x = minutesDialLength x * 0.70
baseDial = rectangleSolid 15
secondsDial x = baseDial (secondsDialLength x)
minutesDial x = baseDial (minutesDialLength x)
hoursDial = baseDial . hoursDialLength
I made the length of our dials variable, and it all depends on the parameter of “baseLength”, which is “w”. This is going to be the width of our clockface. You can see that three FUNCTIONS define the dials, so the dial lengths are variable, but the proportion between the dials is fixed by us… Besides that, all dials have the same width, so that’s why we defined “baseDial = rectangleSolid 15”. We are able to “fix” or “bind” one parameter of rectangleSolid to 15, thus “baseDial” needs only one more parameter, which is the dial’s length. This ability to “bind” functions partially is a very nice feature of Haskell called “currying”. Chech it out here if you want to know more: Currying.
hoursDial = baseDial . hoursDialLength
The definition of “hoursDial” in the last line of the block seems to be very different from the others, doesn’t it? hoursDial is still a function, but where is it’s length (“x”) parameter?? Well, we don’t need it, because hoursDial is defined as a composition of the functions hoursDialLength and baseDial. Remember high school?
(f ∘ g)(x) = f(g(x))
So now compare the two definition of hoursDial below, substituting “baseDial” for “f” and “hoursDialLength” for “g”. Nice, hun?
hoursDial x = baseDial (hoursDialLength x)
hoursDial = baseDial ∘ hoursDialLength
Very often in Haskell we can use function composition to make code look less verbose and easier to read.
Now, onwards with our clock! Until now, we have got 3 nice static dials. Let’s make them move :) In gloss, we can make our pictures move by adding an extra parameter to the picture definition: the time. Take a look at this:
movingSquare t = rotate (90 * t) (rectangleSolid 100 100)
Our “movingSquare” is now an animation. Animations have type Float → Picture, i.e, they are functions giving a picture (frame) for each instant of time! Cool idea, don’t you think? The time parameter (“t”) is the number of seconds elapsed since the beginning of the animation. Thus, in the example above, we are rotating our little square 90 degrees per second…
But the seconds dial of our clock needs to turn 6 degrees per second (in order to turn 360 degrees in one minute). And the minutes dial needs to be 60 times slower than the seconds dial, and so on… Thus, that’s how we define the speeds of our dials:
secondsRotationSpeed t = 6 * t
minutesRotationSpeed t = secondsRotationSpeed t / 60
hoursRotationSpeed t = minutesRotationSpeed t / 60
Our speed functions define how much a dial is turned (degrees), according to the time (seconds) elapsed. And now, with the speeds defined, we can very easily define our “moving objects” - our animated clock dials:
seconds x t = rotate (secondsRotationSpeed t) (secondsDial x)
minutes x t = rotate (minutesRotationSpeed t) (minutesDial x)
hours x t = rotate (hoursRotationSpeed t) (hoursDial x)
They still take that length parameter (“x”), but now they also take a time parameter (“t”). And the “core of the movement” here is the function “rotate”. The amount to rotate depends on time, and the picture to be rotated is parameterized by the length we give (“x”).
Well, well, well… We can have now our clock. It will be very minimalistic (Apple style :P), so it will ONLY contain three dials and nothing more. If you have some Pictures, and want to build one single Picture out of them all, the appropriately named “pictures” function from gloss can help you:
clock x t = pictures [dial x t | dial <- [seconds, minutes, hours]]
Our clock is an animation, so it has that trailing time (“t”) parameter, plus it has a variable size, which we pass as the “x” parameter. We take each of our animated dials, and apply the SAME parameters to all of them. What it means, of course, is that “global time” “runs” at the same speed for all dials. This is not the case when the clock travels close to the speed of light, but I digress… :P
As a final touch of elegance, however, I wanted to add the possibility of accelerating time :) With the function “variableSpeedClock” below, we can make a second in our animation take less (or more) than a second in the real world. Take a look:
variableSpeedClock speed x t = clock x (t * speed)
We multiply t by the speed parameter, so (quite obviously) the larger the value of speed, the quicker one “second” will be.
That’s it, that’s our FULL funky minimalistic analog clock. We can now RUN the animation by calling the (also appropriately called) animate function from gloss:
animate (InWindow "GlossClock" (size, size) (200,200)) white (variableSpeedClock speed' size')
In case you want to run the code and see the beauty for yourself, there are two possibilities:
1. Install the Haskell Platform in your computer (for Ubuntu: “sudo apt-get install haskell-platform”), and afterwards install gloss (typing cabal install gloss). You can them grab the full code here, compile it with “ghc –make Clock.hs” and RUN!
2. Using this BEAUTIFUL GLOSS WEB INTERFACE: Copy the code from HERE, paste it into the text field on the page and click “run”! | 2019-02-20 22:26:33 | {"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": 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": 0.5000272989273071, "perplexity": 2254.3198593360157}, "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-2019-09/segments/1550247496694.82/warc/CC-MAIN-20190220210649-20190220232649-00071.warc.gz"} |
https://answerbun.com/matter-modeling/what-ab-initio-methods-are-better-suited-to-modelling-disorder-in-materials/ | What ab initio methods are better suited to modelling disorder in materials?
Matter Modeling Asked on November 7, 2021
In contrast to the perfect periodic bulk materials, now computational approaches are moving towards modeling ‘real’ materials or solid solutions with dopants, dislocations, grain boundaries, and interfaces [1]. For example, the special quasirandom structures (SQS) approach is one way of modeling random alloys with statistical site occupations [2].
So my question is, what are other ab initio/theoretical approaches of modeling disorder in materials and what disorder are they good at modeling?
References
[1] Yang, Y., Chen, C., Scott, M. et al. Deciphering chemical order/disorder and material properties at the single-atom level. Nature 542, 75–79 (2017). https://doi.org/10.1038/nature21042
[2] Alex Zunger, S.-H. Wei, L. G. Ferreira, and James E. Bernard Special quasirandom structures Phys. Rev. Lett. 65, 353 – Published 16 July 1990
Coherent Potential Approximation (CPA) Method.
This method is implemented successfully in the first-principles matter modeling packages Questaal and EMTO.
You can find a general introduction about CPA on Wikipedia.
Answered by Jack on November 7, 2021
A very neat recent development for the modelling of disorder is the so-called localization landscape theory, first introduced here.
Consider a general Hamiltonian $$hat{H}=-frac{hbar^2}{2m}nabla^2+V$$. The landscape function $$u$$ is defined as the solution to this equation:
$$hat{H}u=1.$$
This innocent-looking landscape function has been shown to encode a great deal of information for studying the complex energy landscapes associated with disorder. For example, it can be shown that (i) the eigenstates $$psi(mathbf{r})$$ and eigenenergies $$E$$ of the Hamiltonian obey $$|psi(mathbf{r})|leq Eu(mathbf{r})$$, so that the localization landscape function $$u$$ delimits localization regions in space, or that (ii) the function $$W(mathbf{r})=1/u(mathbf{r})$$ acts as a confining potential that controls quantities like the exponential decay of Anderson localized states.
The localization landscape theory has recently been adopted in materials simulations of disorder. This series of three papers provides a very comprehensive introduction to the theory, and a demonstration of what is possible:
1. Basic theory and modelling: paper.
2. Urbach tails of disordered InGaN alloy quantum well layers: paper.
3. Carrier transport and recombination in light emitting diodes: paper.
Answered by ProfM on November 7, 2021
Better late than never for a partial answer.
I have worked with modeling the disorder of coverage on surfaces of materials. Recently we published a paper where we give a very general approach to modeling adsorbate-adsorbate interactions which we have seen is a problem in current literature. By using non-ideal buildups of coverage (where each adsorbate is not equally spaced), we have been able to identify better models of NO coverage on Pt3Sn. You can see below that at accessible high coverages, the structure is heavily disordered and would be incredibly difficult to find.
A large problem with modeling disorder is getting any useful information out of it. This problem needs to be solved before very disordered models will give better results than ideal models. In contrast to the comment by Camps, I don't feel making a good disordered model is difficult, the hard part is understanding what part of the disorder is actually relevant to the result.
Answered by Tristan Maxson on November 7, 2021
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2 Asked on August 19, 2021 | 2022-09-25 12:01:56 | {"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 8, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.47190946340560913, "perplexity": 2605.597932757126}, "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/1664030334528.24/warc/CC-MAIN-20220925101046-20220925131046-00139.warc.gz"} |
http://ibmaths4u.com/viewtopic.php?f=3&t=431&view=print | Page 1 of 1
Permutations Combinations IB maths HL
Posted: Wed Jan 22, 2014 7:40 pm
Permutations Combinations - IB Math HL _ IB mathematics Higher Level
In a class of 12 students, 4 are male and 8 are female. How can we find the number of committees of 5 members can be formed containing 2 males and 3 females?
Thanks
Re: Permutations Combinations IB maths HL
Posted: Wed Jan 22, 2014 7:41 pm
Permutations, Combinations - IB Maths HL
The males can be chosen by $^4\mathrm{C}_2$ ways
The females, can be chosen by $^8\mathrm{C}_3$ ways
So, the members of this committee can be chosen by
$^4\mathrm{C}_2 \cdot ^8\mathrm{C}_3 =$
$=6 \cdot 56 =336$ ways.
Hope these help!!
Re: Permutations Combinations IB maths HL
Posted: Wed Jan 22, 2014 7:41 pm
elizabeth wrote:Permutations, Combinations - IB Maths HL
The males can be chosen by $^4\mathrm{C}_2$ ways
The females, can be chosen by $^8\mathrm{C}_3$ ways
So, the members of this committee can be chosen by
$^4\mathrm{C}_2 \cdot ^8\mathrm{C}_3 =$
$=6 \cdot 56 =336$ ways.
Hope these help!!
Thank you Elisabeth!! | 2018-07-19 01:45:46 | {"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": 8, "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": 0.9378331899642944, "perplexity": 3350.779456579724}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676590443.0/warc/CC-MAIN-20180719012155-20180719032155-00225.warc.gz"} |
https://casite-746751.cloudaccess.net/docs/6kocc/b38a8c-iron-carbon-phase-diagram | # iron carbon phase diagram
Why is only one proton transferred in the reaction between sulfuric acid and sodium chloride. rev 2020.11.24.38066, The best answers are voted up and rise to the top, Chemistry Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. \begin{align} $\ce{HF}$ is similar in acidity to $\ce{HSO4-}$ (in water), so it's not as readily explainable in such a qualitative analysis. \ce{2 NaBr + H2SO4 + 2H+ &-> Br2 + SO2 + 2H2O + 2Na+} Magnesium oxalate will react with sulfuric acid to form magnesium sulfate and oxalic acid. Should recorded lectures be provided for students when teaching a math course online? Stronger acids have higher free energy content mostly due to enthalpy. I am surprised by your reaction "going crazy" especially because I do not expect a reaction netween these two acids. Why do sodium halides react so differently with sulfuric acid? No. The reaction was carried out simply by grinding SiO2/H2SO4 mixture with o-phenylenediamine, and ketone in the mortar; then the mixture was poured out into a sealed flask. Bundle of optical fibers composed of high purity silica.Silicon dioxide(SiO2) reacts in heated reflux under dinitrogen with ethylene glycol and an alkali metal base to produce highly reactive, pentacoordinate silicates which provide access to a wide variety of new silicon compounds. But difference is, to the hydrolysis of alkyne, HgSO 4 is required with H 2 SO 4. This property is used to grow single crystals of quartz in a hydrothermal process where natural quartz is dissolved in superheated water in a pressure vessel that is cooler at the top. Free energy is what thermodynamically determines the paths of reaction, and it is a combination of enthalpy changes and entropy changes. Both compounds can be hydrolyzed. Page 1 of 1. Would sodium metal react differently with aqueous solutions than with pure water? In aqueous solutions, the order of acidity is $\ce{HF} \ll \ce{HCl} < \ce{HBr} < \ce{HI}$. A detailed analysis of free energies of reactions should agree with the change in the mechanism since it is unlikely these reactions are kinetically controlled. Watch. This kind of reaction is about volatility, not "acid strength"! sodium oxide, potassium oxide, lead(II) oxide, zinc oxide, or mixtures of oxides, forming silicates and glasses as the Si-O-Si bonds in silica are broken successively).As an example the reaction of sodium oxide and SiO2 can produce sodium orthosilicate, sodium silicate, and glasses, dependent on the proportions of reactants: Examples of such glasses have commercial significance, e.g. Why does Lovecraft write that Mount Nansen (approx. What are the full requirements to get a best buddy in Pokemon Go? The reaction is also used in blast furnaces to remove sand impurities in the ore by neutralisation with calcium oxide, forming calcium silicate slag. I have the following equation, based on the reaction of $\ce{SiO2}$ with $\ce{HF}:$ $$\ce{SiO2 + HCl -> SiCl4 + 2 H2O}$$ I want to know if this reaction is indeed feasible, what conditions may be needed to induce it, or if it isn't possible why that's the case. Therefore, it becomes strongly exothermic for the $H^{+}$ to migrate to another substance that can better accept it, creating a weaker acid. The silicates are essentially insoluble in all polar solvent except methanol. SiO2 dissolves in hot concentrated alkali or fused hydroxide, as described in this idealized equation: Silicon dioxide(SiO2) will neutralise basic metal oxides (e.g. As it does not contain any hydrogen, it cannot act as a Br?nsted–Lowry acid. Silica is converted to silicon by reduction with carbon.Fluorine reacts with silicon dioxide(SiO2) to form SiF4 and O2 whereas the other halogen gases (Cl2, Br2, I2) are essentially unreactive. Equivalently, $\ce{H2SO4}$ is not as strong an oxidising agent to reduce $\ce{NaF}$ and $\ce{NaCl}$. “…presume not God to scan” – what does it mean? Asking for help, clarification, or responding to other answers. Think of NaCl + SiO2 makes Na2SiO3 + HCl if temperature is sufficient. #1 Report Thread starter 11 years ago #1 And I though it was supposed to go to NO2+ to be used in electrophilic subsitution? Why do sodium halides react so differently with sulfuric acid? Why iron reacts differently with concentrated and dilute sulfuric acid? Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. $\ce{HI}$ is somewhat close in acidity to $\ce{H2SO4}$ since neither conjugate base is very coordinating. \tag{1a}\label{NaF}\\ The inverse reaction is highly endothermic, and disfavoured. Sorry, but I don't quite understand why stronger acids have lower entropy (that's what I assume makes it more thermodynamically favourable). Not at all. How can I find the area of an overlayer structure? What is this hole above the intake of engines of Mil helicopters? Reacts with nonmetals, metals, amphoteric oxides and hydroxides. If a person is dressed up as non-human, and is killed by someone who sincerely believes the victim was not human, who is responsible? Silicon (II) oxide react with sulfuric acid to produce silicon dioxide, sulfur dioxide and water. H2SO4 is a diprotic acid, so it will lose one proton (hydrogen). Conventional explanation: $\ce{NaI}$ is a strong enough reducing agent to reduce the sulfur, and $\ce{NaBr}$ is a little less strong, so sulfur is not reduced as much. Krishna visiting Sudra's home or touching a Sudra. Why is "threepenny" pronounced as THREP.NI? \ce{8 NaI + H2SO4 + 8H+ &-> 4 I2 + H2S + 4 H2O + 8Na+} Hello, I'm facing a problem with this question: My book says that SiO 2 reacts with hot, and concentrated Sodium Hydroxide to form Sodium Silicate, Na 2 SiO 3 and water. ( II ) oxide react with bases under certain conditions write that Mount Nansen ( approx:. Bike trainer over a stationary bike every creature in existence the original Trek! Oxalic acid H 1 8 N 2, NaHSO4 will be a likely product that Mount Nansen approx! Magnesium oxalate will react with sulfuric acid to produce CH 3 CH 3 propanone. Rss reader HNO3 act as a Lux-Flood acid, SO it will lose one proton in. Creature in existence Mil helicopters of chemistry the name of this brick refers to )! Sample of sulfuric acid and sodium chloride terms of service, privacy policy and cookie policy by acid being... Wrong answers in here strong bases will react with sulfuric acid and sodium chloride dioxide and water it can act... And cookie policy react with sulfuric acid of Mil helicopters learn True Polymorph do! And students in the field of chemistry policy and cookie policy when teaching a math online. 4 / HgSO 4 to produce CH 3, propanone in water, some strong bases will react bases... 4 sio 2 + SO 2 does sio2 react with h2so4 H 2 O a ethical hacker to know C. Dioxide does n't react with sulfuric acid acids have higher free energy is what thermodynamically the... Me to find the name of this brick to alkali ), neutralized by acid SO... Great answers Polymorph, do you learn True Polymorph, do you learn True Polymorph do! First unread Skip to page: irfy Badges: 13 on writing great answers to silicon by with. To form magnesium sulfate and oxalic acid on the mixture, NaHSO4 be. Is about volatility, not a hydrogen halide NaHSO4 will be a product! Electronic applications when teaching a math course online propyne, CH 3 CH 3,.! Silicates are essentially insoluble in all polar solvent except methanol our tips writing! On silica a certain speech as unacceptable ” – what does it mean not expect reaction... Copy and paste this URL into your RSS reader alkyne, HgSO to... Area of an overlayer structure acidity '' is something which happens in aqueous solutions only and chloride. How can I find the name of this brick in plastic bottles as a base likely product to compound... Energy content mostly due to enthalpy quartz for use in electronic applications plastic bottles as a theme one. Why iron reacts differently with concentrated and dilute sulfuric acid to produce silicon dioxide, dioxide..., it can not act as a result refers to alkali ), neutralized by acid, reacts with,! Network former or lattice former thermodynamically determines the paths of reaction, and students in the H2SO4! Magnesium sulfate and oxalic acid does not contain any hydrogen, it not... A Keytone is only soluble in sulfuric acid produce sodium hydrogen sulfate instead of sodium halides react SO differently sulfuric! You saw looking for a ethical hacker to does sio2 react with h2so4 the C language in depth nowadays H2SO4 is a and! Xinxiang Yellow River Fine Chemical Industry Co., Ltd plastic bottles as a result going crazy especially! Trainer over a stationary bike but difference is, to the hydrolysis of alkyne, HgSO 4 required... Differently with aqueous solutions only as a Br? nsted–Lowry acid see our tips on writing great.... Mixture, NaHSO4 will be a likely product silicon dioxide does n't react with acid! This questions says that silicon dioxide, sulfur dioxide and water creature in existence proton hydrogen! And cookie policy to react with sulfuric acid ( H2SO4 ) silicon reduction! Regards to reciprocating Thanksgiving dinner invitations netween these two acids a bike trainer a! The original Star Trek series ever tackle slavery as a theme in of. Reaction between sulfuric acid tackle slavery as a result Post your answer ”, you agree to our of! Acid and sodium chloride 2 SO 4 instead of sodium halides react differently! Happens in aqueous solutions than with pure water ( refers to alkali ), neutralized does sio2 react with h2so4,. Nucleophilic substitution of a haloalkane with ammonia produce ammonium, not a hydrogen halide writing great answers that was you... From reacting with HgSO 4 is required with H 2 SO 4 / HgSO 4 required... Reaction H2SO4 + HNO3, why does Lovecraft write that Mount Nansen (.! Exchange is a question and answer site for scientists, academics, teachers, and it a! Sudra 's home or touching a Sudra acid ( H2SO4 ) be provided for students when teaching math... The properties of basic oxides ( refers to alkali ), neutralized by acid, it. The properties of basic oxides ( refers to alkali ), neutralized by acid, reacts ammonia. Start new discussion reply endothermic, and disfavoured the field of chemistry → +! | 2022-06-26 05:00:24 | {"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": 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": 0.44177117943763733, "perplexity": 6463.927534114623}, "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/1656103037089.4/warc/CC-MAIN-20220626040948-20220626070948-00420.warc.gz"} |
https://stats.libretexts.org/Bookshelves/Probability_Theory/Book%3A_Probability%2C_Mathematical_Statistics%2C_and_Stochastic_Processes_(Siegrist)/07%3A_Point_Estimation/7.05%3A_Best_Unbiased_Estimators | # 7.5: Best Unbiased Estimators
### Basic Theory
Consider again the basic statistical model, in which we have a random experiment that results in an observable random variable $$\bs{X}$$ taking values in a set $$S$$. Once again, the experiment is typically to sample $$n$$ objects from a population and record one or more measurements for each item. In this case, the observable random variable has the form $\bs{X} = (X_1, X_2, \ldots, X_n)$ where $$X_i$$ is the vector of measurements for the $$i$$th item.
Suppose that $$\theta$$ is a real parameter of the distribution of $$\bs{X}$$, taking values in a parameter space $$\Theta$$. Let $$f_\theta$$ denote the probability density function of $$\bs{X}$$ for $$\theta \in \Theta$$. Note that the expected value, variance, and covariance operators also depend on $$\theta$$, although we will sometimes suppress this to keep the notation from becoming too unwieldy.
#### Definitions
Suppose now that $$\lambda = \lambda(\theta)$$ is a parameter of interest that is derived from $$\theta$$. (Of course, $$\lambda$$ might be $$\theta$$ itself, but more generally might be a function of $$\theta$$.) In this section we will consider the general problem of finding the best estimator of $$\lambda$$ among a given class of unbiased estimators. Recall that if $$U$$ is an unbiased estimator of $$\lambda$$, then $$\var_\theta(U)$$ is the mean square error. Mean square error is our measure of the quality of unbiased estimators, so the following definitions are natural.
Suppose that $$U$$ and $$V$$ are unbiased estimators of $$\lambda$$.
1. If $$\var_\theta(U) \le \var_\theta(V)$$ for all $$\theta \in \Theta$$ then $$U$$ is a uniformly better estimator than $$V$$.
2. If $$U$$ is uniformly better than every other unbiased estimator of $$\lambda$$, then $$U$$ is a Uniformly Minimum Variance Unbiased Estimator (UMVUE) of $$\lambda$$.
Given unbiased estimators $$U$$ and $$V$$ of $$\lambda$$, it may be the case that $$U$$ has smaller variance for some values of $$\theta$$ while $$V$$ has smaller variance for other values of $$\theta$$, so that neither estimator is uniformly better than the other. Of course, a minimum variance unbiased estimator is the best we can hope for.
#### The Cramér-Rao Lower Bound
We will show that under mild conditions, there is a lower bound on the variance of any unbiased estimator of the parameter $$\lambda$$. Thus, if we can find an estimator that achieves this lower bound for all $$\theta$$, then the estimator must be an UMVUE of $$\lambda$$. The derivative of the log likelihood function, sometimes called the score, will play a critical role in our anaylsis. A lesser, but still important role, is played by the negative of the second derivative of the log-likelihood function. Life will be much easier if we give these functions names.
For $$\bs{x} \in S$$ and $$\theta \in \Theta$$, define \begin{align} L_1(\bs{x}, \theta) & = \frac{d}{d \theta} \ln\left(f_\theta(\bs{x})\right) \\ L_2(\bs{x}, \theta) & = -\frac{d}{d \theta} L_1(\bs{x}, \theta) = -\frac{d^2}{d \theta^2} \ln\left(f_\theta(\bs{x})\right) \end{align}
In the rest of this subsection, we consider statistics $$h(\bs{X})$$ where $$h: S \to \R$$ (and so in particular, $$h$$ does not depend on $$\theta$$). We need a fundamental assumption:
We will consider only statistics $$h(\bs{X})$$ with $$\E_\theta\left(h^2(\bs{X})\right) \lt \infty$$ for $$\theta \in \Theta$$. We also assume that $\frac{d}{d \theta} \E_\theta\left(h(\bs{X})\right) = \E_\theta\left(h(\bs{X}) L_1(\bs{X}, \theta)\right)$ This is equivalent to the assumption that the derivative operator $$d / d\theta$$ can be interchanged with the expected value operator $$\E_\theta$$.
Proof
Note first that $\frac{d}{d \theta} \E\left(h(\bs{X})\right)= \frac{d}{d \theta} \int_S h(\bs{x}) f_\theta(\bs{x}) \, d \bs{x}$ On the other hand, \begin{align} \E_\theta\left(h(\bs{X}) L_1(\bs{X}, \theta)\right) & = \E_\theta\left(h(\bs{X}) \frac{d}{d \theta} \ln\left(f_\theta(\bs{X})\right) \right) = \int_S h(\bs{x}) \frac{d}{d \theta} \ln\left(f_\theta(\bs{x})\right) f_\theta(\bs{x}) \, d \bs{x} \\ & = \int_S h(\bs{x}) \frac{\frac{d}{d \theta} f_\theta(\bs{x})}{f_\theta(\bs{x})} f_\theta(\bs{x}) \, d \bs{x} = \int_S h(\bs{x}) \frac{d}{d \theta} f_\theta(\bs{x}) \, d \bs{x} = \int_S \frac{d}{d \theta} h(\bs{x}) f_\theta(\bs{x}) \, d \bs{x} \end{align} Thus the two expressions are the same if and only if we can interchange the derivative and integral operators.
Generally speaking, the fundamental assumption will be satisfied if $$f_\theta(\bs{x})$$ is differentiable as a function of $$\theta$$, with a derivative that is jointly continuous in $$\bs{x}$$ and $$\theta$$, and if the support set $$\left\{\bs{x} \in S: f_\theta(\bs{x}) \gt 0 \right\}$$ does not depend on $$\theta$$.
$$\E_\theta\left(L_1(\bs{X}, \theta)\right) = 0$$ for $$\theta \in \Theta$$.
Proof
This follows from the fundamental assumption by letting $$h(\bs{x}) = 1$$ for $$\bs{x} \in S$$.
If $$h(\bs{X})$$ is a statistic then
$\cov_\theta\left(h(\bs{X}), L_1(\bs{X}, \theta)\right) = \frac{d}{d \theta} \E_\theta\left(h(\bs{X})\right)$
Proof
First note that the covariance is simply the expected value of the product of the variables, since the second variable has mean 0 by the previous theorem. The result then follows from the basic condition.
$$\var_\theta\left(L_1(\bs{X}, \theta)\right) = \E_\theta\left(L_1^2(\bs{X}, \theta)\right)$$
Proof
This follows since $$L_1(\bs{X}, \theta)$$ has mean 0 by the theorem above.
The following theorem gives the general Cramér-Rao lower bound on the variance of a statistic. The lower bound is named for Harold Cramér and CR Rao:
If $$h(\bs{X})$$ is a statistic then $\var_\theta\left(h(\bs{X})\right) \ge \frac{\left(\frac{d}{d \theta} \E_\theta\left(h(\bs{X})\right) \right)^2}{\E_\theta\left(L_1^2(\bs{X}, \theta)\right)}$
Proof
From the Cauchy-Scharwtz (correlation) inequality, $\cov_\theta^2\left(h(\bs{X}), L_1(\bs{X}, \theta)\right) \le \var_\theta\left(h(\bs{X})\right) \var_\theta\left(L_1(\bs{X}, \theta)\right)$ The result now follows from the previous two theorems.
We can now give the first version of the Cramér-Rao lower bound for unbiased estimators of a parameter.
Suppose now that $$\lambda(\theta)$$ is a parameter of interest and $$h(\bs{X})$$ is an unbiased estimator of $$\lambda$$. Then $\var_\theta\left(h(\bs{X})\right) \ge \frac{\left(d\lambda / d\theta\right)^2}{\E_\theta\left(L_1^2(\bs{X}, \theta)\right)}$
Proof
This follows immediately from the Cramér-Rao lower bound, since $$\E_\theta\left(h(\bs{X})\right) = \lambda$$ for $$\theta \in \Theta$$.
An estimator of $$\lambda$$ that achieves the Cramér-Rao lower bound must be a uniformly minimum variance unbiased estimator (UMVUE) of $$\lambda$$.
Equality holds in the previous theorem, and hence $$h(\bs{X})$$ is an UMVUE, if and only if there exists a function $$u(\theta)$$ such that (with probability 1) $h(\bs{X}) = \lambda(\theta) + u(\theta) L_1(\bs{X}, \theta)$
Proof
Equality holds in the Cauchy-Schwartz inequality if and only if the random variables are linear transformations of each other. Recall also that $$L_1(\bs{X}, \theta)$$ has mean 0.
The quantity $$\E_\theta\left(L^2(\bs{X}, \theta)\right)$$ that occurs in the denominator of the lower bounds in the previous two theorems is called the Fisher information number of $$\bs{X}$$, named after Sir Ronald Fisher. The following theorem gives an alternate version of the Fisher information number that is usually computationally better.
If the appropriate derivatives exist and if the appropriate interchanges are permissible then $\E_\theta\left(L_1^2(\bs{X}, \theta)\right) = \E_\theta\left(L_2(\bs{X}, \theta)\right)$
The following theorem gives the second version of the Cramér-Rao lower bound for unbiased estimators of a parameter.
If $$\lambda(\theta)$$ is a parameter of interest and $$h(\bs{X})$$ is an unbiased estimator of $$\lambda$$ then
$\var_\theta\left(h(\bs{X})\right) \ge \frac{\left(d\lambda / d\theta\right)^2}{\E_\theta\left(L_2(\bs{X}, \theta)\right)}$
Proof
This follows from the results above.
#### Random Samples
Suppose now that $$\bs{X} = (X_1, X_2, \ldots, X_n)$$ is a random sample of size $$n$$ from the distribution of a random variable $$X$$ having probability density function $$g_\theta$$ and taking values in a set $$R$$. Thus $$S = R^n$$. We will use lower-case letters for the derivative of the log likelihood function of $$X$$ and the negative of the second derivative of the log likelihood function of $$X$$.
For $$x \in R$$ and $$\theta \in \Theta$$ define \begin{align} l(x, \theta) & = \frac{d}{d\theta} \ln\left(g_\theta(x)\right) \\ l_2(x, \theta) & = -\frac{d^2}{d\theta^2} \ln\left(g_\theta(x)\right) \end{align}
$$L^2$$ can be written in terms of $$l^2$$ and $$L_2$$ can be written in terms of $$l_2$$:
1. $$\E_\theta\left(L^2(\bs{X}, \theta)\right) = n \E_\theta\left(l^2(X, \theta)\right)$$
2. $$\E_\theta\left(L_2(\bs{X}, \theta)\right) = n \E_\theta\left(l_2(X, \theta)\right)$$
The following theorem gives the second version of the general Cramér-Rao lower bound on the variance of a statistic, specialized for random samples.
If $$h(\bs{X})$$ is a statistic then
$\var_\theta\left(h(\bs{X})\right) \ge \frac{\left(\frac{d}{d\theta} \E_\theta\left(h(\bs{X})\right) \right)^2}{n \E_\theta\left(l^2(X, \theta)\right)}$
The following theorem give the third version of the Cramér-Rao lower bound for unbiased estimators of a parameter, specialized for random samples.
Suppose now that $$\lambda(\theta)$$ is a parameter of interest and $$h(\bs{X})$$ is an unbiased estimator of $$\lambda$$. Then $\var_\theta\left(h(\bs{X})\right) \ge \frac{(d\lambda / d\theta)^2}{n \E_\theta\left(l^2(X, \theta)\right)}$
Note that the Cramér-Rao lower bound varies inversely with the sample size $$n$$. The following version gives the fourth version of the Cramér-Rao lower bound for unbiased estimators of a parameter, again specialized for random samples.
If the appropriate derivatives exist and the appropriate interchanges are permissible) then $\var_\theta\left(h(\bs{X})\right) \ge \frac{\left(d\lambda / d\theta\right)^2}{n \E_\theta\left(l_2(X, \theta)\right)}$
To summarize, we have four versions of the Cramér-Rao lower bound for the variance of an unbiased estimate of $$\lambda$$: version 1 and version 2 in the general case, and version 1 and version 2 in the special case that $$\bs{X}$$ is a random sample from the distribution of $$X$$. If an ubiased estimator of $$\lambda$$ achieves the lower bound, then the estimator is an UMVUE.
### Examples and Special Cases
We will apply the results above to several parametric families of distributions. First we need to recall some standard notation. Suppose that $$\bs{X} = (X_1, X_2, \ldots, X_n)$$ is a random sample of size $$n$$ from the distribution of a real-valued random variable $$X$$ with mean $$\mu$$ and variance $$\sigma^2$$. The sample mean is $M = \frac{1}{n} \sum_{i=1}^n X_i$ Recall that $$\E(M) = \mu$$ and $$\var(M) = \sigma^2 / n$$. The special version of the sample variance, when $$\mu$$ is known, and standard version of the sample variance are, respectively, \begin{align} W^2 & = \frac{1}{n} \sum_{i=1}^n (X_i - \mu)^2 \\ S^2 & = \frac{1}{n - 1} \sum_{i=1}^n (X_i - M)^2 \end{align}
#### The Bernoulli Distribution
Suppose that $$\bs{X} = (X_1, X_2, \ldots, X_n)$$ is a random sample of size $$n$$ from the Bernoulli distribution with unknown success parameter $$p \in (0, 1)$$. In the usual language of reliability, $$X_i = 1$$ means success on trial $$i$$ and $$X_i = 0$$ means failure on trial $$i$$; the distribution is named for Jacob Bernoulli. Recall that the Bernoulli distribution has probability density function $g_p(x) = p^x (1 - p)^{1-x}, \quad x \in \{0, 1\}$ The basic assumption is satisfied. Moreover, recall that the mean of the Bernoulli distribution is $$p$$, while the variance is $$p (1 - p)$$.
$$p (1 - p) / n$$ is the Cramér-Rao lower bound for the variance of unbiased estimators of $$p$$.
The sample mean $$M$$ (which is the proportion of successes) attains the lower bound in the previous exercise and hence is an UMVUE of $$p$$.
#### The Poisson Distribution
Suppose that $$\bs{X} = (X_1, X_2, \ldots, X_n)$$ is a random sample of size $$n$$ from the Poisson distribution with parameter $$\theta \in (0, \infty)$$. Recall that this distribution is often used to model the number of random points in a region of time or space and is studied in more detail in the chapter on the Poisson Process. The Poisson distribution is named for Simeon Poisson and has probability density function $g_\theta(x) = e^{-\theta} \frac{\theta^x}{x!}, \quad x \in \N$ The basic assumption is satisfied. Recall also that the mean and variance of the distribution are both $$\theta$$.
$$\theta / n$$ is the Cramér-Rao lower bound for the variance of unbiased estimators of $$\theta$$.
The sample mean $$M$$ attains the lower bound in the previous exercise and hence is an UMVUE of $$\theta$$.
#### The Normal Distribution
Suppose that $$\bs{X} = (X_1, X_2, \ldots, X_n)$$ is a random sample of size $$n$$ from the normal distribution with mean $$\mu \in \R$$ and variance $$\sigma^2 \in (0, \infty)$$. Recall that the normal distribution plays an especially important role in statistics, in part because of the central limit theorem. The normal distribution is widely used to model physical quantities subject to numerous small, random errors, and has probability density function $g_{\mu,\sigma^2}(x) = \frac{1}{\sqrt{2 \, \pi} \sigma} \exp\left[-\left(\frac{x - \mu}{\sigma}\right)^2 \right], \quad x \in \R$
The basic assumption is satisfied with respect to both of these parameters. Recall also that the fourth central moment is $$\E\left((X - \mu)^4\right) = 3 \, \sigma^4$$.
$$\sigma^2 / n$$ is the Cramér-Rao lower bound for the variance of unbiased estimators of $$\mu$$.
The sample mean $$M$$ attains the lower bound in the previous exercise and hence is an UMVUE of $$\mu$$.
$$\frac{2 \sigma^4}{n}$$ is the Cramér-Rao lower bound for the variance of unbiased estimators of $$\sigma^2$$.
The sample variance $$S^2$$ has variance $$\frac{2 \sigma^4}{n-1}$$ and hence does not attain the lower bound in the previous exercise.
If $$\mu$$ is known, then the special sample variance $$W^2$$ attains the lower bound above and hence is an UMVUE of $$\sigma^2$$.
If $$\mu$$ is unknown, no unbiased estimator of $$\sigma^2$$ attains the Cramér-Rao lower bound above.
Proof
This follows from the result above on equality in the Cramér-Rao inequality.
#### The Gamma Distribution
Suppose that $$\bs{X} = (X_1, X_2, \ldots, X_n)$$ is a random sample of size $$n$$ from the gamma distribution with known shape parameter $$k \gt 0$$ and unknown scale parameter $$b \gt 0$$. The gamma distribution is often used to model random times and certain other types of positive random variables, and is studied in more detail in the chapter on Special Distributions. The probability density function is $g_b(x) = \frac{1}{\Gamma(k) b^k} x^{k-1} e^{-x/b}, \quad x \in (0, \infty)$ The basic assumption is satisfied with respect to $$b$$. Moreover, the mean and variance of the gamma distribution are $$k b$$ and $$k b^2$$, respectively.
$$\frac{b^2}{n k}$$ is the Cramér-Rao lower bound for the variance of unbiased estimators of $$b$$.
$$\frac{M}{k}$$ attains the lower bound in the previous exercise and hence is an UMVUE of $$b$$.
#### The Beta Distribution
Suppose that $$\bs{X} = (X_1, X_2, \ldots, X_n)$$ is a random sample of size $$n$$ from the beta distribution with left parameter $$a \gt 0$$ and right parameter $$b = 1$$. Beta distributions are widely used to model random proportions and other random variables that take values in bounded intervals, and are studied in more detail in the chapter on Special Distributions. In our specialized case, the probability density function of the sampling distribution is $g_a(x) = a \, x^{a-1}, \quad x \in (0, 1)$
The basic assumption is satisfied with respect to $$a$$.
The mean and variance of the distribution are
1. $$\mu = \frac{a}{a+1}$$
2. $$\sigma^2 = \frac{a}{(a + 1)^2 (a + 2)}$$
The Cramér-Rao lower bound for the variance of unbiased estimators of $$\mu$$ is $$\frac{a^2}{n \, (a + 1)^4}$$.
The sample mean $$M$$ does not achieve the Cramér-Rao lower bound in the previous exercise, and hence is not an UMVUE of $$\mu$$.
#### The Uniform Distribution
Suppose that $$\bs{X} = (X_1, X_2, \ldots, X_n)$$ is a random sample of size $$n$$ from the uniform distribution on $$[0, a]$$ where $$a \gt 0$$ is the unknown parameter. Thus, the probability density function of the sampling distribution is $g_a(x) = \frac{1}{a}, \quad x \in [0, a]$
The basic assumption is not satisfied.
The Cramér-Rao lower bound for the variance of unbiased estimators of $$a$$ is $$\frac{a^2}{n}$$. Of course, the Cramér-Rao Theorem does not apply, by the previous exercise.
Recall that $$V = \frac{n+1}{n} \max\{X_1, X_2, \ldots, X_n\}$$ is unbiased and has variance $$\frac{a^2}{n (n + 2)}$$. This variance is smaller than the Cramér-Rao bound in the previous exercise.
The reason that the basic assumption is not satisfied is that the support set $$\left\{x \in \R: g_a(x) \gt 0\right\}$$ depends on the parameter $$a$$.
### Best Linear Unbiased Estimators
We now consider a somewhat specialized problem, but one that fits the general theme of this section. Suppose that $$\bs{X} = (X_1, X_2, \ldots, X_n)$$ is a sequence of observable real-valued random variables that are uncorrelated and have the same unknown mean $$\mu \in \R$$, but possibly different standard deviations. Let $$\bs{\sigma} = (\sigma_1, \sigma_2, \ldots, \sigma_n)$$ where $$\sigma_i = \sd(X_i)$$ for $$i \in \{1, 2, \ldots, n\}$$.
We will consider estimators of $$\mu$$ that are linear functions of the outcome variables. Specifically, we will consider estimators of the following form, where the vector of coefficients $$\bs{c} = (c_1, c_2, \ldots, c_n)$$ is to be determined: $Y = \sum_{i=1}^n c_i X_i$
$$Y$$ is unbiased if and only if $$\sum_{i=1}^n c_i = 1$$.
The variance of $$Y$$ is $\var(Y) = \sum_{i=1}^n c_i^2 \sigma_i^2$
The variance is minimized, subject to the unbiased constraint, when $c_j = \frac{1 / \sigma_j^2}{\sum_{i=1}^n 1 / \sigma_i^2}, \quad j \in \{1, 2, \ldots, n\}$
Proof
Use the method of Lagrange multipliers (named after Joseph-Louis Lagrange).
This exercise shows how to construct the Best Linear Unbiased Estimator (BLUE) of $$\mu$$, assuming that the vector of standard deviations $$\bs{\sigma}$$ is known.
Suppose now that $$\sigma_i = \sigma$$ for $$i \in \{1, 2, \ldots, n\}$$ so that the outcome variables have the same standard deviation. In particular, this would be the case if the outcome variables form a random sample of size $$n$$ from a distribution with mean $$\mu$$ and standard deviation $$\sigma$$.
In this case the variance is minimized when $$c_i = 1 / n$$ for each $$i$$ and hence $$Y = M$$, the sample mean.
This exercise shows that the sample mean $$M$$ is the best linear unbiased estimator of $$\mu$$ when the standard deviations are the same, and that moreover, we do not need to know the value of the standard deviation. | 2020-07-06 04:27:25 | {"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": 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": 4, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9817169308662415, "perplexity": 107.42634270670942}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655890105.39/warc/CC-MAIN-20200706042111-20200706072111-00244.warc.gz"} |
https://hurmanblirrikepio.firebaseapp.com/1942/74274.html | What are medians of triangles? In today's video math lesson, we go over just that! We define medians of triangles, and go over a couple of interesting proper
Höjd median Bissectrix Triangle Grade 7. Bissectrix, median, triangelhöjd. Geometri klass 7. ✪ 7: e klass, 17 lektion, medianer, bisector och
No.3 Sulphur obscure, shining and sparsely punctate median line, punctures on each side minute and vague, triangle dull and impunctate, becoming somewhat rugose minus minsta tal) 12, medelvärde 10, och median 7. Vad är The spiral below is created by connecting 30-60-90 triangles with squares. Each triangle has. Område i en triangel; Median av en triangel; Mediatrix av en triangel Hesselbach triangel; Kärlekstriangel; Vanliga frågor om Triangle; Vad The measures of the ocellar triangle equilateral triangle, POL: LOL: OOL= 17:13: 14. part of metanotum short with a median ridge, laterally broadened and Anatomy Of The Encephalon In Situation In The Cranial Box Median Section, material och ikoner med abstract triangle shaped backgrounds set hard-coal 13.1.1 Significance of taint (triangle and extended triangle test) .
2020-10-14 | 26 min To mathematically prove this, we need to introduce a median line, a line constructed from an interior angle to the midpoint of the opposite side. To Prove :- ∠B median. Uttal: US [ˈmidiən] UK [ˈmiːdiən]. n.Medianen; (triangel) Mellersta; adj.Median, mellanliggande; i mitten genom mittpunkten av den; WebAntalet 1. average, mean; middle number or value in any given set of numbers or values (Mathematics); straight line from the vertex of a triangle to the middle of the Sub-skills approach and extensive approach to reading in As English language teachers, only by understanding what the different approaches are, what raised impunctate median line extending across supraclypeal area to median quite densely and finely punctate, triangle smooth and impunctate, surface Tyngdpunktens avstånd från ett hörn i triangeln utgör sålunda 2/3 av motsvarande medians hela längd. Omskriven cirkel.
A triangle has three medians, and they all cross over at a special point called the "centroid".
Beräkning av medelvärde och median | TI-84 Plus CE-T Python Edition. Ma 1, Ma 2 | Programmeringsexempel av medelvärde och median. Publisher: Texas Instruments Sverige Tags: Algebra, Computer Algebra, Integral calculus, Triangle.
Visit https://StudyForce.com/index.php?board=33.0 to start asking questions.Q. Determine To find the equation of the median of a triangle we examine the following example: Consider the triangle having vertices $$A\left( { - 3,2} \right)$$, B\left( {5,4 2020-02-25 · The medians are of equal length on every side of an equilateral triangle. When the median is drawn from the vertices of the equal angles in an isosceles triangle, then the median is of equal length. Though, the median can never be equal in a scalene triangle.
What are medians of triangles? In today's video math lesson, we go over just that! We define medians of triangles, and go over a couple of interesting proper
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Median of a Triangle Properties - 6:42 Trigonometry & Geometry Papapodcasts. Kurser. Lyssna på Apple Podcasts. Avsnittets webbplats · Fler avsnitt. Equilateral, scalene, right and isosceles triangle calculator.
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Ma 1, Ma 2 Tags: Algebra, Computer Algebra, Integral calculus, Triangle. Beräkning av medelvärde och median | TI-84 Plus CE-T Python Edition. Ma 1, Ma 2 | Programmeringsexempel av medelvärde och median. Publisher: Texas Instruments Sverige Tags: Algebra, Computer Algebra, Integral calculus, Triangle. I det här avsnittet ska vi lära oss om trianglar, olika typer av trianglar och hur vi beräknar en triangels omkrets och area.
Median Triangeln kallas ett segment som förbinder triangelns vertex från mitten av motsatt sida. Bevis: Median Triangle delar upp i hälften (välj ett av svaren). the line drawn from an angle of a triangle to the middle of the opposite side; any line having the nature of a diameter. median plane (anat.), the mesial plane.
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A line segment from a vertex (corner point) to the midpoint of the opposite side. A triangle has three medians, and they all cross over at a special point called the "centroid" Try moving points A, B or C:
A line segment from a vertex of a triangle to the midpoint of its opposite side is called the median of the triangle. There are three medians of a triangle. Medians of a triangle are concurrent i.e. they passes through a single point which is called the centroid of the triangle. For medians of a right triangle the following relation is valid: In a right triangle, the sum of the squares of the lengths of medians drawn to the legs is equal to five quarters of the square of the length of the hypotenuse. Solution for A median of a triangle is a segment connecting a vertex of a triangle to the midpoint of the opposite side. Let T be the triangle with vertices (0,… A median of a triangle is a line segment that joins the vertex of a triangle to the midpoint of the opposite side.
2020-02-25 · The medians are of equal length on every side of an equilateral triangle. When the median is drawn from the vertices of the equal angles in an isosceles triangle, then the median is of equal length. Though, the median can never be equal in a scalene triangle. The median is always in the interior of the triangle. Each median divides a triangle
A median of a triangle is a cevian of the triangle that joins one vertex to the midpoint of the opposite side.
På grundval av detta kan Ant. border: Median line of neck. Post. border: ant. border of scm. Inferior border: Superior Sternum. Splits into: Submandibular tirangle, Submental triangle and Episode 3: Will the median worker do better or worse in the future? | 2022-12-02 00:24:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.436604380607605, "perplexity": 7614.209911857314}, "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-2022-49/segments/1669446710870.69/warc/CC-MAIN-20221201221914-20221202011914-00125.warc.gz"} |
https://www.physicsforums.com/threads/space-time-interval-question.206353/ | # Space-time interval question
1. Dec 28, 2007
### bernhard.rothenstein
Please tell me if the following statement is correct:
In
cc(dt)^2-(dx)^2=cc(dt')^2-(dx')^2
dx and dx' represent proper length dt and (dt') representing non-proper time intervals.
2. Dec 29, 2007
### Fredrik
Staff Emeritus
It would be easier to read your posts if you used LaTeX or at least the "sup" vbtag, like this: dx2.
The integral of $$\sqrt{|-dt^2+dx^2|}$$ (units such that c=1) along a curve is called "proper time" if the curve is timelike and "proper length" if the curve is spacelike.
3. Dec 29, 2007
### bernhard.rothenstein
Thanks. Is there a simpler way to answer my question?
4. Dec 29, 2007
### Gokul43201
Staff Emeritus
There should exist only one frame where you measure the proper length of an object, no?
5. Dec 29, 2007
### Staff: Mentor
No, dx dx' dt dt' are all coordinate intervals.
6. Dec 29, 2007
### JesseM
It's correct if you're talking about two events where the spatial separation between them is dx and the time interval between them is dt in your first inertial coordinate system, while in your second inertial coordinate system the spatial separation is dx' and the time interval is dt'. In SR this will work for two events that are arbitrarily far apart, in a curved spacetime of GR it only works locally (basically considering two events that are infinitesimally close together). Also, note that if you are using three spatial dimensions x, y and z as opposed to just one, then the equation would have to be written as:
c^2 * (dt)^2 - (dx)^2 - (dy)^2 - (dz)^2 = c^2 * (dt')^2 - (dx')^2 - (dy')^2 - (dz')^2
of course, since by the pythagorean theorem we know the total distance dL between two spatial coordinates must be equal to sqrt(dx^2 + dy^2 + dz^2), the above equation reduces to:
c^2 * (dt)^2 - (dL)^2 = c^2 * (dt')^2 - (dL')^2
...where dL is the total spatial separation between the events in the first coordinate system and dL' is the total spatial separation between the events in the second coordinate system.
7. Dec 29, 2007
### Fredrik
Staff Emeritus
I'm pretty sure that's what he meant, but it's not a great idea to call dx and dx' proper lengths. Proper length is a coordinate-independent quantity that's only defined along spacelike curves.
8. Dec 29, 2007
### Fredrik
Staff Emeritus
If you want a yes/no answer rather than an explanation of what proper length is, then there is a simple answer: The dx, dx', dt and dt' are all coordinate-dependent quantities, but proper length is something coordinate-independent.
9. Dec 29, 2007
### robphy
An interval is measurement involving [the spacetime displacement between] two events.
Note that "proper length" is more correctly defined as the "distance" between two parallel timelike-lines on a spacetime diagram. So, some additional structure needs to be specified to make any reference to a "proper length".
It is [often] helpful to consider the Euclidean analogue of the problem.
As others have mentioned, dx,dx',dt,dt' are (in general) coordinate-dependent quantities.
They are essentially the legs of some "right"-triangle where the hypotenuse joins the two events. The freedom to choose among these right-triangles is essentially the coordinate-dependence.
10. Dec 29, 2007
### bernhard.rothenstein
Thanks to all. I rephrase my question. Consider the Lorentz transformations
$$\Deltax$$=$$\gamma$$[$$\Deltax'$$+V$$\Deltat'$$]
It is correct to say that $$\Deltax$$ and $$\Deltax'$$ are proper lengths
$$\Deltat$$ and $$\Deltat'$$ representing non-proper time intervals.
Please consider the concept of proper length and non-proper time interval as defined
say in
Thomas A. Moore A Travelers Guide to Spacetime McGraw-Hill,Inc 1995 pp. 126 and 46.
V in the Lorentz transformation is measured as a quotient between a proper length and a nonproper time interval.
11. Dec 29, 2007
### JesseM
Your LaTeX code wasn't working, so I fixed it in the above quote. Anyway, I don't know of any definition of "proper length" and "proper time" such that your statement would be true. Normally "proper length" refers to the length of an object in its own rest frame, but you are just talking about the coordinate distance between two events. Perhaps you could quote the definition of proper length given in "A Travelers Guide to Spacetime" since I don't have a copy of that book?
Last edited: Dec 29, 2007
12. Dec 29, 2007
### bernhard.rothenstein
space time interval question
I quote from Moore:
"We might say that to measure the "true" length of an object in space-time we should measure its length in the inertial frame in which it is at rest. This "true length" (sometimes called proper length) of the object will be longer than the value measured in any other inertial reference frame. For clarity's sake, let us refer to this length as the rest length of the object."
Are dx and dx' in the LT proper length in accordance with that definition?
I quote again from Moore:
"The time between two events measured by any clock present at both events is called a proper time. A proper time measured by a given clock is an absolute quantity independent of reference frame."
Are dt and dt' in the LT proper time intervals or they are non-proper or distorted time intervals?
Is V in LT measured as a quotient between a proper length and a nonproper time interval?
13. Dec 29, 2007
### Staff: Mentor
In any SR inertial frame, if two events have a spatial coordinate separation of dx and a temporal coordinate separation of dt then:
1a) if the events are timelike separated the proper time between them is sqrt(dt2-dx2/c2)
1b) |dt| is the proper time between the events iff dx=0
2a) if the events are spacelike separated the proper distance between them is sqrt(dx2-c2dt2))
2b) |dx| is the proper distance between the events iff dt=0
14. Dec 29, 2007
### country boy
There is an ambiguity in Moore's use of "proper" in "proper length." Unlike "proper time," which is the difference in time between two events (the measurements of a clock at rest), the length of a rod is not the difference between two unique events. If I refer to a rod at rest and define "proper length" to be the distance between the two ends of the rod as measured simultaneously, those measurements are indeed two unique events. [The measured distance then corresponds to the "proper distance" as defined above by DaleSpam.] However, the confusion comes when this "proper distance" is used as Moore's "proper length," determined by measurements performed at arbitrary times. The length of the rod at rest is not a "proper" distance in the same sense as "proper" time.
So, what about cc(dt)^2-(dx)^2=cc(dt')^2-(dx')^2 ? Both dx and dx' can be equal to a proper length by Moore's definition. If they refer to the same rod then dx=dx', and the equation can only be true if dt=dt'. If dx and dx' are equal to the proper lengths of two different rods then dx may not equal dx', but the equation can still be true for special combinations of dt and dt'. But if the equation refers to a measurement of DaleSpam's "proper distance" between two events, then |dx|=|dx'|=ds and dt=dt'=0.
15. Dec 29, 2007
### bernhard.rothenstein
space time interval question
Thanks. Let consider the Lorentz transformations for the space-time coordinates of the same event. They relate dx,dx',dt and dt'. As detected from I and I' respectively dx and dx' represent proper lengths and theit magnitude does not depend on the way in which the corresponding observers measure it. dt and dt' measured from I and I' respectively are nonproper time intervals measured in Einstein's approach being measured as a difference between the readings of two distant clocks synchronized in accoprdance with the procedure he proposed. The speed V in the LT is measured as a quotient between a proper length and a nonproper time interval.
Other synchronization procedures and other measurement conventions lead to changes in the physical meaning of the physical quantities involved in the LT. So a time transformation could relate two nonproper time intervals, a poper and a nonproper time interval or even two proper time intervals. So a "everyday clock synchronization procedured' proposed by Leubner leads to absolute simultaneity and length dilation. Such situations can be found in the case of a photographic detection of moving rods and clocks. An interesting situation is presented when a convention is made that the moving observer measures distances on a chart devised on Earth and time using his wrist watch. In all cases the Authors reinvent Einstein's special relativity and space-time diagrams addapted to the imposed conditions. What I try to do is to show that the LT as presented by Einstein account for all the possible and faisible scenarios.
As a lonely physicist I would highly appreciate your opinion telling me where I am wrong in the lines above.
Thanks in advance. Please take into account that English is not my first language and that I prepare the paper for readers who know SR from standard texts.
16. Dec 30, 2007
### Staff: Mentor
It is very unproductive to use a standard formula, standard variables, and standard terminology in your initial post and then wait until much later to mention the fact that you are using non-standard definitions of more fundamental things like space and time. If you want to use anything other than standard definitions of things you should point that out explicitly in your original post.
17. Dec 30, 2007
### country boy
As I see it, the problem is still with the meaning of "proper length." Any measurement of distance using a stationary ruler can be called a proper length. Likewise, any measurement of elapsed time using a stationary clock can be called a "proper time." But these measurements, if performed at different times or places from the events themselves will not be the space-time coordinate differences that appear in the Lorentz transformations. | 2016-10-22 12:07:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.7714740633964539, "perplexity": 922.3676087304626}, "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/1476988718957.31/warc/CC-MAIN-20161020183838-00449-ip-10-171-6-4.ec2.internal.warc.gz"} |
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# 4.5: Measurement of Gaseous Emissions from Animal Housing
Mélynda Hassouna
UMR Sol, Agro et hydrosystèmes et Spatialisation, INRAe, Agrocampus, France
Salvador Calvet
Institute of Animal Science and Technology, Universitat Politècnica de València, Valencia, Spain
Richard S. Gates
Agricultural & Biosystems Engineering, and Animal Sciences, Iowa State University, Ames, Iowa, USA
Enda Hayes
Air Quality Management Resource Centre, Department of Geography and Environmental Management, University of the West of England, Bristol, United Kingdom
Sabine Schrade
Ruminants Research Unit, Federal Department of Economic Affairs, Education and Research EAER, Agroscope, Ettenhausen, Switzerland
Key Terms
Emission processes Mass balance Ammonia Measurement techniques Validation Greenhouse gases Sampling Ventilation
### Introduction
Animal housing and manure storage facilities are two principal on-farm sources of gaseous emissions to the atmosphere. The most important pollutants emitted are ammonia (NH3), methane (CH4), and nitrous oxide (N2O).
Ammonia (NH3) is a colorless gas with a pungent smell that can have impacts on environmental and human health (Figure 4.5.1). Ammonia is emitted by many agricultural activities, including crop production as well as animal production. Ammonia plays a key role in the formation of secondary particulate matter (PM) by reacting with acidic species such as sulfur dioxide (SO2) and nitrogen oxides (NOx) to form fine aerosols, and is thus called a particulate precursor. The PM created by the reaction of NH3 and acidic species in the atmosphere contributes to poor air quality including regional haze. These particles have an aerodynamic diameter of less than 2.5 microns and are generally referred to as “PM-fine.” They are readily inhaled and populations exposed to PM-fine have greater respiratory and cardiovascular health risks such as asthma, bronchitis, cardiac arrythmia and arrest, and premature death. Some emitted NH3 is subsequently deposited on land and water downwind of facilities, and can acidify soils and freshwater. The addition of available nitrogen (N) to low-nutrient ecosystems disturbs their balance and can alter the relative growth and abundance of plant species.
Nitrous oxide and CH4 are potent greenhouse gases (GHGs) which contribute to global warming. The global warming potential (GWP) is a factor specific to each GHG and allows comparisons of the relative global warming impacts between different GHGs. This factor indicates how much heat a given gas traps over a certain time horizon (usually 100 years), compared with an equal mass of carbon dioxide (CO2). Nitrous oxide GWP for a 100-year time horizon is 265 with a lifetime in the atmosphere of 114 years. Methane GWP for a 100-year time horizon is 28 with a lifetime in the atmosphere of 12 years.
Nitrous oxide, also known as “laughing gas,” is colorless and odorless, and contributes to the destruction of the atmospheric ozone layer. In agriculture, the main source of N2O emissions is soil, from crop fertilizer use, soil cultivation, and spreading of urine and manure. Other sources include industrial processes, and natural processes involving soils and oceans.
Methane is a volatile organic compound, odorless and flammable. In agriculture, the main sources are enteric fermentation (fermentation that takes place in the digestive systems of animals) and the degradation of manure. Methane contributes to ozone formation in the lower atmosphere, and to ozone layer depletion in the upper atmosphere.
Researchers and engineers have developed different approaches to reliably measure and quantify emissions of NH3, N2O, and CH4 from animal production facilities. The implementation of these methods helps to understand the production processes, to identify the influencing factors, and to develop mitigation techniques or practices. The specific characteristics of animal housing and the variability of the houses and animal production systems make the development and implementation of the different methods a real challenge.
### Concepts
#### Animal Houses
Animal housing is designed to provide shelter and protection with control of feed consumption, diseases, parasites, and the interior thermal environment. An animal house is designed to take into account animal heat and moisture production, building characteristics (e.g., insulation and volume), and outdoor climate. Inside the house, animals produce the following critical components that affect emissions:
• • sensible heat that is transferred to the interior air by means of convection, conduction, and radiation, and causes an increase of air temperature;
• • latent heat that is generated through the evaporation of moisture from the lungs, skin, urine, and fecal material, and through increases of air humidity;
• • mixtures of feces and urine, which become a source of gaseous emissions (NH3, N2O, CO2, water vapor, CH4), and heat; and
• • CO2 from animal respiration.
The control of temperature, moisture, gas concentrations and dust concentrations inside the house is essential to achieve optimal conditions for animal growth and production. The optimal conditions vary as a function of animal age, species, and breed, and rely on the implementation of ventilation systems. The ventilation system partly controls the rate and total emissions from the building. There are two types of ventilation systems: mechanical and natural. The RAMIRAN European network (RAMIRAN, 2011) defines the two systems as:
• • mechanical ventilation, which is ventilation of a building, usually for pigs, poultry, or calves, through the use of electrically powered fans in the walls or roof that are normally controlled by the temperature in the building; and
• • natural ventilation, which is ventilation of a building, e.g., for cattle, by openings or gaps designed into the roof and/or sides of the building.
#### NH3, N2O, and CH4 Emissions Processes in Animal Housing and Influencing Factors
Ammonia is volatilized as a gas during manure management. It is mainly derived from the urea excreted in urine (or uric acid in the feces, in the case of poultry). The process of forming NH3 from urea is relatively fast and is outlined in Figure 4.5.2. Once excreted, urea is decomposed within a few hours to a few days into ammonium (NH4) by means of the enzyme urease, which is widely present in feces and soils. Ammonium is in equilibrium with dissolved NH3 enhanced at high pH values. A second physical equilibrium exists between dissolved and free NH3 in the manure matrix. Finally, the free NH3 can be released to the atmosphere. This is a mass transfer process affected by air velocity, diffusion from beneath the surface, and exposure to air on the surface of the manure.
This is a continuous process that starts in the animal housing itself and continues during manure management and land application. Several factors are involved in the amount of NH3 emitted to the atmosphere:
• • manure composition; the most relevant factors are the amount of urea excreted by the animals, the pH of the manure, and its moisture content;
• • the environmental conditions, particularly temperature and wind speed above the emitting surface;
• • the facilities for animal housing and manure management; and
• • management practices, particularly those altering the contact of manure with air, and urine with feces, by reducing time of exposure or contact surface.
On farms, N2O originates from the management of manure and its application to land as fertilizer. Emission of N2O occurs from the successive nitrification and denitrification of NH4. A first aerobic phase is required for the nitrification, while anoxic conditions are required for denitrification (Figure 4.5.3). These conditions are characteristic of the following situations:
• • composting with alternative wetting, mixing, and drying periods;
• • aerobic treatment of slurry;
• • air cleaners at the air exhaust of the animal house based on biological scrubbing of air; and
• • application of manure to soil and subsequent drying-wetting events.
Methane is produced during the anaerobic decomposition of organic matter. This occurs mainly during digestion in ruminants and decomposition of manure. In this process, the microbial breakdown of organic matter occurs in different stages, from more complex molecules to the simplest. The main mechanism is presented in Figure 4.5.4. Apart from the presence of organic matter and anoxic conditions, time (a few weeks) is needed to complete the process, and the process may be inhibited due to certain conditions, such as NH3 accumulation. In contrast to NH3, CH4 has very low solubility in water and, once produced, is released to the atmosphere through a characteristic bubbling, in the case of slurries.
For enteric fermentation in ruminants, key factors are feed composition and genetics. More digestible feeds reduce the amount of CH4. Feed constituents such as lipids or essential oils may reduce CH4 production through inhibition. Genetics also influence the amount of CH4 produced and can be modified through animal genetic selection.
CH4 emission during manure management is due to the presence of organic matter subjected to anaerobic conditions for sufficient time (about one month, at least) for methanogenic bacteria to develop. The amount and composition of organic matter determines the maximum potential for CH4 formation. Manure management practices that interrupt anaerobic conditions, reduce the load of organic matter, or feature biogas capture are potentially effective for mitigating emissions.
#### Measuring Emissions from Animal Housing
The most common approach used to determine gas emission rates from animal houses is based on quantifying ventilation rates and inlet and outlet concentrations of the gas (Figure 4.5.5). The mass flow rate of emitted gas (emission rate or ER) is proportional to the ventilation rate and the concentration difference between exhaust and outside air. Several different techniques are available to measure gas concentrations inside and outside the house and the ventilation rate.
##### Gas Concentration Measurement Techniques
The techniques used most often to measure NH3, CH4, and N2O concentrations of animal houses are either physical (optical, gas chromatography, chemiluminescence) or chemical (acid traps, active colorimetric tubes).
###### Optical Techniques
Optical techniques are based on the Beer-Lambert absorption law, which indicates that the quantity of light of a given wavelength absorbed is related to the number of gas molecules in the light’s path that are able to absorb it.
Optical techniques rely on the use of a light source, a chamber to contain the air sample during measurement, and a detector to quantify the target gas absorption. The main techniques used in animal houses are infrared (IR) spectroscopy (photoacoustic or Fourier transform), tunable diode laser absorption spectroscopy (TDLAS), off-axis integrated cavity output spectroscopy (OA-ICOS), and cavity ring-down spectroscopy (CRDS). Differences between these techniques include the detection principle of absorption and the type and wavelength of light sources (quantum cascade laser, tunable diode laser, or IR source). Techniques using lasers (a monochromatic source with a narrow band of wavelengths) are more selective, accurate, and stable than techniques with a polychromatic IR source (i.e., large band of wavelengths) because selection of the absorption of a specific wavelength from a polychromatic IR source is difficult to achieve.
One main advantage of optical techniques is that they make monitoring of concentration dynamics in near real time possible, including monitoring several gases with different concentrations at the same time (Powers and Capelari, 2016). Advantages of optical instruments include linear responses over a wide range of concentrations and the ability to measure concentrations both inside (where there could be a high concentration level) and outside (low concentration level) the animal house with the same instruments. Most optical instruments have response times adapted to measurement in animal houses. They are portable and can be used on site. Nevertheless, they can be expensive, must be calibrated, and still require accurate estimation of ventilation rate.
###### Gas Chromatography
A gas chromatograph separates components in the sample and measures their concentrations. The equipment has four basic elements: an injector, a column, an oven surrounding the column, and a detector. The sample is vaporized in the injector and swept by the carrier gas through a heated column. The column separates each compound according to its polarity and boiling point. The detector identifies and quantifies the compounds separated. The detectors include a flame ionization detector for CH4 and CO2 and an electron capture detector for N2O. This technique is accurate if the detector has been calibrated for the range of concentrations measured. It requires use of a carrier gas and regular calibration, which makes on-site implementation and continuous measurement difficult. It is often used to measure previously collected samples.
###### Chemiluminescence
Chemiluminescence is used to measure NH3 concentration. NH3 in the sample is first oxidized to N2O by a catalytic converter, and then the N2O is further oxidized to nitric oxide (NO) at high temperature and an elevated energy state. As the molecules return to a lower energy state, they release electromagnetic radiation at a specific wavelength, which is measured and quantified.
###### Acid Traps
An acid trap is a standard reference technique for measuring NH3. A known volume of air is pumped through an acid solution and recorded (Figure 4.5.6). The acid solution is later analyzed in the laboratory with a colorimetric or photometric method (Hassouna et al., 2016) to estimate the amount of NH3 trapped in the solution, as:
$NH_{3,trapped} = [N-NH_{4}^{+}]_{acid\ solution} \times m_{acid\ solution}$
where NH3,trapped = amount of NH3 trapped in the solution (kg)
[N-NH4+]acid solution = concentration of ammonium in the acid solution (kg kg−1)
macid solution = mass of the solution (kg)
From this, NH3 concentration in the air sample ($$C_{N-NH_{3}, air}$$ in kg m−3) can be calculated as:
$C_{N-NH_{3, air}} = \frac{NH_{3, trapped}}{V_{sample}}$
where Vsample (m3) is the volume of air that passed through the solution.
Strong acid solutions are used, such as boric acid, orthophosphoric acid, nitric acid, and sulfuric acid. The trap can be used for a few hours or a few days depending on the NH3 concentrations in the incoming air, the acid concentration, and volume of acid solution in the vials. Sampling time and concentration should be determined before the experiment as a function of the expected NH3 concentrations. Two vials with acid solution are used sequentially to avoid saturating a single solution. This technique provides a mean NH3 concentration over the sampling period and thus is not suitable for studies that require monitoring dynamics of NH3 concentrations in a house. Nevertheless, as it is not expensive or too time-consuming, it can be used to check the consistency of measurements made, for instance, with optical techniques.
###### Active Colorimetric Tubes
Active colorimetric tubes are a manual technique that can be used to estimate NH3, NO, volatile organic compounds (VOC), and CO2 concentrations. Tubes are manufactured to react to a specific range of concentrations of a specific target gas. Before measurement, both ends of a sealed test tube are cut open. The tube is connected tightly to a hand pump, which draws air through the tube. If present in the air, the target gas reacts with reagents in the tube. The strength of the reaction is proportional to the concentration of the gas in the air. A graduated scale is used to read the degree of color change in the tube, which indicates the concentration of the target gas. Many of the reactions used are based on pH indicators, such as bromophenol blue to measure NH3 concentrations. Gas concentration is expressed in ppm or mL m−3. This technique is reliable and simple to use. It can be used to estimate concentrations, but not to measure them continuously or accurately.
##### Ventilation Rate
In mechanically ventilated houses with modern ventilation control systems, ventilation rate could be one of many data recorded continuously. In such situations, the data are thus easily available for emission calculations. For other houses, or if the time step of recording is not suitable or recorded data are not reliable, the ventilation rate must be measured or assessed. Different methods to estimate the ventilation rate have been evaluated and described in the literature (Ogink et al., 2013; Wang et al., 2016). The method chosen depends on the type of ventilation (natural or mechanical), the accessibility of the exhaust to make physical measurements, the level of ventilation rate, and the desired degree of accuracy. Some techniques are indirect (tracer gas, heat balance), while others are direct (fan wheel anemometer, specialized instruments).
###### Tracer Gas Techniques Using Artificial Tracer Gases
Tracer gas techniques are commonly used to quantify the ventilation rate in many kinds of houses, but mainly those with natural ventilation. An external tracer gas should be safe, inert, measurable, not produced in the house, and inexpensive (Phillips et al., 2001; Sherman, 1990). The most common tracer gas used in animal houses is sulfur hexafluoride (SF6) (Mohn et al., 2018). A critical requirement is that of near-perfect air mixing inside the animal house to ensure that the tracer gas and the targeted gas (for emission calculations) being measured both disperse in a similar way. Air can be mixed artificially using a purpose-built ventilation duct (Figure 4.5.7). Tracer gases can be dosed automatically using a mass flow controller and critical orifices (Figure 4.5.8).
The basic principle for tracer gas techniques is conservation of mass (of both target gas and tracer gas). By monitoring the dosed mass flow and concentration at the sampling points of the tracer gas, the ventilation rate can be determined (Figure 4.5.9). A tracer gas release technique is chosen based on the ventilation rate, the detection limit of the device used to monitor tracer gas concentration, and the ability to control and monitor the dosed mass flow accurately. According to Ogink et al. (2013), three tracer gas release techniques can be distinguished:
• • constant injection method: tracer gas is injected at a constant rate, and its concentration is measured directly over a period of time and used to estimate the ventilation rate;
• • decay method: tracer gas is injected until its concentration stabilizes, then injection is stopped and the decay in concentration is used to calculate the ventilation rate; and
(a)
(b)
Figure $$\PageIndex{8}$$: (a) Tracer-gas dosing by steel tubes with critical orifices protected by steel elements next to the floors in a dairy housing; (b) gas bottles with mass-flow controller.
• • concentration method: tracer gas is distributed in the air of a house to a certain concentration to be constant.
Only the constant injection method and the decay method are common for measurements in animal houses.
To calculate the emission or mass flow of the target gas (e.g., NH3, CH4), a background correction of the concentration (Cx) must first be calculated for the target gases and the tracer gases:
$C_{x} = C_{x, id} - C_{x, bgd}$
where x = T (tracer gas) or G (target gas)
Cx,id = indoor gas concentration (μg m−3)
Cx,bgd = background gas concentration (μg m−3)
The ratio of the background concentrations of emitted (target) gas, CG, and tracer gas, CT, then corresponds to the ratio of their mass flow rates ($$\dot{m}$$, g d−1):
$\frac{\dot{m}_{G}}{\dot{m}_{T}} = \frac{C_{G}}{C_{T}}$
and thus
$\dot{m}_{G} = \frac{\dot{m}_{T}C_{G}}{C_{T}}$
###### Carbon Dioxide (CO2) Mass Balance or Tracer Gas Methods Using an Internal Tracer
The CO2 mass balance method is sometimes considered a tracer gas technique in which CO2 is used as an internal tracer, that is, not dosed but produced by animal respiration and manure. It can be used in naturally or mechanically ventilated houses. It is based on the hypothesis that ventilation rate determines the relationship between CO2 production in the house and the difference in CO2 concentrations between the inside and outside of the house. This method has been widely described (Blanes and Pedersen, 2005; Estellés et al., 2011; Samer et al., 2012) and is more accurate in buildings with no litter and no gas heating system.
The ventilation rate for the house can be calculated as:
$VR = \frac{\text{total heat per house}\times \text{ventilation flow per hpu}}{1000}$
where the total heat produced for the entire house is expressed in heat production units (hpu; 1 hpu is 1 kW of total animal metabolic heat production at 20°C) and the ventilation flow per hpu is in m3 h−1 hpu−1.
The International Commission of Agricultural Engineering provides a method to calculate total heat production (sensible plus latent) for different animal categories (Pedersen and Sällvik, 2002). For instance, for fattening pigs, the total heat produced for the entire house is calculated by multiplying the total heat per animal (in W animal−1) by the number of animals and converting to heat production units as:
$\text{total heat per house} = \text{total heat per animal} \times \text{number of animals}$
$\text{total heat per animal} = (5.09\times m^{0.75} +[1-(0.47+0.003 \times m)] \times [(n \times 5.09 \times m^{0.75}) - (5.09\times m^{0.75})]$
where m = animal mass (kg)
n = feed energy in relation to the heat dissipation due to maintenance (g d−1)
Ventilation flow per hpu varies as a function of animal activity at different times of the day and difference between indoor and outdoor CO2 concentrations:
$\text{Ventilation flow per hpu} = \frac{c \times (\text{relative animal activity})}{(CO_{2,indoors}-CO_{2,outdoors})\times 10^{-6}}$
where c = CO2 production (m3 h−1 hpu−1); varies as a function of animal type (Pedersen and Sällvik, 2002; Pedersen et al., 2008).
$$CO_{2, indoors} \text{ and } CO_{2, outdoors} = \text{measured indoor and outdoor } CO_{2} \text{ concentrations at time h (mL m}^{-3})$$
Relative animal activity is calculated as:
$\text{Relative animal activity} = 1-a\ sin[(\frac{2\pi}{24} \times (h+6-h_{min})]$
where a = constant expressing amplitude with respect to the constant value 1, which is a scaling factor based on empirical observation and which varies depending on the animal type (Pedersen et al., 2008)
h = time at sampling (this should be a decimal number 0 ≤ h ≤ 24), e.g., (2:10 = 2.2)
hmin = activity factor that relates to the time of day with minimum activity (hours after midnight) (Pedersen and Sällvik, 2002)
###### Use of Sensors
Fan wheel or hot wire anemometers can be used to quantify ventilation rate in mechanically ventilated houses that draw outlet air through ducts or exhaust fans. One important requirement is having access to exhaust flow where the measurements are to be made, which is not possible in many animal houses.
The anemometer measures air velocity, and ventilation rate (VR) is calculated as follows:
$VR=sA$
where s = mean airspeed (m h−1)
A = cross-sectional area of the ventilation duct or air stream (m2)
Proper methods must be utilized to obtain representative mean air velocity over the flow area, for example by selecting a sufficient number of measurement points and applying either log-linear or log-Tchebycheff rules (ISO 3966, 2008) for measurement points spacing.
Use of anemometers is not recommended in naturally ventilated houses because of their rapid change in air fluxes and large size of the open area, which would require many sensors to obtain a representative estimate of the ventilation rate.
In mechanically ventilated houses, continuous monitoring of the static pressure differential and the operating status (on-off) of each fan can be used to estimate the fan’s ventilation rate based on its theoretical or measured performance characteristics. Ideally, the in situ performance of each fan is determined first, and the house ventilation rate can be estimated by summing all operating fan flow rates. For example, Gates et al. (2004, 2005) developed and improved a fan assessment numeration system (FANS) to measure the in situ performance curve of ventilation fans operating in a negative pressure mechanically ventilated animal house (Figure 4.5.10). This unit is placed either against a fan on the inside of the house, or at the fan exterior with appropriate flexible ducting (Morello et al., 2014) to direct all airflow through the unit. A series of anemometers traverse the entire flow area to obtain a single mean air velocity, which is multiplied by the calibrated unit cross sectional area. A series of these measurements taken at different building static pressures provides an empirical fan performance curve, obtained, for example, from the regression equation of measured flow on building static pressure. Then, measurements of fan run-time and concurrent static pressure can be used to determine reasonably accurate airflow rates for each fan, and their sum is the building ventilation rate. Previous work has clearly shown that neglecting to account for building ventilation by means of direct measurement results in substantial loss in accuracy of estimates for ER, due to the variation among fans.
##### The Mass Balance Approach for a Global Estimation of N and C Emissions and Emissions Measurement Validation
A mass balance approach estimates emissions based on changes in livestock over time, without the need to measure emissions directly (Figure 4.5.11). The approach estimates total N or C emissions rather than emissions of specific gases (e.g., N-NH3, N-N2O, N2, C-CH4, C-CO2) or emission dynamics. The accuracy of mass balance calculations depends on the technical and livestock management data available, characterization of the manure and feed, and, in certain cases, the length of the period considered. To test the validity of the data used to calculate an N or C mass balance, those of non-volatile elements such as phosphorus (P) and potassium (K) must be calculated. As the latter elements are non-volatile, their mass balance deficits (difference between inputs and outputs) should be zero, but the data used in calculations will have uncertainty, especially under commercial conditions. If the mass balance deficit for P and K is too high (e.g., > 15%), then the estimates of total N and C emissions must be reconsidered.
The estimation of N, C, or water emissions (X emissions, where X is for N, C, or water) over the production period can be calculated according to the following equation:
$X_{inputs}-X_{outputs} = X_{emissions}$
Xinputs and Xoutputs are the quantity of X in all inputs and outputs. The estimation of these quantities requires careful data collection (quantities, chemical compositions) concerning animals, feed, eggs or milk (a function of animal production), litter, manure, and animal mortality. Models should be used to estimate the quantity of X in animals as a function of their weight.
### Applications
Implementing the different emissions measurement methods in an animal house requires the development of a protocol based on the objectives of the project, the specifics of the animal house, the interior environment, and outdoor weather. Three important points that should be considered in a protocol concern the sampling, the data acquisition, and the validity check of the measurements.
#### Sampling and Sensor Locations
Evaluation of gas emissions requires measurement of inlet and outlet gas concentration. Sampling at air inlets or outlets is recommended if they can be identified, if their locations are fixed over the measurement period, and if they can be easily reached. When these conditions cannot be fulfilled, multiple locations inside the house are usually selected to provide a mean indoor concentration to accommodate spatial variability of indoor concentration. The same should be done for outdoor concentration.
The presence of animals inside the animal area makes installation of gas sensors or sampling tubes more complicated. Ideally, they should be installed when no animals are in the house, such as during an outdoor or vacancy period. They should be located where animals cannot bite, bump, or move them, and should be carefully protected from animals (Figure 4.5.12). Successful sensor placement requires a trade-off between minimizing animal disturbance and maximizing the representativeness of measurements.
The environment inside animal housing is generally harsh for sensors and the air sampling system. Direct exposure to the combination of humidity, NH3, and suspended particulate matter can damage the sensors (Figure 4.5.13). Furthermore, indoor air is generally warmer and more humid than outdoor air because of animal heat production or the use of a heating or cooling system inside the house. These differences should be considered when sampling indoor air; for example, sampling tubes should be heated and insulated if air samples are analyzed in a cooler place and condensation within sampling lines might be expected.
#### Data Acquisition
During the production period, emissions can vary greatly during a 24-hour cycle and over longer time intervals. Variability is due to the same parameters that affect spatial variability, changes in animal behavior, their excretion patterns and quantity, and whether or not they have outdoor access. For instance, fattening pigs and poultry will excrete more total ammonia nitrogen (TAN) each day as they grow, yielding ever higher potential for NH3 emissions. These two kinds of temporal dynamics (daily and production period) must be considered when measuring gas emissions.
Information concerning the production conditions (number of animals, feed and water consumption, animal mortality) and outdoor climate are also required for validation of measurements and comparison with emissions data already published. All operations made by the farmers or operators (for instance, feeding changes, litter supply, or cooling system implementation) and specific events (for instance, electric power shutdown) during the measurement period should be noted because they will be helpful for data analysis and interpretation.
#### Validation of Measurements
In order to achieve good quality measurements, data validation steps are necessary at several levels:
• • Validation measurements for parts or the whole measuring setup should be carried out in advance, especially if the setup, single components, and/or the measurement objective (e.g., housing system) was not measured in this configuration before.
• • Calibration of analytical devices and sensors has to be performed according to their specifications. For some analytical instruments, measurements with a reference method (e.g., acid traps for NH3) are recommended.
• • Frequent checks of operational mode and measurement values as well as housing and management conditions are necessary.
• • Plausibility checks of raw data and emission values (e.g., comparison of courses of gas concentrations and wind speed, data check in view of a predetermined plausible range based upon user scientific and technological knowledge) help to find outliers, non-logical values, etc. These incorrect values have to be eliminated according to predefined criteria.
• • Redundancy in measurements can enhance the reliability of the values. For instance, CO2 concentrations could be measured with both a gas chromatograph and an optical gas analyzer during startup or periodically over the project.
• • Comparison of the cumulative emissions for N and carbon (C) with N and C mass balance deficits over the measuring period.
### Examples
Example $$\PageIndex{1}$$
Example 1: Calculate ammonia (NH3) emission rate from a mechanically ventilated pig house using the carbon dioxide mass balance approach
Problem:
The following case study is based on a project undertaken on behalf of the Irish Environmental Protection Agency to test the suitability of existing NH3 emission factors currently being used for different pig life stages and to explore the potential impact on Natura 2000 sites (i.e., Special Areas of Conservation and Special Protection Areas that may be sensitive to N). The monitoring equipment is described in detail by Kelleghan et al. (2016).
The house used mechanical ventilation, with air inlets along the side of the house and ceiling exhaust fans, but no access to the exhaust stream for direct measurement of ventilation rate. Gas concentrations were measured at 10 a.m. in a house with 406 fattening pigs of 81.25 kg animal−1 reared on a fully slatted floor. Indoor NH3 concentrations in the house were measured using a Los Gatos Research ultraportable ammonia analyzer (UAA) in combination with an eight-inlet multiport unit allowing for multiple sampling points. Outdoor concentrations were not measured; for the purposes of this study and the calculations, they were assumed to be zero. Indoor and outdoor CO2 was measured in the sample gas drawn by the UAA using a K30 CO2 sensor (Senseair, Sweden). Measured gas concentrations are presented in Table 4.5.1. Additional parameters adapted for finishing pigs are included in Table 4.5.2. Calculate the building ventilation rate and the NH3 emission rate.
Solution
Calculate the building ventilation rate using the CO2 mass balance approach expressed by Equation 4.5.6:
Measured CO2,indoors 915 ppm (mL m−3) CO2,outdoors 403 ppm (mL m−3) NH3,indoors 3.73 (mg m3) NH3,outdoors 0 (mg m3)
$$VR=\frac{\text{total heat per house} \times \text{ventilation flow per hpu}}{1000}$$ (Equation $$\PageIndex{6}$$)
To calculate the total heat per house, first, calculate the relative animal activity with Equation 4.5.10 using the given values:
Table $$\PageIndex{2}$$: Specific parameters useful for calculations.
Parameter Value for This Experiment
a (Equation 4.5.11)
0.53
hmin (Equation 4.5.11)
1:40 AM = 1.7 in equation
n (Equation 4.5.9)
3.38 (g day−1)
c (Equation 4.5.10)
m3 h−1 hpu−1
$$\text{Relative animal activity} = 1-a\ sin[(\frac{2\pi}{24})\times(h+6-h_{min})]$$ (Equation $$\PageIndex{10}$$)
$$\text{Relative animal activity} = 1-0.53\times sin[(\frac{2\pi}{24})\times(10+6-1.7)] = 1.2994$$
Next, calculate the total heat production per animal with Equation 4.5.8:
$$\text{Total heat per animal} = 5.09 \times m^{0.75} + [1-(0.47+0.003\times m)]\times [(n \times 5.09\times m^{0.75})-(5.09\times m^{0.75})]$$ (Equation $$\PageIndex{8}$$)
$$\text{Total heat per animal} = 5.09 \times 81.25^{0.75} + [1-(0.47+0.003\times 81.25)]\times [(0.00338 \times 5.09\times 81.25^{0.75})-(5.09\times 81.25^{0.75})] = 231.6\ W \text{ animal}^{-1}$$
Thus, with 406 pigs, the total heat production for the house is:
$$406 \times 231.6 = 94026.5\ W$$
Next, calculate ventilation flow per heat producing unit (hpu) with Equation 4.5.9:
$$\text{Ventilation flow per hpu} = \frac{c \times (\text{relative animal activity})}{CO_{2,indoors}-CO_{2,outdoors} \times 10^{-6}}$$ (Equation $$\PageIndex{9}$$)
$$\text{Ventilation flow per hpu} = \frac{0.185 \times (1.2994)}{(915-403) \times 10^{-6}} = 469.5$$
Ventilation flow per hpu will equal 469.5 m3 h−1 hpu−1.
Finally, substitute the computed values into Equation 4.5.6 to calculate the building ventilation rate:
$$VR = \frac{\text{total heat per house} \times \text{ventilation flow per hpu}}{1000}$$ (Equation $$\PageIndex{6}$$)
$$VR = \frac{94026.5 \times 469.5}{1000} = 44,145.2$$
The building ventilation rate is 44145.2 m3 h−1.
To calculate NH3 emission rate, note that the NH3 emission rate for the house is proportional to the difference between indoor and outdoor NH3 concentrations multiplied by the ventilation rate:
$$\frac{(3.73-0)\times 44,145.2}{3600} = 45.7$$
The NH3 emission rate for this example is 45.7 mL s−1. This equates to 9.7 g day−1 animal−1.
Example $$\PageIndex{2}$$
Example 2: Calculate methane (CH4) emissions from a naturally ventilated dairy cattle house using tracer gas (SF6) measurement data
Problem:
This example is based on investigations carried out in experimental dairy housing for emission measurements (Mohn et al., 2018). The housing consists of two experimental compartments, each for 20 dairy cows, and a central section for milking, technical installations, an office, and analytics. The experimental compartments are naturally ventilated without thermal insulation and with flexible curtains as facades.
The diluted tracer gas, SF6, was dosed continuously through steel tubes with critical capillaries (every third meter) next to the aisles to mimic the emission sources. Stainless steel tubes and critical orifices were protected with metal profiles from damage by animals and contamination with excrement. To adjust analyzed tracer gas concentration in an optimal range (> 0.05 μg m−3, < 1.5 μg m−3SF6), tracer gas flow was set according to meteorological and ventilation conditions (e.g., curtains open/closed) by mass flow controllers. Integrative air samples at a height of 2.5 m with a piping system consisting of Teflon tubes and critical glass orifices (every second meter) allow a representative sample. Teflon filters protected the critical orifices from dust and insects. Flow rates for individual orifices of the dosing and sampling systems were monitored before and after every measuring period using mass flow meters. The analytical instrumentation for CH4 (cavity ring-down spectrometer, CRDS, Picarro Inc., Santa Clara, CA, USA) and SF6 analysis (GC-ECD, Agilent, Santa Clara, CA, USA) was located in an air-conditioned trailer in the central section. The two compartments were sampled alternately for 10 min. each. Further, once per hour, the background (approximately 25 m from housing, unaffected by the housing) was sampled, so at least two 10-min samples per compartment were obtained every hour.
To describe the measurement situation, relevant accompanying parameters such as housing and outdoor climate, animal parameters (e.g., live weight, milk yield, milk composition, milk urea content, urine urea content), and feed (quality and quantity, amount of trough residue) were recorded.
Calculate the CH4 emissions during one 10-min measurement taken on 23 September at 12:40 p.m. in a compartment with perforated floors holding 20 cows. The measured gas concentrations are presented in Table 4.5.3.
Solution
Calculate the background correction (Cx) according to Equation 4.5.3 using measured concentrations of SF6 $$(C_{SF6})$$ and CH4 $$(C_{CH4})$$:
Table $$\PageIndex{3}$$: Concentrations of CH4 and SF6 at 12:40 p.m., 23 September.
Parameter Value
SF6 mass flow
2.879 g d−1
SF6 background
0.052 μg m−3
SF6 housing, sampling points
1.820 μg m−3
CH4 background
1384.2 μg m−3
CH4 housing, sampling points
9118.6 μg m−3
$$C_{x}=C_{x,sp}-C_{x,bgd}$$ (Equation $$\PageIndex{3}$$)
$$C_{CH4}=9118.6-1384.2=7734.4\ \mu g \ m^{-3}$$
$$C_{SF6}=1.820-0.052=1.768\ \mu g \ m^{-3}$$
Calculate emission or mass flow calculation of CH4 using Equation 4.5.5:
$$\dot{m}_{G} = \frac{\dot{m}_{T} \times C_{G}}{C_{T}}$$ (Equation $$\PageIndex{5}$$)
$$\dot{m}_{CH4} = \frac{2.879\ g \ d^{-1} \times 7734.4\ \mu g\ m^{-3}}{1.768\ \mu g\ m^{-3}} = 12,597.9\ g\ d^{-1}$$
ER per cow (20 cows per compartment): 629.7 g d−1
### Image Credits
Figure 1. Department for Environment Food & Rural Affairs. (CC By 4.0). (2018). Environmental impacts of ammonia emissions. Retrieved from https://www.gov.uk/government/publications/code-of-good-agricultural-practice-for-reducing-ammonia-emissions/code-of-good-agricultural-practice-cogap-for-reducing-ammonia-emissions
Figure 2. Calvet, S. (CC By 4.0). (2014). Process leading to ammonia emission and contributing factors. Adapted from Snoek et al.
Figure 3. Calvet, S. (CC By 4.0). (2001). Reactions leading to N2O emissions: nitrification (in green) and denitrification (in blue). Adapted from Wrage et al.
Figure 4. Calvet, S. (CC By 4.0). (2020). Process and microorganisms involved in methane formation.
Figure 5. Calvet, S. (CC By 4.0). (2020). Direct emission measurement in an animal house. The difference in concentration between indoors and outdoors and the ventilation rate must be measured (adapted from Calvet et al., 2013).
Figure 6. Hassouna, M. (CC By 4.0). (2020). Acid trap configuration.
Figure 7. Hassouna, M. (CC By 4.0). (2020). Duct for dispersing a tracer gas within an animal house.
Figure 8. Agroscope, Schrade, S. (CC By 4.0). (2020). (a) Tracer-gas dosing by steel tubes with critical orifices protected by steel elements next to the floors in a dairy housing; (b) gas bottles with mass-flow controller. Photos adapted from Agroscope. [Fair Use].
Figure 9. Agroscope, Schrade, S. (CC By 4.0). (2020). Principle of the tracer gas method.
Figure 10. Gates, R. (CC By 4.0). (2020). Fan Assessment Numeration System (FANS) developed by Gates et al. (2005) to describe the performance curve of a ventilation fan in an animal house. Note the unit can be used on either the inlet or exhaust side of the ventilation fan.
Figure 11. Hassouna, M. (CC By 4.0). (2016). The mass balance approach in animal houses.
Figure 12. Right. Agroscope, Schrade, S. (CC By 4.0). (2020). Sensors must be installed beyond the reach of animals.
Figure 12 Left. Hassouna, M. (CC By 4.0). (2020). Sensors must be installed beyond the reach of animals.
Figure 13. Hassouna, M. (CC By 4.0). (2020). Clogging of a sampling tube and its dust filter after three months of measurement in a dairy cattle house.
### References
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Calvet, S., Gates, R. S., Zhang, G., Estelles, F., Ogink, N. W. M., Pedersen, S., & Berckmans, D. (2013). Measuring gas emissions from livestock buildings: A review on uncertainty analysis and error sources. Biosyst. Eng., 116(3), 221-231. https://doi.org/10.1016/j.biosystemseng.2012.11.004.
Estelles, F., Fernandez, N., Torres, A. G., & Calvet, S. (2011). Use of CO2 balances to determine ventilation rates in a fattening rabbit house. Spanish J. Agric. Res., 9(3), 8. doi.org/10.5424/sjar/20110903-368-10.
Gates, R. S., Casey, K. D., Xin, H., Wheeler, E. F., & Simmons, J. D. (2004). Fan assessment numeration system (FANS) design and calibration specifications. Trans. ASAE, 47(5), 1709-1715. https://doi.org/10.13031/2013.17613.
Gates, R. S., Xin, H., Casey, K. D., Liang, Y., & Wheeler, E. F. (2005). Method for measuring ammonia emissions from poultry houses. J. Appl. Poultry Res., 14(3), 622-634. https://doi.org/10.1093/japr/14.3.622.
Hassouna, M., Eglin, T., Cellier, P., Colomb, V., Cohan, J.-P., Decuq, C., . . . Ponchant, P. (2016). Measuring emissions from livestock farming: Greenhouse gases, ammonia and nitrogen oxides. France: INRA-ADEME.
ISO 3966. (2008). Measurement of fluid flow in closed conduits. Velocity area method for regular flows using Pitot static tubes.
Kelleghan, D. B., Hayes, E. T., & Curran, T. P. (2016). Profile of ammonia and water vapor in an Irish broiler production house. ASABE Paper No. 162461252. St. Joseph, MI: ASABE. https://doi.org/10.13031/aim.20162461252.
Mohn, J., Zeyer, K., Keck, M., Keller, M., Zahner, M., Poteko, J., . . . Schrade, S. (2018). A dual tracer ratio method for comparative emission measurements in an experimental dairy housing. Atmospheric Environ., 179, 12-22. https://doi.org/10.1016/j.atmosenv.2018.01.057.
Morello, G. M., Overhults, D. G., Day, G. B., Gates, R. S., Lopes, I. M., & Earnest Jr., J. (2014). Using the fan assessment numeration system (FANS) in situ: A procedure for minimizing errors during fan tests. Trans. ASABE, 57(1), 199-209. https://doi.org/10.13031/trans.57.10190.
Ogink, N. W. M., Mosquera, J., Calvet, S., & Zhang, G. (2013). Methods for measuring gas emissions from naturally ventilated livestock buildings: Developments over the last decade and perspectives for improvement. Biosyst. Eng., 116(3), 297-308. https://doi.org/10.1016/j.biosystemseng.2012.10.005.
Pedersen, S., & Sallvik, K. (2002). 4th Report of working group on climatization of animal houses. Heat and moisture production at animal and house levels. Research Centre Bygholm, Danish Institute of Agricultural Sciences.
Pedersen, S., Blanes-Vidal, V., Jorgensen, H., Chwalibog, A., Haeussermann, A., Heetkamp, M. J., & Aarnink, A. J. (2008). Carbon dioxide production in animal houses: A literature review. Agric. Eng. Int.: CIGR J.
Phillips, V. R., Lee, D. S., Scholtens, R., Garland, J. A., & Sneath, R. W. (2001). A review of methods for measuring emission rates of ammonia from livestock buildings and slurry or manure stores, Part 2: Monitoring flux rates, concentrations and airflow rates. J. Agric. Eng. Res., 78(1), 1-14. https://doi.org/10.1006/jaer.2000.0618.
Powers, W., & Capelari, M. (2016). Analytical methods for quantifying greenhouse gas flux in animal production systems. J. Animal Sci., 94(8), 3139-3146. https://doi.org/10.2527/jas.2015-0017.
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Samer, M., Ammon, C., Loebsin, C., Fiedler, M., Berg, W., Sanftleben, P., & Brunsch, R. (2012). Moisture balance and tracer gas technique for ventilation rates measurement and greenhouse gases and ammonia emissions quantification in naturally ventilated buildings. Building Environ., 50, 10-20. https://doi.org/10.1016/j.buildenv.2011.10.008.
Sherman, M. H. (1990). Tracer-gas techniques for measuring ventilation in a single zone. Building Environ., 25(4), 365-374. https://doi.org/10.1016/0360-1323(90)90010-O.
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• Was this article helpful? | 2021-12-02 00:57:01 | {"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": 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": 0.4810473620891571, "perplexity": 5282.419483839801}, "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-2021-49/segments/1637964361064.58/warc/CC-MAIN-20211201234046-20211202024046-00113.warc.gz"} |
https://www.usgs.gov/media/images/lunar-apennine-mountains | # Lunar Apennine Mountains
### Detailed Description
Top down view of the Apennine Mountains. Taken during the Apollo 15 mission. Originally thought to be rough and jagged by map makers Arnold Mason and Robert Hackman, it was discovered that they had been leveled and rounded by micrometoroid impacts over billions of years.
Public Domain. | 2022-01-18 03:01:36 | {"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": 0.8000699877738953, "perplexity": 10864.023499156809}, "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/1642320300658.84/warc/CC-MAIN-20220118002226-20220118032226-00264.warc.gz"} |
https://stats.stackexchange.com/questions/411491/bic-in-practice-with-gaussian-distribution | # BIC in practice with gaussian distribution
I am considering a gaussian distribution: $$$$y \sim N(net(x,w), \sigma^2).$$$$ $$net()$$ is just the output of some neural net with weights $$w$$ and input $$x$$.
The log-likelihood is $$$$\log L = -\frac{n}{2} (\log(2\pi) + \log(\sigma^2)) - \frac{1}{2\sigma^2} \sum (y_i - net(x,w))^2$$$$ and the BIC is \begin{align} BIC &= -2 \log L + \log(n) \cdot d \\ &= n(\log(2\pi) + \log(\sigma^2)) + \frac{1}{\sigma^2} \sum (y_i - net(x,\hat{w}))^2 + \log(n) \cdot d\\ &\approx n/\sigma^2 (MSE + \frac{\log(n)\cdot d \cdot \sigma^2}{n}), \end{align} where $$d$$ is the number of parameters. What I wonder is, how do I estimate $$\sigma^2$$ in practice? My intuition was to estimate it with usual MLE which is the MSE, i.e. $$\hat{\sigma}^2 = 1/n \sum(y_i - net(x, \hat{w}))^2$$, but then the first term would just cancel out... And does the variance count as a parameter in $$d$$? I am really confused how to use this in practice.
I don't see any particular reason for not estimating $$\sigma$$ as in any Gaussian process, with sample standard deviation given a sample of outputs from the neural network for some fixed $$x, \omega$$
But make sure $$\sigma$$ is indeed independent from those!
And yes, $$\sigma$$ is a parameter in its own right!
• Thank you! If I am getting you right, this is exactly the formula to estimate $\sigma^2$ that I have written down (which is the MSE). So you say that is okay that the is $BIC = n + \log(n) \cdot d$? That seems weird to me – msloryg Jun 4 '19 at 16:10 | 2021-06-14 15:46:57 | {"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": 14, "wp-katex-eq": 0, "align": 1, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9998241066932678, "perplexity": 632.349623399756}, "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-2021-25/segments/1623487612537.23/warc/CC-MAIN-20210614135913-20210614165913-00239.warc.gz"} |
http://clay6.com/qa/69722/the-diameter-of-the-pupil-of-human-eye-is-about-2mm-human-eye-is-the-most-s | # The diameter of the pupil of human eye is about $2mm$. Human eye is the most sensitive to the wave length $555\;nm$ Find the limit of resolution of human eye.
$\begin{array}{1 1} 33.9 \times 10^{-4} rad \\ 3.39 \times 10^4 \;rad \\ 33.9\times 10^4 \;rad \\ 3.39\times 10^4\;rad \end{array}$ | 2020-07-16 18:11:54 | {"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": 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": 0.8231562376022339, "perplexity": 332.54079094739166}, "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/1593657172545.84/warc/CC-MAIN-20200716153247-20200716183247-00153.warc.gz"} |
https://www.physicsforums.com/threads/riemann-integrable.109599/ | # Riemann Integrable
1. Feb 6, 2006
Is it possible to show by induction that f:[a,b]->R, a bounded function, is Riemann integrable if f has a countable number of discontinuities? I'm told this is usually done with Lebesgue integrals, but I don't see why an inductive proof of this using Riemann/Darboux integrals can't work.
2. Feb 6, 2006
### matt grime
What are you inducting on? It can't be the unmber of discontinuities. showing something for a finite number of things in no way says anything about the infinite case.
3. Feb 6, 2006
### NateTG
No.
Consider, for example, the characteristic function of the irrationals.
f(x) is 1 if x is irrational, and 0 if x is rational.
This has a countable number of discontinuities -- one at every rational number -- but is most certainly not Riemann integrable as the lower sum will always be 0, and the upper sum will always be 1.
Last edited: Feb 6, 2006
4. Feb 6, 2006
### matt grime
A 'rational' number of discontinuities? No, it has an uncountable number of discontinuities. It is discontinuous at every point at of the interval [0,1]
5. Feb 6, 2006
Well, f is integrable if it has 1 discontinuity, and whenever it has n discontinuities and is integrable, it implies that having n+1 discontinuities is integrable.
6. Feb 6, 2006
### matt grime
and that tells you nothing (as given) about countably (but not finitely) many discontinuities. (Proof... well, according to this logic, the set of natural numbers is finite by 'induction')
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http://loppao.com/0ft2me/how-to-find-tangent-in-physics-18ebfe | Anything pulled, hung, supported, or swung from a rope, string, cable, etc. Math & Physics forum @ gamedev.net foxmanx_7 Author. A tangent to a curve is a line that touches the curve at one point and a normal is a line perpendicular to a tangent to the curve. Linear Speed (Tangential Speed): Linear speed and tangential speed gives the same meaning for circular motion. I have made an attempt involving bisecting c2-p1 at M, and performing trigonometric operations to find measure of angle TMC2. Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. Sine, Cosine and Tangent. A similar method can be used to measure μ k. To do that you give the top object a push as you increase the angle. Example question: Find m at the point (9, 3). In this section, we are going to see how to find the slope of a tangent line at a point. Below is the simple online Tangential and Radial acceleration calculator. In this article, we will discuss how to find the tangent and normal to a circle. The equation of a tangent to the circle at (x 1, y 1) is given by xx 1 + yy 1 = a 2. b. C2 and P1 are known points. The tangent function, along with sine and cosine, is one of the three most common trigonometric functions.In any right triangle, the tangent of an angle is the length of the opposite side (O) divided by the length of the adjacent side (A).In a formula, it is written simply as 'tan'. The short question: Is there any simple way in Nape to calculate the points of tangency with a Nape body object or shape given a point outside that body? Then I was asked to find the phase constant. if a flat plane were constructed with the same normal from the reference point, the tangent vector would be coplanar with that plane). Steps to find Tangent and Normal to a Circle. m = (9-5)/(3-2.3) = 4/.7 = … Usually when you’re doing a problem like this, you will be given a function whose tangent line you need to find.And you will also be given a point or an x value where the line needs to be tangent to the given function.. Thus, it can also be called as tangential speed, distance taken in a In this case we use again same definition. For a given angle θ each ratio stays the same no matter how big or small the triangle is. In one dimension motion we define speed as the distance taken in a unit of time. The geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. In this non-linear system, users are free to take whatever path through the material best serves their needs. I tried a few things but finally gave up and asked Mastering Physics for the answer, which is: $\phi_0=2.62$ rad. Solution: Solving Problems with the Tangent Ratio Examples: 1. Step 1. Tangential and Radial Acceleration Calculator. We can plug in the slope for "m" and the coordinates of the point for x and y: In the graph above the tangent line is again drawn in red. Example: Calculate the length of the side x, given that tan θ = 0.4 . Tan is sin/cos. Thus, a particle in circular motion with a tangential acceleration has a total acceleration that is the vector sum of … theta = tan –1 (y/x). So you are actually using the derivative for this. Given: Equation = x 2 + 3x + 1 x = 2. is subject to the force of tension. 20 m north or minus 50 feet). These unique features make Virtual Nerd a viable alternative to private tutoring. a. The tangent touches the curve at (2.3, 5). Find an equation of the tangent to the curve at the given point by both eliminating the parameter and without eliminating the parameter. From basic algebra to complex calculus, … When a current is passed through the circular coil, a magnetic field (B) is produced at the center of the coil in a direction perpendicular to the plane of the coil. Learn how differentiation used to find equations of the tangent … That line would be the line tangent to the curve at that point. Thus, for our triangle, we know: Using your calculator, solve for : This is . In physics, tension is the force exerted by a rope, string, cable, or similar object on one or more objects. Radius of circle C2 is also constant and known. You find the tangent line of a function by finding the derivative, the slope, of that function at a specific point. One common application of the derivative is to find the equation of a tangent line to a function. Solution: Step 1: To find the y value, substitute the x value in given equation. Once we have the point from the tangent it is just a matter of plugging the values into the formula. Like all forces, tension can accelerate objects or cause them to deform. I am trying to find point T to eventually construct line p1-t, which is tangent to circle c2. The velocity of an object at any given moment is the slope of the tangent line through the relevant point on its x … Example: Draw the tangent line for the equation, y = x 2 + 3x + 1 at x=2. The working of tangent galvanometer is based on the tangent law. Hi, i am trying to code a function that calculates the vertexes tangent for a model, but it still looking flat and i don't know why :/ If somebody know how to do this and find any errors in my code, please give me a hand! If x 2 + y 2 = a 2 is a circle, then. That point is called the point of tangency. 122 September 25, 2009 12:32 PM. Like the inverse of sin, the inverse of tan is also restricted to quadrants 1 and 4. The tangent line will be perpendicular to the line going through the points and , so it will be helpful to know the slope of this line: Since the tangent line is perpendicular, its slope is . If you've plotted the displacement-time graph (a parabola) and can draw the tangents to this curve at the two time instants given, just find the slopes = (delta D / delta t ) of these two tangent lines. A Tangent vector is typically regarded as one vector that exists within the surface's plane (for a flat surface) or which lies tangent to a reference point on a curved surface (ie. Sine, Cosine and Tangent (often shortened to sin, cos and tan) are each a ratio of sides of a right angled triangle:. Find the adjacent side given the opposite side of a right triangle. The answer is -pi/4 Alright, archtan / tan^-1(x) is the inverse of tangent. In SI units, it is measured in radians per second squared (rad/s 2 ), and is usually denoted by the Greek letter alpha ($\alpha$). We know that the tangent of an angle is equal to the ratio of the side adjacent to that angle to the opposite side of the triangle. However, in this case the direction of motion is always tangent to the path of the object. Angular acceleration is the rate of change of angular velocity. Plug in the numbers for this example to get The sine, cosine and tangent are used to find the degrees of a right angle triangle. To calculate them: Divide the length of one side by another side Suppose that the coordinates of the vector are (3, 4). If we extend this line, we can easily calculate the displacement of distance over time and determine our velocity at that given point. How to use the tangent ratio to find missing sides or angles? The direction of velocity vector is tangent to the curve (so it's same as the unit vector computed). Now, take the decimal portion in order to find … The equation of normal to the circle at (x 1, y … So in this sense the derivative actually recreates the curve you are given. To write the equation in the form , we need to solve for "b," the y-intercept. Since I had this equation in my notes, Substitute that point and the derivative into the slope intercept formula, y=mx+b, to find the y-intercept. You can find the angle theta as the tan –1 (4/3) = 53 degrees.. You can use the Pythagorean theorem to find the hypotenuse — the magnitude, v — of the triangle formed by x, y, and v:. If y = f(x) is the equation of the curve, then f'(x) will be its slope. What is the first law in physics? We are basically being asked the question what angle/radian does tan(-1) equal. The slope of the graph at the two time instants IS the same thing as the slope of the tangent lines at these two time instants. Using the unit circle we can see that tan(1)= pi/4. Find the opposite side given the adjacent side of a right triangle. The tangent vector is at any point of the curve parametrized by t can be found by differentiation: dx/dt = <3, 6 t, 6t> If x(t) is the position vector of a particle following this path, then this derivative is the velocity vector (which by definition is tangent to the path). Its working is based on the principle of the tangent law of magnetism. High School Physics: ... Find the tangential velocity of a bicycle whose wheels have an angular velocity of 10 pi radians per second and a radius of 12 inches. Knowing this we are solving for the inverse of tan -1. To accomplish this, what you actually do is making use of a lot of tangent lines! So, the coefficient of static friction is equal to the tangent of the angle at which the objects slide. The direction of tangential acceleration is tangent to the circle whereas the direction of centripetal acceleration is radially inward toward the center of the circle. tangential acceleration: The acceleration in a direction tangent to the circle at the point of interest in circular motion. We may obtain the slope of tangent by finding the first derivative of the equation of the curve. 2. Determine the slope of the line 6x+2y=1 Slope of a line perpendicular to 6x+2y=1 is the opposite reciprocal of the line's slope. The unit vector (towards the tangent at this point) is given by $$\hat{v}=\cos\theta\hat{i}+\sin\theta\hat{j}$$ where $\theta$ is angle from x-axis( can be computed from the angle that is given). The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century. 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Of angle TMC2 users are free to take whatever path through the material best serves their needs to the! -1 ) equal ( x ) will be its slope motion we define speed as the taken! ( Remember that the coordinates of the vector are ( 3, 4 ) am trying to tangent.
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https://tex.stackexchange.com/questions/42199/rendering-mathml-in-latex | # Rendering MathML in LaTeX?
I have MathML that I need to render in my LaTeX document. Is there a recommended way of doing so?
So far, from googling, I've found MathParser - a Java converter. But I was hoping there might be a better way to do it.
• How elaborate are the MathML fragments that you need to render? It might be possible to write a simple converter if they only use a limited subset of MathML. – Bruno Le Floch Jan 24 '12 at 23:28
• @BrunoLeFloch Are MathML and LaTeX similar enough to easily write a simple converter for? It may be complex fragments (data from a client) but it might still be manageable to write one. – mifrai Jan 25 '12 at 0:27
• Is it content math or presentation mathml – Aditya Aug 26 '13 at 14:09
• googlecode is down, but pmml2tex is available on github: github.com/davidcarlisle/web-xslt/tree/master/pmml2tex – ShreevatsaR Nov 9 '17 at 2:19
• @DavidCarlisle I've successfully used your stylesheet (and some other tools available online) - and agree with the comment. – nam Sep 3 at 22:02 | 2019-09-19 14:31:24 | {"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.544356644153595, "perplexity": 2367.974559658988}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573533.49/warc/CC-MAIN-20190919142838-20190919164838-00025.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=211221 | # Polynomial with two unknowns
by AFG34
Tags: polynomial, solved, unknowns
P: 128 1. The problem statement, all variables and given/known data The graph of f(x)= 3x^4 + 14x^3 + px^2 + qx + 24 has x-intercepts -4 and 2. Determine the function. 3. The attempt at a solution I could solve it if there were only one unknown but i don't know how to do it if there are two unknowns. What i did so far is plug -4 for x and 0 for the output, got an expression = 0 did the same thing for 2 since both equal 0, i set them equal to each other, simplified and got: 2p-q=13.3 Don't know what to do next.
P: 1,754 Rather then setting them equal to each other. Solve for p with one of your x-intercepts then plug it in your other set with the other x-intercept.
P: 128 but there are two unknowns in each expression
Emeritus
PF Gold
P: 9,772
Polynomial with two unknowns
Quote by AFG34 but there are two unknowns in each expression
What roco is getting at is that you can create a system of two simultaneous equations thus;
$$f(-4) = 0$$
$$f(2) = 0$$
P: 128 yes i know that, that is what i initially did. So you get 2 equations, both equal to 0, both have q and p in them. But i don't know what to do next.
Emeritus
PF Gold
P: 9,772
Quote by AFG34 yes i know that, that is what i initially did. So you get 2 equations, both equal to 0, both have q and p in them. But i don't know what to do next.
Have you never solved simultaneous equations before?
P: 128 no, i haven't.
Emeritus
PF Gold
P: 9,772
Quote by AFG34 i don't think so.
Okay, in that case if you post the two equations you obtain I shall walk you through the process.
P: 128 1) 0 = 16p - 4q - 104 2) 0 = 4p +2q + 184 then i divided both by 2: 1) 0 = 8p - 2q - 52 2) 0 = 2p + q + 92
Emeritus Sci Advisor PF Gold P: 9,772 Good, so now multiply (2) by 2 and then add the two equations.
P: 1,754 Did you ever learn to solve matrices in Algebra class?
P: 128 8p + 132 ok so p = -11, q = -70 thnx
P: 128
Quote by rocophysics Did you ever learn to solve matrices in Algebra class?
nope
P: 1,754
Quote by AFG34 nope
I'll go step by step. Let me type this up.
Emeritus
PF Gold
P: 9,772
Quote by rocophysics Did you ever learn to solve matrices in Algebra class?
If the OP hasn't met simultaneous equations it's pretty safe to say that they haven't been introduced to linear algebra (which IMHO is over-kill for a question such as this).
Quote by AFG34 8p + 132 ok so p = -11
Firstly, what you have written is an expression, not an equation. Secondly, you might want to check your coefficent of p.
P: 128 0 = 12p + 132 is an equation ya i typed it wrong checked the back of the book, got the right answer (f(x)= 3x^4 + 14x^3 - 11x^2 - 70x + 24). ok i know how to do them now, thnx hootenanny, funny name
P: 1,754
Quote by AFG34 0 = 12p + 132 is an equation ya i typed it wrong checked the back of the book, got the right answer. ok i know how to do them now, thnx
Want to learn the fast way?
P: 128 sure got a quick question: is the graph of y = log3^(x+4) the same as the graph of y = log3^x+4? *the base is 3 not 10 I think the second one is y=log3^x moved up by 4 units.
Related Discussions Calculus & Beyond Homework 5 Calculus & Beyond Homework 2 Introductory Physics Homework 13 Precalculus Mathematics Homework 11 Precalculus Mathematics Homework 12 | 2014-09-02 21:22:09 | {"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": 0.4287625849246979, "perplexity": 785.6189555031402}, "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-2014-35/segments/1409535922871.14/warc/CC-MAIN-20140901014522-00053-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://en.m.wikipedia.org/wiki/Johann_Balmer | # Johann Jakob Balmer
(Redirected from Johann Balmer)
Johann Jakob Balmer (1 May 1825 – 12 March 1898) was a Swiss mathematician and mathematical physicist.
Johann Jakob Balmer
Born1 May 1825
Died12 March 1898 (aged 72)
NationalitySwitzerland
Alma materUniversity of Basel
Scientific career
FieldsMathematics
## Biography
Balmer was born in Lausen, Switzerland, the son of a Chief Justice also named Johann Jakob Balmer. His mother was Elizabeth Rolle Balmer, and he was the oldest son. During his schooling he excelled in mathematics, and so decided to focus on that field when he attended university.
He studied at the University of Karlsruhe and the University of Berlin, then completed his Ph.D. from the University of Basel in 1849 with a dissertation on the cycloid. Johann then spent his entire life in Basel, where he taught at a school for girls. He also lectured at the University of Basel. In 1868 he married Christine Pauline Rinck at the age of 43. The couple had a total of six children.
Despite being a mathematician, Balmer is best remembered for his work on spectral series. His major contribution (made at the age of sixty, in 1885) was an empirical formula for the visible spectral lines of the hydrogen atom, the study of which he took up at the suggestion of Eduard Hagenbach also of Basel.[1][2] Using Ångström's measurements of the hydrogen lines, he arrived at a formula for computing the wavelength as follows:
${\displaystyle \lambda \ ={\frac {hm^{2}}{m^{2}-n^{2}}}}$
for n = 2, h = 3.6456×10−7 m, and m = 3, 4, 5, 6, and so forth.
In his 1885 notice, he referred to h (now known as the Balmer constant) as the "fundamental number of hydrogen." Balmer then used this formula to predict the wavelength for m = 7 and Hagenbach informed him that Ångström had observed a line with wavelength 397 nm. Two of his colleagues, Hermann Wilhelm Vogel and William Huggins, were able to confirm the existence of other lines of the series in the spectrum of hydrogen in white stars.
Balmer's formula was later found to be a special case of the Rydberg formula, devised by Johannes Rydberg in 1888.
${\displaystyle {\frac {1}{\lambda }}\ ={\frac {4}{h}}\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)=R_{H}\left({\frac {1}{n_{1}^{2}}}-{\frac {1}{n_{2}^{2}}}\right)}$
with ${\displaystyle R_{H}}$ being the Rydberg constant for hydrogen, ${\displaystyle n_{1}=2}$ for Balmer's formula, and ${\displaystyle n_{2}>n_{1}}$ .
A full explanation of why these formulas worked, however, had to wait until the presentation of the Bohr model of the atom by Niels Bohr in 1913.
Johann Balmer died in Basel. He was 72 when he died.
## References
1. ^ Balmer, J.J. (1885). "Notiz über die Spectrallinien des Wasserstoffs" [Note on the spectral lines of hydrogen]. Annalen der Physik und Chemie. 3rd series (in German). 25: 80–87.
2. ^ Magie, William Francis (1969). A Source Book in Physics. Cambridge, Massachusetts: Harvard University Press. p 360
3. ^ | 2019-02-20 15:15:13 | {"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": 0.7618536949157715, "perplexity": 2429.394516434139}, "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/1550247495147.61/warc/CC-MAIN-20190220150139-20190220172139-00583.warc.gz"} |
http://openstudy.com/updates/4facebe2e4b059b524f87759 | ## milliex51 3 years ago How do I prove (1-cos2x)(1+tan2x)=tan2x?
1. milliex51
do I solve it like this: $(\sin ^{2}x)(1+\tan ^{2x})$ $\sin ^{2x}\div1\times(sinx \div cosx)$ $\sin ^{2x}cosx \div sinx$
2. milliex51
x is suppose to be down, oops.
3. FoolForMath
Are you sure that is true: http://www.wolframalpha.com/input/?i=+%28%281-cos+2x%29%281%2Btan+2x%29%29+%3D%3D+tan+2x
4. experimentX
5. milliex51
false? .. oh.
6. milliex51
how would I know when it's false on a test?
7. experimentX
1# wolfram says so 2# it is giving solution ... this is only true in certain points. not on every ponits
8. milliex51
i don't understand..
9. experimentX
10. experimentX
sin(x) = cos(pi/2 - x) <---- put any value of x, you will always find it true!!
11. FoolForMath
I don't know how this TrueQ works: http://www.wolframalpha.com/input/?i=TrueQ%5Bsin+%5E2+x+%2B+cos+%5E2+x+%3D+1%5D
12. experimentX
LOL .. i found it a little while ago
13. TransendentialPI
Why not just expand the left side and do some cancelling?
14. milliex51
that's what I tried but then there's going to be no sinx at the bottom?
15. experimentX
It is equivalent to .. proving $$\-cos 2x ( 1 +\tan2x) - 1 = 0$$ $$\implies \cos2x + \sin2x + 1 = 0$$ <--- which is a false identity!!
16. experimentX
$$(\tan2x + 1) - (\tan2x+1)\cos2x = \tan 2x$$ $$\implies 1 - (\tan2x+1)\cos2x = 0$$ | 2015-05-23 02:51:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7703346610069275, "perplexity": 4157.250257072921}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207927104.48/warc/CC-MAIN-20150521113207-00100-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://brilliant.org/problems/its-going-to-be-hard/ | # It's going to be hard!
Geometry Level 2
The length of string between a kite and a point on the ground is 90 meters.If the string makes an angle $$\theta$$ with the ground level such that $$tan$$$$\theta=\frac{15}{8}$$, how high is the kite?Assume that there is no slack in the string.
× | 2017-07-28 15:19:07 | {"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": 0.5229434370994568, "perplexity": 401.72327030955597}, "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-2017-30/segments/1500550969387.94/warc/CC-MAIN-20170728143709-20170728163709-00507.warc.gz"} |
https://en.m.wikibooks.org/wiki/Abstract_Algebra/Algebras | # Abstract Algebra/Algebras
In this section we will talk about structures with three operations. These are called algebras. We will start by defining an algebra over a field, which is a vector space with a bilinear vector product. After giving some examples, we will then move to a discussion of quivers and their path algebras.
## Algebras over a FieldEdit
Definition 1: Let ${\displaystyle F}$ a field, and let ${\displaystyle A}$ be an ${\displaystyle F}$ -vector space on which we define the vector product ${\displaystyle \cdot \,:\,A\times A\rightarrow A}$ . Then ${\displaystyle A}$ is called an algebra over ${\displaystyle F}$ provided that ${\displaystyle (A,+,\cdot )}$ is a ring, where ${\displaystyle +}$ is the vector space addition, and if for all ${\displaystyle a,b,c\in A}$ and ${\displaystyle \alpha \in F}$ ,
1. ${\displaystyle a(bc)=(ab)c}$ ,
2. ${\displaystyle a(b+c)=ab+ac}$ and ${\displaystyle (a+b)c=ac+bc}$ ,
3. ${\displaystyle \alpha (ab)=(\alpha a)b=a(\alpha b)}$ .
The dimension of an algebra is the dimension of ${\displaystyle A}$ as a vector space.
Remark 2: The appropriate definition of a subalgebra is clear from Definition 1. We leave its formal statement to the reader.
Definition 2: If ${\displaystyle (A,+,\cdot )}$ is a commutative ring, ${\displaystyle A}$ is called a commutative algebra. If it is a division ring, ${\displaystyle A}$ is called a division algebra. We reserve the terms real and complex algebra for algebras over ${\displaystyle \mathbb {R} }$ and ${\displaystyle \mathbb {C} }$ , respectively.
The reader is invited to check that the following examples really are examples of algebras.
Example 3: Let ${\displaystyle F}$ be a field. The vector space ${\displaystyle F^{n}}$ forms a commutative ${\displaystyle F}$ -algebra under componentwise multiplication.
Example 4: The quaternions ${\displaystyle \mathbb {H} }$ is a 4-dimensional real algebra. We leave it to the reader to show that it is not a 2-dimensional complex algebra.
Example 5: Given a field ${\displaystyle F}$ , the vector space of polynomials ${\displaystyle F[x]}$ is a commutative ${\displaystyle F}$ -algebra in a natural way.
Example 6: Let ${\displaystyle F}$ be a field. Then any matrix ring over ${\displaystyle F}$ , for example ${\displaystyle \left({\begin{array}{cc}F&0\\F&F\end{array}}\right)}$ , gives rise to an ${\displaystyle F}$ -algebra in a natural way.
## Quivers and Path AlgebrasEdit
Naively, a quiver can be understood as a directed graph where we allow loops and parallell edges. Formally, we have the following.
Definition 7: A quiver is a collection of four pieces of data, ${\displaystyle Q=(Q_{0},Q_{1},s,t)}$ ,
1. ${\displaystyle Q_{0}}$ is the set of vertices of the quiver,
2. ${\displaystyle Q_{1}}$ is the set of edges, and
3. ${\displaystyle s,t\,:\,Q_{1}\rightarrow Q_{0}}$ are functions associating with each edge a source vertex and a target vertex, respectively.
We will always assume that ${\displaystyle Q_{0}}$ is nonempty and that ${\displaystyle Q_{0}}$ and ${\displaystyle Q_{1}}$ are finite sets.
Example 8: The following are the simplest examples of quivers:
1. The quiver with one point and no edges, represented by ${\displaystyle 1}$ .
2. The quiver with ${\displaystyle n}$ point and no edges, ${\displaystyle 1\quad 2\quad ...\quad n}$ .
3. The linear quiver with ${\displaystyle n}$ points, ${\displaystyle 1\,{\stackrel {a_{1}}{\longrightarrow }}\,2\,{\stackrel {a_{2}}{\longrightarrow }}\,...\,{\xrightarrow {a_{n-1}}}\,n}$ .
4. The simplest quiver with a nontrivial loop, ${\displaystyle 1{\underset {a}{\stackrel {b}{\leftrightarrows }}}2}$ .
Definition 9: Let ${\displaystyle Q}$ be a quiver. A path in ${\displaystyle Q}$ is a sequence of edges ${\displaystyle a=a_{m}a_{m-1}...a_{1}}$ where ${\displaystyle s(a_{i})=t(a_{i-1})}$ for all ${\displaystyle i=2,...,m}$ . We extend the domains of ${\displaystyle s}$ and ${\displaystyle t}$ and define ${\displaystyle s(a)\equiv s(a_{0})}$ and ${\displaystyle t(a)\equiv t(a_{m})}$ . We define the length of the path to be the number of edges it contains and write ${\displaystyle l(a)=m}$ . With each vertex ${\displaystyle i}$ of a quiver we associate the trivial path ${\displaystyle e_{i}}$ with ${\displaystyle s(e_{i})=t(e_{i})=i}$ and ${\displaystyle l(e_{i})=0}$ . A nontrivial path ${\displaystyle a}$ with ${\displaystyle s(a)=t(a)=i}$ is called an oriented loop at ${\displaystyle i}$ .
The reason quivers are interesting for us is that they provide a concrete way of constructing a certain family of algebras, called path algebras.
Definition 10: Let ${\displaystyle Q}$ be a quiver and ${\displaystyle F}$ a field. Let ${\displaystyle FQ}$ denote the free vector space generated by all the paths of ${\displaystyle Q}$ . On this vector space, we define a vector porduct in the obvious way: if ${\displaystyle u=u_{m}...u_{1}}$ and ${\displaystyle v=v_{n}...v_{1}}$ are paths with ${\displaystyle s(v)=t(u)}$ , define their product ${\displaystyle vu}$ by concatenation: ${\displaystyle vu=v_{n}...v_{1}u_{m}...u_{1}}$ . If ${\displaystyle s(v)\neq t(u)}$ , define their product to be ${\displaystyle vu=0}$ . This product turns ${\displaystyle FQ}$ into an ${\displaystyle F}$ -algebra, called the path algebra of ${\displaystyle Q}$ .
Lemma 11: Let ${\displaystyle Q}$ be a quiver and ${\displaystyle F}$ field. If ${\displaystyle Q}$ contains a path of length ${\displaystyle |Q_{0}|}$ , then ${\displaystyle FQ}$ is infinite dimensional.
Proof: By a counting argument such a path must contain an oriented loop, ${\displaystyle a}$ , say. Evidently ${\displaystyle \{a^{n}\}_{n\in \mathbb {N} }}$ is a linearly independent set, such that ${\displaystyle FQ}$ is infinite dimensional.
Lemma 12: Let ${\displaystyle Q}$ be a quiver and ${\displaystyle F}$ a field. Then ${\displaystyle FQ}$ is infinite dimensional if and only if ${\displaystyle Q}$ contains an oriented loop.
Proof: Let ${\displaystyle a}$ be an oriented loop in ${\displaystyle Q}$ . Then ${\displaystyle FQ}$ is infinite dimensional by the above argument. Conversely, assume ${\displaystyle Q}$ has no loops. Then the vertices of the quiver can be ordered such that edges always go from a lower to a higher vertex, and since the length of any given path is bounded above by ${\displaystyle |Q_{0}|-1}$ , there dimension of ${\displaystyle FQ}$ is bounded above by ${\displaystyle \mathrm {dim} \,FQ\leq |Q_{0}|^{2}-|Q_{0}|<\infty }$ .
Lemma 13: Let ${\displaystyle Q}$ be a quiver and ${\displaystyle F}$ a field. Then the trivial edges ${\displaystyle e_{i}}$ form an orthogonal idempotent set.
Proof: This is immediate from the definitions: ${\displaystyle e_{i}e_{j}=0}$ if ${\displaystyle i\neq j}$ and ${\displaystyle e_{i}^{2}=e_{i}}$ .
Corollary 14: The element ${\displaystyle \sum _{i\in Q_{0}}e_{i}}$ is the identity element in ${\displaystyle FQ}$ .
Proof: It sufficed to show this on the generators of ${\displaystyle FQ}$ . Let ${\displaystyle a}$ be a path in ${\displaystyle Q}$ with ${\displaystyle s(a)=j}$ and ${\displaystyle t(a)=k}$ . Then ${\displaystyle \left(\sum _{i\in Q_{0}}e_{i}\right)a=\sum _{i\in Q_{0}}e_{i}a=e_{j}a=a}$ . Similarily, ${\displaystyle a\left(\sum _{i\in Q_{0}}e_{i}\right)=a}$ .
To be covered:
- General R-algebras | 2016-10-24 20:12:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 109, "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": 0.9709470868110657, "perplexity": 764.3583141113361}, "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-2016-44/segments/1476988719754.86/warc/CC-MAIN-20161020183839-00248-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://asa2.silverchair.com/anesthesiology/article/126/1/173/660/Current-Status-of-Neuromuscular-Reversal-and?searchresult=1 | Postoperative residual neuromuscular block has been recognized as a potential problem for decades, and it remains so today. Traditional pharmacologic antagonists (anticholinesterases) are ineffective in reversing profound and deep levels of neuromuscular block; at the opposite end of the recovery curve close to full recovery, anticholinesterases may induce paradoxical muscle weakness. The new selective relaxant-binding agent sugammadex can reverse any depth of block from aminosteroid (but not benzylisoquinolinium) relaxants; however, the effective dose to be administered should be chosen based on objective monitoring of the depth of neuromuscular block.
To guide appropriate perioperative management, neuromuscular function assessment with a peripheral nerve stimulator is mandatory. Although in many settings, subjective (visual and tactile) evaluation of muscle responses is used, such evaluation has had limited success in preventing the occurrence of residual paralysis. Clinical evaluations of return of muscle strength (head lift and grip strength) or respiratory parameters (tidal volume and vital capacity) are equally insensitive at detecting neuromuscular weakness. Objective measurement (a train-of-four ratio greater than 0.90) is the only method to determine appropriate timing of tracheal extubation and ensure normal muscle function and patient safety.
“We cannot solve our problems with the same thinking we used when we created them.”
Albert Einstein
IT is widely recognized that the introduction of neuromuscular blocking agents into clinical care has revolutionized surgery and facilitated significant medical advances in the last century. In their 2015 article, Game changers: the 20 most important anesthesia articles ever published, Barash et al.1 list the seminal article by Griffith and Johnson2 on the use of curare in general anesthesia as number 13. Appropriately, the other “top-20 contender” article is the study by Beecher and Todd3 of surgical deaths during anesthesia that, although rebutted at the time, features a “special discussion of muscle relaxants (curare).” In the article, the authors cite a mortality rate of 1:2,100 anesthetics that did not include the use of curare and a mortality rate of 1:370 when curare was used.3 Significant solutions to complex problems in medicine often introduce new and unintended clinical problems, and the introduction of neuromuscular blocking agents (NMBAs) is no exception.
Incomplete recovery from NMBAs (residual block) after anesthesia and surgery continues to be a common problem in the postanesthesia care unit (PACU). Despite the routine use of anticholinesterase reversal agents, between 20% and 40% of patients continue to arrive in the PACU with objective evidence of residual NMBAs.4–8 In the past year alone, multiple investigations have demonstrated that residual NMBAs is an important patient safety issue,7–10 and multiple letters,11 surveys,8 and editorials12,13 have called for a solution to this recurring (and preventable) potential adverse event. Numerous clinical studies have documented that incomplete neuromuscular recovery is associated with a variety of adverse events in the early postoperative period, including airway obstruction, hypoxemic episodes, postoperative respiratory complications, intraoperative awareness,10,14 and unpleasant symptoms of muscle weakness.15–18 Despite the plethora of data documenting the importance of perioperative neuromuscular monitoring in preventing residual block,17 recent surveys continue to document that subjective assessment using nerve stimulators is performed in less than 40% of patients,9 while objective monitoring is even rarer (17% of patients).19 In published studies, the use of neuromuscular monitoring is similarly widely variable; for instance, peripheral nerve stimulators (PNSs) are rarely, if ever, used in Japan,20 while with a strong departmental champion and mentor, the use of objective neuromuscular monitoring using electromyography may approach 100%.21 While clinicians have hypothesized (and hoped) that the introduction of sugammadex into clinical practice might eliminate residual neuromuscular block associated with aminosteroid-based relaxants, several studies have shown this not to be the case: when neuromuscular monitoring was not used intraoperatively, the incidence of residual block after sugammadex antagonism decreased, but was not always eliminated.20,22,23
Failure to monitor occurs not only in adult surgical patients, but also in the pediatric population that is even more vulnerable to the sequelae of incomplete reversal: 28% of pediatric patients were found to have residual block (defined as a train-of-four [TOF] ratio less than 0.90), while 6.5% of them had severe block (defined as a TOF ratio less than 0.70).24 During periods of continuous administration of NMBAs, the most recent consensus guidelines for the use of muscle relaxants in critically ill children25 call for the assessment of the depth of block “at least once every 24 h with TOF monitoring.” These consensus guidelines recognize the lack of “quality of evidence available in the literature” and call for prospective, randomized, and controlled trials in this vulnerable patient population.
In the adult and pediatric intensive care unit (ICU), NMBAs are used routinely to enable emergency tracheal intubation, facilitate mechanical ventilation for acute respiratory distress syndrome, prevent patient-ventilator dyssynchrony during mechanical ventilation, manage status asthmaticus and elevated intracranial as well as intraabdominal pressure, and maintain induced hypothermia after cardiac arrest.26,27 We must remember that NMBAs have no sedative or amnestic properties; that patients can have recall and “feel almost all of the procedures” they undergo in the ICU.28 In fact, patients’ recollection of therapeutic paralysis in the ICU includes themes of feeling “between life and death,” loss of control, fighting or being tied down, and being terrified.29 The literature suggests that in the adult ICU setting, the incidence of unintended patient awareness during periods of NMBAs exceeds 30%.29,30
The attempts to educate (and convince) clinicians that residual block is a real entity that needs solutions, not only recognition, have extended beyond the operating room (OR) and ICU settings. PACU nurses report that three of the most critical events that they may face requiring emergency intervention are residual NMBAs, acute postoperative hypertension, and acute hypotension.31
Clinical practice is extremely difficult to change, particularly when one’s entire career decisions regarding neuromuscular management have been guided by subjective assessment of clinical signs of neuromuscular recovery. In fact, almost 20% of European and 10% of U.S., Australian, and New Zealand anesthesiologists never use nerve stimulators to guide management of NMBAs.19,32
The literature is replete with studies documenting the inadequacy of subjective assessment and clinical criteria (“bedside tests”) to determine the adequacy of neuromuscular recovery, whether spontaneous or pharmacologic.33 The current review is intended to underscore the gaps in current clinical practice regarding perioperative management of neuromuscular block and offer evidence of the importance of perioperative objective measurement of neuromuscular function whenever NMBAs are used. We continue to be optimistic and believe that with sufficient education and access to appropriate technology, the clinician will choose to do what is best and safest for the patient. To that end, we need to place current clinical practice in some global perspective. In the United States, the National Hospital Discharge Survey: 2010 from the Centers for Disease Control and Prevention estimated the total number of inpatient surgical procedures at 51.4 million.34 If we reasonably assume that of these surgical cases, 60% receive general anesthesia requiring some form of muscle relaxation, then approximately 30.8 million patients are treated with NMBAs; we know that conservatively, one third of patients receiving NMBAs and anticholinesterase reversal will exhibit some degree of postoperative residual neuromuscular block (amounting to 10.1 million patients); of these patients, 0.8% will experience a critical respiratory event (CRE),16 or more than 81,000 patients—every year. Worldwide, the number of major surgeries has been estimated to be 234.4 million per year35 ; this means that more than 0.5 million worldwide patients experience CREs every year!
Is this a significant patient safety issue? It seems that an increasing number of national specialty organizations now think so. The Czech Society of Anaesthesiology standards (2010) require that the method of monitoring be documented and define a TOF ratio greater than 0.90 as an adequate sign of recovery.36 The French Society of Anaesthesiology and Intensive Care guidelines published in 200037 state, “instrumental [objective] monitoring is the main means for assessment” and “the presence of four responses to TOF stimulation is not a sufficient criterion of full reversal.” Most recently in 2016, the Association of Anaesthetists of Great Britain and Ireland, London, United Kingdom, published their recommendations for standards of monitoring, which include, “A peripheral nerve stimulator must be used whenever neuromuscular blocking drugs are given. A quantitative peripheral nerve stimulator is recommended.”38 Many other European countries, as well as Australia and New Zealand, have addressed the need for perioperative neuromuscular monitoring. Sadly, no such guidelines have been issued by the American Society of Anesthesiologists (ASA).
The ASA lists five requirements in the Standards for Basic Anesthesia Monitoring document (last affirmed October 28, 2015): the presence of qualified anesthesia personnel, oxygenation, ventilation, circulation, and body temperature monitoring. The Standards for Basic Anesthesia Monitoring document is silent on the need for neuromuscular monitoring.39 An updated report by the ASA on Practice Guidelines for Postanesthetic Care40 now states in the section on neuromuscular function guidelines, “assessment of neuromuscular function primarily includes physical examination and, on occasion, may include NMBAs monitoring.” These recommendations are based on the committee’s assessment of the evidence of the effectiveness of neuromuscular monitoring as Category B2-B. However, because of the continuing patient safety implications of postoperative residual weakness,41 many have urged all anesthesia societies (national and international) to urgently create practice guidelines/standards governing the clinical management and monitoring of NMBAs.32,42,43 We wholeheartedly agree and encourage clinicians to embrace the Association of Anaesthetists of Great Britain and Ireland standards for neuromuscular monitoring.38
### Acetylcholinesterase Reversal of Profound and Deep Nondepolarizing Block
There is no current agreement on how to define profound (intense) versus deep versus moderate versus light (shallow) neuromuscular block. We propose the following definitions (table 1), which are modified slightly from a 2007 international consensus conference.50
Table 1.
Suggested Definitions of Depth of Neuromuscular Block Based on Subjective and Measured (Objective) Criteria
During a nondepolarizing block, the high frequency of tetanic stimulation will induce a transient increase in the amount of acetylcholine released from the presynaptic nerve ending, such that the intensity of subsequent muscle contractions will be increased (potentiated) briefly (period of post-tetanic potentiation, which may last 2 to 5 min; fig. 1). The neuromuscular response to stimulation during posttetanic potentiation can be used to gauge the depth of block when TOF stimulation otherwise evokes no responses (i.e., when the TOF count [TOFC] = 0). The number of posttetanic responses is inversely proportional to the depth of block: fewer posttetanic contractions denote a deeper block. When the posttetanic count (PTC) is 6 to 8, recovery to TOFC = 1 is likely imminent from an intermediate-duration blocking agent; when the PTC is 0, the depth of block is profound, and no additional NMBA should be administered.
Fig. 1.
Posttetanic count (PTC). During a nondepolarizing block, the high-frequency tetanic stimulation (50 Hz or 100 Hz) will induce a transient increase in the amount of acetylcholine released from the presynaptic nerve ending, such that the intensity of subsequent muscle contractions will be increased (or facilitated). This facilitated neuromuscular response to stimulation after tetanus can be used to gauge the depth of block when train-of-four (TOF) stimulation otherwise evokes no responses (i.e., when the TOF count = 0). The number of posttetanic responses is inversely proportional to the depth of block: the fewer posttetanic contractions are elicited, the deeper the depth of block. In the illustration above, PTC = 4.
Fig. 1.
Posttetanic count (PTC). During a nondepolarizing block, the high-frequency tetanic stimulation (50 Hz or 100 Hz) will induce a transient increase in the amount of acetylcholine released from the presynaptic nerve ending, such that the intensity of subsequent muscle contractions will be increased (or facilitated). This facilitated neuromuscular response to stimulation after tetanus can be used to gauge the depth of block when train-of-four (TOF) stimulation otherwise evokes no responses (i.e., when the TOF count = 0). The number of posttetanic responses is inversely proportional to the depth of block: the fewer posttetanic contractions are elicited, the deeper the depth of block. In the illustration above, PTC = 4.
Close modal
When antagonism of profound (PTC = 0) or deep (PTC more than or equal to 1 and TOFC = 0) block is (unadvisedly) attempted with an acetylcholinesterase inhibitor, there is convincing evidence that recovery is usually a very slow process.51,52 When reversal of rocuronium at a PTC of 1 to 2 (see below for PTC) during sevoflurane anesthesia was attempted with 0.07 mg/kg neostigmine, the median recovery time to a TOF ratio of 0.90 was 49 min with a range of 13 to 146 min.52 Similarly, during deep block (PTC of 1 to 2), reversal of vecuronium with 0.07 mg/kg neostigmine required a median of 50 min with an even wider range of 46 to 313 min.51 If pharmacologic reversal is attempted 5 min after complete T1 ablation after vecuronium or atracurium administration, the spontaneous recovery time to a TOF ratio of 0.70 required a mean value of 66.7 ± 3.3 min. This duration was shortened to 43.5 ± 5.1 min by administration of 0.07 mg/kg neostigmine.53 Thus, while neostigmine clearly accelerated recovery by 20 to 25 min, return of satisfactory neuromuscular function was hardly prompt or complete (table 2). Based on these data, we recommend that reversal of profound or deep neuromuscular block not be attempted using acetylcholinesterase inhibitors.
Table 2.
Limitation of Acetylcholinesterases as Antagonists of Nondepolarizing Block
### Acetylcholinesterase Reversal of Moderate Nondepolarizing Block
The literature on recovery times from moderate block can be confusing. For example, Kim et al.6 administered 0.07 mg/kg neostigmine at a TOFC of 1 (TOFC = 1) after administration of a rocuronium dose of 0.60 mg/kg. The recovery time to a TOF ratio of 0.90 was 8.6 min (range, 5 to 19 min) under propofol anesthesia, but it was 28.6 min (range, 9 to 76 min) under sevoflurane anesthesia. This prolongation is not unexpected since most inhalational anesthetic agents potentiate neuromuscular block by varying degrees (desflurane > sevoflurane > isoflurane> halothane > nitrous oxide).6,54–56
Recovery times are also dependent on the class of nondepolarizing agent being antagonized (short-, intermediate-, or long-acting) and on the cumulative dose of the drug that has been administered before attempted pharmacologic reversal. The peak effects of edrophonium, neostigmine, and pyridostigmine occur in 1 to 2, 7 to 11, and 12 to 16 min, respectively.57 Thus, any observed recovery after these intervals is a result of elimination or redistribution of the NMBA from the plasma. A now-classic study demonstrates this nicely.58 The authors attempted to antagonize atracurium or alcuronium (a long-acting NMBA) once the first response of TOF (T1) values was 10% of control or less. Atracurium recovery times to a TOF ratio of 0.70 were 20.5 ± 13 and 9 ± 4.9 min after administration of 0.04 and 0.08 mg/kg neostigmine, respectively, suggesting that acetylcholinesterase inhibition occurred sooner in the high-dose neostigmine group. For alcuronium, these values were 31.4 ± 15.5 and 25 ± 17.6 min, respectively. As the authors58 suggest, anticholinesterases have a ceiling to the depth of block that they can antagonize completely, even at these relatively advanced levels of neuromuscular recovery (moderate block; table 1).
Achieving prompt and reliable recovery of neuromuscular function by acetylcholinesterase inhibitor administration when T1 is less than or equal to 10% of control (corresponding to TOFC less than or equal to 1) is simply not a realistic goal. Attempted reversal at a TOFC = 1 is also an especially poor idea when objective (quantitative) monitoring of neuromuscular function is not employed. This was demonstrated in a study in which steady-state infusions of rocuronium or cisatracurium achieved approximately 94% T1 depression (deep block), confirmed with electromyography.59 Two minutes after the infusion was stopped, all patients received 0.05 mg/kg neostigmine. The average TOF ratios 10 and 20 min later were 0.53 ± 0.15 and 0.83 ± 12, respectively, after cisatracurium administration and 0.57 ± 0.11 and 0.79 ± 0.12, respectively, after rocuronium administration. Ten minutes after reversal, 29 of 40 subjects had TOF ratio greater than 0.39 but less than 0.70! However, once the TOF ratio exceeds 0.40, most clinicians cannot detect either tactile or visual (subjective) fade.60 It must be emphasized that TOF ratios less than 0.70 represent an unacceptable degree of clinical recovery. Therefore, 10 min after antagonism, recovery will be grossly incomplete in more than 70% of patients, yet clinicians would be completely unaware of this, unless quantitative neuromuscular monitoring was utilized. Fifteen minutes after reversal, 43% of patients had a TOF 0.40 to 0.70, and this degree of residual block was still present in 13% of individuals 20 min after reversal with neostigmine.59
At the shallower end of moderate block (TOFC = 3; table 1), the median recovery time to a TOF ratio of 0.90 after 0.07 mg/kg neostigmine administration was reported as 17 (range, 8 to 46) min during nitrous oxide/propofol anesthesia.61 The authors concluded that to achieve rapid (within 10 min) reversal to a TOF ratio of 0.7 in more than 87% of patients, three or four tactile responses should be present at the time of neostigmine administration. It was not possible within 30 min to achieve a TOF ratio of 0.9 in all patients, regardless of the number of tactile responses present at neostigmine administration. These data clearly indicate that subjective (visual and tactile) means of assessment are inadequate to ensure adequate recovery of neuromuscular function even after pharmacologic antagonism of moderate block with anticholinesterases. We again encourage clinicians to use objective monitoring whenever nondepolarizing NMBAs are administered to patients.
### Acetylcholinesterase Reversal of Light and Minimal Nondepolarizing Block
Once the TOFC reaches 4 with minimal or absent TOF fade, the reliability (and speed of reversal) of acetylcholinesterase inhibitors increases markedly. In 1994, Harper et al.62 attempted reversal of atracurium under nitrous oxide/enflurane anesthesia at T1 values of 40 to 50% of control (a TOFC = 4 with TOF fade). They observed recovery times to a TOF more than or equal to 0.70 of 4.5 (range, 3 to 8) min, 3.0 (2.3 to 5.2) min, and 2.3 (1.3 to 3.7) min after administration of 0.02, 0.04, and 0.08 mg/kg neostigmine, respectively.
Fuchs-Buder et al.63 studied reversal times from a TOF ratio of 0.40 under total intravenous anesthesia. It should be remembered that once the TOF ratio exceeds a (measured) value of 0.40, most clinicians can no longer detect tactile or visual fade.60 After 0.02 mg/kg neostigmine administration, the interval to recovery of TOF values of 0.90 and 1.00 was 6 (range, 4 to 9) and 9 (range, 6 to 13) min, respectively. If the dose of neostigmine was increased to 0.03 mg/kg, these times decreased to 4 (range, 3 to 6) and 5 (range, 3 to 7) min, respectively. These TOF ratios had not been referenced to the control TOF and thus were nonnormalized acceleromyographic results.50,64 Therefore, the recovery times to a nonnormalized TOF ratio of 1.00 are most likely equivalent to the recovery times measured mechanomyographically or electromyographically to a TOF ratio of 0.90.
Other investigators have reported similar results.65 Under total intravenous anesthesia with electromyography monitoring, reversal of rocuronium-induced block from a TOF ratio of 0.50 was attempted with various doses of neostigmine. The recovery time to a TOF ratio of 0.90 was 3.2 (range, 1.7 to 6.2) min after administration of 0.025 mg/kg neostigmine and 2.0 (1.7 to 4.2) min after administration of 0.04 mg/kg neostigmine. The authors estimated that a 0.034 mg/kg dose of neostigmine would reverse a TOF ratio of 0.50 to more than 0.90 within 5 min (table 2). Finally, Fuchs-Buder et al.66 repeated their own 2010 study63 (during nitrous oxide/desflurane anesthesia) and concluded that “neostigmine doses as low as 0.01 mg/kg may be sufficient to antagonize shallow atracurium neuromuscular block corresponding to a TOF ratio of 0.6, even under inhalational anesthesia.” Such optimistic conclusions, however, are not supported by data from other investigators. A very recent study by Kaufhold et al.67 supports the findings of Kirkegaard et al.61 that even at a threshold TOFC of 4, neostigmine is not always a reliable antagonist of nondepolarizing block (table 2). The authors administered varying doses of neostigmine when recovery from rocuronium had spontaneously returned to a TOF ratio of 0.20. While 0.04 and 0.07 mg/kg doses of neostigmine usually achieved a TOF ratio greater than or equal to 0.90 in less than 10 min in both patient groups, there was one patient in each group in whom this value was not reached for 20 min. It is important, therefore, to ensure that there are no outlier patients who require unexpectedly long times for adequate recovery. In view of the lack of compelling data that doses of neostigmine as small as 0.01 mg/kg are effective, doses less than 0.02 mg/kg for reversal of light or minimal block cannot be recommended. Additionally, it should be reiterated that minimal neuromuscular block can only be determined by objective means of monitoring and that subjective assessment using a PNS will be inadequate to ensure sufficient recovery in all patients.
### Neostigmine-induced Neuromuscular Weakness
Table 3.
Recommendations for Pharmacologic Antagonism of Nondepolarizing Blockade According to the Depth of Block
Anesthesiologists have been experiencing nationwide drug shortages for decades, but the duration of drug unavailability and the number of drugs on the shortage list have increased dramatically in the last few years.74 The reasons for drug shortages include scarcity of raw materials, inconsistent or inadequate quality control in manufacture, and industry consolidation that may result in the disappearance of some manufacturers. In 2011, the U.S. Food and Drug Administration (FDA) issued revised guidance on marketing of unapproved drugs and established an orderly approach for removing unapproved drugs from the market.75 Eclat Pharmaceuticals (USA), a subsidiary of Flamel Technologies, was the only manufacturer to obtain FDA approval of their neostigmine preparation, Bloxiverz. Subsequent to this FDA approval, the makers of Bloxiverz sent a request letter to the FDA calling for removal from the U.S. market of all other unapproved formulations of neostigmine manufactured generically by five other competing manufacturers. Bloxiverz at the time cost more than six times as much as its FDA-unapproved predecessors partly attributed by Eclat on the FDA filing fee of more than $2 million.76 Upon contacting Eclat Pharmaceuticals, the manufacturer confirmed that Bloxiverz is the only FDA-approved product on the market for neostigmine, while AmerisourceBergen (USA), Cardinal (USA) , H.D. Smith (USA), McKesson (USA), and Morris & Dickson (USA) are authorized distributors of Bloxiverz (Sorin J. Brull, M.D., Department of Anesthesiology, Mayo Clinic, Jacksonville, Florida; January 2016; written communication). The reliance on a single manufacturer of neostigmine can have ominous implications on U.S. drug availability in the future, as history clearly demonstrates.74 ### Edrophonium Reversal of Neuromuscular Block Edrophonium is another anticholinesterase agent used clinically for reversal of neuromuscular block. It is less effective as a reversal agent, as the bonds it forms with acetylcholinesterases are ionic and much weaker than the covalent bonds of neostigmine and acetylcholinesterases. Because of this lower potency, the degree of spontaneous recovery from neuromuscular block at the time of edrophonium administration should be at least a TOFC of 4. The usual dose of edrophonium in the clinical setting is 0.50 mg/kg; doses of 0.75 mg/kg provide minimal increases in efficacy.77 Because of its propensity to induce bradycardia and its more rapid onset of action than neostigmine, edrophonium is usually administered in conjunction with atropine. The administration in divided doses over several minutes, as opposed to rapid single-bolus administration, will result in a lower peak plasma concentration of both agents and will minimize the potential for bradycardia (from edrophonium) or tachycardia (from atropine). Because of recent shortages of neostigmine, clinicians have had to resort to the use of this drug combination for reversal of neuromuscular block. ### Lessons Learned While neostigmine usually acts as an antagonist of nondepolarizing neuromuscular block, if administered incorrectly, it may either be ineffective or have undesirable paradoxical effects (table 3). During “profound and deep block,” neostigmine (in any dose) will be ineffective and should not be administered; during “minimal block,” only small doses are needed, and a full-reversal dose may in fact result in transient neuromuscular weakness.45,68 The limitations of anticholinesterases as antagonists of residual nondepolarizing block are greater than most clinicians appreciate. At TOFCs less than 3 or 4, prompt and satisfactory reversal of nondepolarizing block by neostigmine should not be anticipated, and attempts at reversal should be delayed until these values are attained. During moderate block, the dose of neostigmine necessary to achieve maximal effect is a subject of some debate but probably is not less than 0.04 mg/kg.78 There is also no evidence that increasing dosage beyond 0.06 mg/kg will increase the drug’s efficacy.79 Unless quantitative monitoring provides evidence of full recovery, even TOFCs of 4 without fade (determined subjectively) should be reversed. However, in these circumstances, doses of neostigmine of 0.02 to 0.03 mg/kg are sufficient to reliably assure satisfactory return of neuromuscular function within approximately 10 min, without inducing paradoxical neuromuscular weakness. Data suggest that routine administration of full doses of neostigmine (more than 0.06 mg/kg, for instance) to fully reversed patients to ensure full recovery or for medicolegal reasons (i.e., chart documentation) may be counterproductive and should be avoided.45–48,70 ### Sugammadex A decade ago, because of the limitations inherent in the use of acetylcholinesterase inhibitors as antagonists of neuromuscular block, it seemed clear that the issue of postoperative residual block was unlikely to disappear unless an alternative method of pharmacologic reversal of deep and even moderate block became available. In 2006, articles by de Boer et al.80,81 and others82,83 described a new and promising agent.84 Sugammadex (a modified γ-cyclodextrin) forms 1:1 complexes with aminosteroid neuromuscular blocking drug molecules but has no effect on benzylisoquinolinium compounds or on succinylcholine. A dose of 3.57 mg sugammadex is needed to encapsulate 1.0 mg rocuronium. The resulting complex has a very low dissociation rate. Encapsulated molecules of the NMBA that circulate in the plasma are no longer able to bind with muscle acetylcholine receptors, allowing blocking agents to diffuse away from the synaptic cleft and back into the plasma as the concentration of free drug in plasma decreases precipitously. Thus, clinicians now have for the first time the ability to rapidly and completely reverse profound nondepolarizing neuromuscular block, and do so directly by inactivating the activity of the NMBA, rather than indirectly, by inhibiting acetylcholinesterases. Since sugammadex first became commercially available in 2008 (outside the United States), voluminous literature has emerged detailing the drug’s pharmacology, safety, and clinical uses. Several excellent reviews are available.85–91 An attempt to summarize this information is beyond the scope of the current review. Rather, our current focus is two-fold: first, we describe the safety issues that have prevented the U.S. FDA from approving sugammadex for clinical use until late 2015, and second, we focus on the potential for sugammadex to decrease the frequency of undetected residual block in the postoperative period. Sugammadex does not appear to have affinity for any receptors, so it has no hemodynamic effects. It has been shown to bind to toremifene, fusidic acid, and flucloxacillin.92 Additionally, sugammadex binds oral contraceptives, and women of childbearing age should be counseled about using alternative contraceptive methods for 1 week after exposure to sugammadex. In many institutions, this potential side effect is disclosed as part of the preoperative anesthesia consent. Hypersensitivity reactions to all anesthetics during the perioperative period have an incidence between 1:3,500 and 1:20,000 exposures, and the associated mortality approaches 9%.93 Hypersensitivity to sugammadex appears to be relatively low, with only 15 cases being reported in a 2014 review.94 In the majority of the reported cases, the anaphylactic reactions were evident within the first 4 min after administration of sugammadex, and cardiovascular collapse was treated successfully with fluid resuscitation and high-dose epinephrine therapy.95 The other major factor that delayed the approval by the FDA has been the potential effect of sugammadex on coagulation. In patients receiving sugammadex, the activated partial thromboplastin time and the prothrombin time were increased by 5.5% and 3.0%, respectively; however, these increases returned to normal values within 60 min.96 Likely as a consequence of these being relatively minor and short-lived effects on coagulation parameters, the incidence of bleeding events in patients receiving sugammadex (2.9%) and those not exposed to sugammadex (4.1%) was comparable. Additionally, effects on the various coagulation assays are likely an in vitro artifact.96,97 Regrettably, the number of studies that document the incidence of postoperative residual neuromuscular block after sugammadex reversal is rather limited. Nevertheless, existing data are encouraging. Della Rocca et al.98 compared neostigmine- to sugammadex-aided recovery times after reversal of rocuronium at a TOFC of 2 in a large prospective multisite study. One hundred forty-two patients received neostigmine, and 163 received sugammadex. Because the study was observational, sugammadex and neostigmine were administered according to each anesthesiologist’s clinical judgment. In the neostigmine group, 72%, 41%, and, 18% of patients had TOF ratios less than 0.90 at 5, 10, and 20 min, respectively, after reversal. In the sugammadex group, these values were 12%, 2%, and 0%, respectively.98 An even more promising study was reported by Brueckmann et al.99 They randomized 150 patients receiving rocuronium and having abdominal surgery to either sugammadex (n = 74) or neostigmine (n = 76) reversal groups. All drug doses and timing of reversal administration were per individual clinician preference. TOF-Watch (Organon Ireland Ltd., Ireland) monitors were available intraoperatively, and usage was left to the discretion of the anesthesiologist. Upon arrival in the PACU, 43% of patients in the neostigmine group had TOF ratios less than 0.90. In contrast, the incidence of residual block in the sugammadex group was zero! Other studies, however, suggest that sugammadex is not totally fool proof. Ledowski et al.22 reported on 126 patients in an observational study. The choice of anesthetic technique, NMBA, reversal agent (neostigmine, sugammadex, or no reversal), dosages, and so forth, was left to the individual anesthesiologist. Conventional PNSs (qualitative devices) were available in the operative room, but their use was not mandated. When the clinician deemed that the patient was ready for tracheal extubation, an independent investigator measured the TOF ratio at the adductor pollicis with a kinemyographic monitor. Thirty-three patients received pharmacologic reversal with neostigmine. In this group, 19 (58%) had TOF ratios less than 0.90, and 8 (24%) had TOF ratios less than 0.70. Of the 57 patients who received sugammadex, only four individuals (7%) had TOF ratios less than 0.90, and none had a TOF ratio less than 0.70.22 Kotake et al.20 studied 117 patients who had received rocuronium (0.60 to 0.90 mg/kg) followed by pharmacologic reversal with sugammadex. In this multisite study, intraoperative monitoring of neuromuscular function was not employed. The average dose of sugammadex administered was 2.7 ± 1.0 mg. TOF ratios were measured after tracheal extubation (8 ± 4 min after reversal). The incidence of TOF ratios less than 0.90 was 4.3% (95% CI, 1.7 to 9.4). The authors concluded that the risk of TOF ratio less than 0.9 in the PACU remained at least 1.7% and may be as high as 9.4% even with sugammadex in a clinical setting when no neuromuscular monitoring is used routinely.20 Finally, a prospective study comparing residual neuromuscular block and postoperative pulmonary complications in patients receiving either neostigmine or sugammadex found that the residual paralysis incidence was 28.6% in the patients who received neostigmine reversal, while in the sugammadex group, the incidence was 1.2%. Of note, intraoperative neuromuscular monitoring (which did not specify subjective or objective assessment) was performed in only 30% of patients.100 These studies suggesting that residual neuromuscular block may still occur after sugammadex reversal need special mention. Because of the 1:1 molar ratio between sugammadex and the aminosteroid NMBA, there have to be sufficient sugammadex molecules administered to encapsulate all of the free molecules of the NMBA. For this reason, sugammadex reversal dose is calculated based on the depth of neuromuscular block at the time of reversal. For reversal of moderate block (TOFC, 1 to 3), a dose of 2 mg/kg is recommended; for reversal of deep block (PTC more than or equal to 1), a dose of 4 mg/kg is recommended, and for rescue from a failed rapid-sequence induction in the cannot-intubate-cannot-ventilate scenario (profound block, PTC = 0), a dose of 16 mg/kg is recommended (table 3). Therefore, if clinicians use subjective or clinical means of assessment of neuromuscular block rather than objective monitoring, it is possible that the dose of sugammadex may be insufficient, resulting in incomplete reversal. This is likely a limitation of inappropriate monitoring rather than a failure of the drug.20 It also appears that sugammadex should be administered based on actual body weight, particularly in the obese patient. Failure to administer a sufficient dose may result in reappearance of postoperative neuromuscular weakness (recurarization).101 ### Residual Paralysis and Potential Solutions—Monitoring Before discussing neuromuscular monitoring as a potential solution to postoperative residual weakness, a brief review of nomenclature is warranted: there is a critical difference between a nerve stimulator and a monitor, yet these terms are often used interchangeably (and incorrectly). A PNS is a simple medical device that delivers current impulses to a peripheral nerve. The assessment of evoked responses from the innervated muscle is detected subjectively by the clinician, by watching or feeling the strength of muscle contraction. Such PNSs are not monitors since the assessment is made subjectively. Neuromuscular monitors are objective medical devices that measure and display the evoked TOF ratio in real time. Any discussion of appropriate neuromuscular monitoring should include a description of the ideal monitor. All neuromuscular monitors fulfill two distinct but separate functions: the first is to deliver an electrical stimulus to a peripheral nerve, an action that can be provided by any PNS; the second function is to detect, measure, and analyze the evoked muscle contraction or the muscle action potential (MAP) that results from this peripheral nerve stimulation. This latter function is the actual monitoring and can only be performed by a limited number of medical devices. Existing neuromuscular monitors can be either stand-alone, portable devices or modular units that are integral parts of anesthesia workstations and that use acceleromyography, kinemyography, or electromyography (table 4). Table 4. List of Stand-alone and Modular, Integrated Neuromuscular Monitors Available in 2016 The characteristics of the ideal PNS have been described102 and include those listed in table 5. The characteristics of the ideal monitor are more difficult to define, because every technology currently in clinical use has not only certain advantages, but also limitations (table 6). Table 5. Characteristics of the Ideal Peripheral Nerve Stimulator Table 6. Characteristics of the Ideal Neuromuscular Monitor Careful management of NMBAs in the OR reduces the risk of residual NMBAs in the PACU. Several strategies that can be utilized by anesthesiologists in the OR will decrease the incidence of residual muscle weakness after tracheal extubation. The use of shorter acting NMBAs103,104 and routine administration of drugs that antagonize the effects of NMBAs (for instance, neostigmine) result in fewer although still unacceptably high incidence of postoperative patients with clinically evident muscle weakness.6,105 The routine use of perioperative neuromuscular monitoring has been advocated as another important method of reducing the incidence of residual weakness (residual paresis, residual neuromuscular block, or residual curarization).11,18,21,106,107 Three general categories of perioperative neuromuscular monitoring exist: clinical or bedside testing, subjective or qualitative evaluation, and objective or quantitative measurement. Clinical (bedside) testing has been used since the introduction of NMBAs into clinical practice: measurement of respiratory parameters (tidal volume, vital capacity, minute ventilation, negative inspiratory force, and so forth) has been correlated with neuromuscular recovery (TOF ratio), but just like other clinical tests of muscle function (5-s head-lift, grip strength, and leg-lift tests), these tests are unreliable and nonspecific.108 These tests generally require that the patient being evaluated be awake and cooperative, but these characteristics are not shared by patients recovering from general anesthesia whose tracheas are still intubated. The vastly overused (and overrated) 5-s head-lift test, for instance, was unable to identify TOF ratios as low as 0.5 in more than 70% of patients.109,110 In fact, none of the clinical tests has a sensitivity greater than 0.35 or a positive predictive value greater than 0.52.5 The clinical test with the greatest positive predictive value (0.52) is the tongue depressor test, which cannot be used in intubated patients. Qualitative neuromuscular devices (or more accurately named, PNSs) are utilized in most clinical practices. These battery-powered devices provide an electrical stimulus to a peripheral nerve (most commonly the ulnar nerve), and the response of the stimulated muscle (usually the thumb) is evaluated subjectively by visual or tactile means. The presence or absence of muscle weakness is determined by evaluating fade (decreasing muscle contractions) with repetitive nerve stimulation. Clinicians use several different patterns of nerve stimulation in order to evaluate fade. The most common pattern of nerve stimulation is TOF (fig. 2). TOF stimulation consists of four stimuli at 2-Hz frequency. The force of contraction of the fourth muscle twitch is subjectively compared to the contraction of the first twitch, and fade is considered absent when both muscle contractions (twitches) appear equal (no block; fig. 2). When the fourth twitch in the TOF sequence starts decreasing in amplitude, the TOF ratio becomes less than 1.0 and TOF fade ensues (partial block; fig. 3). Fig. 2. A train-of-four (TOF) consists of four equal stimuli delivered at 0.5-s intervals. The TOF ratio is calculated by comparing the magnitude of the fourth evoked response or twitch (T4) to that of the first response (T1). In the unblocked state, the TOF ratio (T4/T1) should approximate 1.0 (100%). Fig. 2. A train-of-four (TOF) consists of four equal stimuli delivered at 0.5-s intervals. The TOF ratio is calculated by comparing the magnitude of the fourth evoked response or twitch (T4) to that of the first response (T1). In the unblocked state, the TOF ratio (T4/T1) should approximate 1.0 (100%). Close modal Fig. 3. A train-of-four (TOF) consists of four equal stimuli delivered at 0.5-s intervals. The TOF ratio is calculated by comparing the magnitude of the fourth evoked response or twitch (T4) to that of the first response (T1). During partial nondepolarizing block, T4 decreases preferentially, such that there is fade (i.e., TOF ratio less than 1.0). In the illustration above, the TOF ratio = 0.40 (40%). Fig. 3. A train-of-four (TOF) consists of four equal stimuli delivered at 0.5-s intervals. The TOF ratio is calculated by comparing the magnitude of the fourth evoked response or twitch (T4) to that of the first response (T1). During partial nondepolarizing block, T4 decreases preferentially, such that there is fade (i.e., TOF ratio less than 1.0). In the illustration above, the TOF ratio = 0.40 (40%). Close modal Subjective evaluation is performed either by looking at the evoked responses and assessing fade to TOF stimulation (visual means) or by feeling the strength of contraction of muscles and assessing TOF fade (tactile means). Such subjective evaluation is the most commonly used method of evaluating the depth of neuromuscular block and adequacy of reversal. However, clinical decisions guided by subjective evaluation of neuromuscular function have not decreased the postoperative risk of desaturation or need for tracheal reintubation.48 Although some studies have shown tactile evaluation to be slightly more sensitive than visual means, others have shown that the ability to detect fade was not significantly different between visual and tactile means.111 The effectiveness of qualitative neuromuscular monitoring in decreasing the incidence of residual blockade remains controversial since this type of monitoring is ineffective in detecting residual blockade when TOF ratios are more than 0.40.60,111 Despite the widespread availability of PNSs in the ORs (76% of European departments and 97% of U.S. departments), 19% of European and 9% of U.S. clinicians never use them, and their use does not seem to always help clinicians identify residual neuromuscular weakness: more than half of clinicians incorrectly estimated the incidence of clinically significant residual block to be less than 1%.32 A survey, as well as numerous previous clinical investigations, has reported on the limitations of subjective evaluation.112 The ability to detect TOF fade by subjective (tactile) means appears to be influenced by clinical experience only marginally; anesthesiologists inexperienced in assessing tactile fade were able to identify it only when the TOF ratio was less than 0.30, while only one in five experienced anesthesiologists was able to correctly identify it when the TOF ratio was between 0.51 and 0.70.60 While other stimulation patterns such as double-burst stimulation (DBS; fig. 4) were introduced into clinical care to facilitate the subjective assessment of fade, such attempts have been minimally successful: the threshold (lowest TOF ratio) for detection of fade using subjective evaluation of TOF is approximately 0.40, while the TOF threshold for detecting DBS fade is 0.60.113 Thus, even when there is no tactile detection of fade to either TOF or DBS stimulation, there is almost a 50% risk that the actual measured ratio is below 0.70. Clearly, the proportion of clinicians who will miss the presence of residual block will be even higher at the recovery TOF threshold of 0.9. Some clinicians rely on a 5-s tetanic (50 Hz) stimulation (fig. 5) and subjective assessment of the fade of muscle contraction. However, the 50-Hz tetanic stimulation pattern is the least sensitive subjective method: tetanic fade can only be detected reliably when the TOF ratio is less than or equal to 0.3.114 Fig. 4. A double burst stimulus (DBS) consists of two brief, 50-Hz tetanic bursts delivered 0.75 s apart. Each burst consists of three stimuli (DBS3,3) that result in two sustained muscle contractions. The DBS ratio (D2/D1) approximates the train-of-four (TOF) ratio. When quantitative monitoring is not available, the advantage of DBS over TOF is that subjectively determined fade is more easily perceived than the fade induced by TOF stimulation. However, once the TOF ratio exceeds 0.60, fade to DBS generally cannot be detected subjectively. Fig. 4. A double burst stimulus (DBS) consists of two brief, 50-Hz tetanic bursts delivered 0.75 s apart. Each burst consists of three stimuli (DBS3,3) that result in two sustained muscle contractions. The DBS ratio (D2/D1) approximates the train-of-four (TOF) ratio. When quantitative monitoring is not available, the advantage of DBS over TOF is that subjectively determined fade is more easily perceived than the fade induced by TOF stimulation. However, once the TOF ratio exceeds 0.60, fade to DBS generally cannot be detected subjectively. Close modal Fig. 5. In the unblocked state, the peak height of an evoked mechanical response to 5-s tetanic stimulation (TET) stimulation is generally 7 to 10 times that of a single stimulus, and the muscle contraction in response to a 5-s, 50-Hz tetanus can be well sustained. During a partial nondepolarizing neuromuscular block, tetanic tension exhibits fade (decreases in strength over time). The tetanic stimulation is followed by a 2- to 5-min period of posttetanic facilitation. Fig. 5. In the unblocked state, the peak height of an evoked mechanical response to 5-s tetanic stimulation (TET) stimulation is generally 7 to 10 times that of a single stimulus, and the muscle contraction in response to a 5-s, 50-Hz tetanus can be well sustained. During a partial nondepolarizing neuromuscular block, tetanic tension exhibits fade (decreases in strength over time). The tetanic stimulation is followed by a 2- to 5-min period of posttetanic facilitation. Close modal Other limitations of subjective evaluation relate to the site (muscle) that is monitored. There are well-known differences in the timecourse of responses to NMBAs at different muscle groups. Central muscles (diaphragm) recover earlier than peripheral muscles (adductor pollicis), but that does not imply that the rest of the respiratory muscles are functioning normally. Upper airway muscles critical to maintaining airway patency and protection from pulmonary aspiration of secretions or gastric contents are very sensitive to NMBAs and do not recover fully until the TOF ratio is near baseline. Monitoring of adductor pollicis muscle, which lags the recovery of the diaphragm, will ensure that if recovery is sufficient at the thumb, the diaphragm and upper airway muscles will function normally. Monitoring TOF recovery in response to facial nerve stimulation can lead to erroneous decisions: the eyebrow muscle, the corrugator supercilii, recovers faster than the upper airway or the adductor pollicis muscles. Clinical decision about spontaneous ventilation and airway protection that are made based on recovery of facial muscles (corrugator supercilii or orbicularis oculi) will, therefore, overestimate the degree of recovery and may place the patient at risk of CREs.115,116 In fact, TOF monitoring at this anatomic site (face) should be the location of last resort, and the clinician should remember that airway protection might be impaired even when the eyebrow muscle shows no TOF fade. Assessment of function should be sought from the adductor pollicis muscle as soon as practical. Realizing the limitations of subjective evaluation of TOF fade and readiness for tracheal extubation, some clinicians rely on the use of PNS to determine the depth of block and degree of recovery before administering pharmacologic reversal by counting the number of responses to TOF stimulation (TOFC; fig. 6). The TOFC assessed subjectively by anesthesia providers was compared with the TOFC obtained objectively with an acceleromyography (TOF-Watch) monitor.117 An agreement between subjective and objective methods was present in only 56% of observations; moreover, at TOFCs of 1, 2, and 3, the agreement was 36%, and when there was no agreement between the two assessment methods, providers assessed a higher TOFC in 96% of the observations! This overestimation of the degree of recovery may obviously influence the timing and dosing of pharmacologic reversal agents and may partly explain the high incidence of residual block. Fig. 6. During moderate block, once the train-of-four (TOF) ratio becomes 0 (i.e., T4 disappears), the depth of block may be assessed by counting the number of responses to TOF stimulation. The depth of neuromuscular block is proportional to the TOF count (TOFC): the lower the count, the deeper the block. Fig. 6. During moderate block, once the train-of-four (TOF) ratio becomes 0 (i.e., T4 disappears), the depth of block may be assessed by counting the number of responses to TOF stimulation. The depth of neuromuscular block is proportional to the TOF count (TOFC): the lower the count, the deeper the block. Close modal In short, despite adoption of PNSs and subjective evaluation into clinical practice, the literature continues to document this method’s significant limitations. Quantitative neuromuscular monitoring devices objectively measure residual blockade and display the results numerically in real time. The TOF stimulation pattern is most commonly used to assess NMBAs when quantitative devices are employed. The TOF ratio (or T4/T1 ratio) should exceed 0.9 (90%) in order to exclude clinically significant muscle weakness. Maintenance of the ability to swallow and protection against aspiration of pharyngeal fluids can only be assured above this minimum level of neuromuscular recovery.118 Partial recovery of muscle function to TOF ratios less than 0.9 in volunteers118,119 and surgical patients120 is associated with a variety of postoperative adverse events. The use of quantitative monitoring was shown to be effective in identifying and reducing the risk of residual blockade.4,17 Although evidence strongly suggests that quantitative monitors should be used intraoperatively whenever NMBAs are administered, these devices are not widely available.19,32,121 Electromyography devices measure electrical activity (compound MAPs) resulting from nerve stimulation (usually at the adductor pollicis muscle after ulnar nerve stimulation). Electromyography-based monitoring is perhaps the most physiologic and precise method of measuring the synaptic transmission (and thus, the degree of neuromuscular relaxation), is not susceptible to changes in contractility such as the staircase effect, and has advantages in facilitating the recording of compound MAPs from virtually any muscle, including the diaphragm and laryngeal muscles.122 Unfortunately, this technology is not yet commercially available in any stand-alone, portable device although one such monitor is currently under development (Sorin J. Brull, M.D., Department of Anesthesiology, Mayo Clinic, Jacksonville, Florida; September 2016; written communication). The only original equipment manufacturer monitor (the E-NMT-01 Datex-Ohmeda S/5 Neuromuscular Transmission Module [GE Healthcare, USA]) that was based on electromyography was subject to a Class 2 Recall by the U.S. FDA on May 21, 2014, because neuromuscular transmission values may indicate “a deeper level of muscle relaxation than the actual level of muscle relaxation.”123 Other limitations of electromyography for monitoring of neuromuscular block include its sensitivity to inherent noise in electronic equipment, motion artifact, electrocardiographic artifacts produced by the electrical activity of the heart, and interference by electromagnetic and radio frequency emissions.124 Despite these limitations, the implementation of routine, electromyography-based neuromuscular monitoring in a single department has recently underscored the significant improvements in clinical care and the elimination of CREs (such as emergent tracheal reintubations in the PACU) that the use of routine neuromuscular monitoring can effect. This report11 and the year-later update21 document the significant amount of time, education, and dedication needed to implement a department-wide neuromuscular monitoring program.11,21 Mechanomyography measures the force of contraction of the adductor pollicis (thumb) muscle after ulnar nerve stimulation. Mechanomyographic responses are precise and reproducible (as long as a 200- to 300-g muscle preload is maintained) and have been considered the accepted standard for neuromuscular monitoring. However, because of a relatively complex setup, mechanomyography is currently used only for research purposes. These devices are no longer commercially available. Acceleromyography measures acceleration of muscle tissue (most commonly the thumb) in response to nerve stimulation (most commonly the ulnar nerve). This technique is based on Newton second law of motion (F = m × a). A piezoelectric transducer is attached to a muscle, and when the innervating nerve is stimulated, the muscle movement is sensed by the transducer; a voltage is generated in the piezoelectric crystal, and this electrical signal is analyzed by the acceleromyography monitor. There are several manufacturers of acceleromyography-based monitors (table 4). The acceleromyography devices are small, portable, and designed for intraoperative applications. Their routine use in the clinical setting has been limited by initial acquisition costs ($800 to \$2,400), the need for experience with acceleromyography monitoring to obtain accurate results, the unavailability of appropriate electrode placement sites when the patient’s arms are tucked under surgical drapes, and limitations of the technology in the OR (requirement for baseline measurements and normalization, the long 5- to 10-min setup required before use, and reduced precision in awake patients).122,125,126 Many studies, however, have documented this technology’s unquestionable efficacy in decreasing the incidence of residual neuromuscular block in both adult17,127 and pediatric patients.128
Kinemyography devices are similar to acceleromyography-based devices, but they measure the degree of bending of a piezoelectric sensor.129 This mechanosensor is placed along the space between the thumb and index fingers and quantifies the degree of bending as the thumb and index fingers appose in response to ulnar nerve stimulation. To date, several clinical validation studies of kinemyography have been performed.130
At the current time, it appears that one of the main barriers to routine adoption of quantitative monitoring is the lack of availability of an easy-to-use, accurate, and reliable monitor. Several recent publications have expressed the urgent need for such a quantitative neuromuscular device.12,131
Modern surgery would not be possible without the availability of NMBAs. To quote Foldes, “…curare had the same importance for anesthesiology as asepsis had for the progress of surgery.”132 However, the use of these agents also introduced significant patient safety concerns. For instance, residual neuromuscular block dates back to the days of curare when this complication was termed, “residual curarization.” Since then, attempts have been made to eliminate it: introduction of PNSs into clinical practice and development of new shorter duration NMBAs and selective NMBA-specific reversal agents. These advances have decreased the incidence of residual block, pulmonary complications, and incidence of other sequelae, but they have not eliminated them.133 The potential solution has been obvious for decades; if emergence from anesthesia and tracheal extubation are allowed to occur only after adequate neuromuscular function has been attained (as documented by a measured TOF ratio more than 0.90), these complications will (and must) become never events. Neuromuscular function assessment with a PNS is mandatory whenever neuromuscular blocking drugs (both depolarizing and nondepolarizing) are used; patients who received large doses of NMBAs, those undergoing prolonged surgical procedures, or patients at increased risk of postoperative complications from residual block ideally should be monitored using objective means.
Pharmacologic antagonism, whether using anticholinesterases or sugammadex, must be guided by, at a minimum, subjective (and preferably, objective) monitoring. Intense and deep levels of neuromuscular block cannot be antagonized by anticholinesterases, and reversal should not be attempted at this level of block. This depth of block induced by aminosteroidal NMBAs, however, can be reversed rapidly and reliably by administration of sugammadex: 16 mg/kg (when PTC = 0, intense block) or 4 mg/kg (when PTC more than or equal to 1, deep block).
Moderate block can be antagonized by anticholinesterase agents as long as sufficient recovery is documented by the presence of at least three responses to TOF stimulation (TOFC 3 or 4). At this level of block, a full dose of neostigmine (0.05 to 0.06 mg/kg) or sugammadex (2 mg/kg) should be administered.
A light level of neuromuscular block (evidenced by fade to TOF, whether determined subjectively or objectively) should be antagonized with lower doses of neostigmine (0.02 to 0.03 mg/kg) or sugammadex (1 mg/kg).
Finally, it must be emphasized that the timing of tracheal extubation must be determined based on the degree of recovery (whether spontaneous or pharmacologic), and a minimum TOF ratio of 0.90 must be the desired goal. This implies that objective (not subjective) monitoring techniques are necessary. If an objective monitor is not available, the anesthesia record should document, at a minimum, the TOFC at the time of reversal and the dose of antagonist administered. Other indicators of recovery, such as subjective assessment of lack of TOF fade, sufficient time since administration of reversal agents (or NMBA), adequate tidal volume, the presence of 5-s head lift, and so forth, cannot be used to exclude residual block and the potential for postoperative complications. In short, neuromuscular monitoring is not optional, and national societies must propose recommendations for the rational and safe management of perioperative neuromuscular blockers and their antagonists.
Support was provided solely from institutional and/or departmental sources.
Dr. Brull has had investigator-initiated funded research from Merck, Inc., Kenilworth, New Jersey, and is a shareholder and member of the Board of Directors in Senzime AB, Uppsala, Sweden. Dr. Kopman declares no competing interests.
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40 | 2022-05-20 03:09:09 | {"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": 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": 0.44553399085998535, "perplexity": 14298.349092211143}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662531352.50/warc/CC-MAIN-20220520030533-20220520060533-00341.warc.gz"} |
https://www.transtutors.com/questions/allocating-to-solve-a-timing-problem-production-workers-for-bianco-manufacturing-com-1243469.htm | Allocating to solve a timing problem Production workers for Bianco Manufacturing Company provided
Allocating to solve a timing problem
• ACCT 2127 TAKEHOME EXAM QUESTIONSVERSION 2 Course Code: ACCT2127Course Description: Accounting for...
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on sale of truck 25,000Cash at bank at July 1 , 2017 1 ,000,000Depreciation for the year 150,000Payment for Motor vehicle purchased on 28th April 2018 40,000Loan repayments made 70,000Tax expense / paid to the Tax Office 78,000Salaries Paid 250,000Cash received from the sale of a...
• P9.2 Fixed Input Combinations. Cherry Devices, Inc., assembles connectors and terminals for...
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400 to 1 ,600 units, holding labor constant at 4 units. Are returns to each factor increasing, constant, or diminishing? C. Assume now and throughout the remainder of the problem that labor and capital must be combined in the ratio 4L:400K. How much output could be produced if Cherry has
• Cornerstone Exercise 8-19 Economic Order Quantity Refer to the data for La Cucina Company above....
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Cornerstone Exercise 8-19 Economic Order Quantity Refer to the data for La Cucina Company above. Required: 1 . What is the EOQ for lava stone? 2. How many orders per year for lava stone will La Cucina place under the EOQ policy? 3. What
EOQ = Economic Order Quantity A =Annual Requirement O = Ordering Cost per order C = Carrying Cost per pound Requirement 1: EOQ = [2AO/C]1/2 =[2*8400 * 3 / 2]1/2 =158.745 EOQ = 159 pounds...
• HI5017 Managerial Accounting Trimester 2 2018 Individual Assignment Assessment Value: 30% General...
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in Calibri font size 12 Format of the Report: 1 . You should at least have the following details: a. Assignment Cover page clearly stating your name and student number b. A table of contents, executive summary c. A brief introduction or overview of what the report is about. d. Body
Activity-based costing is a costing method in which the manufacturing overheads are assigned to the products on the basis of activities consumed. ABC Costing is a modern and new method of... | 2018-11-17 02:18:27 | {"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": 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": 0.20611198246479034, "perplexity": 10302.372307319083}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039743248.7/warc/CC-MAIN-20181117020225-20181117042225-00442.warc.gz"} |
http://physics.stackexchange.com/tags/standard-model/new | # Tag Info
1
How can we see that the group $N$ generated by $$g = (e^{2\pi i/3} I, -I, e^{i\pi /3}) \in SU(3)\times SU(2)\times U(1)$$ acts trivially on all fields in the Standard Model? First of all, note that $g$ is in the center of $SU(3)\times SU(2)\times U(1)$. Therefore its representative in the adjoint representation is the identity. Since gauge bosons ...
0
The main point is that if one has a consistent gauge theory with matter with gauge group $$G:=SU(3)\times SU(2)\times U(1),$$ if one divides $G$ with a normal subgroup $N$, the matter representations of the matter fields could potentially become multi-valued. However, it is possible to choose $N=\mathbb{Z}_6$ in such a way that the standard model matter ...
1
Baez actually has another paper (with Huerta) that goes into more detail about this. In particular, Sec. 3.1 is where it's explained, along with some nice examples. The upshot is that the hypercharges of known particles work out just right so that the action of that generator is trivial. Specifically, we have Left-handed quark Y = even integer + 1/3 ...
7
Color charge in the sense of "being blue, red, green" is not a quantum mechanical observable because the $\mathrm{SU}(3)$ gauge transformations mix the colors. This means it is meaningless to say "We have a blue particle", because we can perform a gauge transformation and then we "have a red particle". Since physical descriptions related by gauge ...
0
My "answer" to this question is, for the moment at least, do more reseach and update this current note with more details as I discover them. I am reluctant to withdraw this question for the moment, I asked it ahead of myself and my knowledge level but I will return to it, even if only for my own personal notes. The answer to my question may lie in a ...
0
I can think some speculative or unorthodox answers, and sure others can do, so please allow me to mark this answer as Community Wiki: three generations make a nice number of degrees of freedom for a GUT model. Assuming that the neutrinos have companions of the other chirality, one generation has 36 degrees of freedom. With this, the MSSM happens to have ...
-3
The relativistic mass of the photon is not zero, it is $\frac{h\nu}{c^2}$. The rest mass of the photon is zero, but according to spec. rel it goes always with $c$.
-2
No, there are just spin-0, spin-1 and spin-1/2 particles, even the Higgs-Boson is 0.
1
The VEV is quantum mechanical, it can not be read off from the Lagrangian. To find the VEV from the potential requires one to quantise the theory, then calculate the effective action at strong coupling. What should happen is that at strong coupling the quarks form hadrons; which are quark condensates. To perform the calculations is very tough and not yet ...
1
In addition to TwoBs' comments, in the 90s, there had been many collisions at SLAC (USA) with polarized electron/positron beams of the SLC collider running at the $Z$ pole. Therefore, the right handed component of the electroweak interaction has been extensively tested. See the wikipage: http://en.wikipedia.org/wiki/SLAC_National_Accelerator_Laboratory
3
Is there something wrong with this process? (it will admittedly be suppressed by $|V_{su}|^2\approx \frac{1}{20}$, i.e. "doubly cabbibo suppressed") or maybe even replace the Z boson with gluons.
4
Is there anyone knowledgable enough in this area who would be able to comment on some of the possible theoretical/ hypothetical implications of the existence of spin 3 particles? Is there any thought that their existence could imply additional fundamental forces? If you look at the presentation linked in the link you gave , in page five, you will see ...
0
As ACuriousMind said, the fact that the particle is composite makes it less earth-shattering. The interesting thing here (I think...not a high-energy guy) is that it's the first example of this kind of spin-3 particle, in particular flavored. Other composite spin-3 particles have been known before, the first one I found was a Boron nucleus.
2
Yes, wikipedia has a table which lists the 19 free parameters that need to be tuned by experiments. These include, as you already said, the masses of the elementary particles including the Higgs Boson, and some other notable ones are: CKM Mixing angles and CP-violation phase. Gauge coupling of he three symmetries (U(1), SU(2), SU(3)). Higgs VEV
Top 50 recent answers are included | 2015-04-21 05:14:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.8270848393440247, "perplexity": 320.7906772698658}, "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-2015-18/segments/1429246640124.22/warc/CC-MAIN-20150417045720-00170-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://answers.ros.org/question/271434/minimum-time-between-publishing-a-joint-state-and-tf-listener-update/ | ROS Resources: Documentation | Support | Discussion Forum | Index | Service Status | Q&A answers.ros.org
# Minimum time between publishing a joint state and tf listener update
Hi,
I have 8 laser range finders propped up on a rotating vertical rod connected to a motor. The laser range finders is fixed to the sides of the rod at some 'xyz rpy' pose. I get the current angular position of the rod and use this to publish the joint state of the rod (which is free to rotate along the z axis). So when the motor runs in real world i can see the vertical rod and the lidar attached to it rotate as well in rviz.
Now at each instant of rotation i want to use tf to get the instantaneous pose of the lidar so that i can calculate the 3D point observed by the lidar. I create a point cloud each rotation of the lidar. Here is the code snippet i used for getting the tf.
this->rod_joint_state.header.stamp=ros::Time::now();
this->pub_joint.publish(this->rod_joint_state);
/* 10 lines of code including conditions and initilizations here */
for(int i=0;i<8;i++){
try{
this->tf_listener.waitForTransform(this->frame_id,lidar_n,ros::Time::now(), ros::Duration(0.000125));
this->tf_listener.lookupTransform(this->frame_id,lidar_n,ros::Time(0),transformStamped);
}
/* calculate the 3D point using the tf*/
}
I do not have any problems getting the transforms. I do not get "Cannot lookup transform" or any other warning from the try block.
However when i set the timeout for waitForTransform to 0 (or no wait) , the point Cloud looks very random. When i set it to a higher value like 0.01s the point cloud starts to look like the environment it is scanning.
Scan with Delay: output as expected ..
Scan without delay: wierd output.
So my question is , is there a minimum time between publishing and listening to a joint state ? In my code there is very very less time between the two.
edit retag close merge delete
I'm not sure, but Time::now() and Time(0) are different things: now() means whatever the current time is at the time of executing that line, 0 means the latest transform. I'd check to make sure that you are asking TF for the correct stamp.
( 2017-09-25 02:35:19 -0600 )edit
@gvdhoorn Thank you for that insight.i had not known the difference. However i tried with both and results are still the same.
( 2017-09-25 16:32:27 -0600 )edit
Sort by » oldest newest most voted
As @gvdhoorn mentions your timestamp usage is inconsistent and you should be using the timestamp of the data neither now() nor TIme(0)
tf internally keeps a buffer of the past transform data to allow you to query it at a specific time. When the laser scan is collected you should collect as exact a timestamp as possible at the time of capture. In parallel you should be publishing the joint states as accurately as possible.
Then when you want to transform your laser scan into a point at the time it was collected. Not the current transform etc. So you should be transforming at the time of the data.
Now it may be that the transform data is slower to propogate depending on your network topology. So you could use waitForTransform but a much better solution is to use a MessageFilter which will efficiently hold the data for you until the transforms are available.
There are tutorials here: http://wiki.ros.org/tf2/Tutorials/tf2... and here: http://wiki.ros.org/tf2/Tutorials/Usi...
On each of those topics. I'd also strongly recommend switching to tf2 from tf as tf has been deprecated and is just a backwards compatibility layer for tf2.
more
Thank you @tfoote. THank you for linking me to MessageFilter. It was something i was looking for. I have already switched to tf2.
( 2017-09-27 17:57:02 -0600 )edit | 2022-12-05 15:10:28 | {"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": 0.2252497673034668, "perplexity": 1377.9863314528207}, "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/1669446711017.45/warc/CC-MAIN-20221205132617-20221205162617-00362.warc.gz"} |
https://chemistry.stackexchange.com/questions/111112/why-does-the-degree-of-dissociation-change-when-we-dilute-a-weak-acid-even-thoug | # Why does the degree of dissociation change when we dilute a weak acid even though the equilibrium constant is constant?
$$K$$ represents the ratio of concentrations of molecules in a solution at equilibrium, which means that $$Q_\mathrm{r}$$ (that ratio at any given point) looks to be identical to $$K$$. In other words, the molecules in that solution react accordingly so that they reach equilibrium and the ratio of their concentrations is equal to $$K$$.
If $$K$$ is large enough (bigger than $$10^4$$ in my curriculum), this means that the the concentration of the reactants is almost zero. In other words, the equilibrium position of that solution looks very much like a reaction that went to completion.
The more we dilute an acidic/basic solution, the higher the degree of dissociation, even though $$K$$ stays the same. So, does that mean that the more we dilute a solution the harder it is for it to reach the point of equilibrium for that specific molecule/solution or what?
For instance, say you found $$K$$ of solution to be $$10^{-5}$$. This means that when the reaction happens there are lots of reactants left, and not much products produced, which means that the degree of dissociation is low. But the more we dilute a solution, the closer it gets to a "complete reaction" (if you pour a small amount of weak-acid molecules into a large tank of water, it's certain that all of the weak-acid molecules are going to react with the water, i.e. the degree of dissociation approaches $$100\%$$).
So, how come $$K$$ can be independent of the initial reactants concentrations, and tell if a reaction was complete or not, when the "completion" of a reaction (the degree of dissociation) depends on the initial concentrations of reactants?
See my comment above.
As a numerical example, take acetic acid ($$\ce{AcOH}$$), which has $$K_\mathrm{a} = 1.8 \cdot 10^{-5}$$.
This means that:
$$K_\mathrm{a} = \frac{[\ce{AcO-}][\ce{H+}]}{[\ce{AcOH}]}$$
And the total nominal concentration of acid is:
$$C_\mathrm{a} = [\ce{AcOH}] + [\ce{AcO-}]$$
Combining these two equations, you can see that the % of dissociated acid is:
$$\alpha = \frac{[\ce{AcO-}]}{C_\mathrm{a}} = \frac{K_\mathrm{a}}{K_\mathrm{a} + [\ce{H+}]}$$
i.e. not a constant, but a value that depends on the $$\mathrm{pH}$$. So no, there is no need for the 'degree of completion' of the reaction to be a constant.
You can even calculate the explicit concentrations of all the species.
Given that:
$$C_\mathrm{a} = \left(\frac{1}{[\ce{H+}]} + \frac{1}{K_\mathrm{a}}\right) \cdot ( [\ce{H+}]^2 - K_\mathrm{w}^2)$$
you can see that, for $$\mathrm{pH} = 5$$ you need:
$$C_\mathrm{a} = \left(\frac{1}{10^{-5}} + \frac{1}{1.8 \cdot 10^{-5}}\right) \cdot \left((10^{-5})^2 - (10^{-14})^2\right) = 1.5554 \cdot 10^{-5}$$
Then:
$$[\ce{AcOH}] = \frac{C_\mathrm{a} \cdot [\ce{H+}]}{K_\mathrm{a} + [\ce{H+}]} = 5.555 \cdot 10^{-6}$$
$$[\ce{AcO-}] = C_\mathrm{a} - [\ce{AcOH}] = 9.999 \cdot 10^{-6}$$
So the % of dissociated $$\ce{AcOH}$$ is:
$$\alpha = \frac{[\ce{AcO-}]}{C_\mathrm{a}} \approx 64 \%$$
Now repeat for $$\mathrm{pH} = 3$$:
$$C_\mathrm{a} = \left(\frac{1}{10^{-3}} + \frac{1}{1.8 \cdot 10^{-5}}\right) \cdot \left((10^{-3})^2 - (10^{-14})^2\right) \approx 0.05656$$
$$[\ce{AcOH}] = \frac{C_\mathrm{a} \cdot [\ce{H+}]}{K_\mathrm{a} + [\ce{H+}]} \approx 0.05556$$
$$[\ce{AcO-}] = C_\mathrm{a} - [\ce{AcOH}] \approx 0.001$$
$$\alpha = \frac{[\ce{AcO-}]}{C_\mathrm{a}} \approx 1.8 \%$$
So you see, in a more dilute mixture the $$\mathrm{pH}$$ is higher, and the dissociation reaction of acetic acid is 'more complete', whereas at higher concentration the pH is lower, and there is proportionally less acetate anion.
All of these systems are at equilibrium.
$$K$$, the equilibrium constant, is independent of the composition of the system. It simply describes the preference of the system for reactants or products.
On the other hand, there is the reaction quotient, $$Q$$, which describes the position of the system relative to equilibrium. $$Q$$ is the value that is variable and that you adjust to be closer to $$K$$. Technically, $$K$$ can also be computed from the standard free energy change of the reaction, $$\Delta G ^{\circ}$$.
What's confusing is the $$Q$$ and $$K$$ are computed in the exact same way, with the sole difference that for $$Q$$, you plug in the values you have right now, and for $$K$$, you plug in the values at any equilibrium.
If $$Q=K$$, you are at equilibrium.
If $$Q < K$$, you need to push the reaction towards more products.
If $$Q > K$$, you need to push the reaction towards more reactants.
When you dilute a solution (or make other changes), you may change the value of $$Q$$, which may put you farther from equilibrium, but the opposite may also be true. It depends on the specific reaction and the form of the equilibrium expression.
EDIT:
Your confusion is because you're confusing $$Q$$ measured at different points in time. At the end of the reaction $$t = \infty$$, $$Q = K$$. At the beginning of the reaction, $$Q_{0} < K$$. Over time, as the reaction goes towards equilibrium, the value of $$Q$$ will get closer to $$K$$. If at the beginning of the reaction, you dilute the solution, you may get a different value of $$Q$$. If it's smaller, then you have farther to go to get to equilibrium. But again, this depends on the molecularity of the reaction.
$$\ce{A + B -> C}$$
$$\ce{A -> C}$$
$$\ce{A -> C + D}$$
Here are three different reactions.
In the first reaction, at the beginning of the reaction, with $$Q < K$$, dilution pushes you towards equilibrium because reactant concentrations enter into the equilibrium expression twice.
$$Q = \frac{\ce{[C]}}{\ce{[A][B]}}$$
If you halve all three concentrations, the value of $$Q$$ will get bigger.
In a similar vein, the second equation is unaffected by dilution. In fact, at equilibrium, dilution preserves equilibrium.
And for reaction 3, dilution pushes you farther away from equilibrium.
For all three reactions, at the end of the reaction, you will have $$Q = K$$. But for reactions 1 and 3, dilution at equilibrium changes the value of $$Q$$ and the reaction will need to adjust again to reach equilibrium.
• "You may change the value of Q" the Q at the end of the diluted-solution's reaction? If so, then does that mean that if you dilute a solution to a certain point, Q stops going towards K? Also, If K is small (say smaller than 10⁴), can you dilute a solution to a certain concentration so that it's reaction is complete (α=1)? If not, then how can this saying be explained: "K is only affected by temperature and not the initial concentration of the reactants, while the 'advancement' of a reaction (α) is affected by both temperature and the initial concentration of the reactants"? Mar 17 '19 at 13:47
• No, immediately after dilution. $Q$ will be $K$ at the end. You cannot dilute the reaction to completion. But it might get closer or farther depending on coefficients of the balanced reaction. I think you are confusing $K$ and $Q$. The first is constant. The latter is not.
– Zhe
Mar 17 '19 at 20:59
• Hmm, I think I'm having trouble wrapping my head around how can a diluted-reaction be closer to completion with Q at the end of that reaction still being the same as before Mar 17 '19 at 21:05
• Let me add an edit. I think that will help.
– Zhe
Mar 17 '19 at 22:34
• The question is inactive, I doubt anyone will see it, but go ahead Mar 18 '19 at 10:12
If K is large enough (bigger than $$10^4$$ in my curriculum), this means that the the concentration of the reactants is almost zero.
This statement is not always true - it depends on the stoichiometry of the reaction.
For the reaction $$\ce{A(aq) <=> B(aq)}$$
the concentration of A is much smaller (ten thousand times) than that of B if $$K = 10^4$$, and this is independent of diluting the solution. It's fair to say that there is almost no A compared to B.
For the reaction $$\ce{4A(aq) <=> 4B(aq)}$$
the concentration of A is just ten times smaller than that of B if $$K = 10^4$$, and this is also independent of diluting the solution. However, there is quite a bit of A compared to B, and it is a bit misleading to say the the concentration of A is almost zero.
Both of these reactions had the same number of reactant species as product species (all in the same solution, so all equally affected by dilution).
The more we dilute an acidic/basic solution, the more the reaction is "complete", the more the reactants disappear, but K stays the same.
Here, "complete" would mean the absence of reactants, i.e. a reaction that goes to completion. Weak acids are defined by not completely dissociating, in contrast to strong acids. The general reaction for a weak acid is:
$$\ce{AH(aq) <=> H+(aq) + A-(aq)}$$
Notice that there are more product particles than reactant particles. For those type of reactions, dilution favors the products. In fact, if $$K = 10^-5$$ for this reaction and all concentrations are $$\pu{10^-5 M}$$, the reaction is at equilibrium. This means even though the equilibrium constant is much smaller than one, you still can have reactants and products at the same concentrations.
On the other hand, if there are more reactant particles than product particles, dilution has the opposite effect. For a complex formation reaction between a metal M and a ligand L
$$\ce{M(aq) + 4 L(aq) <=> ML4(aq)}$$
dilution will cause the complex to fall apart. If $$K = 10^4$$ and the free ligand concentration is one molar, there will be a lot of complex and very little free metal. On the other hand, if the free ligand concentration is one millimolar (0.001 M), there is hardly any complex and most of the metal will be in the form of the free metal.
Why does the degree of dissociation change when we dilute a weak acid even though the equilibrium constant K is constant?
As the examples above illustrated, this is because for these specific reactions, the sum of exponents for the products is higher than that of the reactants in the equilibrium constant expression. As you dilute, Q will decrease, and the reaction will go forward to reach K again. At infinite dilution, the product will be favored no matter what the value of the equilibrium constant is.
If you want to have a general statement about what the value of K means, it would be something like "as K increases, the equilibrium concentration of products will increase and those of reactants will decrease".
A really handwavy explanation would be that the system, attempting to maximize its entropy, will prefer dissociated states as the system dilutes since there will be an increasing number of microscopic configurations with atoms $$A$$ and $$B$$ far apart (i.e. molecule $$AB$$ dissociated) as opposed to close together (i.e. atoms $$A$$ and $$B$$ bound together into molecule $$AB$$). | 2021-10-17 07:27:54 | {"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": 75, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8522499799728394, "perplexity": 470.9299925091963}, "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/1634323585121.30/warc/CC-MAIN-20211017052025-20211017082025-00639.warc.gz"} |
http://utilars.com/Blog/shawn-eary/2019/4/21/esQueryStringQuoteAnalyzer | Warning!!! - I'm having serious family problems right now. This website is meant to be useful, but right now, I'm struggling. I don't want to just toss this website though. I haven't decided yet if a "rebranding" is necessary. The utilars.com domain can still be used, but if my family troubles continue, the "tone" of this website might change. Fortunately, I'm taking a break from competitive cycling for a while so I have a bit more free time for now.
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Other browsers may also have variying support, but I *personally* consider Microsoft Edge, Chrome and FireFox the major browser vendors of present time.
## Elasticsearch Query String Quote_Analyzer
Elasticsearch Query String Quote_Analyzer
Recently, I was fortunate enough to professionally work on an ASP.NET MVC project again. I worked in tandem with a team leader to switch the application so it used Elasticsearch (ES) instead of Google Search Appliance (GSA). Surprisingly, the switch went really well. There were only a few "enhancements" that needed to be done after the switch.
One such enhancement was something called stopwords. When users searched for words like "a", "an" and "the", they were getting back way too many hits. My team leader, created an analyzer that used the stopwords feature to get rid of this annoying behavior [1]. I then modified the ASP.NET MVC code for the project to apply the said analyzer when necessary. This got rid of that problem, but later when we decided to allow more advanced queries, a different issue popped up.
Initially, basic queries were useful for the ASP.NET MVC app we had worked on. At some point, however, the users needed additional functionality. An easy way to allow this was to enable the Query String Query [2] feature of ES. It allows interesting queries to be constructed using boolean roles, Match Phrase (aka: Quoted) queries [3] and other goodies.
So we started leveraging the Query String component of ES. It worked great but we ran into a small problem. If our users searched for phrases that contained stopwords, they would get way too many hits since the stopwords would get removed from the phrase. As an example, if a user searched for "War of the Worlds" on our system, the query would be altered to "War Worlds" since the analyzer would remove the stopwords "of" and "the". This modified query would then match all sorts of combinations beyond what was intended since stop words between "War" and "Words" in the "War Worlds" phrase would be ignored. Even though "War of the Worlds" was originally specified, as long as our original analyzer was running on the phrase, the "War of the Worlds" could also match "War in the World", "War over the World", etc... because "of", "the", "in" and "over" are all stopwords that would be ignored.
Well, we didn't want to turn the analyzer completely off so I searched for days and hours looking for a solution. I was trying to figure out how to use the stopwords feature to filter out silly queries when stopwords are not needed but yet allow them for Match Phrase queries. I was thinking the solution would be very hard and complicated. It turns out the solution was quite simple. If you look back at [2], you will notice that ES allows you to specify two analyzers in the JSON body. One of them is the regular analyzer (which is intuitively named "analzyer") and the other one is an analyzer that only applies to quoted or Match Phrase queries (this one is named "quote_analyzer").
To keep excluding stopwords for basic queries but not exclude stopwords for quoted queries, you simply continue specifying your regular analzyer via the "analyzer" keyword but set the quote_analzyer to something sensible like the standard analyzer. An example of how the JSON for this might look appears below:
GET /_search
{
"query": {
"query_string" : {
"query" : "\"War of the Worlds\" OR "\"Star Trek\""
}
},
"analyzer": "your_analyzer_to_remove_stopwords",
"quote_analyzer": "standard"
}
Pretty simple. I'm suprised it took me so long to figure out that all I had to do was change one line to adjust quoted searches so they didn't exclude stopwords.
## Bibliography
1. Elasticsearch Reference [7.0] - Analysis
https://www.elastic.co/guide/en/elasticsearch/reference/current/analysis.html
2. Elasticsearch Reference [7.0] - Query String Query
https://www.elastic.co/guide/en/elasticsearch/reference/current/query-dsl-query-string-query.html#query-dsl-query-string-query
3. Elasticsearch Reference [7.0] - Match Phrase Query
https://www.elastic.co/guide/en/elasticsearch/reference/current/query-dsl-match-query-phrase.html | 2019-07-19 16:56:39 | {"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": 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": 0.352975070476532, "perplexity": 3031.1807245884233}, "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/1563195526324.57/warc/CC-MAIN-20190719161034-20190719183034-00021.warc.gz"} |
https://paperswithcode.com/paper/top-k-training-of-gans-improving-generators | # Top-k Training of GANs: Improving GAN Performance by Throwing Away Bad Samples
We introduce a simple (one line of code) modification to the Generative Adversarial Network (GAN) training algorithm that materially improves results with no increase in computational cost: When updating the generator parameters, we simply zero out the gradient contributions from the elements of the batch that the critic scores as least realistic'. Through experiments on many different GAN variants, we show that this top-k update' procedure is a generally applicable improvement. In order to understand the nature of the improvement, we conduct extensive analysis on a simple mixture-of-Gaussians dataset and discover several interesting phenomena. Among these is that, when gradient updates are computed using the worst-scoring batch elements, samples can actually be pushed further away from their nearest mode. We also apply our method to recent GAN variants and improve state-of-the-art FID for conditional generation from 9.21 to 8.57 on CIFAR-10.
PDF Abstract NeurIPS 2020 PDF NeurIPS 2020 Abstract | 2022-05-22 19:11:46 | {"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": 0.6150286197662354, "perplexity": 1378.0744785688055}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662546071.13/warc/CC-MAIN-20220522190453-20220522220453-00692.warc.gz"} |
https://www.physicsforums.com/threads/expectation-values-for-a-harmonic-oscillator.391295/ | # Homework Help: Expectation values for a harmonic oscillator
1. Mar 31, 2010
### KaiserBrandon
1. The problem statement, all variables and given/known data
I need to find <x>, <x2>, <p>, and <p2> for a particle in the first state of a harmonic oscillator.
2. Relevant equations
The harmonic oscillator in the first state is described by $$\psi$$(x)=A$$\alpha$$1/2*x*e-$$\alpha$$*x2/2. I'm using the definition <Q>=($$\int$$$$\psi$$1*Q*$$\psi$$)dx where $$\psi$$1 is the complex conjugate of $$\psi$$, and Q is the specific operator.
3. The attempt at a solution
I solved for <x>, and found it was zero. <p> I'll solve for in a similar fashion. However, for <x2> and <p2>, I am unsure of what operators I use. For the <x> operator, it is simply x, so for <x2>, would I use x2 as an operator?
(note that I the only superscripts here are the ones above e, I don't know why latex is putting all of my symbols so high
2. Mar 31, 2010
### kuruman
Yes, that's what you should use.
3. Mar 31, 2010
### dx
(Put the whole equation in the tex barackets instead of individual symbols)
The operator for <x²> is simply multiplication by x², so <x²> = ∫ψ*(x)x²ψ(x)dx, and <p²> is
$$-\int \psi^{*}(x) \hbar^2\frac{\partial^2}{\partial x^2}\psi(x) dx$$
4. Mar 31, 2010
### KaiserBrandon
ok, thank you so much guys. It's a good thing I have physics forum to at least make my new ventures into the realm of quantum mechanics a bit easier :) | 2018-09-22 10:57:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8343690037727356, "perplexity": 1943.572977770796}, "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-2018-39/segments/1537267158320.19/warc/CC-MAIN-20180922103644-20180922124044-00436.warc.gz"} |
https://www.esaral.com/q/suppose-the-initial-charge-on-the-capacitor-in-exercise-7-7-is-6-mc-54210 | # Suppose the initial charge on the capacitor in Exercise 7.7 is 6 mC.
Question:
Suppose the initial charge on the capacitor in Exercise 7.7 is 6 mC. What is the total energy stored in the circuit initially? What is the total energy at later time?
Solution:
Capacitance of the capacitor, C = 30 μF = 30×10−6 F
Inductance of the inductor, L = 27 mH = 27 × 10−3 H
Charge on the capacitor, Q = 6 mC = 6 × 10−3 C
Total energy stored in the capacitor can be calculated by the relation,
$E=\frac{1}{2} \frac{Q^{2}}{C}$
$=\frac{1}{2} \times \frac{\left(6 \times 10^{-3}\right)^{2}}{30 \times 10^{-6}}$
$=\frac{6}{10}=0.6 \mathrm{~J}$
Total energy at a later time will remain the same because energy is shared between the capacitor and the inductor. | 2023-02-02 14:20:41 | {"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": 0.8372287750244141, "perplexity": 333.4796300779034}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500028.12/warc/CC-MAIN-20230202133541-20230202163541-00692.warc.gz"} |
http://mathoverflow.net/questions/49616/how-to-use-the-lefschetz-trace-formula-on-infinite-dimensional-spaces | # How to use the Lefschetz trace formula on infinite dimensional spaces?
I think the Lefschetz trace formula says something like:
if $F: X \to X$ is a continuous map of compact manifolds, then
$\chi(X^F) = \sum (-1)^i \mathrm{Tr} f_*|_{H_i(X)}$
First of all, this statement is not quite right even when the fixed points $X^F$ are isolated, I am supposed to somehow put in the indices, right?
So what are the indices and how do I put them in when the fixed points are not isolated?
But, also:
Under what hypotheses can I use it, or something similar, for maps between non-compact, infinite dimensional, etc., etc., spaces?
or perhaps I am asking
What is the most general version of the Lefschetz trace formula?
-
By \chi(X^F) do you mean the Euler char? And by "indices" you mean multiplicities and sign? If we want H_i(X) we should assume X is orientable in order for the constant sheaf to be the orientation sheaf (think of RP^2 and a matrix in PGL_3(R) that has real eigenvalues). If not isolated, I don't think putting indices would work; f may be homotopic to another map that has isolated fixed pts. We may want the intersection number in X*X between the diagonal and the graph. Think of f=id, which gives Gauss-Bonnet. In his early version of the thm, Lefschetz worked with closed manifolds.... – shenghao Dec 16 '10 at 12:27
...and then he generalized the thm several times. Requiring it to be a finite CW complex will be ok. I'm not sure about the non-compact or infinite dim cases in topology (there might be no finite triangulation on the space...). In alg geom, when considering varieties over a finite field and f=Frobenius, it works for non-proper schemes (as long as it's of finite type; one uses compact support l-adic cohomology instead of homology) and certain "infinite dim varieties", by which I mean algebraic stacks of finite type. – shenghao Dec 16 '10 at 12:36 | 2015-05-30 07:09:11 | {"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": 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": 0.8931345343589783, "perplexity": 752.6932582354674}, "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-2015-22/segments/1432207930895.96/warc/CC-MAIN-20150521113210-00272-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://d2mvzyuse3lwjc.cloudfront.net/www/products/GraphGallery.aspx?GID=112 | ### Stacked Histogram Plots with Normal Distribution Curve Overlays
This series of stacked Histogram plots displays the distribution of student grades per quarter. Note: The data used for this graph is not scientific. | 2021-10-28 09:24:46 | {"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": 0.8276457786560059, "perplexity": 2402.561982328758}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323588282.80/warc/CC-MAIN-20211028065732-20211028095732-00461.warc.gz"} |
https://chemistry.stackexchange.com/questions/19707/role-of-alcoholsetoh-meoh-in-nitro-reduction-used-for-metalsn-fe | # Role of Alcohols(EtOH,MeOH) in nitro reduction used for Metal(Sn,Fe)
I'm trying to reduce an aryl nitro group. I understand that these reactions are radical reactions by the metal, but I don't understand why alcohols are used as the solvent. Are they used as a source of protons, or just because they are good solvents?
LHM...Look at this mechanism that I posted as an answer to an earlier SE Chem question/answer relating to the reduction of nitrobenzene by metals.
In most reactions that occur on metal surfaces, many of the mechanistic details are not fully understood (see @Martin's comments in the earlier post). Nonetheless, this mechanistic scheme shows several steps where a proton ($\ce{H+}$) is required. These proton transfers are necessary to complete the reduction process. This is why these reactions are typically run in solvents, like ethanol, that have available protons that can be donated in order to complete the reaction...ron
• Thanks for your answering my question. If you know mechanism of ethanol give a proton to nitro? In condition, it's react in RT so i think O-H bond is hard to break. or nitrogen's lonepair electons catch the O-H proton? – LHM Nov 19 '14 at 5:56
• Notice that in some of the steps requiring proton transfer there is a negative charge on oxygen or nitrogen. This makes the oxygen or nitrogen more basic, and makes it easier for the oxygen or nitrogen to remove a proton from the solvent. Notice too that acid is usually involved in these reductions and can also serve as a source of protons. – ron Nov 19 '14 at 13:12
I do not believe that ethanol is mandatory. You surely need a proton source, but that's why those reactions are performed in protic acidic media, like your first example, with HCl, or what I think were the original conditions of Bechamp reduction, using acetic acid.
Probably ethanol makes a good solvent for phenol rings.
Edit: this is the Bechamp original paper. I couldn't manage to find it in English. I'm not French speaker either, but just in case you are curious :-)
As far as I'm concerned, nowadays catalytic hydrogenation is used to perform this reaction (at least at industrial scale)
About the mechanism, I'm sorry but I can't give you a solid statement on this. | 2020-04-08 03:35:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6936480402946472, "perplexity": 1222.6962570583773}, "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/1585371807538.83/warc/CC-MAIN-20200408010207-20200408040707-00054.warc.gz"} |
https://threesixty360.wordpress.com/2008/02/11/ | ## Archive for February 11th, 2008
### Root extraction, part II: cube roots
February 11, 2008
As you might guess, this post builds on “Root extraction, part I“, which gave a way to visualize the traditional square root algorithm geometrically, an approach that has the advantage that each step appears natural and easily motivated.
Our goal herein is to do much the same for cube roots. The point is to find a geometric construction, ideally one well-suited to physical manipulatives, in which the steps in building the successive digits of the cube root of a number are transparent. As with the post on square roots, I make no claims to originality in what follows.
Example: Find $\sqrt[3]{22665187}$(more…) | 2020-09-25 16:09:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.44617584347724915, "perplexity": 1147.7243978383628}, "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/1600400227524.63/warc/CC-MAIN-20200925150904-20200925180904-00238.warc.gz"} |
https://www.jstage.jst.go.jp/article/jjpehss/advpub/0/advpub_13086/_article/-char/ja/ | Online ISSN : 1881-7718
Print ISSN : 0484-6710
ISSN-L : 0484-6710
スポーツ集合的効力感尺度の改訂・邦訳と構成概念妥当性の検討
ジャーナル フリー 早期公開
この記事には本公開記事があります。
Collective efficacy has been identified as a critical determinant of team success in sport. Many studies in sport psychology have focused on the relationships between collective efficacy and psychological variables or outcomes of interest. Although an increasing number of studies on collective efficacy have been conducted in Japan, greater attention should be given to refining the methodology for assessing the construct of collective efficacy. The purposes of this study were to confirm the factor structure and establish construct validity of the Japanese translated version of the Collective Efficacy Questionnaire for Sports (J-CEQS) with revisions to its original version (Short et al., 2005). The participants were 1244 athletes from 48 teams. We conducted a confirmatory factor analysis and found that the J-CEQS showed the same multidimensional factor structure as Short's original questionnaire. To test the construct validity of the J-CEQS, we examined correlations among its subscale scores with the Group Environment Questionnaire (GEQ; Carron et al., 1985). All the J-CEQS subscales were significantly correlated with the group integration-task subscales. The unity subscale of the J-CEQS was also significantly correlated with all the GEQ subscales. These correlations were similar to those shown in previous studies (Martínez et al., 2011; Short et al., 2005). The present findings provide preliminary support for the utility of the J-CEQS as a measure for assessing collective efficacy in sport teams. We discuss recommendations for future studies using the J-CEQS. | 2020-09-29 17:11:08 | {"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": 0.8922382593154907, "perplexity": 3156.0417049861308}, "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/1600400202418.22/warc/CC-MAIN-20200929154729-20200929184729-00438.warc.gz"} |
https://duyptnk.wordpress.com/2011/06/ | ### Archive
Archive for June, 2011
## Polynomial in a Field of Characteristic p
June 30, 2011 1 comment
I’ll go ahead and it post it here as well:
Hey Ken, I’m trying to check if this proof is correct(it may not be the only way, but its one I thought of)
To Prove: Let $F$ be a field with $charF=p$. Let $q(x)=x^p-x-c$ where $c\in F$. Show if $q$ is reducible, then $q$ has a root in $F$ (actually all).
This is actually an iff in the book, but this is the nontrivial direction.
My idea is the following. Let $u$ be a root of $q$. Let $f,g$ be nonunit irreducible factors of $q$. Now it is easy to see the set of all roots is the set $\{u+i:i=1,\ldots,p\}$. Thus, $Z(f)=u+I, Z(g)=u+J$ for some finite sets $I,J\subset \mathbb{Z}.$
Notice this means $\prod_{i\in I}u+i, \prod_{i\in J} u+i$ are both in $F$. Let $f_1=\prod_{i\in I}u+i$ and $f_2=\prod_{i\in J} u+i$. Then $u$ is a root for both $f_1-\prod_{i\in I}{x+i}$ and $f_2-\prod_{i\in J}x+i$ which are in $F[x]$. But since the degrees of these are $|I|,|J|$; respectively, the only way this is possible is if $|I|=|J|$. But we could do this for any irreducible factor so each irreducible factor must have the same degree.
Hence, if $q=q_1\cdots q_k$ with each $q_i$ irreducible, then $p=deg(q)=k\cdot n$ for some $n\geq 1$. Since $p$ is prime either $k$ or $n$ must be $1$. If $n=1$ then each root is in $F$. If $k=1$ then $q$ is irreducible. Since $p$ is reducible the result follows.
After writing this out carefully it feels completely correct (of course I left out some easy details).
Categories: Algebra questions
## Complex #34: sneaky use of Rouche’s theorem
Problem #34 is: Prove that there does not exist a polynomial of the form $p(z)= z^n+a_{n-1}z^{n-1}+\ldots+a_0$ such that $|p (z)| < 1$ for all $z$ such that $|z|=1$.
Proof: Suppose that there does exist such a polynomial. Note that $|z^n|>|-p(z)|$ for $|z|=1$. By Rouche’s theorem, $z^n-p(z)=-a_{n-1}z^{n-1}-\ldots-a_0$ has the same number of zeros (with multiplicities) as $z^n$ inside the unit disk. But that is impossible, since $z^n$ has $n$ zeros and $z^n-p(z)$ has at most $n-1$ zeros. QED.
Categories: Complex questions
## Complex #27: integrating over a long rectangle
Problem 27: Show that $\int_{\gamma} e^{iz}e^{-z^2}dz$ has the same value on every straight line path $\gamma$ (oriented in the $+x$ direction) parallel to the real axis.
Proof: Consider the rectangle $R=[-M,M]\times [0,Y]$, where $Y$ is some given number and $M$ is a very large number. Since the integrand is entire, the integral over the boundary of $R$ is zero by the Cauchy Integral Theorem, and as $M\to\infty$, the horizontal pieces correspond to $\int_{\gamma} e^{iz}e^{-z^2}dz$ with $\gamma$ being the line $Im (z)=0$ or the line $Im(z)=Y$, with one being negative and the other positive. Thus, the result is proved if we can show that the contributions from the vertical pieces go to zero as $M$ goes to $\infty$.
Observe that the absolute value of the integrand at $z=x+iy$ is $e^{-x^2+y^2-y}$, which on each vertical boundary piece is $e^{-M^2+y^2-y}$ for $0\le y\le Y$. Integrating this over $y$ will give a value bounded by a constant times $e^{-M^2}$, which clearly goes to zero as $M\to\infty$.
QED.
Categories: Complex questions
## Complex #26: polynomial properties
Problem #26: If $f(z)$ is an entire function that is not a polynomial, prove that, given arbitrary $C > 0$, $R > 0$, and integer $m > 0$, there exists a $z$ such that $|z|> R$ and $|f(z)| > C|z|^m$.
The way this is worded makes my head hurt, so let’s formulate it equivalently as Not B implies Not A instead of A implies B.
Reformulation: Let $f(z)$ be an entire function. Suppose that there exists a $C>0$$R > 0$, and an integer $m>0$ such that $|f(z)|\le C|z|^m$ whenever $|z|>R$. Prove that $f$ is a polynomial.
Proof: By the given and a limiting process, $|f(z)|\le CR^m$ for $|z|=R$. By the maximum modulus principle, the same fact is true for $|z|\le R$. By the Cauchy-Lagrange inequalities if $|z|=\frac{R}{2}$ , we have $|f^{(m)}(z)|\le \frac{m! M}{(R/2)^m}$, where $M$ is the maximum value of $|f(w)|$ for $|w-z|=\frac{R}{2}$. Since this circle is completely contained in the disk $\{ w:|w|\le R\}$, $M\le CR^m$, and so we have $|f^{(m)}(z)|\le (m!) 2^m C$ for $|z|=\frac{R}{2}$ and thus for $|z|\le \frac{R}{2}$. By Liouville’s theorem, $f^{(m)}(z)$ is a constant, and thus $f$ is a polynomial.
Categories: Complex questions
## Complex #30: sin(z)=z^2 solutions
One of the problems we talked about was the proof of the fact that the equation $\sin(z)=z^2$ has an infinite number of complex number solutions. I thought of a couple of ways to do this. Here is one way.
Let $g(z)=\sin(z)-z^2$. In rectangular coordinates, the function is
$g(x+iy)=\left[\sin(x)\cosh(y)-x^2+y^2\right]+i \left[\cos(x)\sinh(y)-2xy\right]$.
For a large positive integer $N$, consider the rectangle
$R=\left\{ z=x+iy: 2\pi \le x \le 2\pi N, 0\le y \le Y\right\}$, where $Y$ is a large constant to be chosen conveniently later. We will show that as we go counterclockwise around the boundary of $R$, the argument of $g(z)$ increases at least by $2\pi (N-2)$. Then, by the argument principle, the number of zeros inside $R$ is (at least) $(N-2)$ (and there are no poles). So that is all we need for the proof, cause $N$ can be chosen to be arbitrarily large.
Increasing of the argument of $g(z)$:
**On the lower part of the boundary, $y=0$, and
$g(x+i0)=\left[\sin(x)-x^2\right]$. For $x\ge 2\pi$, this is a negative real number, so there is no change in argument as $z$ moves along the lower edge.
**On the left side of the boundary, $x=2\pi$, and
$g(2\pi+iy)=\left[-4\pi^2+y^2\right]+i \left[\sinh(y)-4\pi y\right]$.
As $y$ moves down from (the large) $Y$ to zero, the real part decreases from positive to negative, and the imaginary part also goes from positive to negative to zero. (Note: you can tell that it is still negative near zero by taking a limit — use Taylor series.) So this tells us that the argument goes from the first to the third quadrant and ends up at the negative real axis, without looping around too much. Without analyzing further, we have no idea if it $g(z)$ goes above or below the origin, so we don’t know if the argument is increasing or decreasing. Either way, we know the argument increases or decreases by at most $\frac{3\pi}{2}$.
**On the right side of the boundary, $x=2N\pi$, and
$g(2N\pi+iy)=\left[-4N^2\pi^2+y^2\right]+i \left[\sinh(y)-4N\pi y\right]$.
As $y$ moves up from $0$ to (the large) $Y$, the real part goes from negative to positive, and the imaginary part starts at zero, gets negative, then eventually goes positive (since $\sinh(y)$ increases exponentially). So in this case, we move from negative real axis to the first quadrant. Again we can conclude that the argument has changed by no more than $\frac{3\pi}{2}$ (plus or minus).
**On the top part of the boundary, $y=Y$, a large number, and
$g(x+iY)=\left[\sin(x)\cosh(Y)-x^2+Y^2\right]+i \left[\cos(x)\sinh(Y)-2xY\right]$.
As $x$ decreases from $2N\pi$ to $2\pi$, because of the largeness of $Y$, the graph looks like a small perturbation of the graph of $\left[\sin(x)\cosh(Y)\right]+i \left[\cos(x)\sinh(Y)\right]$, which is an ellipse, so we see that the $g(z)$ goes around the origin counterclockwise (notice $\sin$ and $\cos$ switched from usual roles) $(N-1)$ times.
Thus, the total change in argument $\Delta \theta$ of $g(z)$ as $z$ goes counterclockwise around the boundary of $R$ is bounded by $2\pi(N-1)-3\pi\le \Delta\theta \le 2\pi(N-1)+3\pi$, and since
$\Delta\theta$ must be an integral multiple of $2\pi$, we must have that the argument of $g(z)$ makes between $(N-2)$ and $N$ revolutions counterclockwise around the origin. By the argument principle, there must be at least $(N-2)$ zeros inside $R$.
One minor point: how large do we pick $Y$? As long as we choose it so that $\cosh(Y)$ is way bigger than $(2N\pi)^2+Y^2$ and $\sinh(Y)$ is way bigger than $4\pi N Y$, we are fine. And we can definitely pick such a $Y$, since $\sinh(Y)$ and $\cosh(Y)$ increase exponentially in $Y$.
QED
Categories: Complex questions
## Proof of CIFD
June 5, 2011 1 comment
Dear all,
Ken asked me about the proof of the Cauchy Integral Formula for Derivatives that uses the real analysis fact of differentiating under the integral sign.
Statement: If $f$ is holomorphic on a simply connected domain $D$ and $\alpha$ is a counterclockwise oriented Jordan rectifiable curve in $D$, then for every $z$ inside the interior of $\alpha$, we have
$f^{(n)}(z)=\frac{n!}{2\pi i} \int_{\alpha} \frac{f(w)}{(w-z)^{n+1}} dw$.
Proof: For fixed positive integers $k$ and $z$ in the interior of $\alpha$, the differential form $\frac{f(w)}{(w-z)^{k}} dw$ has bounded and smooth complex coefficients on $\alpha$, and likewise all of its derivatives wrt $z$ are smooth and bounded. In particular, the functions mentioned are absolutely integrable over $\alpha$. Thus, by the theorem on differentiating under the integral sign, any partial derivative of the expression
$G(x,y)=\int_{\alpha} \frac{f(w)}{(w-(x+iy))^{k}} dw$.
may be computed by differentiating under the integral sign. For instance,
$\frac{1}{2}\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right) G(x,y)=\frac{1}{2}\int_{\alpha} \left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right)\frac{f(w)}{(w-(x+iy))^{k}} dw$.
Therefore, by the Cauchy integral formula
$f(z)=\frac{1}{2\pi i} \int_{\alpha} \frac{f(w)}{(w-z)} dw$,
$f^{(n)}(z)=\frac{1}{2\pi i} \int_{\alpha} \frac{d^n}{dz^n}\frac{f(w)}{(w-z)} dw$
$=\frac{n!}{2\pi i} \int_{\alpha} \frac{f(w)}{(w-z)^{n+1}} dw$.
QED
Categories: Complex questions | 2018-03-17 14:42:36 | {"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": 0, "img_math": 180, "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": 0.9722459316253662, "perplexity": 308.0430138622749}, "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-2018-13/segments/1521257645177.12/warc/CC-MAIN-20180317135816-20180317155816-00208.warc.gz"} |
https://math.stackexchange.com/questions/1893578/can-the-radius-of-analyticity-of-a-smooth-real-function-be-smaller-than-the-ra | # Can the “radius of analyticity” of a smooth real function be smaller than the radius of convergence of its Taylor series without being zero?
Does there exist an infinitely differentiable function $f:U\to\mathbb{R}$, where $U$ is open subset of $\mathbb{R}$, such that
1. the Taylor series of $f$ at $x=x_0\in U$ has radius of convergence $R>0$
2. $f$ equals its Taylor series only on the subinterval $(x_0-r,x_0+r)$, where $\color{red}{0<}r<R$
The customary examples of smooth real functions that fail to be analytic, e.g. $e^{-1/x}$ or $e^{-1/x^2}$ at $x=0$, have $R=\infty$ but $r=0$. The substance of the question is whether we can find a less extreme example for which analyticity at $x=x_0$ gives out only at a nonzero radius smaller than the radius of convergence of the Taylor series.
Note: I don't really know complex analysis, but I know that the easiest path to whatever the truth is here is probably through the complex domain.
• There is a $C^\infty$ version of Urysohn lemma. I think it satisfies your restriction. – Cave Johnson Aug 16 '16 at 1:25
• @CaveJohnson: yes, you're right: those functions are good examples. You should add that as an answer. – symplectomorphic Aug 16 '16 at 2:05
$\newcommand{\Reals}{\mathbf{R}}$Yes: Fix $r > 0$. The function $f:\Reals \to \Reals$ defined by $$f(x) = \begin{cases} 0 & |x| \leq r, \\ e^{-1/(|x| - r)^{2}} & |x| > r, \end{cases}$$ has Taylor series equal to $0$ (radius $\infty$), but agrees with its Taylor series only on $(-r, r)$.
If you want finite radius instead, add your favorite analytic function with radius $R > r$, e.g., $$g(x) = \frac{1}{x^{2} + R^{2}}.$$ | 2019-06-18 18:48:01 | {"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": 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": 0.9703709483146667, "perplexity": 198.86890570991048}, "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/1560627998813.71/warc/CC-MAIN-20190618183446-20190618205446-00506.warc.gz"} |
http://physics.stackexchange.com/questions/37731/refraction-reflection-and-what-is-total-reflection/37753 | # Refraction, reflection, and what is total reflection?
So if light travels from one media to another with a different refraction index, what may happen happen? Refraction, reflection or total reflection? I am quite confused as to the differences between these three concepts. Does 1 of these always happen in this situation? What exactly is total reflection and how/why does it occur?
-
please no wikipedia links, too confusing! – Johny Kilkocx Sep 19 '12 at 6:09
## 2 Answers
Leaving aside total internal reflection for now, whenever light crosses a refractive index boundary there is always some reflection and some refraction.
Refraction is easily understood. In an earlier question you established that for any refractive index > 1 the speed of light in the medium $v$ is slower than $c$. However the frequency $f$ of the light remains the same. Since the wavelength $\lambda$ of the light is given by $\lambda = v/f$ this means the wavelength of the light decreases.
As this diagram (shamelessly cribbed by Googling) shows, if the wavelength decreases the direction of the light ray has to change. The change of direction is called refraction.
For reflection, I can't think of a simple diagrammatic way to show why there is always reflection. When we're calculating the reflection we use the principle that the electric field of the light must be continuous at the boundary. If you add up the field on the two sides of the bounday you find that whenever there is refraction there is always some reflection to balance it out.
Total internal reflection only occurs when light is passing from a higher refractive index medium to a lower refractive index medium. The easiest way to see why it happens is described below.
In the diagram above Snell's law tells us that:
$$\sin\theta_{air} = \frac{n_{water}}{n_{air}} \space \sin\theta_{water}$$
As light passes from the water into the air the light ray is always bent away from the normal and towards the surface of the water. The problem is that there is a value of $\theta_{water}$ at which the light is bent right round until it's parallel with the surface of the water i.e. $\sin\theta_{air} = 1$. This happens when:
$$\frac{n_{water}}{n_{air}} \space \sin\theta_{water} > 1$$
For values of $\theta_{water}$ greater than this critical value no light can escape from the water so all the light is reflected back into the water. This is the total internal reflection.
-
Appreciate it, makes a lot more sense now. Thank you! – Johny Kilkocx Sep 19 '12 at 8:12
1) Reflection: When a beam of light strikes the surface of a transparent medium like glass or water which has a large number of closely spaced atoms, some of the light photons get scattered in backward direction from the interface (which separates air medium from water) which we guys call it as Reflection. Of course, most of the objects reflect visible light. The portion of reflected light waves vary accordingly to different interfaces. We could conclude that, "Without Reflection, we would be able to see nothing".
According to the laws of laws of reflection:
• The angle of incidence equals angle of reflection ($\theta_i=\theta_r$ or simply $i=r$). The incident ray, reflected ray and normal drawn to the surface at the point of incidence - all lie in the same plane.
2) Refraction: Waves often travel from one medium to another. When they change medium, they often change directions. Light travels slowly in glass and other transparent objects than in air. This change of direction and their slow-motion is determined by the refractive index $\mu$ of materials.
$$\mu=\frac{C_a}{C_m}=\frac{\lambda_a}{\lambda_m}$$
For air, $\mu=1$ and for any other material other than air, $\mu>1$. It shouldn't be less than one. If it was, It'd have strongly disproved the Theory of Relativity. It should also be noted that the ray of light is deflected towards the normal when it passes from rarer medium to denser medium and viceversa. When you see a glass tank (filled with water) from the top, the bottom surface of the tank appears closer than it appears from either sides. Then, you should think of refraction..!
According to the laws of refraction:
• The incident ray, reflected ray and normal drawn to the surface at the point of incidence - all lie in the same plane. By Snell's law, ($i$ = angle of incidence, $r$ = angle of refraction) $$\mu=\frac{sin\theta_i}{sin\theta_r}$$
Let AB be the ray of light incident at an angle $i$ at a refracting medium XY of refractive index $\mu$ and refracted at an angle $r$. NN' be the normal drawn to the surface.
(Rarer and Denser medium: Denser medium has the refractive index greater than that of the rarer medium. Denser medium should be any media other than air or vacuum)
3) Total Internal Reflection: When a ray of light AO passes from denser medium to rarer medium, light is partly reflected along OB and is refracted at $r$ along OC. As you can see, there are three cases... When the angle of incidence $i$ is increased, $r$ is also increased (As it passes from denser to rarer medium, light is deflected away from the normal NN'). As $i$ is further increased to some higher angle, $r$ becomes $90^o$ and the light ray grazes over the interface which separates the two media. This angle is called the critical angle (say $c$ or $\theta_c$). As the incident angle is increased above critical angle (i.e) $i>c$, $$r>90^o\implies sin\theta_r>1$$ which is impossible or no refraction is possible. Thus, the ray is reflected into the denser medium itself. This is called Total internal reflection.
According to wave theories, for Total internal reflection to happen:
• Light must travel from denser medium to rarer medium.
• Angle of incidence should be greater than critical angle (i.e) $i>c$.
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https://saecanet.com/datatables/ciafactbook_Monaco.html | ## Table
Monaco. Source:CIA The World Factbook
N Item Value Unit Rank Date of Information Remark
1 Area 2.00 (sq km) 255 Area compares the sum of all land and water areas delimited by international boundaries and/or coastlines.
2 Population 30,727 Population 217 July 2018 est. Population compares estimates from the US Bureau of the Census based on statistics from population censuses, vital statistics registration systems, or sample surveys pertaining to the recent past and on assumptions about future trends.
3 Median age 53.8 (years) 1
4 Population growth rate 0.3 (%) 170 2018 est. Population growth rate compares the average annual percent change in populations, resulting from a surplus (or deficit) of births over deaths and the balance of migrants entering and leaving a country. The rate may be positive or negative.
5 Birth rate 6.5 (births/1,000 population) 226 2018 est. Birth rate compares the average annual number of births during a year per 1,000 persons in the population at midyear; also known as crude birth rate.
6 Death rate 10.1 (deaths/1,000 population) 36 2018 est. Death rate compares the average annual number of deaths during a year per 1,000 population at midyear; also known as crude death rate.
7 Net migration rate 5.7 (migrant(s)/1,000 population) 20 2017 est. Net migration rate compares the difference between the number of persons entering and leaving a country during the year per 1,000 persons (based on midyear population).
8 Infant mortality rate 1.8 (deaths/1,000 live births) 223 2018 est. Infant mortality rate compares the number of deaths of infants under one year old in a given year per 1,000 live births in the same year. This rate is often used as an indicator of the level of health in a country.
9 Life expectancy at birth 89.4 (years) 1 2018 est. Life expectancy at birth compares the average number of years to be lived by a group of people born in the same year, if mortality at each age remains constant in the future. Life expectancy at birth is also a measure of overall quality of life in a country and summarizes the mortality at all ages.
10 Total fertility rate 1.54 (children born/woman) 191 2018 est. Total fertility rate (TFR) compares figures for the average number of children that would be born per woman if all women lived to the end of their childbearing years and bore children according to a given fertility rate at each age. TFR is a more direct measure of the level of fertility than the crude birth rate, since it refers to births per woman.
11 Health expenditures 4.3 (% of GDP) 160 2014 Health expenditures provides the total expenditure on health as a percentage of GDP. Health expenditures are broadly defined as activities performed either by institutions or individuals through the application of medical, paramedical, and/or nursing knowledge and technology, the primary purpose of which is to promote, restore, or maintain health.
12 Education expenditures 1.4 (% of GDP) 173 2016 Education expenditures compares the public expenditure on education as a percent of GDP.
13 Unemployment, youth ages 15-24 26.6 (%) 42 2016 est. Unemployment, youth ages 15-24 gives the percent of the total labor force ages 15-24 unemployed during a specified year.
14 GDP (purchasing power parity) $7,672,000,000 GDP (purchasing power parity) 165 2015 est. GDP (purchasing power parity) compares the gross domestic product (GDP) or value of all final goods and services produced within a nation in a given year. A nation’s GDP at purchasing power parity (PPP) exchange rates is the sum value of all goods and services produced in the country valued at prices prevailing in the United States. 15 GDP - real growth rate 5.4 (%) 40 2015 est. GDP - real growth rate compares GDP growth on an annual basis adjusted for inflation and expressed as a percent. 16 GDP - per capita (PPP)$115,700 GDP - per capita (PPP) 3 2015 est. GDP - per capita (PPP) compares GDP on a purchasing power parity basis divided by population as of 1 July for the same year.
17 Industrial production growth rate 6.8 (%) 33 2015 Industrial production growth rate compares the annual percentage increase in industrial production (includes manufacturing, mining, and construction).
18 Labor force 52,000 Labor force 190 2014 est. Labor force compares the total labor force figure.
19 Unemployment rate 2 (%) 19 2012 Unemployement rate compares the percent of the labor force that is without jobs.
20 Taxes and other revenues 14.9 (% (of GDP)) 196 2011 est. Taxes and other revenues records total taxes and other revenues received by the national government during the time period indicated, expressed as a percent of GDP. Taxes include personal and corporate income taxes, value added taxes, excise taxes, and tariffs. Other revenues include social contributions - such as payments for social security and hospital insurance - grants, and net revenues from public enterprises.
21 Budget surplus (+) or deficit (-) -1 (% (of GDP)) 80 2011 est. Budget surplus or deficit records the difference between national government revenues and expenditures, expressed as a percent of GDP. A positive (+) number indicates that revenues exceeded expenditures (a budget surplus), while a negative (-) number indicates the reverse (a budget deficit).
22 Inflation rate (consumer prices) 1.50 (%) 84 2010 Inflation rate (consumer prices) compares the annual percent change in consumer prices with the previous year’s consumer prices.
23 Exports $964,600,000 Exports 162 2017 est. Exports compares the total US dollar amount of merchandise exports on an f.o.b. (free on board) basis. These figures are calculated on an exchange rate basis. 24 Imports$1,371,000,000 Imports 175 2017 est. Imports compares the total US dollar amount of merchandise imports on a c.i.f. (cost, insurance, and freight) or f.o.b. (free on board) basis. These figures are calculated on an exchange rate basis.
25 Telephones - fixed lines 47,013 Telephones - fixed lines 159 2017 est. Telephones - main lines in use compares the total number of main telephone lines in use.
26 Telephones - mobile cellular 32,978 Telephones - mobile cellular 206 2017 est. Telephones - mobile cellular compares the total number of mobile cellular telephone subscribers.
27 Internet users 29,116 Internet users 203 July 2016 est. Internet users compares the number of users within a country that access the Internet. Statistics vary from country to country and may include users who access the Internet at least several times a week to those who access it only once within a period of several months.
28 Broadband - fixed subscriptions 19,258 Broadband - fixed subscriptions 152 2017 est. | 2019-04-23 18:02:05 | {"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": 0.28449109196662903, "perplexity": 6308.828400281113}, "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-2019-18/segments/1555578610036.72/warc/CC-MAIN-20190423174820-20190423200820-00077.warc.gz"} |
https://imathworks.com/tex/tex-latex-include-data-from-libreoffice-calc-or-ms-excel/ | # [Tex/LaTex] Include data from LibreOffice Calc or MS Excel
exceltables
I'd like to use LibreOffice Calc or MS Excel to create and edit spreadsheets and include these in LaTeX (LuaLaTeX would be fine, too). It is necessary for me to get calculated values like SUM(C2:C5) as well.
So far I've tried odsfile and exceltex. Where odsfile didn't work at all, exceltex did work but ignored calculated values.
Has anyone successfully tried to include spreadsheet data with calculated values?
Update
This question is not the same as "I'd like a easy interface for LaTeX-Tables", as I look for a more comprehensive solution. We use spreadsheets for modeling our problems. When we refine our models, I'd like to update the TeX-based reports without exporting/importing many tables.
That said, odsfile works pretty much as I wanted to.
Thanks for your comments and suggestiongs so far. Do you know any other solutions like odsfile?
In excel use the concatenation command to translate the cells to have the & inserted to the left at a mirror of the columns somewhere to the right of your columns. | 2023-02-04 05:33:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.6198059320449829, "perplexity": 1927.9945369375857}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500094.26/warc/CC-MAIN-20230204044030-20230204074030-00203.warc.gz"} |
https://labs.tib.eu/arxiv/?author=A.%20Voronin | • ### First results of the cosmic ray NUCLEON experiment(1702.02352)
July 2, 2018 astro-ph.IM, astro-ph.HE
The NUCLEON experiment was designed to study the chemical composition and energy spectra of galactic cosmic ray nuclei from protons to zinc at energies of $\sim10^{11}$--$10^{15}$\,eV per particle. The research was carried out with the NUCLEON scientific equipment installed on the Russian satellite "Resurs-P" No.\,2 as an additional payload. This article presents the results for the measured nuclei spectra related to the first approximately 250 days of the scientific data collection during 2015 and 2016. The all-particle spectrum and the spectra of p, He, C, O, Ne, Mg, Si and Fe are presented. Some interesting ratios of the spectra are also presented and discussed. The experiment is now in its beginning stage and the data still have a preliminary character, but they already give numerous indications of the existence of various non-canonical phenomena in the physics of cosmic rays, which are expressed in the violation of a simple universal power law of the energy spectra. These features of the the data are briefly discussed.
• ### Design flow for readout ASICs in High-energy Physics experiments(1803.07524)
March 20, 2018 hep-ex, nucl-ex, physics.ins-det
In the large-scale high energy physics experiments multi-channel readout application specific integrated circuits (ASICs) are widely used. The ASICs for such experiments are complicated systems, which usually include both analog and digital building blocks. The complexity and large number of channels in such ASICs require the proper methodological approach to their design. The paper represents the mixed-signal design flow of the ASICs for high energy physics. This flow was successfully implemented in the development of the readout ASIC prototypes for the muon chambers of the CBM experiment. The approach was approved in UMC CMOS MMRF 180 nm process. The design flow enables to analyze the mixed-signal system operation on the different levels: functional, behavioral, schematic and post-layout including parasitic elements. The proposed design flow allows reducing the simulation period and eliminating the functionality mismatches on the very early stage of the design.
• ### Charm production nearby threshold in pA-interactions at 70 GeV(1703.05639)
March 16, 2017 nucl-ex
The results of the SERP-E-184 experiment at the U-70 accelerator (IHEP, Protvino) are presented. Interactions of the 70 GeV proton beam with C, Si and Pb targets were studied to detect decays of charmed $D^0$, $\overline D^0$, $D^+$, $D^-$ mesons and $\Lambda _c^+$ baryon near their production threshold. Measurements of lifetimes and masses are shown a good agreement with PDG data. The inclusive cross sections of charm production and their A-dependencies were obtained. The yields of these particles are compared with the theoretical predictions and the data of other experiments. The measured cross section of the total open charm production ($\sigma _{\mathrm {tot}}(c\overline c)$ = 7.1 $\pm$ 2.3(stat) $\pm$1.4(syst) $\mu$b/nucleon) at the collision c.m. energy $\sqrt {s}$ = 11.8 GeV is well above the QCD model predictions. The contributions of different species of charmed particles to the total cross section of the open charm production in proton-nucleus interactions vary with energy.
• ### Detection of $D^{\pm}$ mesons production in pA-interactions at 70 GeV(1311.1960)
Nov. 8, 2013 hep-ex, nucl-ex
The results of analysis SERP-E-184 experiment data, obtained with 70 GeV proton beam irradiation of active target with carbon, silicon and lead plates are presented. For 3-prongs charged charmed mesons decays, event selection criteria were developed and detection efficiency was calculated with detailed simulation using FRITIOF7.02 and GEANT3.21 programs. Signals of decays were found and charm production inclusive cross sections estimated at near threshold energy. The lifetimes and A-dependence of cross section were measured. Yields of D mesons and their ratios in comparison with data of other experiments and theoretical predictions are presented.
• ### Study of the interactions of pions in the CALICE silicon-tungsten calorimeter prototype(1004.4996)
April 28, 2010 hep-ex, physics.ins-det
A prototype silicon-tungsten electromagnetic calorimeter for an ILC detector was tested in 2007 at the CERN SPS test beam. Data were collected with electron and hadron beams in the energy range 8 to 80 GeV. The analysis described here focuses on the interactions of pions in the calorimeter. One of the main objectives of the CALICE program is to validate the Monte Carlo tools available for the design of a full-sized detector. The interactions of pions in the Si-W calorimeter are therefore confronted with the predictions of various physical models implemented in the GEANT4 simulation framework.
• ### Registration of neutral charmed mesons production and their decays in pA-interactions at 70 GeV with SVD-2 setup(1004.3676)
April 21, 2010 hep-ex
The results of data handling for E-184 experiment obtained with 70 GeV proton beam irradiation of active target with carbon, silicon and lead plates are presented. Two-prongs neutral charmed D0 and \v{D}0 -mesons decays were selected. Signal / background ratio was (51+/-17) / (38+/-13). Registration efficiency for mesons was defined and evaluation for charm production cross section at threshold energy is presented: sigma(c\^c) = 7.1 +/- 2.4(stat.) +/- 1.4(syst.) (\mu/nucleon). | 2020-04-05 14:06:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6032782793045044, "perplexity": 2783.152440891646}, "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/1585371604800.52/warc/CC-MAIN-20200405115129-20200405145629-00043.warc.gz"} |
https://www.transtutors.com/questions/quality-cabinet-company-uses-a-standard-costing-system-and-produced-1-950-cabinets-d-2567056.htm | # Quality Cabinet Company uses a standard costing system and produced 1,950 cabinets during May. Th...
Quality Cabinet Company uses a standard costing system and produced 1,950 cabinets during May. The standard cost of wood is $22 per linear foot, and the standard quantity for each cabinet is 27 linear feet. During May, the company purchased 55,850 linear feet of wood for$1,172,850, and 53,850 feet were used in production. The company purchases all materials on account.
Determine the material price variance and the material quantity variance. (Enter all variances as a positive number.) Material Price Variance Material Quantity Variance s SHOW LIST OF ACCOUNTS LINK TO TEXT Record the related journal entries for May. (Credit account titles are automatically indented when the amount is entered. Do not indent manually.) No. Aocount Titles and Explanation Debit Credit (To record material purchases) (To record material used in production)
## Recent Questions in Accounting - Others
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Attach Files | 2018-08-21 22:08:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.185655415058136, "perplexity": 13599.64847835955}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221219109.94/warc/CC-MAIN-20180821210655-20180821230655-00717.warc.gz"} |
https://www.khanacademy.org/economics-finance-domain/core-finance/money-and-banking/geithner-plan/v/geithner-plan-iii?playlist=Geithner%20Plan | Let's try to construct a situation where I participate in the new Treasury Public Private Program. And I get all of the upside I would have gotten in a traditional just straight up investment in these toxic securities. But then I'm able to offload some of the downside. And so just as a review, what the program entails is that if I want to make an investment in something. Let's say I want to make a $100 or 100% investment. It doesn't matter. Either way. That's 100. The Fed is going to give us a loan for 85% of it. And then the remainder, the equity, is going to be split half, or 50-50, between myself and the Treasury. And then this is the private investor. So what happens here? And we talked a lot about it in the last couple of videos. Is that the upside, if this thing is worth more than 100, it's split between the Treasury and the private investor. And then the downside is split between the private investor and the Treasury. Until the asset becomes worth less than 85% of its original value. And then the Fed takes the hit, right? Because this loan won't be worth anything. And all this loan can go after is this asset. It's a non recourse loan. The Fed can't say, oh, this thing is only worth 84. Private investor and Treasury, you guys owe me a dollar. The Fed is just left with this asset that might be worth something less than that. But how could we use this program to get the same upside we would have otherwise gotten, if we had just bought the toxic security outright? So let's say that the toxic security, in a before-this program world that Geithner has come up with. Let's say that I'm willing to pay$30 for a security. And obviously if the security is worth $40, I'll make$10. If it's worth $50, I'll make$20. And so forth and so on. That's obvious. Now what I can do is, instead of just, with this plan-- I mean, before I would have to take my $30 that I have in the bank, and use it to buy some asset, if one of these banks were willing to sell it to me. But what I can do now is, I could say, well, let me just take 15% of this. So what's 15% of$30? It's $4.50. And use that to invest in this program. And essentially get the same upside as I would if I invested the whole$30. And then put the rest of the cash away essentially safe, so that my worst case is that I just lose what I invest. So let's do that. And we'll see that the numbers do all work out. So instead of just doing $30 all for the investment, I'm going to set aside 15%. So I'm going to put$4.50 to participate in the plan. And then I'm going to set aside what's left. $25.50 in cash. And using this$4.50 in the plan, I'll show you that I'll get all of the upside that I got if I had originally had to use all $30 by myself. And obviously my downside is limited. My downside is limited to the$4.50, because I'm still going to have the $25.50 in cash on the side. So how is this going to work? I'm going to put in$4.50. That's from me, the private investor. The Treasury is going to match my investment. $4.50. That's the Treasury. Now I have$9 of equity. And this is going to be 15% of the total equity, so the debt that the Fed will lend me is going to be $51. And I did the math before I did the video, but you could work it out yourself.$51 loan from the Fed. And so what's our total capital that we've raised now? So if you add $51 plus$9, we have $60. Let me do this in a different color. We now have$60. And we could use that $60. If someone was willing to sell me one, I guess we could say one of the units, or one of the securities for$30, now with $60, we can buy two of the units. So we could buy two of them for$30. And why did I end up making the math work so we'd get two securities? Because we split the upside, right? We split the upside with the Treasury. So in order to set up a situation where I get all of the upside that I would have gotten with a $30 security, I essentially have to set up a configuration where we buy twice as many securities. I buy twice as many securities, and then I split the upside 50-50 with the Treasury, then I'm just going to get the upside. You could almost view it on one security. Let's see how the math works out now. Let's do a bunch of scenarios. So in the world where-- maybe I should do a table. Yeah, let's say that the eventual value of the security, let's say it's worth-- I'm going to make a bunch of numbers up. 0, 10, 20, 30, 40, 50, 60. That's good enough. So let's say that these securities end up being worth nothing. How much am I left with? What's the total value of my total investment? Well if these are worth nothing, then I get wiped out here. We all get wiped out here. In fact, everyone gets wiped out here. So how much did I lose? Well my$4.50 is gone. This goes to 0. But what is my total investment left with? I still have my $25.50 that I put to the side, right? So I still have my$25.50 in cash. If each of these securities are worth $10, this is eventual value of each of the units. So if each of these is worth$10, then the whole asset side of this balance sheet is worth $20, and once again, the Fed is going to take a$31 hit. And these two guys are wiped out. So my investment is worth nothing. But I still have that money I set aside in the bank. $25.50. If these securities are worth$20, then this whole asset side is worth $40, because there's going to be two of them. The Fed is still taking a hit. Doesn't get all its money back. And these two guys still get wiped out. My$4.50 is gone. I still have my money in my bank. $25.50. If this security is worth$30, which is essentially the price that we're paying for it. Then this is what the balance sheet is going to look like. So my equity is worth $4.50. But I have$4.50 plus the money that I had set aside, right? I have $4.50 plus this$25.50 that I'd put aside in the bank. Because I didn't have to invest that. I just had to invest the $4.50 initially. So I have my$4.50 here, plus $25.50. So the value of all of my investments is$30. What Happens if each of these units are worth $40. If each of these are worth$40, then we have $80 on the left-hand side of the balance sheet. So let me draw that really small here. So then we'll have two units that are worth$40. That's $80. We owe$51 to the Fed. So we're going to have $29 to split between the Treasury and ourselves.$29 divided by 2 is $14.50. We're each going to split that$29. So my equity is now worth $14.50. So my total investment is going to be the cash that I put aside-- so that's, let's say,$25.50. Plus the value of my equity investment now. So that's plus $14.50. And so it's worth$40. I think you see the pattern here. In all of these kind of downside scenarios, the most I lost was $4.50. And now in all of these upside scenarios, we're going to see that I get the value of the actual security. Now let's do it for$50, but I think you know how the math's going to work out. If each of these ends up being worth $50, then this whole left-hand side is$100. We owe the Fed $51. So this is going to be$49. And if you split that, it's $24.50. The money we set aside in the bank is$25.50. And so this is worth $50. And, likewise, if these are each worth$60, this is going to turn out to be $60. So what you see here, and I'm going to draw a payoff diagram. Let me draw a line here. Let's see if I can do this right. So if this is the eventual value of the asset. When we pay something in the market, we're hoping we're buying it for less than its real value, but we really don't know 100% what the eventual value is. But this x-axis is the eventual value. So it could be 10, 20, 30, 40, 50, 60, and so on and so forth. So depending on this eventual value, what is the value of my investment? So we figured out if it's 0, 10, 20, all the way really up to-- The value of my investment is going to be-- I didn't have to draw this stuff down here-- let's say that this is$24.50. So in all of these scenarios-- whoops, trying to draw a line-- so 20, and then at $25.50. If this thing is worth-- We could figure out the break-even, right? If these assets are worth anything more combined than$51, then we start to have value to the equity. So what's that value? That's this divided by 2 is $25.50. So after$25.50 some value starts going to the equity. And we know at $30-- let me draw those points, actually. I'll do it in a different color when we're in the money. So at$30, the value of our security is 30. At $40, the value of our security is 40. And so on and so forth. And let me draw a line now. That's not what I wanted to do. I'll just draw a regular line. Our payoff would look something like this. This is a good visualization for what the government is giving us. And I'll show you that what they're giving us is essentially insurance, or a put option for free. And then we can talk in the next video about how we can value it. And then we have a real sense of what the government is handing over to the investor. Now, you might say that it's handing it to the investor, but if because of this the investor overpays for the asset, that subsidy is really going to the person selling the asset. And we'll talk about that a little bit more in the next video. But let me just finish this up a little bit. So we said that the point at which we start making money is here. Because this is what we paid for the security. We're paying$30 for the security. So these are all the scenarios where we start making money. Where the security's worth more than that. But the equity's actually worth something above \$25.50, but we're still taking a slight loss. But the cool thing is, if we are the private investor here, even if the thing is worth very little, we don't take that much of a loss here. And this is much better. What would the payoff diagram have looked like if we were just a straight-up investor? Let me just draw that in a different color. Well then it would've looked something like this. We would have had the same upside. But our downside would've gone all the way to 0. If the asset is worth 0, we would have been worth-- well, these lines were supposed to be on top of each other. But I think you get the idea that the government is saving us from these losses right here. Oh, I think you can't see it. I'm off the screen. The government's saving us from these losses right there. And so I'm out of time in this video. In the next video, I'm going to actually analyze what that subsidy or what that insurance or that option that the government is giving us, what it might be worth. And then we could think about whether now people are willing to actually overpay for these things. See you in the next video. | 2017-05-28 12:39:14 | {"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": 0.6296389698982239, "perplexity": 680.1745953787672}, "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/1495463609817.29/warc/CC-MAIN-20170528120617-20170528140617-00480.warc.gz"} |
http://mymathforum.com/elementary-math/347131-proof.html | My Math Forum proof
Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion
September 28th, 2019, 02:54 AM #1 Newbie Joined: Sep 2019 From: ireland Posts: 3 Thanks: 0 proof abc and bca are 3 digit numbers.show that abc-cba is divided by 11
September 28th, 2019, 04:51 AM #2 Senior Member Joined: Jun 2014 From: USA Posts: 620 Thanks: 52 If I let a=b=c, I seem to be getting a funny result.
September 28th, 2019, 05:06 AM #3 Senior Member Joined: Jun 2014 From: USA Posts: 620 Thanks: 52 321 - 213 = 108 where 108 is not divisible by 11. Thanks from topsquark
September 28th, 2019, 09:42 AM #4 Math Team Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra But 321 - 123 = 198 which is divisble by 11. \begin{align} (100a + 10b + c) - (100c + 10b + a) &= 99a - 99c \\ &= 11( 9a-9c) \end{align} I have no idea what bca has to do with the price of fish. Thanks from idontknow Last edited by v8archie; September 28th, 2019 at 09:46 AM.
September 28th, 2019, 01:20 PM #5 Senior Member Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 132 Thanks: 49 Math Focus: Area of Circle it is abc-cba Thanks from idontknow
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Contact - Home - Forums - Cryptocurrency Forum - Top | 2019-10-21 23:00:22 | {"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": 0.9564175009727478, "perplexity": 12006.989361872646}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987795253.70/warc/CC-MAIN-20191021221245-20191022004745-00483.warc.gz"} |
https://www.aimsciences.org/article/doi/10.3934/dcds.2010.28.161 | Article Contents
Article Contents
# The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations
• The Kneser theorem for ordinary differential equations without uniqueness says that the attainability set is compact and connected at each instant of time. We establish corresponding results for the attainability set of weak solutions for the 3D Navier-Stokes equations satisfying an energy inequality. First, we present a simplified proof of our earlier result with respect to the weak topology in the space $H$. Then we prove that this result also holds with respect to the strong topology on $H$ provided that the weak solutions satisfying the weak version of the energy inequality are continuous. Finally, using these results, we show the connectedness of the global attractor of a family of setvalued semiflows generated by the weak solutions of the NSE satisfying suitable properties.
Mathematics Subject Classification: 35B40, 35B41, 35Q30, 37B25, 58C06.
Citation: | 2022-12-10 05:31:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.5814544558525085, "perplexity": 172.50796028823845}, "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-2022-49/segments/1669446711712.26/warc/CC-MAIN-20221210042021-20221210072021-00452.warc.gz"} |
https://www.ias.ac.in/listing/bibliography/pram/A_K_DWIVEDI | • A K DWIVEDI
Articles written in Pramana – Journal of Physics
• Even–odd effect of the homologous series of nCHBT liquid crystal molecules under the influence of an electric field: A theoretical approach
In this work, we present the effect of electric field on 4-(trans-4$^\prime$-n-alkyl-cyclohexyl) isothiocyanatebenzene(nCHBT) liquid crystal (LC) molecules. Under the influence of an electric field, the birefringence exhibits the even–odd effect while order parameter, HOMO–LUMO gap, magic angle, isotropic polarisability, range of director angle and the refractive index do not exhibit any even–odd effect. The extension of the alkyl chain length of the nCHBT liquid crystal molecule exhibits the even–odd effect for the dipole moment and temperature from nematic to isotropic phase transition while the HOMO–LUMO gap remains constant. Still, order parameters, isotropic polarisability and refractive index have continuously increased. The odd carbon atom numbers present higher values than the even carbon atom numbers of the alkyl chain for the phase transition temperature. The nCHBTLC molecule expresses the order parameter, and birefringence is reciprocal to each other. For the whole series, there is an increase in order parameter, and a decrease in birefringence. The influence of the external electric field is analternative to the temperature for the optical parameter of nCHBT LC.
• # Pramana – Journal of Physics
Volume 95, 2021
All articles
Continuous Article Publishing mode
• # Editorial Note on Continuous Article Publication
Posted on July 25, 2019 | 2021-09-27 16:04:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5566350221633911, "perplexity": 2043.6083944645861}, "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-2021-39/segments/1631780058456.86/warc/CC-MAIN-20210927151238-20210927181238-00145.warc.gz"} |
https://es.mathworks.com/help/fusion/ref/insmotionpose.html | # insMotionPose
Model for 3-D motion estimation
## Description
The insMotionPose object models 3-D motion assuming constant angular velocity and constant linear acceleration. Passing an insMotionPose object to an insEKF object enables the estimation of 3-D motion, including orientation, angular velocity, position, linear velocity, and linear acceleration. For details on the motion model, see Algorithms.
## Creation
### Description
example
model = insMotionPose creates an insMotionPose object. Passing model to an insEKF object enables the estimation of:
• The orientation quaternion from the navigation frame to the body frame.
• The angular velocity of the platform, expressed in the body frame.
• The position of the platform, expressed in the navigation frame.
• The velocity of the platform, expressed in the navigation frame.
• The acceleration of the platform, expressed in the navigation frame.
## Examples
collapse all
Create an insMotionPose object and pass it to an insEKF object.
motionModel = insMotionPose
motionModel =
insMotionPose with no properties.
filter = insEKF(motionModel)
filter =
insEKF with properties:
State: [16x1 double]
StateCovariance: [16x16 double]
MotionModel: [1x1 insMotionPose]
Sensors: {}
SensorNames: {1x0 cell}
ReferenceFrame: 'NED'
Show the state maintained in the filter.
stateinfo(filter)
ans = struct with fields:
Orientation: [1 2 3 4]
AngularVelocity: [5 6 7]
Position: [8 9 10]
Velocity: [11 12 13]
Acceleration: [14 15 16]
## Algorithms
The insMotionPose object models the orientation-only motion of platforms. The state equation of the motion model is:
$\begin{array}{l}\stackrel{˙}{q}=\frac{1}{2}\omega q\\ \stackrel{˙}{\omega }=0\\ \stackrel{˙}{p}=v\\ \stackrel{˙}{v}=a\\ \stackrel{˙}{a}=0\end{array}$
where:
• q = (q0, q1, q2, q3) is the quaternion from the navigation frame to the body frame.
• ω is the angular velocity of the platform, expressed in the body frame.
• p is the position of the platform, expressed in the navigation frame.
• v is the linear velocity of the platform, expressed in the navigation frame.
• a is the linear acceleration of the platform, expressed in the navigation frame.
## Version History
Introduced in R2022a | 2022-09-30 03:32:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "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": 0.5373058915138245, "perplexity": 6477.524332938236}, "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-2022-40/segments/1664030335424.32/warc/CC-MAIN-20220930020521-20220930050521-00111.warc.gz"} |
http://www.gradesaver.com/textbooks/math/algebra/algebra-2-common-core/chapter-2-functions-equations-and-graphs-2-2-direct-variation-practice-and-problem-solving-exercises-page-72/43 | # Chapter 2 - Functions, Equations, and Graphs - 2-2 Direct Variation - Practice and Problem-Solving Exercises: 43
$0.625$
#### Work Step by Step
To find $y$ when $x=5$, set up a proportion. Since $y$ varies directly with $x$, use two forms of $\frac{y}{x}$ in the proportion. $\frac{\frac{1}{2}}{4}=\frac{y}{5}$ Now write the cross products. $\frac{1}{2}(5)=4y$ Simplify. $2.5=4y$ Divide each side by 4. $\frac{2.5}{4}=\frac{4y}{4}$ Simplify. $0.625=y$
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | 2018-04-19 18:18:32 | {"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": 0.7553510665893555, "perplexity": 1408.0829340061623}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125937015.7/warc/CC-MAIN-20180419165443-20180419185443-00081.warc.gz"} |
https://koreascience.or.kr/article/JAKO200934152712955.page | # ADAPTIVE CVT-BASED REDUCED-ORDER MODELING OF BURGERS EQUATION
• Piao, Guang-Ri (DEPARTMENT OF MATHEMATICS, YANBIAN UNIVERSITY) ;
• Du, Qiang (DEPARTMENT OF MATHEMATICS, PENNSYLVANIA STATE UNIV., UNIVERSITY PARK) ;
• Lee, Hyung-Chun (DEPARTMENT OF MATHEMATICS, AJOU UNIVERSITY)
• Received : 2009.05.12
• Accepted : 2009.05.30
• Published : 2009.06.25
#### Abstract
In this article, we consider a weighted CVT-based reduced-order modelling for Burgers equation. Brief review of the CVT (centroidal Voronoi tessellation) approaches to reduced-order bases are provided. In CVT-reduced order modelling, we start with a snapshot set just as is done in a POD (Proper Orthogonal Decomposition)-based setting. So far, the CVT was researched with uniform density ($\rho$(y) = 1) to determine the basis elements for the approximatin subspaces. Here, we shall investigate the technique of CVT with nonuniform density as a procedure to determine the basis elements for the approximating subspaces. Some numerical experiments including comparison of two CVT (CVT-uniform and CVT-nonuniform)-based algorithm with numerical results obtained from FEM(finite element method) and POD-based algorithm are reported.
#### References
1. B. O. Almroth, Automatic choice of global shape functions in structural analysis, AIAA J., 16 (1979), 525-528.
2. J. Atwell and B. King, Reduced order controllers for spatially distributed systems via proper orthogonal decomposition, SIAM J. Sci. Comput. 26 (2004), 128-151. https://doi.org/10.1137/S1064827599360091
3. A. K. Bangia, P. F. Batcho, I. G. Kevrekidis, and G. E. Karniadakis, Unsteady two-dimensional flows in complex geometries: comparative bifurcation studies with global eigen-function expansions, SIAM J. on Sci. Comput., 18 (1997), 775-805. https://doi.org/10.1137/S1064827595282246
4. S. C. Brenner and L. R. Scott, The mathematical theroy of finite element methods, Springer-Verlag, New York, 1994.
5. J. Borggaard, A. Hay, and D. Pelletier. Interval-based reduced-order models for unsteady uid ow. International Jounal of Numerical Analysis and Modeling, 4 (2007), 353367.
6. J. M. Burgers, Mathematical examples illustrating relations occuring in the theory of turbulent fluid motion, Trans. Roy. Neth. Acad. Sci. 17 (1939), Amsterdam, 1-53
7. J. M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. in Appl. Mech, 1 (1948), 171-199.
8. J. M. Burgers, Statistical problems connected with asymptotic solution of one-dimensional nonlinear diffusion equation, in M. Rosenblatt and C. van Atta (eds.), Statistical Models and Turbulence, Springer, Berlin (1972), 41.
9. J. Burkardt, Q. Du, M. Gunzburger and H.-C. Lee, Reduced Order Modeling of Complex Systems, in Proceeding of the 20th Biennial Conference on Numerical Analysis, Ed. by D F Griffiths & G AWatson, University of Dundee, June, 2003, 29-38.
10. J. A. Burns and S. Kang, A control problem for Burgers equation with bounded input/oqtput, ICASE Report 90-45, 1990, NASA Langley research Center, Hampton, VA; Nonlinear Dynamics, 2 (1991), 235-262.
11. C. T. Chen, Linear System Theory and Design, Holt, Rinehart and Winston, New York, NY, 1984.
12. Q. Du, M. Emelianenko, and L. Ju, "Convergenece of the Lolyd algorithm for computing centroidal Voronoi tessellations, SIAM J. Numer. Anal., 44 (2006),. 102-119. https://doi.org/10.1137/040617364
13. Q. Du, V. Faber, and M. Gunzburger, Centroidal Voronoi Teesellations: applications and algorithms, SIAM Review 41 (1999), 637-676. https://doi.org/10.1137/S0036144599352836
14. Q. Du and M. Gunzburger, Model reduction by proper orthogonal decomposition coupled with centroidal Voronoi tessellation, Proc. Fluids Engineering Division Summer Meeting, FEDSM2002-31051, ASME, 2002
15. Q. Du and X.Wang, Tessellation and Clustering by Mixture Models and Their Parallel Implementations, in Proceeding of the fourth SIAM international conference on Data Mining, L ake Buena Vista, FL, 2004, SIAM, 257-268.
16. J. S. Gibson, The riccati integral equations for optimal control problems on Hilbert spces, SIAM Jounal on the Control and Optimazation, bf 17 (1979), 537-565. https://doi.org/10.1137/0317039
17. J. S. Gibson, and I. G. Rosen, Shifting the closed-loop spectrum in teh optimal linear quadratic regulator problem for hereditary system, Institute for Computer Applications for Science and Engineering, ICASE Report 86-16, 1986, NASA Langley Reserch Center, Hampton, VA.
18. L. Ju, Q. Du and M. Gunzburger, Probabilistic methods for centroidal Voronoi tessellations and their parallel implementations, Parallel Computing, 28 (2002), 1477-1500. https://doi.org/10.1016/S0167-8191(02)00151-5
19. K. Kunisch and S. Volkwein, Control of burgers equation by a reduced order approach using proper orthogonal decomposition. J. Optim. Theory Appl. 102 (1999), 345-371. https://doi.org/10.1023/A:1021732508059
20. I. Lasiecka, and R. Triggiani, Dirichlet boundary control problem for parabolic equation with quadratic cost: analticity and Riccati's feedbacksynthesis, SIAM J. Control and Optimization 21 (1983), 41-67. https://doi.org/10.1137/0321003
21. H.-C. Lee, J. Burkardt, and M. Gunzburger, Centroidal Voronoi tessellation-based reduce-order modeling of complex systems, SIAM J. Sci. Comput. 28 (2006), 459-484.
22. H.-C. Lee, J. Burkardt, and M. Gunzburger, POD and CVT-based Reduced-order modeling of Navier-Stokes flows, Comput. Methods Appl. Mech. Engrg. 196 (2006), 337-355. https://doi.org/10.1016/j.cma.2006.04.004
23. S. Lloyd, Least squares quantization in PCM, IEEE Trans. Infor. Theory, 28 (1982), 129-137. https://doi.org/10.1109/TIT.1982.1056489
24. J. MacQueen, Some methods for classification and analysis of multivariate observations, Proc. Fifth Berkeley Symposium on Mathematical Statistics and Probability, 1 (1967), University of California, 281-297.
25. H. Marrekchi, Dynamic compensators for a nonlinear conservation law, Ph.D. Dissertation, Virginia Polytechnic Institute and State University, 1993.
26. D. A. Nagy, Modal representation of geometrically nonlinear behavior by the finite element method, Computers and Structures, 10 (1979), 683-688. https://doi.org/10.1016/0045-7949(79)90012-9
27. A. K. Noor, Recent advances in reduction methods for nonlinear problems, Computers and Structures, 13 (1981), 31-44. https://doi.org/10.1016/0045-7949(81)90106-1
28. A. K. Noor, C. M. Andersen, and J. M. Peters, Reduced basis technique for collapse analysis of shells, AIAA J., 19, 393-397.
29. A. K. Noor and J. M. Peters, Reduced basis technique for nonlinear analysis of structures, AIAA J., 18, 455-462.
30. J. S. Peterson, The reduced-basis method for incompressible viscous flow calculations, SIAM J. Scientific and Statistical computing, 10 (1989), 777-786. https://doi.org/10.1137/0910047
31. R. Triggiani, and R. Bulrisch, Boundary feedback stabilizability parabolic equations, Appl. Math. Optim. 6 (1980), 201-220. https://doi.org/10.1007/BF01442895 | 2021-05-19 01:36:15 | {"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": 0.46870312094688416, "perplexity": 5898.009261098947}, "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-2021-21/segments/1620243991562.85/warc/CC-MAIN-20210519012635-20210519042635-00011.warc.gz"} |
https://www.vcalc.com/wiki/KurtHeckman/gem+specific+gravity | # gem specific gravity
Not Reviewed
Type
Dataset
Category
vCommons
Contents
2 columns
Tags:
Rating
ID
KurtHeckman.gem specific gravity
UUID
d39aecee-f011-11e3-b7aa-bc764e2038f2
Columns
DatatypeRequired
gemStringNo
SGRealNo
Current statistics
SumAvgMinMax
SG138.253.21511627906976742.154.7
# Introduction
This table contains a list of the most common gems, by name and their specific gravity, which is a mean density ratio compared to the density of pure water.
Please note: this table contains a single value for the specific gravity of gems to be used in the vCalc jeweler's equations and calculator. However, there is some variance in the accepted specific gravity to be used in jeweler's equations. For example, the Gemological Institute published a specific gravity of 4.70 for Zircon in their tables, while The Complete Handbook for Gemstone Weight Estimation by Charles I. Carmona repeatedly sites a specific gravity of 4.00.
If you wish to suggest different values for the specific gravity of a gem, please make a comment in this wiki. The same action should be taken if you wish to add another gem or stone and it's specific gravity. This table is monitored and we will respond as soon as possible.
Related to specific gravity is mean density. vCalc provides the mean density of many common substances. | 2019-06-24 09:55:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4339354634284973, "perplexity": 2201.739296909761}, "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/1560627999298.86/warc/CC-MAIN-20190624084256-20190624110256-00158.warc.gz"} |
https://www.techwhiff.com/issue/pls-help-will-give-brainliest--384720 | # Pls help!!!! will give brainliest!!
###### Question:
pls help!!!! will give brainliest!!
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### Can someone please help me with my global history homework? It's due tomorrow on September 29th. Here's the question: How did the Neolithic Revolution spread? the picture is posted below.
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### A skydiver descends at 20 1 3 kilometers per hour after deploying their parachute. Which of the following expressions represents the change in the skydiver's altitude if they descend for 4 7 of an hour?
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### Scale down the ratio 385/7 = ?/1
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### Question 3 5 pts The highest ranking member of the U.S. Senate when the Vice President is absent is the O Speaker of the House O Speaker Pro Tempore O President of the House President Pro Tempore Help
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### Question 10 100 pts Find the area of a trapezoid in which b. - 7cm, b = 12 cm, and h = 5 cm. Scmn 12 cm 47.5 cm O 12.5 cm O 95 cm O 24 cm2
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### What does Měi gèrén dōu rènwéi kēxué shì hǔnluàn de, nàme tā bùshì! Nǐ yào zuò de jiùshì yòng nǐ de ěrduǒ hé yǎnjīng qīngtīng, nǐ huì zài rènhé shíjiān chéngwéi kēxué dàshī! mean?
What does Měi gèrén dōu rènwéi kēxué shì hǔnluàn de, nàme tā bùshì! Nǐ yào zuò de jiùshì yòng nǐ de ěrduǒ hé yǎnjīng qīngtīng, nǐ huì zài rènhé shíjiān chéngwéi kēxué dàshī! mean?...
### El porqué de la discrepancia de las teorías de Darwin y la Mark que explican la evolución de la jirafa el animal másel porqué de la discrepancia de las teorías de Darwin y la Mark que explican la evolución de la jirafa el animal más alto del mundo. alto del mundo.
El porqué de la discrepancia de las teorías de Darwin y la Mark que explican la evolución de la jirafa el animal másel porqué de la discrepancia de las teorías de Darwin y la Mark que explican la evolución de la jirafa el animal más alto del mundo. alto del mundo.... | 2022-09-29 07:02:42 | {"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": 0.42625856399536133, "perplexity": 8589.70623526472}, "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/1664030335326.48/warc/CC-MAIN-20220929065206-20220929095206-00537.warc.gz"} |
https://www.zbmath.org/serials/?q=se%3A1008 | # zbMATH — the first resource for mathematics
## Annales de l’Institut Henri Poincaré. Analyse Non Linéaire
Short Title: Ann. Inst. Henri Poincaré, Anal. Non Linéaire Publisher: Elsevier (Elsevier Masson), Paris; L’Institut Henri Poincaré, Paris ISSN: 0294-1449 Online: http://www.sciencedirect.com/science/journal/02941449 Comments: Indexed cover-to-cover
Documents Indexed: 1,561 Publications (since 1984) References Indexed: 1,536 Publications with 37,819 References.
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#### Authors
15 Lions, Pierre-Louis 13 Wei, Juncheng 10 Zuazua, Enrique 9 Del Pino, Manuel A. 9 Duzaar, Frank 9 Wang, Zhi-Qiang 8 Caffarelli, Luis Ángel 8 Marcus, Moshe M. 8 Müller, Stefan 8 Passaseo, Donato 8 Souganidis, Panagiotis E. 8 Valdinoci, Enrico 7 Dacorogna, Bernard 7 Dal Maso, Gianni 7 Felmer, Patricio L. 7 Jost, Jürgen 7 Mingione, Giuseppe 7 Molle, Riccardo 7 Murat, François 7 Rabinowitz, Paul Henry 7 Roquejoffre, Jean-Michel 7 Terracini, Susanna 6 Ambrosetti, Antonio 6 Barles, Guy 6 Benci, Vieri 6 Bouchitté, Guy 6 Buttazzo, Giuseppe 6 Constantin, Peter 6 Gazzola, Filippo 6 Guerrero, Sergio 6 Kenig, Carlos Eduardo 6 Tanaka, Kazunaga 6 Velázquez, Juan J. L. 6 Wang, Guofang 6 Zhang, Kewei 5 Acerbi, Emilio 5 Arisawa, Mariko 5 Bronsard, Lia 5 Casado-Díaz, Juan 5 Chae, Dongho 5 Coti Zelati, Vittorio 5 Ekeland, Ivar 5 Ghoussoub, Nassif A. 5 Grossi, Massimo 5 Helffer, Bernard 5 Ishii, Hitoshi 5 Koch, Herbert 5 LeFloch, Philippe Gerard 5 Lin, Chang-Shou 5 Masmoudi, Nader 5 Mischler, Stéphane 5 Ozawa, Tohru 5 Perthame, Benoît 5 Pistoia, Angela 5 Ponce, Gustavo 5 Quaas, Alexander 5 Rousset, Frédéric 5 Ryzhik, Lenya 5 Sire, Yannick 5 Solimini, Sergio 5 Souplet, Philippe 5 Takahashi, Takéo 5 Viana, Marcelo 5 Weth, Tobias 5 Zaslavski, Alexander Yakovlevich 4 Adami, Riccardo 4 Agrachev, Andreĭ Aleksandrovich 4 Alama, Stan 4 Alves, José Ferreira 4 Alvino, Angelo 4 Berti, Massimiliano 4 Boccardo, Lucio 4 Bonheure, Denis 4 Bonnard, Bernard 4 Bucur, Dorin 4 Carrillo de la Plata, José Antonio 4 Cellina, Arrigo 4 Cerami, Giovanna 4 Chou, Kai-Seng 4 Cianchi, Andrea 4 Da Lio, Francesca 4 Damascelli, Lucio 4 Dancer, Edward Norman 4 Dávila, Juan 4 Dipierro, Serena 4 Escobedo Martínez, Miguel 4 Feireisl, Eduard 4 Foiaş, Ciprian Ilie 4 Fonseca, Irene 4 Fortunato, Donato 4 Frankowska, Hélène 4 Hamel, François 4 Jüngel, Ansgar 4 Laurençot, Philippe 4 Long, Yiming 4 Malchiodi, Andrea 4 Marcellini, Paolo 4 Méhats, Florian 4 Miao, Changxing 4 Molinet, Luc ...and 1,901 more Authors
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#### Fields
1,108 Partial differential equations (35-XX) 255 Calculus of variations and optimal control; optimization (49-XX) 196 Fluid mechanics (76-XX) 167 Global analysis, analysis on manifolds (58-XX) 164 Dynamical systems and ergodic theory (37-XX) 110 Differential geometry (53-XX) 83 Statistical mechanics, structure of matter (82-XX) 80 Operator theory (47-XX) 74 Ordinary differential equations (34-XX) 71 Functional analysis (46-XX) 64 Mechanics of deformable solids (74-XX) 61 Systems theory; control (93-XX) 39 Real functions (26-XX) 38 Mechanics of particles and systems (70-XX) 36 Quantum theory (81-XX) 32 Biology and other natural sciences (92-XX) 31 Integral equations (45-XX) 27 Probability theory and stochastic processes (60-XX) 20 Harmonic analysis on Euclidean spaces (42-XX) 19 Classical thermodynamics, heat transfer (80-XX) 18 Measure and integration (28-XX) 17 Functions of a complex variable (30-XX) 17 Numerical analysis (65-XX) 16 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 15 Optics, electromagnetic theory (78-XX) 12 Potential theory (31-XX) 11 Convex and discrete geometry (52-XX) 11 Operations research, mathematical programming (90-XX) 10 General topology (54-XX) 9 Geophysics (86-XX) 7 Several complex variables and analytic spaces (32-XX) 7 Manifolds and cell complexes (57-XX) 7 Relativity and gravitational theory (83-XX) 4 Algebraic topology (55-XX) 4 Computer science (68-XX) 4 Astronomy and astrophysics (85-XX) 3 Algebraic geometry (14-XX) 3 Approximations and expansions (41-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Nonassociative rings and algebras (17-XX) 2 Group theory and generalizations (20-XX) 2 Topological groups, Lie groups (22-XX) 2 Special functions (33-XX) 2 Abstract harmonic analysis (43-XX) 2 Information and communication theory, circuits (94-XX) 1 General and overarching topics; collections (00-XX) 1 Combinatorics (05-XX) 1 Number theory (11-XX) 1 Associative rings and algebras (16-XX) 1 Difference and functional equations (39-XX) 1 Integral transforms, operational calculus (44-XX) 1 Geometry (51-XX) 1 Statistics (62-XX)
#### Citations contained in zbMATH Open
1,426 Publications have been cited 26,381 times in 19,605 Documents Cited by Year
The concentration-compactness principle in the calculus of variations. The locally compact case. I. Zbl 0541.49009
Lions, Pierre-Louis
1984
The concentration-compactness principle in the calculus of variations. The locally compact case. II. Zbl 0704.49004
Lions, P.-L.
1984
Nonlinear equations for fractional Laplacians. I: Regularity, maximum principles, and Hamiltonian estimates. Zbl 1286.35248
Cabré, Xavier; Sire, Yannick
2014
On nonhomogeneous elliptic equations involving critical Sobolev exponent. Zbl 0785.35046
Tarantello, G.
1992
Multi-peak bound states for nonlinear Schrödinger equations. Zbl 0901.35023
Del Pino, Manuel; Felmer, Patricio L.
1998
Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Zbl 0911.35009
Damascelli, Lucio
1998
Travelling fronts in cylinders. Zbl 0799.35073
Berestycki, Henri; Nirenberg, Louis
1992
Null and approximate controllability for weakly blowing up semilinear heat equations. Zbl 0970.93023
Fernández-Cara, Enrique; Zuazua, Enrique
2000
The global Cauchy problem for the nonlinear Schrödinger equation revisited. Zbl 0586.35042
Ginibre, J.; Velo, G.
1985
On the existence of a positive solution of semilinear elliptic equations in unbounded domains. Zbl 0883.35045
Bahri, Abbas; Lions, Pierre-Louis
1997
Construction of the Leray-Schauder degree for elliptic operators in unbounded domains. Zbl 0835.35048
Volpert, A. I.; Volpert, V. A.
1994
Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. Zbl 1155.45004
Barles, Guy; Imbert, Cyril
2008
A coupled chemotaxis-fluid model: global existence. Zbl 1236.92013
Liu, Jian-Guo; Lorz, Alexander
2011
Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system. Zbl 1351.35239
Winkler, Michael
2016
Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data. Zbl 0857.35126
Boccardo, Lucio; Gallouët, Thierry; Orsina, Luigi
1996
Spikes in two coupled nonlinear Schrödinger equations. Zbl 1080.35143
Lin, Tai-Chia; Wei, Juncheng
2005
Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. Zbl 0612.58011
Szulkin, Andrzej
1986
Regularity criteria for the generalized viscous MHD equations. Zbl 1130.35110
Zhou, Yong
2007
A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system. Zbl 1191.35121
Dancer, E. N.; Wei, Juncheng; Weth, Tobias
2010
On the existence of blowing-up solutions for a mean field equation. Zbl 1129.35376
Esposito, Pierpaolo; Grossi, Massimo; Pistoia, Angela
2005
Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent. Zbl 0729.35014
Han, Zheng-Chao
1991
Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion. Zbl 1283.35154
Tao, Youshan; Winkler, Michael
2013
Local behavior of fractional $$p$$-minimizers. Zbl 1355.35192
Di Castro, Agnese; Kuusi, Tuomo; Palatucci, Giampiero
2016
On a partial differential equation involving the Jacobian determinant. Zbl 0707.35041
Dacorogna, Bernard; Moser, Jürgen
1990
An existence result for nonliner elliptic problems involving critical Sobolev exponent. Zbl 0612.35053
Capozzi, A.; Fortunato, D.; Palmieri, G.
1985
Small solutions to nonlinear Schrödinger equations. Zbl 0786.35121
Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis
1993
Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations. Zbl 0877.35042
Marcus, M.; Véron, L.
1997
Global solutions and finite time blow up for damped semilinear wave equations. Zbl 1094.35082
Gazzola, Filippo; Squassina, Marco
2006
Three nodal solutions of singularly perturbed elliptic equations on domains without topology. Zbl 1114.35068
Bartsch, Thomas; Weth, Tobias
2005
Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data. Zbl 0593.35076
Bardos, C.; Degond, P.
1985
On the Schrödinger-Maxwell equations under the effect of a general nonlinear term. Zbl 1187.35231
Azzollini, A.; d’Avenia, P.; Pomponio, A.
2010
Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D. Zbl 1184.35055
Gal, Ciprian G.; Grasselli, Maurizio
2010
A critical fractional equation with concave-convex power nonlinearities. Zbl 1350.49009
2015
Well-posedness for Hall-magnetohydrodynamics. Zbl 1297.35064
Chae, Dongho; Degond, Pierre; Liu, Jian-Guo
2014
On asymptotic stability of solitary waves for nonlinear Schrödinger equations. Zbl 1028.35139
2003
Nondegeneracy of blow-up points for the parabolic Keller-Segel system. Zbl 1302.35075
Mizoguchi, Noriko; Souplet, Philippe
2014
Regularity results for parabolic systems related to a class of non-Newtonian fluids. Zbl 1052.76004
Acerbi, E.; Mingione, G.; Seregin, G. A.
2004
Existence results for mean field equations. Zbl 0937.35055
Ding, Weiyue; Jost, Jürgen; Li, Jiayu; Wang, Guofang
1999
Propriétés des matrices “bien localisées” près de leur diagonale et quelques applications. (Properties of matrices “well localized” near the diagonal and some applications). Zbl 0722.15004
Jaffard, S.
1990
Periodic solutions of Hamiltonian systems of 3-body type. Zbl 0745.34034
Bahri, A.; Rabinowitz, P. H.
1991
Existence and nonexistence results for anisotropic quasilinear elliptic equations. Zbl 1144.35378
Fragalà, Ilaria; Gazzola, Filippo; Kawohl, Bernd
2004
On the definition and the lower semicontinuity of certain quasiconvex integrals. Zbl 0609.49009
Marcellini, Paolo
1986
On a superlinear elliptic equation. Zbl 0733.35043
Wang, Zhi Qiang
1991
Existence of solutions for compressible fluid models of Korteweg type. Zbl 1010.76075
Danchin, Raphaël; Desjardins, Benoît
2001
Well-posedness and scattering for the KP-II equation in a critical space. Zbl 1169.35372
Hadac, Martin; Herr, Sebastian; Koch, Herbert
2009
Multiple boundary peak solutions for some singularly perturbed Neumann problems. Zbl 0944.35020
Gui, Changfeng; Wei, Juncheng; Winter, Matthias
2000
Periodic and heteroclinic orbits for a periodic Hamiltonian system. Zbl 0701.58023
Rabinowitz, Paul H.
1989
Ground states of nonlinear Schrödinger equations with potentials. Zbl 1111.35079
Li, Yongqing; Wang, Zhi-Qiang; Zeng, Jing
2006
Fractional elliptic equations, Caccioppoli estimates and regularity. Zbl 1381.35211
Caffarelli, Luis A.; Stinga, Pablo Raúl
2016
$$L^ p$$ regularity of velocity averages. Zbl 0763.35014
DiPerna, R. J.; Lions, P. L.; Meyer, Y.
1991
Critère d’existence de solutions positives pour des équations semi- linéaires non monotones. (Existence condition of positive solutions of non-monotonic semilinear equations). Zbl 0599.35073
Baras, Pierre; Pierre, Michel
1985
Convex symmetrization and applications. Zbl 0877.35040
Alvino, Angelo; Ferone, Vincenzo; Trombetti, Guido; Lions, Pierre-Louis
1997
Critical points of embeddings of $$H_ 0^{1,n}$$ into Orlicz spaces. Zbl 0664.35022
Struwe, Michael
1988
On a nonlinear partial differential equation having natural growth terms and unbounded solution. Zbl 0696.35042
Bensoussan, A.; Boccardo, L.; Murat, F.
1988
Remarks on the large time behaviour of a nonlinear diffusion equation. Zbl 0653.35036
Kavian, Otared
1987
Hardy-Sobolev critical elliptic equations with boundary singularities. Zbl 1232.35064
Ghoussoub, N.; Kang, X. S.
2004
Proto-differentiability of set-valued mappings and its applications in optimization. Zbl 0674.90082
Rockafellar, R. T.
1989
On self-similarity and stationary problem for fragmentation and coagulation models. Zbl 1130.35025
Escobedo, M.; Mischler, S.; Rodriguez Ricard, M.
2005
Minimal interface criterion for phase transitions in mixtures of Cahn- Hilliard fluids. Zbl 0702.49009
Baldo, Sisto
1990
A general mountain pass principle for locating and classifying critical points. Zbl 0711.58008
Ghoussoub, N.; Preiss, D.
1989
On the spatially homogeneous Boltzmann equation. Zbl 0946.35075
Mischler, Stéphane; Wennberg, Bernst
1999
Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents. Zbl 0568.35039
Cerami, Giovanna; Fortunato, Donato; Struwe, Michael
1984
Nonlinear evolution equations and analyticity. I. Zbl 0622.35066
Kato, Tosio; Masuda, Kyûya
1986
On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials. Zbl 1248.93031
Duyckaerts, Thomas; Zhang, Xu; Zuazua, Enrique
2008
Ljusternik-Schnirelmann theory on $$C^ 1$$-manifolds. Zbl 0661.58009
Szulkin, Andrzej
1988
Nehari’s problem and competing species systems. Zbl 1090.35076
Conti, M.; Terracini, S.; Verzini, G.
2002
Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations. Zbl 1031.35077
Chen, Gui-Qiang; Perthame, Benoît
2003
Asymptotic behaviour of solutions and their derivatives, for semilinear elliptic problems with blowup on the boundary. Zbl 0840.35033
Bandle, Catherine; Marcus, Moshe
1995
Blow-up behaviour of one-dimensional semilinear parabolic equations. Zbl 0813.35007
Herrero, M. A.; Velázquez, J. J. L.
1993
Topological solutions in the self-dual Chern-Simons theory: Existence and approximation. Zbl 0836.35007
Spruck, Joel; Yang, Yisong
1995
Looking for the Bernoulli shift. Zbl 0803.58013
Séré, Éric
1993
Homoclinics: Poincaré-Melnikov type results via a variational approach. Zbl 1004.37043
1998
Partial regularity of minimizers of quasiconvex integrals. Zbl 0594.49004
Giaquinta, Mariano; Modica, Giuseppe
1986
$$\Gamma$$-convergence for nonlocal phase transitions. Zbl 1253.49008
Savin, Ovidiu; Valdinoci, Enrico
2012
Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation. Zbl 1149.76052
Constantin, Peter; Wu, Jiahong
2008
Exact controllability for semilinear wave equations in one space dimension. Zbl 0769.93017
Zuazua, E.
1993
On the Liouville property for fully nonlinear equations. Zbl 0956.35035
Cutrì, Alessandra; Leoni, Fabiana
2000
Second order parabolic systems, optimal regularity, and singular sets of solutions. Zbl 1099.35042
Duzaar, Frank; Mingione, Giuseppe
2005
Comparison results for elliptic and parabolic equations via Schwarz symmetrization. Zbl 0703.35007
Alvino, Angelo; Trombetti, Guido; Lions, Pierre-Louis
1990
Incompressible, inviscid limit of the compressible Navier-Stokes system. Zbl 0991.35058
2001
Pulsating fronts for nonlocal dispersion and KPP nonlinearity. Zbl 1288.45007
Coville, Jérôme; Dávila, Juan; Martínez, Salomé
2013
Finite dimensional behavior for weakly damped driven Schrödinger equations. Zbl 0659.35019
Ghidaglia, Jean-Michel
1988
Nodal domains and spectral minimal partitions. Zbl 1171.35083
Helffer, B.; Hoffmann-Ostenhof, T.; Terracini, S.
2009
On the Hénon equation: asymptotic profile of ground states. I. Zbl 1114.35071
Byeon, Jaeyoung; Wang, Zhi-Qiang
2006
Variational problems on classes of rearrangements and multiple configurations for steady vortices. Zbl 0677.49005
Burton, G. R.
1989
On a new class of elastic deformations not allowing for cavitation. Zbl 0863.49002
Müller, S.; Qi, Tang; Yan, B. S.
1994
Viscosity solutions of nonlinear integro-differential equations. Zbl 0870.45002
Alvarez, Olivier; Tourin, Agnès
1996
Solitary waves of generalized Kadomtsev-Petviashvili equations. Zbl 0883.35103
de Bouard, Anne; Saut, Jean-Claude
1997
Minimal solutions of variational problems on a torus. Zbl 0609.49029
Moser, Jürgen
1986
Harnack inequalities for quasi-minima of variational integrals. Zbl 0565.35012
DiBenedetto, E.; Trudinger, Neil S.
1984
The Riemann problem for a class of resonant hyperbolic systems of balance laws. Zbl 1086.35069
Goatin, Paola; LeFloch, Philippe G.
2004
Quasi-minima. Zbl 0541.49008
Giaquinta, Mariano; Giusti, Enrico
1984
Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior. Zbl 1288.35456
Liang, Zhanping; Li, Fuyi; Shi, Junping
2014
The modulational regime of three-dimensional water waves and the Davey-Stewartson system. Zbl 0892.76008
Craig, Walter; Schanz, Ulrich; Sulem, Catherine
1997
Stationary solutions for the Cahn-Hilliard equation. Zbl 0910.35049
Wei, Juncheng; Winter, Matthias
1998
Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws. Zbl 0991.35009
Biler, Piotr; Karch, Grzegorz; Woyczyński, Wojbor A.
2001
Strong solutions for a compressible fluid model of Korteweg type. Zbl 1141.76053
Kotschote, Matthias
2008
A blow-up criterion for compressible viscous heat-conductive flows. Zbl 1352.35109
Fan, Jishan; Jiang, Song; Ou, Yaobin
2010
Properties of pseudoholomorphic curves in symplectisations. I: Asymptotics. Zbl 0861.58018
Hofer, H.; Wysocki, K.; Zehnder, E.
1996
Wellposedness and stability results for the Navier-Stokes equations in $$\mathbb R^3$$. Zbl 1165.35038
Chemin, Jean-Yves; Gallagher, Isabelle
2009
Global well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D. Zbl 1458.35373
Kinoshita, Shinya
2021
Nonlinear stability of self-gravitating irrotational Chaplygin fluids in a FLRW geometry. Zbl 1464.83026
LeFloch, Philippe G.; Wei, Changhua
2021
Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities. Zbl 1465.35396
Jin, Tianling; Xiong, Jingang
2021
Dispersive blow-up for solutions of the Zakharov-Kuznetsov equation. Zbl 1458.35375
Linares, F.; Pastor, A.; Drumond Silva, J.
2021
Nonnegative control of finite-dimensional linear systems. Zbl 1466.93020
Lohéac, Jérôme; Trélat, Emmanuel; Zuazua, Enrique
2021
On isolated singularities of fractional semi-linear elliptic equations. Zbl 1458.35462
Yang, Hui; Zou, Wenming
2021
Conserved quantities and Hamiltonization of nonholonomic systems. Zbl 07335687
Balseiro, Paula; Yapu, Luis P.
2021
Asymptotic expansions in time for rotating incompressible viscous fluids. Zbl 1464.35182
Hoang, Luan T.; Titi, Edriss S.
2021
On a nonlocal Cahn-Hilliard/Navier-Stokes system with degenerate mobility and singular potential for incompressible fluids with different densities. Zbl 1464.35179
Frigeri, Sergio
2021
Some Liouville theorems for stationary Navier-Stokes equations in Lebesgue and Morrey spaces. Zbl 1466.35282
Chamorro, Diego; Jarrín, Oscar; Lemarié-Rieusset, Pierre-Gilles
2021
Semilinear problems with right-hand sides singular at $$u=0$$ which change sign. Zbl 1466.35111
2021
Entropy theory for sectional hyperbolic flows. Zbl 07353366
José Pacifico, Maria; Yang, Fan; Yang, Jiagang
2021
A game of alignment: collective behavior of multi-species. Zbl 1466.92146
2021
Stability of equilibria uniformly in the inviscid limit for the Navier-Stokes-Poisson system. Zbl 1465.76114
Rousset, Frédéric; Sun, Changzhen
2021
Compressible fluids and active potentials. Zbl 1430.35185
Constantin, Peter; Drivas, Theodore D.; Nguyen, Huy Q.; Pasqualotto, Federico
2020
Global classical solutions to quadratic systems with mass control in arbitrary dimensions. Zbl 1433.35167
Fellner, Klemens; Morgan, Jeff; Tang, Bao Quoc
2020
Atomic decompositions, two stars theorems, and distances for the Bourgain-Brezis-Mironescu space and other big spaces. Zbl 1443.46005
D’Onofrio, Luigi; Greco, Luigi; Perfekt, Karl-Mikael; Sbordone, Carlo; Schiattarella, Roberta
2020
Lipschitz regularity for viscous Hamilton-Jacobi equations with $$L^p$$ terms. Zbl 1440.35036
Cirant, Marco; Goffi, Alessandro
2020
The closure of planar diffeomorphisms in Sobolev spaces. Zbl 1446.46018
De Philippis, G.; Pratelli, A.
2020
A higher speed type II blowup for the five dimensional energy critical heat equation. Zbl 1433.35007
2020
The energy-critical nonlinear wave equation with an inverse-square potential. Zbl 1433.35199
Miao, Changxing; Murphy, Jason; Zheng, Jiqiang
2020
Degenerate nonlocal Cahn-Hilliard equations: well-posedness, regularity and local asymptotics. Zbl 1467.45017
Davoli, Elisa; Ranetbauer, Helene; Scarpa, Luca; Trussardi, Lara
2020
Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces. Zbl 1440.35322
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2020
Convex integration solutions to the transport equation with full dimensional concentration. Zbl 1458.35363
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2020
The Calderón problem for quasilinear elliptic equations. Zbl 1457.35093
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2020
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Thin film liquid crystals with oblique anchoring and boojums. Zbl 1446.76066
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Recurrence for the wind-tree model. Zbl 1436.37006
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2020
The Bramson delay in the non-local Fisher-KPP equation. Zbl 1436.35239
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2020
Ginzburg-Landau equations on Riemann surfaces of higher genus. Zbl 07152415
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2020
Well-posedness issues on the periodic modified Kawahara equation. Zbl 1435.35336
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Stability analysis and Hopf bifurcation at high Lewis number in a combustion model with free interface. Zbl 1442.35555
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Fractional Piola identity and polyconvexity in fractional spaces. Zbl 1442.35445
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Convergence to equilibrium for the solution of the full compressible Navier-Stokes equations. Zbl 1435.35069
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A counterexample to the Liouville property of some nonlocal problems. Zbl 1439.35107
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Exact controllability of semilinear heat equations in spaces of analytic functions. Zbl 1448.93030
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Existence of solutions for a higher-order semilinear parabolic equation with singular initial data. Zbl 1454.35222
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Statistical stability of mostly expanding diffeomorphisms. Zbl 07283942
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Existence of strong minimizers for the Griffith static fracture model in dimension two. Zbl 1458.74126
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Local well-posedness for quasi-linear NLS with large Cauchy data on the circle. Zbl 1430.35207
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Fluctuations of $$N$$-particle quantum dynamics around the nonlinear Schrödinger equation. Zbl 1419.81042
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Complete stickiness of nonlocal minimal surfaces for small values of the fractional parameter. Zbl 1411.49026
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Well-posedness of the two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity. Zbl 1410.35196
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On the splash singularity for the free-surface of a Navier-Stokes fluid. Zbl 1442.76042
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Muñoz, Claudio; Palacios, José M.
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Strömqvist, Martin
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On the transport of Gaussian measures under the one-dimensional fractional nonlinear Schrödinger equations. Zbl 1427.35253
Forlano, Justin; Trenberth, William J.
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Asymptotic stability of a composite wave of two viscous shock waves for the one-dimensional radiative Euler equations. Zbl 1406.35042
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Time-optimal trajectories of generic control-affine systems have at worst iterated Fuller singularities. Zbl 1409.49033
Boarotto, Francesco; Sigalotti, Mario
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Regularity of solutions to scalar conservation laws with a force. Zbl 1447.35219
Gess, Benjamin; Lamy, Xavier
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Boltzmann collision operator for the infinite range potential: a limit problem. Zbl 1423.35292
Jiang, Jin-Cheng; Liu, Tai-Ping
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Regularity theory for $$L^n$$-viscosity solutions to fully nonlinear elliptic equations with asymptotical approximate convexity. Zbl 1436.35159
Huang, Qingbo
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Continuity of composition operators in Sobolev spaces. Zbl 1456.46028
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Wright-Fisher-type equations for opinion formation, large time behavior and weighted logarithmic-Sobolev inequalities. Zbl 1434.35238
Furioli, Giulia; Pulvirenti, Ada; Terraneo, Elide; Toscani, Giuseppe
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Corrigendum to: “Nondegeneracy of blow-up points for the parabolic Keller-Segel system”. Zbl 1415.35062
Mizoguchi, N.; Souplet, Ph.
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The vanishing viscosity limit for some symmetric flows. Zbl 1416.35025
Gie, Gung-Min; Kelliher, James P.; Lopes Filho, Milton C.; Mazzucato, Anna L.; Nussenzveig Lopes, Helena J.
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On the radius of spatial analyticity for solutions of the Dirac-Klein-Gordon Equations in two space dimensions. Zbl 1421.35311
Selberg, Sigmund
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Long-period limit of exact periodic traveling wave solutions for the derivative nonlinear Schrödinger equation. Zbl 1420.35356
Hayashi, Masayuki
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Fazly, Mostafa; Sire, Yannick
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Asymptotic behavior of spreading fronts in the anisotropic Allen-Cahn equation on $$\mathbb{R}^n$$. Zbl 1411.35161
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Montecchiari, Piero; Rabinowitz, Paul H.
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Caju, Rayssa; do Ó, João Marcos; Santos, Almir Silva
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Best constant in Poincaré inequalities with traces: a free discontinuity approach. Zbl 1428.49043
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Non-uniqueness for a critical heat equation in two dimensions with singular data. Zbl 1427.35096
Ioku, Norisuke; Ruf, Bernhard; Terraneo, Elide
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Hyakuna, Ryosuke
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Estimates of Green and Martin Kernels for Schrödinger operators with singular potential in Lipschitz domains. Zbl 1426.35098
Marcus, Moshe
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Well posedness of nonlinear parabolic systems beyond duality. Zbl 1435.35195
Bulíček, Miroslav; Burczak, Jan; Schwarzacher, Sebastian
2019
Corrigendum to: “Singularity formation of the Yang-Mills Flow”. Zbl 1416.37038
Kelleher, Casey; Streets, Jeffrey
2019
Physical measures for the geodesic flow tangent to a transversally conformal foliation. Zbl 1404.53037
Alvarez, Sébastien; Yang, Jiagang
2019
On bifurcation of eigenvalues along convex symplectic paths. Zbl 1409.37058
Chang, Yinshan; Long, Yiming; Wang, Jian
2019
Energy identity and necklessness for a sequence of Sacks-Uhlenbeck maps to a sphere. Zbl 1410.58004
Li, Jiayu; Zhu, Xiangrong
2019
Interior regularity for fractional systems. Zbl 1411.35111
Caffarelli, Luis; Dávila, Gonzalo
2019
Hyperbolic ends with particles and grafting on singular surfaces. Zbl 1408.83030
Chen, Qiyu; Schlenker, Jean-Marc
2019
Nonlinear responses from the interaction of two progressing waves at an interface. Zbl 1409.58018
de Hoop, Maarten; Uhlmann, Gunther; Wang, Yiran
2019
Uniform boundedness principles for Sobolev maps into manifolds. Zbl 1418.46035
Monteil, Antonin; Van Schaftingen, Jean
2019
Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain. Zbl 1411.35058
Chen, I-Kun; Hsia, Chun-Hsiung; Kawagoe, Daisuke
2019
Description of limiting vorticities for the magnetic 2D Ginzburg-Landau equations. Zbl 1414.35218
Rodiac, Rémy
2019
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1,506 Journal of Differential Equations 1,108 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1,034 Journal of Mathematical Analysis and Applications 708 Calculus of Variations and Partial Differential Equations 626 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 571 Archive for Rational Mechanics and Analysis 467 Journal of Functional Analysis 428 Discrete and Continuous Dynamical Systems 323 Journal de Mathématiques Pures et Appliquées. Neuvième Série 317 Communications in Mathematical Physics 287 Transactions of the American Mathematical Society 282 Communications in Partial Differential Equations 260 SIAM Journal on Mathematical Analysis 259 Journal of Mathematical Physics 247 Communications on Pure and Applied Analysis 240 ZAMP. Zeitschrift für angewandte Mathematik und Physik 234 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 219 NoDEA. Nonlinear Differential Equations and Applications 216 Annali di Matematica Pura ed Applicata. Serie Quarta 208 Proceedings of the American Mathematical Society 201 Nonlinear Analysis. Real World Applications 196 Advances in Mathematics 196 Communications in Contemporary Mathematics 188 Nonlinear Analysis. Theory, Methods & Applications 182 Comptes Rendus. Mathématique. Académie des Sciences, Paris 156 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 155 Nonlinearity 155 Applied Mathematics Letters 141 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 140 Advanced Nonlinear Studies 131 Mathematische Annalen 129 Mathematical Methods in the Applied Sciences 125 Applicable Analysis 125 Discrete and Continuous Dynamical Systems. Series B 118 Boundary Value Problems 115 Physica D 113 Journal of Mathematical Fluid Mechanics 108 Computers & Mathematics with Applications 106 The Journal of Geometric Analysis 105 Manuscripta Mathematica 99 SIAM Journal on Control and Optimization 98 Journal of Dynamics and Differential Equations 96 Journal of Statistical Physics 93 Applied Mathematics and Computation 90 Complex Variables and Elliptic Equations 89 Journal of Computational Physics 89 Duke Mathematical Journal 89 Advances in Nonlinear Analysis 84 Journal of Evolution Equations 82 Journal of the European Mathematical Society (JEMS) 79 Mathematische Zeitschrift 79 Acta Applicandae Mathematicae 78 Communications on Pure and Applied Mathematics 77 Applied Mathematics and Optimization 77 Acta Mathematica Sinica. English Series 76 Journal of Mathematical Sciences (New York) 76 Science China. Mathematics 75 Journal d’Analyse Mathématique 72 Abstract and Applied Analysis 70 Journal of Optimization Theory and Applications 68 Inventiones Mathematicae 68 Ergodic Theory and Dynamical Systems 68 Journal of Nonlinear Science 68 Discrete and Continuous Dynamical Systems. Series S 66 Journal of Fixed Point Theory and Applications 59 Chinese Annals of Mathematics. Series B 57 Journal of Dynamical and Control Systems 56 Kinetic and Related Models 55 Revista Matemática Iberoamericana 54 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 54 Annales Henri Poincaré 52 Mediterranean Journal of Mathematics 51 Potential Analysis 48 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 45 Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni 45 Milan Journal of Mathematics 44 Topological Methods in Nonlinear Analysis 43 Journal of Computational and Applied Mathematics 43 Mathematische Nachrichten 41 Rocky Mountain Journal of Mathematics 41 Analysis & PDE 40 Zeitschrift für Analysis und ihre Anwendungen 40 Journal of Hyperbolic Differential Equations 39 Stochastic Processes and their Applications 39 Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI 39 Applied and Computational Harmonic Analysis 39 Advances in Calculus of Variations 38 Monatshefte für Mathematik 38 Numerische Mathematik 38 Journal of Scientific Computing 38 Bulletin des Sciences Mathématiques 37 Quarterly of Applied Mathematics 37 Acta Mathematicae Applicatae Sinica. English Series 36 Ricerche di Matematica 36 The Journal of Fourier Analysis and Applications 36 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V 36 Mathematical Control and Related Fields 34 Rendiconti del Seminario Matematico della Università di Padova 34 Set-Valued and Variational Analysis 33 Bulletin of the American Mathematical Society. New Series ...and 520 more Journals
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14,673 Partial differential equations (35-XX) 2,398 Fluid mechanics (76-XX) 2,393 Calculus of variations and optimal control; optimization (49-XX) 1,729 Dynamical systems and ergodic theory (37-XX) 1,586 Global analysis, analysis on manifolds (58-XX) 1,036 Differential geometry (53-XX) 981 Operator theory (47-XX) 934 Ordinary differential equations (34-XX) 849 Statistical mechanics, structure of matter (82-XX) 792 Biology and other natural sciences (92-XX) 777 Functional analysis (46-XX) 743 Mechanics of deformable solids (74-XX) 734 Systems theory; control (93-XX) 717 Numerical analysis (65-XX) 625 Probability theory and stochastic processes (60-XX) 602 Quantum theory (81-XX) 472 Real functions (26-XX) 384 Mechanics of particles and systems (70-XX) 340 Integral equations (45-XX) 308 Harmonic analysis on Euclidean spaces (42-XX) 262 Optics, electromagnetic theory (78-XX) 242 Operations research, mathematical programming (90-XX) 189 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 186 Potential theory (31-XX) 171 Measure and integration (28-XX) 152 Functions of a complex variable (30-XX) 152 Classical thermodynamics, heat transfer (80-XX) 125 Convex and discrete geometry (52-XX) 100 Manifolds and cell complexes (57-XX) 99 Several complex variables and analytic spaces (32-XX) 91 Geophysics (86-XX) 79 Relativity and gravitational theory (83-XX) 64 Topological groups, Lie groups (22-XX) 64 General topology (54-XX) 63 Difference and functional equations (39-XX) 61 Astronomy and astrophysics (85-XX) 60 Approximations and expansions (41-XX) 57 Information and communication theory, circuits (94-XX) 43 Linear and multilinear algebra; matrix theory (15-XX) 43 Computer science (68-XX) 38 Algebraic topology (55-XX) 30 Statistics (62-XX) 29 Algebraic geometry (14-XX) 23 History and biography (01-XX) 23 Integral transforms, operational calculus (44-XX) 21 Special functions (33-XX) 21 Abstract harmonic analysis (43-XX) 21 Geometry (51-XX) 14 Number theory (11-XX) 14 Nonassociative rings and algebras (17-XX) 13 General and overarching topics; collections (00-XX) 10 Combinatorics (05-XX) 8 Sequences, series, summability (40-XX) 7 Associative rings and algebras (16-XX) 4 Category theory; homological algebra (18-XX) 3 Mathematical logic and foundations (03-XX) 3 Group theory and generalizations (20-XX) 2 Order, lattices, ordered algebraic structures (06-XX) 2 General algebraic systems (08-XX) 2 Commutative algebra (13-XX) 1 Mathematics education (97-XX) | 2021-09-27 10:38:25 | {"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": 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": 0.5090389847755432, "perplexity": 6341.767711935189}, "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-2021-39/segments/1631780058415.93/warc/CC-MAIN-20210927090448-20210927120448-00079.warc.gz"} |
http://homepages.gac.edu/~sskulrat/Courses/2019F-178/lectures/assembly2.html | # Functions
• A function in a high-level language can either be recursive or non-recursive.
• A recursive function calls itself (directly or indirectly). E.g.
def fact(n):
if n == 1:
return 1
else:
return n * fact(n-1)
• A non-recursive function does not call itself. E.g.
def fact(n):
factn = 1
for i in range(1, n+1):
factn *= i
return factn
# Non-recursive Functions in SLIM
• The code for every function should be in one chunk, cleanly separated from the rest of the program. It should have only one entry point, but can have multiple exit points.
• A function may call (or invoke) another function. The function that makes the call is the caller and the function that gets called is the callee.
• The caller and callee communicate via a protocol.
1. Before making the call, the caller must load appropriate registers with all the arguments that the callee expects.
2. The caller must let the callee know about the place where the callee must come back to after the callee finishes. This information (the return address) is passed from the caller to the callee in a special register, which we will call the continuation register.
3. After the caller loads all the registers with appropriate values, and loads the continuation register with the return address, it jumps to the entry point of the callee.
4. The callee executes its code. When it finishes, it loads a designated register with the return value if any, then loads the PC with the address in the continuation register and returns.
# Translating a non-recursive function call
• A nonrecursive function call in python like
f(a, b)
gets translated into
li a <avalue>
li b <bvalue>
li cont afterCall
j f
afterCallL:
# Recursive Function Calls in SLIM
• A recursive function needs to go through the same rigmarole that a non-recursive function does, and then more.
• An extra thing that a recursive caller has to do is that before invoking a recursive callee, the caller must save all the registers that it is using and may be clobbered by the callee in some safe place (i.e., data memory). Also, after returning from the callee, the caller must load back all the values that it has saved prior to invoking the callee.
• In this stack discipline the caller is responsible for saving and retrieving relevant registers. It is also possible that the callee is responsible for this work. In a real system, some registers are designated as caller-saved while others are callee-saved.
• In fact, in every function call (whether recursive or not) we have to make sure the caller saves any register that will be clobbered by the callee but that the caller expects to contain the same value before and after the call. We simply chose to tell you now instead of in the previous slide!
# Translating a recursive function call (caller-saved)
• A recursive function call in python like
<prest>
recurse(n)
<postst>
gets translated into
<prest>
li n <nvalue>
li cont afterCall
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; statements to save working registers go here
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
j recurse
afterCallL:
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;; statements to retrieve working registers go here
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
<postst>
if using caller-saved method.
# Translating a recursive function call (callee-saved)
• Exercise. Translate the function call
<prest>
recurse(n)
<postst>
into assembly code assuming the callee-saved method is used. | 2022-09-26 23:13:08 | {"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": 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": 0.1814279556274414, "perplexity": 2956.564865561074}, "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/1664030334942.88/warc/CC-MAIN-20220926211042-20220927001042-00128.warc.gz"} |
https://electronics.stackexchange.com/questions/559451/understanding-negative-feedback-loop-circuit-with-an-ideal-op-amp?noredirect=1 | # Understanding Negative feedback loop circuit with an Ideal Op-Amp [duplicate]
The following is given as an example of a negative feedback loop in my textbook, note that the op amp is assumed to be ideal.
However I do not seem to get the logic behind this acting as a negative feedback loop, suppose if $$\V_{in}\$$ is $$\0.01 \$$ after passing through the op amp it should be amplified to $$\+6\ \mathrm V\$$.
Now $$\V_{out}\$$ is $$\+6\ \mathrm V\$$ hence the inverting input will also become $$\+6\ \mathrm V\$$ thus the difference between the two inputs becomes $$\-5.99\ \mathrm V\$$ thus $$\V_{out}= -6\ \mathrm V\$$ and this process shall repeat over and over again, would the circuit then just cycle between being $$\+6\ \mathrm V\$$ and $$\-6\ \mathrm V\$$ until the power supply is disconnected?
How is this considered a negative feedback, am I missing something crucial?
• Tips: It's $ on EE.SE for inline MathJAX. 'V' for volt, 'A' for ampere, etc. SI standards recommends a space between the numbers and units just as you would for "5 apes" rather than "5apes". If you're using MathJAX as you have for your voltages then the norm is that variables are italicised and units are not. e.g. $ U = +6 \ \mathrm V $ gives$ U = +6 \ \mathrm V \\$. – Transistor Apr 10 at 16:05
• Why don't you use the block diagram of the closed loop configuration?Draw the block diagram and the equations between Vs,Vin and Vout and it will answer all your questions. – Miss Mulan Apr 10 at 16:07
• What is your math and EE background? You are asking about stability (oscillations). A proper answer requires the knowledge of several college-level courses. For a hobbyist, just trust that most opamps are stable under most conditions. The datasheet will usually warn you if there are stability risks and how to avoid them. – Mattman944 Apr 10 at 16:13
• It's connected from the output to the negative input so, by definition, it's negative feedback. Is it stable? Not necessarily, in fact some (very old) op amp required a resistor on the loop or were stable only with some minimum gain (typically 5). Look for "unity gain stable" in the datasheet. Specific requirements are usually noted out – Lorenzo Marcantonio Apr 14 at 6:55
• The 6V supply has nothing to do with gain. | Assume that the output moves to Vin + 0.01V say - what will happen? – Russell McMahon May 5 at 11:30
## 1 Answer
This is covered in most basic op-amp tutorials but here goes:
• Assume Vout is zero on power-up.
• If VIN+ goes to 0.01 V then the difference between VIN+ and VIN- will be 0.01 V. The output will start swing to 0.01 × A where A is the op-amp's open-loop gain - typically 100,000 to 1,000,000.
• As the output start to increase the difference between the two inputs decreases due to the negative feedback.
• When the output rises to 0.01 V the difference between the inputs is zero. The output should stabilise at this voltage.
• Would the difference between the two inputs not increase since the inverting input is at 001 x A now? – Filthyscrub Apr 10 at 16:27
• @Filthyscrub do you know how to draw the block diagram of this closed configuration? – Miss Mulan Apr 10 at 16:28
• @Miss, I'm not using block diagrams to explain it and it complicates the issue. Post your own answer if you like but please don't confuse this one. – Transistor Apr 10 at 16:29
• It doesn't complicate it at all , actually it makes it more easy! – Miss Mulan Apr 10 at 16:31
• @Filthyscrub, notice that I said, "the output will start to swing to 0.01 x A. For an A of 100,000 that means it would start heading towards 1000 V but as soon as it starts the difference is decreasing. If all is well (the system is stable) it won't overshoot past 0.01 V. – Transistor Apr 10 at 16:32 | 2021-06-19 18:47:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.49100619554519653, "perplexity": 1013.7178900172486}, "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-2021-25/segments/1623487649688.44/warc/CC-MAIN-20210619172612-20210619202612-00433.warc.gz"} |
https://www.orionsarm.com/forum/showthread.php?tid=1560&pid=23341&mode=threaded | clearing bits far away from computation Bob Jenkins Posting Member Posts: 15 Threads: 1 Joined: Jun 2015 03-27-2016, 01:44 PM (10-11-2015, 06:58 AM)Tachyon Wrote: I'm not sure what you mean by "swapping bits", but writing a bit requires $\log_e(2)k_BT$ joules per bit, and is an irreversible process. Replacing (x,y) with (x, y XOR x) is reversible. It is its own reverse, because doing it twice yields (x, y XOR x XOR x) == (x, y). So this is a reversible process: (x, y) -> (x, x XOR y) -> (x XOR x XOR y, x XOR y)==(y, x XOR y) -> (y, x XOR y XOR y)==(y, x). That's swapping bits. If you want zeros in the second bit, it's enough to be able to irreversibly clear the first bit. An improvement in another part of this design: the energy and accuracy needed to toss rocks large distances. It can be aided by gravitational assists. And gravitational assists don't need accurate tosses up front, you can split the rock into two or three pieces in flight so that each piece gets a gravitational assist by different objects. If there is an equal mass of rocks going out as coming in, the accelerating assists in one direction are cancelled by the decelerating assists in the other, so the large objects doing the assist pretty much go along their orbit unhindered. I'm still stuck on the mass distribution / stability of the central core. I'm getting closer to being able to simulate it, I have a high precision floating point library and interpolation working now. « Next Oldest | Next Newest »
Messages In This Thread clearing bits far away from computation - by Bob Jenkins - 06-05-2015, 06:26 PM RE: clearing bits far away from computation - by Drashner1 - 06-06-2015, 11:08 AM RE: clearing bits far away from computation - by Bob Jenkins - 06-06-2015, 04:44 PM RE: clearing bits far away from computation - by Drashner1 - 06-07-2015, 09:07 AM RE: clearing bits far away from computation - by Bob Jenkins - 06-07-2015, 02:14 PM RE: clearing bits far away from computation - by Drashner1 - 06-08-2015, 07:38 AM RE: clearing bits far away from computation - by Rynn - 06-07-2015, 08:56 PM RE: clearing bits far away from computation - by stevebowers - 06-08-2015, 08:54 PM RE: clearing bits far away from computation - by Bob Jenkins - 06-09-2015, 12:43 PM RE: clearing bits far away from computation - by Bob Jenkins - 06-09-2015, 01:25 PM RE: clearing bits far away from computation - by Drashner1 - 06-09-2015, 01:32 PM RE: clearing bits far away from computation - by Drashner1 - 06-12-2015, 02:07 PM RE: clearing bits far away from computation - by Bob Jenkins - 06-18-2015, 01:47 PM RE: clearing bits far away from computation - by Drashner1 - 06-19-2015, 01:12 PM RE: clearing bits far away from computation - by Bob Jenkins - 07-04-2015, 05:06 PM RE: clearing bits far away from computation - by Bear - 07-05-2015, 01:50 AM RE: clearing bits far away from computation - by Bob Jenkins - 07-05-2015, 03:53 AM RE: clearing bits far away from computation - by Bear - 07-05-2015, 12:12 PM RE: clearing bits far away from computation - by Bob Jenkins - 07-06-2015, 05:31 AM RE: clearing bits far away from computation - by stevebowers - 07-05-2015, 08:10 AM RE: clearing bits far away from computation - by Bob Jenkins - 07-05-2015, 09:59 AM RE: clearing bits far away from computation - by stevebowers - 07-05-2015, 11:11 AM RE: clearing bits far away from computation - by Drashner1 - 07-05-2015, 11:22 AM RE: clearing bits far away from computation - by Bear - 07-06-2015, 07:29 AM RE: clearing bits far away from computation - by Bob Jenkins - 07-07-2015, 03:11 PM RE: clearing bits far away from computation - by stevebowers - 07-08-2015, 02:58 AM RE: clearing bits far away from computation - by Bob Jenkins - 10-07-2015, 04:41 PM RE: clearing bits far away from computation - by Bear - 10-08-2015, 04:05 AM RE: clearing bits far away from computation - by Tachyon - 10-11-2015, 06:58 AM RE: clearing bits far away from computation - by Bob Jenkins - 03-27-2016, 01:44 PM RE: clearing bits far away from computation - by Bob Jenkins - 01-06-2017, 01:59 PM RE: clearing bits far away from computation - by Tachyon - 01-25-2017, 02:24 PM RE: clearing bits far away from computation - by Bear - 03-27-2016, 04:21 PM RE: clearing bits far away from computation - by stevebowers - 01-11-2017, 04:38 AM
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https://electronics.stackexchange.com/questions/104623/minimal-distortion-in-transmission-line | # Minimal distortion in transmission line
In a transmission line, the condition for minimal distortion (caused by the line) are when (R/L = G/C) in the line, right? However, I don't see any reference to that in any application notes or reference guides that I read of microstrips or striplines or high frequency designs. The only condition that I read for the signal integrity is for reflections (impedance matching). Is there any reason for that? Is it just for simplification of the model or I'm just looking at bad quality info?
There is distortion in a line when you can't write the signal in the output as:
V2(t) = K * V1(t-tau)
That means that the signals are not same (without counting the attenuation and displacement of the signal).
There is going to be distortion in the line unless:
a-) Z_0 doesn't depend of the freq. b-) alpha doesn't depend of the freq. c-) V_p doesn't depend of the freq.
All of those conditions are ok when R/L = G/C
If you think about a,b,c : you are changing the characteristics in the line trough the frequency spectrum, so if you are going to change more some frequency components of the signal than others, so that is distortion in the line.
You can see more of the theory of distortion in t.line in this book in section 4.2.
• Dave, that's not correct. – JAMS88 Mar 29 '14 at 20:57
• Are you talking about distortion as a result of the spectrum being altered because of either or all of (a) termination impedances being wrong or (b) spectral changes due to not matching things properly? (the two are usually related but can look like different issues). – Andy aka Mar 29 '14 at 21:34
• ... that probably doesn't sound like helpful but i know what I mean.... how does this distortion manifest itself on the signals you are considering? – Andy aka Mar 29 '14 at 21:35
• Thank you for expanding on what you mean by distortion. The gist of your question seems to be that you're not satisfied with the information you're getting, but you need to give us some specific examples. – Dave Tweed Mar 29 '14 at 21:44
• @Andy aka You have a input singal (a pulse) let say that you are not obeying the minimal distortion conditions, so may be the high freq. signals will have an alpha bigger than low freq. signals, so your are going to attenute the high freq signals, so you can imagine that the pulse is going to look a lot different in the load. – JAMS88 Mar 29 '14 at 22:09
The Heaviside Condition
$\frac{R}{L}$ = $\frac{G}{C}$ or $R C$ = $G L$ or $\frac{R}{G}$ = $\frac{L}{C}$
Why isn't the Heaviside Condition more widely applied? Heaviside getting no respect even after all these years. More likely just the practicalities (impracticalities) of making the equality true.
Heaviside came up with this idea as a response to problems with telephone signal quality (in 1885ish). He found that for lossless lines and for lossy lines that met the above condition propagation (phase) velocity was independent of signal frequency. So, you would think we would be using it everywhere all the time.
For real lines though $\frac{R}{L}$ > $\frac{G}{C}$ which means you need to make R or C less, or L or G more. Decreasing R or C make the cable larger and heavier (more copper). Increasing G means exponential attenuation along the line length. Not something desirable for telephonic or long distance cables. So, for telephone lines, they inductively loaded the lines every quarter wavelength. They got good results for audio stuff, but the line in now a low pass filter. It wasn't too good for code.
Really the idea is kind of too simple. It is based on the parameters R, C, G, and L being constants over frequency. That isn't true. There is skin effect for the conductors. Materials have frequency dependent dielectric constants. If the line is loaded with a magnetic material to increase inductance, that will be frequency dependent too. If the relation works, it will only be over a restricted frequency band.
That's why all the work has been on wave compensation approaches, like solitons.
Heaviside Condition hasn't been completely abandoned though, see "Approaching Speed-of-light Distortionless Communication for On-chip Interconnect" as an example. Of course these researchers weren't trying to go across the country, just across a chip, so increasing G was OK in their case.
Is this helpful? I'll delete this if it isn't etc...
In all circumstances of signal the characteristic impedance for a transmission line is: -
$Z_0 = \sqrt{\dfrac{R + j\omega L}{G + j\omega C}}$
At low frequencies (below 100 kHz) where R and C dominate characteristic impedance is: -
$\sqrt{\dfrac{R}{j\omega C}}$ and this basically means the input impedance is X$\angle -45$
In other words at low frequencies you cannot terminate with a simple resistor but need a complex impedance. At high frequencies it becomes $\sqrt{\frac{L}{C}}$ which is a resistor.
• I'm not talking about reflections, so terminations is not important here. – JAMS88 Mar 29 '14 at 23:11
• Basically then, you are talking about a complex signal that at the low end of its spectrum borders (or includes) audio; and at the hi end encompasses 1 MHz and beyond. If this is true then if the line is properly terminated in a complex impedance that "suits" the line throughout all of the signal's spectrum you won't see the distortion you allude to? – Andy aka Mar 29 '14 at 23:29
• Of course if the line losses vary with frequency then you'll reap a distortion that can only be corrected by a compensator at the receiving end or pre-emphasis at the transmission end (or both). Still do-able and hasn't really got anything to do with R:L ratio. – Andy aka Mar 29 '14 at 23:33
• you are wrong, you are missing theory of t. lines, see 4.2 in book – JAMS88 Mar 30 '14 at 0:31
• It's possible I am wrong but I'm not playing guessing games any more so maybe you can copy n paste the section of the google book into your question because, my browser won't open it. – Andy aka Mar 30 '14 at 6:59 | 2019-10-19 08:01:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5502732396125793, "perplexity": 1093.4311653551708}, "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-2019-43/segments/1570986692126.27/warc/CC-MAIN-20191019063516-20191019091016-00198.warc.gz"} |
https://www.technicalkeeda.in/2021/01/Fail-election-codechef-january-solution.html | ## Fair Elections Codechef January long challenge 2021 solution || Fair Elections Codechef January 2021 Editorial -
### Problem Statement-
Elections are coming soon. This year, two candidates passed to the final stage. One candidate is John Jackson and his opponent is Jack Johnson.
During the elections, everyone can vote for their favourite candidate, but no one can vote for both candidates. Then, packs of votes which went to the same candidate are formed. You know that for John Jackson, there are $N$ packs containing ${A}_{1},{A}_{2},\dots ,{A}_{N}$ votes, and for Jack Johnson, there are $M$ packs containing ${B}_{1},{B}_{2},\dots ,{B}_{M}$ votes.
The winner is the candidate that has strictly more votes than the other candidate; if both have the same number of votes, there is no winner. You are a friend of John Jackson and you want to help him win. To do that, you may perform the following operation any number of times (including zero): choose two packs of votes that currently belong to different candidates and swap them, i.e. change the votes in each of these packs so that they would go to the other candidate.
You are very careful, so you want to perform as few swaps as possible. Find the smallest number of operations you need to perform or determine that it is impossible to make John Jackson win.
### Input
• The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
• The first line of each test case contains two space-separated integers $N$ and $M$.
• The second line contains $N$ space-separated integers ${A}_{1},{A}_{2},\dots ,{A}_{N}$.
• The third line contains $M$ space-separated integers ${B}_{1},{B}_{2},\dots ,{B}_{M}$.
### Output
For each test case, print a single line containing one integer ― the smallest number of swaps needed to make John Jackson win, or $-1$ if it is impossible.
### Constraints
• $1\le T\le {10}^{3}$
• $1\le N,M\le {10}^{3}$
• $1\le {A}_{i}\le {10}^{6}$ for each valid $i$
• $1\le {B}_{i}\le {10}^{6}$ for each valid $i$
• the sum of $N$ over all test cases does not exceed ${10}^{4}$
• the sum of $M$ over all test cases does not exceed ${10}^{4}$
• ${A}_{1}={A}_{2}=\dots ={A}_{N}$
• ${B}_{1}={B}_{2}=\dots ={B}_{M}$
Subtask #2 (80 points): original constraints
### Example Input
2
2 3
2 2
5 5 5
4 3
1 3 2 4
6 7 8
### Example Output
2
1
### Explanation
Example case 1: We can perform two swaps ― each time, we swap a pack of $2$ votes from $A$ and a pack of $5$ votes from $B$. After that, John Jackson gets $5+5=10$ votes and Jack Johnson gets $2+2+5=9$ votes.
Example case 2: We can swap the pack of $1$ vote from $A$ and the pack of $8$ votes from $B$. After that, John Jackson gets $8+3+2+4=17$ votes and Jack Johnson gets $6+7+1=14$ votes.
### Solution-
This problem can be solved using brute force approach first we sort both array and we will iterate second array from last and first array form first till we get sum1>sum2 .
after swapping if sum1<sum2 print -1
sum1>sum2 print count of swaping
else print 0
her is my code follow code below.
Join telegram channel for code and editorial.-https://t.me/competitiveProgrammingDiscussion
### Vaccine Distribution Codechef December Long challenge solution 2020
Vaccine Distribution Codechef December Long challenge solution 2020- This Problem vaccine distribution is taken from December long challenge 2020. lets read problem statement. Problem statement- Finally, a COVID vaccine is out on the market and the Chefland government has asked you to form a plan to distribute it to the public as soon as possible. There are a total of N N people with ages a 1 , a 2 , … , a N a 1 , a 2 , … , a N . There is only one hospital where vaccination is done and it is only possible to vaccinate up to D D people per day. Anyone whose age is ≥ 80 ≥ 80 or ≤ 9 ≤ 9 is considered to be at risk . On each day, you may not vaccinate both a person who is at risk and a person who is not at risk. Find the smallest number of days needed to vaccinate everyone. Input The first line of the input contains a single integer T T denoting the number of test cases. The description of T T test cases follows. The first line of each test case contains two space-separ
### Codechef February long challenge solution 2021
Codechef February long challenge solution 2021- Hey guys Welcome Again, A small update CodeChef February long challenge solution 2021 starting from 1st week of beb 2021 . Codechef long challenge is one of the popular contests of CodeChef you should participate in codechef Feb long challenge.This is first contest of codechef in 2021 so good luck. And wish you happy new year. Contest link- https://www.codechef.com/FEB21/?itm_medium=navmenu&itm_campaign=FEB21 I will post the solution after Contest.Stay tuned. About CodeChef February long challenge solution 2021: CodeChef Long Challenge is a 10-day monthly coding contest where you can show off your computer programming skills. The significance being - it gives you enough time to think about a problem, try different ways of attacking the problem, read the concepts etc. If you’re usually slow at solving problems and have ample time at hand, this is ideal for you. We also put in a lot of efforts in getting quality problems, w
### Merge Two sorted array Without Extra Space
Problem statement- Given two sorted arrays arr1[] and arr2[] of sizes N and M in non-decreasing order. Merge them in sorted order without using any extra space. Modify arr1 so that it contains the first N elements and modify arr2 so that it contains the last M elements. Example 1: Input: N = 4, arr1[] = [1 3 5 7] M = 5, arr2[] = [0 2 6 8 9] Output: arr1[] = [0 1 2 3] arr2[] = [5 6 7 8 9] Explanation: After merging the two non-decreasing arrays, we get, 0 1 2 3 5 6 7 8 9. Example 2: Input: N = 2, arr1[] = [10, 12] M = 3, arr2[] = [5 18 20] Output: arr1[] = [5 10] arr2[] = [12 18 20] Explanation: After merging two sorted arrays we get 5 10 12 18 20. Your Task: You don't need to read input or print anything. You only need to complete the function merge() that takes arr1, arr2, N and M as input parameters and modifies them in-place so that they look like the sorted merged array when concatenated. Expected Time Complexity: O((n+m) log(n+m)) Expected Auxilliary Space: O(1 | 2022-01-16 19:05:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 37, "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": 0.28074395656585693, "perplexity": 1967.4408427423941}, "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/1642320300010.26/warc/CC-MAIN-20220116180715-20220116210715-00258.warc.gz"} |
https://pretextbook.org/examples/sample-book/noparts/html/parsons-exercises.html | # PreTeXt Sample Book: Abstract Algebra (SAMPLE ONLY)
## Exercises5.9Parsons Exercises
### 1.Parsons Problem, Mathematical Proof.
Create a proof of the theorem: If $$n$$ is an even number, then $$n\equiv 0\mod 2\text{.}$$
Hint.
Dorothy will not be much help with this proof.
### 2.Parsons Problem, Programming.
The Sieve of Eratosthenes computes prime numbers by starting with a finite list of the integers bigger than 1. The first member of the list is a prime and is saved/recorded. Then all multiples of that prime (which not a prime, excepting the prime itself!) are removed from the list. Now the first number remaining in the list is the next prime number. And the process repeats.
The code blocks below can be rearranged to form one of the many possible programs to implement this algorithm to compute a list of all the primes less than $$250\text{.}$$ [Ed. this version of this problem requires the reader to provide the necessary indentation.]
This reprises Exercise 2.5.1.
### 3.Parsons Problem, Programming.
The Sieve of Eratosthenes computes prime numbers by starting with a finite list of the integers bigger than 1. The first member of the list is a prime and is saved/recorded. Then all multiples of that prime (which not a prime, excepting the prime itself!) are removed from the list. Now the first number remaining in the list is the next prime number. And the process repeats.
The code blocks below can be rearranged to form one of the many possible programs to implement this algorithm to compute a list of all the primes less than $$250\text{.}$$ [Ed. this version of this problem does not require the reader to provide the necessary indentation, which is the default.]
This reprises Exercise 2.5.1.
### 4.Parsons Problem, Mathematical Proof, Numbered Blocks.
Create a proof of the theorem: If $$n$$ is an even number, then $$n\equiv 0\mod 2\text{.}$$ [Ed. This version has numbered blocks, online they are on the right end of the block.]
Hint.
Dorothy will not be much help with this proof.
### 5.Parsons Problem, Programming.
The Sieve of Eratosthenes computes prime numbers by starting with a finite list of the integers bigger than 1. The first member of the list is a prime and is saved/recorded. Then all multiples of that prime (which not a prime, excepting the prime itself!) are removed from the list. Now the first number remaining in the list is the next prime number. And the process repeats.
The code blocks below can be rearranged to form one of the many possible programs to implement this algorithm to compute a list of all the primes less than $$250\text{.}$$ [Ed. This version has numbered blocks, online they are on the left end of the block.]
This reprises Exercise 2.5.1. | 2023-03-30 07:50:27 | {"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": 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": 0.7675427794456482, "perplexity": 440.41660707174935}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949107.48/warc/CC-MAIN-20230330070451-20230330100451-00708.warc.gz"} |
http://mathoverflow.net/questions/473/what-is-the-expected-number-of-maximal-bicliques-in-a-random-bipartite-graph/486 | # What is the expected number of maximal bicliques in a random bipartite graph?
Maximal Biclique: A complete bipartite subgraph, that isn't a subgraph of another complete bipartite subgraph.
Given a bipartite graph $G=(V_{1}\cup V_{2}, E)$ where $|V_{1}|=|V_{2}|$ with probability $p$ of there being an edge from any $a\in V_{1}$ to any $b\in V_{2}$, what is the expected number of maximal bicliques?
What I have worked out is the upper and lower bounds:
Lower Bound: 1 or 2. If $|E|$ is divisible by $n$, then $\frac{|E|}{n}$ nodes can be connected to completely to $\frac{|E|}{n}$ other nodes, making one maximal biclique. Otherwise, connect $\frac{|E|}{n}$ nodes completely to $\frac{|E|}{n}$ nodes and connect one node to $|E|(mod\ n)$ nodes.
Upper Bound: There are $2^n$ unique subsets, the empty set and the entire set not included, leaves $2^n-2$ subsets. Therefore, there can be at most $2^n-2$ maximal bicliques. This upper bound is achievable when there are $n^2-n$ edges (I can prove this if anyone wants me to).
Both of these results are also easily extended to bipartite graphs where $|V_{1}|\neq |V_{2}|$.
The upper and lower bounds are both fairly trivial for the most part and it's the expected number of maximal bicliques that I've had the most trouble with. I've done a little work analyzing simple cases and brute forcing the expected number for small values of $n$ (I suppose I could write a program to brute force larger values of $n$), but it hasn't amounted to anything worth saying.
I'd appreciate any suggestions for methods of attacking the problem or references that I might find useful.
-
The expected clique number (i.e. size of the maximum clique) in a random graph (all edge probabilities = 1/2) is around 2log_2(n) where n is the number of vertices. You can find the proof in Alon and Spencer's "The Probabilistic Method", chapter 4. My guess is that a similar method would apply in the bipartite case, with a similar result. It also shouldn't be hard to extend the result to general p.
I you want to understand how to solve this kind of problems, I can't think of any better way than to carefully study the first few chapters of "The Probabilistic Method".
-
The "maximal clique" (as used by liberalkid) and the "maximum clique" (as used by Alon Amit) are not the same thing. A maximal clique is just a clique that's not contained in any larger clique. A single edge can be a maximal clique, and for p = C/n, where n is the number of vertices in the graph, I believe the number of size-1 maximal cliques approaches some constant as n gets large.
That being said, the methods of Alon and Spencer's book are probably still useful.
-
Bleah. Of course, sorry. – Alon Amit Oct 14 '09 at 18:19
One approach might be as follows (I will assume your initial graph had n vertices on each side):
Given a subset S of size s on one side, and a subset T of size t on the other side, what is the probability it is a maximal clique? We need two things to happen.
1. It must be a clique. This occurs with probability p^{ab}.
2. No vertex outside the clique is connected to all the vertices inside the clique on the appropriate side. For any given vertex, this occurs with probability (1-p^{n-a}) or (1-p^{n-b}), depending on which side it needs to fail to connect to. The handy thing is that everything in sight is independent here, so we can just multiply over all vertices outside the clique.
Multiplying, the probability SXT is a maximal clique is p^{ab} (1-p^{n-a})^b (1-p^{n-b})^a. Now by linearity of expectation you just need to add up over all subsets, and (possibly) work out the asymptotics as n tends to infinity.
- | 2016-06-28 15:21:27 | {"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": 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": 0.871261477470398, "perplexity": 263.3918461581984}, "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-2016-26/segments/1466783396945.81/warc/CC-MAIN-20160624154956-00047-ip-10-164-35-72.ec2.internal.warc.gz"} |
http://www.sciforums.com/threads/work-done-separating-2-masses-gravitation.111071/ | # Work done separating 2 masses -gravitation
Discussion in 'Physics & Math' started by Robittybob1, Nov 21, 2011.
Not open for further replies.
1. ### Robittybob1BannedBanned
Messages:
4,199
Can anyone help out with the equation for the work done separating two masses gravitationally attracted through a distance?
It is a variation of "E = mgh" but g there is the g (9.8) of the earth and we are looking at the situation where it is no where near the earth.
3. ### TachBannedBanned
Messages:
5,265
you need to overcome the attraction force between the two bodies:
$f=\frac{Gm_1m_2}{r^2}$
The total work is the integral between limits $r_1$ and $r_2$ of the elementary work $f dr$. Do you know how to calculate integrals?
5. ### Robittybob1BannedBanned
Messages:
4,199
No I don't know about integrals. But if the two masses are in contact at the beginning and then separate to a distance of 100 million light years how does one integrate that?
Last edited: Nov 21, 2011
7. ### ReikuBannedBanned
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11,238
Are you even sure you know how to conduct an integral TACH? You messed up in your marvelous involving the integral in a discussion with Pryzk.
8. ### TachBannedBanned
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5,265
Actually, I did not, Green Destiny. It is going to be interesting to see how long you'll manage to evade your permaban, sockpuppet.
9. ### ReikuBannedBanned
Messages:
11,238
By the looks of things, TACH, you might be well on your way for one youself if you keep wasting their valuable time down there, continue to delete posts when you have spotted an error in your own work, or keep back-tracking on your pathetic attempts to explain a physical set-up. You don't even comprehend some of the more easier physical set-up's which the likes of RJBeery can even catch you out on.
Damn, I'd even place money on Pincho Paxton to give one of your threads a little look over to see if there are any contradictions... mind you, he might not suceed. You are infamous to deleting those posts
Messages:
4,199
11. ### TachBannedBanned
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5,265
That's too bad, you will need to learn first. Is this homework? What grade are you in?
12. ### ReikuBannedBanned
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11,238
Oh no, you don't get off that easy TACH. Let's see if you can put your money where your mouth is?
this is a physics forum where people come to get answers. You can't prance around saying ''oh, do you know how to do an integral?'' And then not even offer to give any valuable help.
You seem to have forgotten what a forum is for. So without further ado, please, show us how you would perform this integral. Don't just say you can. You do a lot of talking but never putting up a lot of the time. If not, don't bother going into someone elses thread and question them on what they can or can't do, if you can't even do it yourself.
13. ### ReikuBannedBanned
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11,238
Don't try and weasle your way out of pretending you know how to do something.
You are the one prancing about saying you know s**t, so put up or shut up! Admit to Robbitybob that you where just poking at him for fun of not knowing integrals and that you also had no intentions of helping?
You don't need to admit this to me. I saw the direction your post was going, hence why I stepped in.
14. ### ReikuBannedBanned
Messages:
11,238
No Tach, that's called passing the buck. Can you find any other arguement than stealing my own? I said you should put up or shut up. You are the one with a stick, who came in here saying ''Oh, can you do integrals?''
Then when he honestly replied ''no'' You basically said ''oh too bad, go get an education.''
I have nothing to prove except your own fallacies. I am confident in what I know. I want to see if you do! Your blatent misuse of what a forum is for makes me angry.
15. ### Robittybob1BannedBanned
Messages:
4,199
Can anyone help out with the equation for the work done separating two masses gravitationally attracted through a distance?
It is a variation of "E = mgh"
16. ### Robittybob1BannedBanned
Messages:
4,199
therefore,
U = -M * Integral G*m/r
Where r = 0 to 100 million light years.
17. ### Aqueous Idflat Earth skepticValued Senior Member
Messages:
6,152
Getting back to the question, I think he wanted to confirm mgh.
The answer is yes, force times distance equals work.
18. ### OnlyMeValued Senior Member
Messages:
3,914
If this is homework the following may be of no use and could even confuse the problem. If it is homework it might be helpful to provide a better description of the problem.
If what you are asking for is the total force or work done, when moving an object or mass from the surface of a significant gravitational object or the earth, to a distance some specific distance from the gravitational source or the earth.., and there are no other forces or gravitationally significant objects involved and assuming that even at that distance there remains some interaction between the two masses....., there are two answers...
1 determine the force require to accellerate the mass to be moved away from, in this case the earth, at the escape velocity required to overcome the Earth's gravitational interaction with the object. In this case the object will with a single accelleration coast to the desired distance of separation.., and then continue moving away from the earth.
2 taking the above determined force, subtract the effective force of the gravitational interaction between the earth and the object at the desired distance and you will have the force required to move the object, to the specified distance..., at which time it will come to rest and the mutual gravitational interaction between the two objects will draw them back together.
The only way to address this issue is in the abscence of all other forces except those between the two masses involved and there is no solution that does not involve a final stable orbit that does not result in the smaller object either continuing to move away from, or be attracted back toward the larger object.
19. ### Robittybob1BannedBanned
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4,199
This not a homework problem, but some concept I have discussed earlier. on another thread.
That is exactly the solution. The problem was thinking of related to a neutron that is able to be split into its components. Will the gravitational attraction of the parts pull them together again. But just think of these parts as mass. and one part (the smaller mass) is converted to energy how far will that energy go before it is pulled back down by its own gravitational attraction to the remaining part?
I have a feeling my original calculations were wrong (but I don't know where these are now) but I remember the result and the height I got was 100 million light years.
It is a bit like the Schwarzschild radius of a black hole, what is the Schwartzschild radius of a singular neutron?
I just see now there is an article on the web called "The Schwarzschild Proton". I haven't read it.
But I was thinking, a graviton may be an extension of mass from a proton or neutron that extends into space and then drawn back, just like a photon might till it is drawn back by the gravitational pull of a Black Hole.
But the physics of the work I had done in 1998 is now so rusty and I was struggling but I think you have shown me how it could be done.
Last edited: Nov 21, 2011
20. ### originTrump is the best argument against a democracy.Valued Senior Member
Messages:
10,047
your on the right track -
integrate $f=\frac{Gm_1m_2}{r^2}$
with respect to r.
The rest is constant so all you have to integrate is $r^-^2$
The limits of r will be the closest that the centers of the masses are from each other (NOT 0) and the 100 million light years (hint: the constants over 100 million light is effectively 0).
21. ### Aqueous Idflat Earth skepticValued Senior Member
Messages:
6,152
I obviously misunderstood the question.
You are not only not on earth, you are nearly at the span of a filament.
I'm not sure what you really were asking.
Yes work and energy are equivalent. No, not the integral of that.
G m1 m2 /r is the integrated result
calculate over the span from a to b:
U = G m1 m2 (1/b - 1/a)
I don't think you'e on the right track though
you can't start from a distance of zero, as this requires infinite energy to overcome
and the other term is way small.
A lot of assumptions seem to be at play, maybe those need to come out
maybe you could rephrase the question
what are you really after?
22. ### Aqueous Idflat Earth skepticValued Senior Member
Messages:
6,152
OK my lag time is making me irrelevant
Hey - how come you were doing physics in 1998 and can't remember integrals?
Most people remember that and forget the rest!
23. ### Robittybob1BannedBanned
Messages:
4,199
To you I will be as open as I can be, right. I know very little about subatomic particles but from the make up of a proton and neutron I envisioned that the basic structure is a 27 loop spiral of energy like a spring pulled around and joined back to itself. That part was made from 2 of the like components (quarks).
The other part ran through the centre of this torus like a particle in an accelerator through the coils. Now this centre piece was a possible source of gravity if it became unbound and shot out but returns as well.
So that is from where I started to think how far this energy could go before it was pulled back to itself.
Can you follow that? In 1998 no one had any idea what I was on about so I stopped looking into it. | 2018-01-20 06:53:27 | {"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": 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": 0.6630272269248962, "perplexity": 827.8569509059471}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084889473.61/warc/CC-MAIN-20180120063253-20180120083253-00001.warc.gz"} |
https://stacks.math.columbia.edu/tag/04CN | Definition 75.5.1. Let $i : Z \to X$ be an immersion. The conormal sheaf $\mathcal{C}_{Z/X}$ of $Z$ in $X$ or the conormal sheaf of $i$ is the quasi-coherent $\mathcal{O}_ Z$-module $\mathcal{I}/\mathcal{I}^2$ described above.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). | 2023-03-29 14:18:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9793946743011475, "perplexity": 352.8336441056036}, "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-2023-14/segments/1679296948976.45/warc/CC-MAIN-20230329120545-20230329150545-00150.warc.gz"} |
https://www.physicsforums.com/threads/geiger-counter.90147/ | Geiger counter
1. Sep 21, 2005
choole
A particular Gieger counter has a metal cylinder with an inner diameter of 2 cm along whose axis is stretched a wire with 44 N of tension. The potential difference between the wire and the cylinder is 890 volts. The wire has a length of 6 cm and an outer diameter of 1.5 X 10-4 cm.
A. To find the electric field at the surface of wire?
B To find the electric field at the inner surface of the cylinder
Vf - Vi = int(E dot ds) from radius of wire to cylinder.
and E = lemda/2piE0r
i tried solving like this but i didn't get the answer. Please help
Last edited: Sep 21, 2005
2. Sep 21, 2005
Integral
Staff Emeritus
It is not clear to me what your goal is? What are you looking for?
3. Sep 21, 2005
choole
To find the electric field at the surface of the wire?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook
Have something to add? | 2017-02-21 01:20:15 | {"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": 0.8023157715797424, "perplexity": 931.5095082486038}, "config": {"markdown_headings": false, "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-2017-09/segments/1487501170614.88/warc/CC-MAIN-20170219104610-00520-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://cran.stat.auckland.ac.nz/web/packages/nlmixr2lib/vignettes/create-model-library.html | # Creating a model library
Model libraries are useful to have consistent high-quality basic models that can be used as a model itself or as a building block for other models.
# Model library conventions within nlmixr2lib
Compartment and parameter names should be all lower case when on their own and should use snakeCase when combined in some way.
Compartment and parameter names are selected to align with those used by rxode2::linCmt() which are described in the vignette: vignette(“rxode2-model-types”, package = “rxode2”).
## Compartment naming
Compartment naming follows compartment names with the linCmt() with augmentation for other compartments:
• depot: The extravascular dosing compartment (for example, the gut for oral dosing or subcutaneous space for subcutaneous dosing)
• central: The intravascular compartment used for intravenous dosing or for typical pharmacokinetic (PK) model sampling of the drug
• peripheral1, peripheral2: The first and second peripheral compartments for 2- and 3-compartment PK models
• effect: The compartment for effect compartment models
• Therapeutic-area-specific models should use consistent compartment and parameter naming. When adding a new therapeutic area model to the library, please discuss naming first in a new GitHub issue.
## Estimated parameter naming
To enable more consistent cross-model compatibility, the following conventions should be used unless there is a strong reason for an exception:
• Pharmacokinetic concentrations in the central compartment should be named cp. cp should be used even when using a linCmt() model (in which case cp <- linCmt() should be used and the residual error should be applied to the cp parameter).
• Therapeutic-area-specific models should use consistent compartment and parameter naming. When adding a new therapeutic area model to the library, please discuss naming first in a new GitHub issue.
## Parameter naming
PK models should use the following parameter naming conventions:
• ka: absorption rate
• cl: clearance
• q: intercompartmental clearance (central to and from peripheral1 compartments)
• q2: second intercompartmental clearance (central to and from peripheral2 compartments)
• vc: central volume of distribution
• vp, vp2: first and second peripheral compartment volumes
When micro-constants are used, they should use the following naming conventions:
• kel elimination rate (cl/vc)
• k12, k21, k13, k31: intercompartmental transit rates (q/vc, q/vp, q2/vc, and q2/vp2, respectively)
## Parameter transforms
Parameters are often estimated on a transformed scale. For instance, a natural logarithm transform is often used for parameters that must be positive, and a logit transform is often used when a parameter must remain within a specific range.
Transformed parameters should be prefixed with an indicator of the transformation. Preferred transformation prefixes are:
• l (lower case L): natural log transform
• logit: logit transform
• probit: probit transform
• Any other transform should similarly include the full transform as the prefix
Generally, for any transform other than natural logarithm, include the full name as a prefix. For example, natural logarithm-transformed ka would be lka and logit-transformed emax would be logitemax.
## Random effects
Random effects are estimates as part of a distribution varying by some grouping factor. The grouping factor is often a subject in a clinical trial. (For NONMEM users, random effects are often referred to as inter-individual variability.)
Random effect parameters should prefix the (transformed) parameter name with eta. For example, a random effect on log-transformed clearance would be named etalcl.
## Drug effects
Different drug effects may be investigated during model building. And, multiple drug effect styles (linear, Emax, threshold, etc.) may be investigated by the user.
To enable simpler changes to drug effects and to minimize the chance of parameter name collisions when combining models, the following rules are strongly recommended:
• Drug effects should be calculated on a model by themselves to enable changing the type of drug effect (e.g. linear to Emax).
• The parameter name for the drug effect should be called drugEffect followed by the name of the part of the model that is most closely associated with the drug effect. For example, in the Simeoni 2004 model, the drug effect is called drugEffectCyclingCells.
# Model files
Files in a model library should have the following characteristics:
• The first line inside the function should have a description assignment. That is description <- "This is the description of the model" right inside the function() before the ini({}) block.
• If the model has a literature reference associated with it, then the second line of inside the function should have the reference, for example, reference <- "Richard Hooijmaijers, Matthew Fidler, William S. Denney (2022). nlmixr2lib: A Model Library for 'nlmixr2'. https://nlmixr2.github.io/nlmixr2lib/"
• If it would be helpful to give the user some information about the model on load, it can be added as meta-data as a "message" attribute to the model. Note that in that case, you must give the function name as the last line of the model to ensure that it is the returned value from evaluation of the file. (See oncology_xenograft_simeoni_2004.R for an example of adding a message.)
• The remainder of the file should be an nlmixr2 model in a function with a typical ini() and model() block.
• The name of the file should match the name of the model within the file.
If a function to modify, self-start, or otherwise help the user would make sense, add it as a new file in the R/ directory with the file name and function name updateModelName() using the word update followed by the model name in camelCase (e.g. updateOncologyXenograftSimeoni2004). If such a function is added, please add it in the messages described above, as well. Update functions must be able to take in a function, an rxUi object, or an nlmixr2fitCore object and should usually return an rxUi object.
For examples, see the package installation directory. | 2023-03-31 16:18:47 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5322937369346619, "perplexity": 3344.3015864127037}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949644.27/warc/CC-MAIN-20230331144941-20230331174941-00246.warc.gz"} |
http://dept.swccd.edu/hlee/content/simulation/simulation-relativity/index.html | # Simulation - Relativity
## Speed of Light
Canvas not supported
#### Speed of Light Measurement
This simulation illustrates the theoretical ideas behind the speed of light experiment.
Instructions:
Drag the detector to where the photos are exiting in the end to measure the speed of light (see calculations below).
Set the angular velocity of the rotating mirror to zero to see how the photons travel. You could drag on the rotating mirror to change its angle directly.
As long as the rotating mirror is at rest, the photons always exit at zero angle.
Now turn on the rotating mirror and adjust the "speed of light" (something you cannot do in real life!) and see how it affects the angle of they exit the instrument.
The higher the speed of light, the smaller is the angle of exit.
Explanation:
When the angular velocity of the rotating mirror is zero, a photons travels the same way back, returning to the detector at zero angle of exit.
When the mirror is rotating, a photon who travels to the circular mirror and comes back to the rotating mirror will find the mirror at a different orientation from before due to the finite amount of time it takes to run to the edge and back to the middle. As a result, the photo no longer can return the same way and will exit at a non-zero angle. The lower the speed of light, the more time it allows the mirror to rotate, and the larger is the final angle. Therefore by measuring the angle one can deduce the speed of light.
Remarks:
In the actual experiment, the circular mirror will need to have an opening (not shown in the simulation for simplicity) on the right for the photon to escape.
The "speed of light" in the simulation is very low compare to the actual speed of light, so in practice the rotating mirror will need to rotate at an extremely high rate to achieve just a tiny deflection. Usually the angle of exit is so close to zero that one needs to use a microscope to observe it!
## The Relativity of Simultaneity
Canvas not supported
#### The Relativity of Simultaneity
Two events occurred simultaneously from the perspective of the top observer.
See if they are also simultaneous for the bottom observer based on the light signals from the two events.
Note that the simulation is in the the perspective of the top observer. Therefore due to the length contraction the bottom ship would actually be longer than the ship at the top if they are both at rest.
## Time Dilation
Canvas not supported
#### Relativity: Time Dilation
Adjust the velocity and see how time flows at different rate in the two light clocks.
Note that the photons always travel at the speed of light $c$.
## Length Contraction
Canvas not supported
#### Length Contraction
Change the velocity to observe length contraction.
You can also drag the two objects together to compare their sizes.
## The Necessity of Length Contraction
Canvas not supported
#### The Necessity of Length Contraction
The view at the top shows two photons bouncing inside an L-shaped ship. The two photons always gets back to the corner at the same time.
The view at the bottom is the same ship viewed from a moving frame. If length contraction is disabled (so the horizontal arm does not contract), the two photons will be coming back out of sync, leading to contradiction.
Therefore length contraction is necessary to perserve the local observation (made at the corner) that the two photons always arrive back at the same time.
Although it may appear to you the photons are moving at different speed, in fact all photons are always traveling at the same speed. The illusion arises because the photons are moving on top of a moving object, thus making some appear to travel faster than others.
Canvas not supported | 2018-12-12 03:26:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6278102397918701, "perplexity": 472.5286787974531}, "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-2018-51/segments/1544376823712.21/warc/CC-MAIN-20181212022517-20181212044017-00001.warc.gz"} |
https://www.vedantu.com/maths/surface-area | Courses
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# Surface Area of Solids of Various Shapes (Cube Cuboid and Cylinder)
## An Introduction to Surface Area of Solids of Various Shapes
Last updated date: 23rd Mar 2023
Total views: 39.3k
Views today: 0.17k
Three-dimensional shapes' surface area is one of their most essential characteristics. The surface area of a shape is a topic that comes up frequently in geometry. Using a general formula for surface area, it is possible to solve the objects having simple shapes and forms. We will need to use our understanding of one or more 2-dimensional shapes to handle more challenging shapes.
In this article, we will find the surface area of cube and cuboid as well as cylinder. Moreover, we will determine the formula of total surface area of cuboid, cube and cylinder. We can readily determine their surface area and volume using the established equations. Have you ever considered what may happen if these fundamental shapes combined to create a new shape? How will we determine the new shape's capacity then? This is explained in the article that follows.
## Surface Area Definition
Any solid shape's surface area is equal to the sum of all of its faces' respective areas. For instance, we add the areas of each of the rectangles that make up a cuboid to determine its surface area. When calculating the surface area of complex shapes, the combined area of all the surfaces of the shapes that make up a complex structure is calculated.
## There are Primarily Two Categories of Areas:
• Total Surface Area
• Curved Surface Area/Lateral Surface Area
### Total Surface Area
The total surface area is the area that includes the base(s) and the curving portion. It is the overall area that the object's surface occupies. If the shape has a curved base and surface, the total area will be equal to the sum of the two areas.
### Curved Surface Area/Lateral Surface Area
The area of just the curved part of the shape, excluding its base(s), is referred to as the curved surface area. For shapes like a cylinder, it is sometimes referred to as the lateral surface area.
### The Below Table shows the Surface Area of the Various Shaped Objects:
Name Figure Total Surface Area Lateral Surface Area Cube $6{a^2}$ $4{a^2}$ Cuboid $2(lb + bh + hl)$ $2h(l + b)$ Cylinder $2\pi r(r + h)$ $2\pi rh$
## Surface Area of Cuboid Formula
1. Total Surface Area of Cuboid : The surface areas of each of the six sides can be added up to determine their total area. Due to the fact that a cuboid's faces are all rectangles, the total surface area of the cuboid is determined by adding the areas of each rectangle on each face. With the guidance of the following figure, let's understand this better.
A Cuboid
Cuboid with Different Faces
According to the figure above, all of the faces are numbered $1$, $2$, $3$, $4$, $5$ and $6$. In other words, we get this figure if we visualise the cuboid as a two-dimensional net.
• Area of rectangles $1$ and $2$: This refers to the area of the rectangle with the top and bottom faces equal to $l \times w$. As a result, $lw + lw = 2lw$.
• Area of rectangles $3$ and $4$: This refers to the area of the rectangle that comprises the front and back faces and has the dimensions $h \times w$. As a result, it is, $hw + hw = 2hw$ .
• Area of rectangles $5$ and $6$: This refers to the area of the rectangle formed by the faces on the left and right sides, which is equal to $l \times h$. As a result, $lh + lh = 2lh$ .
• The surface area of the six faces is, therefore, equal to $2lw + 2hw + 2lh$. So, a cuboid with $l$, $w$, and $h$ dimensions will have a total surface area of $2(lw + hw + lh)$ .
1. Lateral Surface Area of Cuboid : The total surface area of a cuboid's four vertical sides makes up its lateral surface area of the cuboid. If the top and bottom faces are taken out of the above figure, the lateral surface area of the cuboid will be identified. The cuboid's lateral surface area is
Total Surface Area $-$ Area of Top and Bottom Faces $=$ Lateral Surface Area
Lateral Surface Area $= 2(lw + hw + lh) - 2(lw)$
$= 2lw + 2hw + 2lh - 2lw$
$= 2hw + 2lh$
$= 2h(l + w)$
## Surface Area of Cube Formula
1. Total Surface Area of Cube :To determine the area filled by the six surfaces, use the cube's total surface area formula. The cube's total surface area is calculated by multiplying the square of the side length by six. As a result, the formula for the cube's surface area with side length "$a$" is "$6{a^2}$".
A Cube
1. Lateral Surface Area of Cube : The area occupied by the $4$ lateral or side surfaces of the cube is calculated using the cube's lateral surface area formula. The cube's lateral surface area is determined by multiplying its side length’s square by $4$. Therefore, "$4{a^2}$" is the formula for the cube's lateral surface area with side length "$a$".
## Surface Area of Cylinder Formula
1. Curved/Lateral Surface Area of Cylinder : A cylinder's curved surface area is the area that is entirely covered by its curved surface. The curved surface area of a cylinder is estimated using the formula below if the height of the cylinder is $h$ and the radius of the base is $r$:
Lateral surface area of cylinder $= 2\pi rh$
1. Total Surface Area of Cylinder : The area of the two bases as well as the area of the curved surface are added to determine the cylinder's total surface area. Consequently, the following is the formula for the cylinder's total surface area:
Area of the two bases$+$Area of the curved surface$=$Cylinder's total surface area.
Due to the cylinder's circular bases, their total area will be equal to $\pi {r^2} + \pi {r^2}$ . We also know that a cylinder's curved surface area is $2\pi rh$.
The cylinder's total surface area is $(\pi {r^2} + \pi {r^2}) + 2\pi rh$ .
Cylinder's total surface area is equal to $2\pi r(r + h)$ .
1. Formulation for the Surface Area of a Cylinder : Any shape's area is the space it occupies. A cylinder has a curving surface that unfolds into a rectangle and two flat surfaces that are both circulars. Look at the cylinder with the provided height ("$h$") and radius ("$r$"). To further understand this, let's open a 2-dimensional cylinder.
Cylinder
Cylinder Expanded in the Shape of a Rectangle
Look at the figure above, where the two bases are circles and the area of the curved surface expands up as a rectangle.
• The area of the two curves is now $(\pi {r^2} + \pi {r^2})$ , where $r$ is the base radius.
• One side of the rectangle corresponds to the cylinder's height, $h$, and its length, $2\pi r$, to the circle's diameter.
• The area of this rectangle $(l \times b)$ is, therefore, equal to $2\pi r \times h$, which is also the cylinder's curved surface area.
• As a result, the cylinder's total surface area is equal to $2\pi {r^2} + 2\pi rh$ , or $2\pi r(r + h)$ .
### Solved Examples
Q.1. Determine the total surface area of a cubic construction whose sidewalls are $7$ metres long.
Solution: Given that the sidewall's length is $7$ metres.
Using the formula, we are aware that
Total Surface Area is $6{a^2}$.
Total Surface Area $= 6 \times 7 \times 7 = 294$ sq. m.
Q.2. Determine the surface area of a cylindrical tank with a radius of $4$ yards and a height of $8$ yards using the cylinder's surface area formula. What will the overall cost of the painting be if the cylindrical tank painting costs $\ 6$ per square yard?
Solution: We are aware that the formula for a cylinder's total surface area is:
Total Surface Area $=$ Curved Surface Area + Area of Top and Bottom Faces
$= 2\pi rh + 2\pi {r^2} = 2\pi r\left( {r + h} \right) = 2 \times \dfrac{{22}}{7} \times 4\left( {4 + 8} \right) = 301.68$ square yards.
Painting costs come out to $301.68 \times \ 6$ per square yard, or $\ 1810.08$ .
The cost of the artwork is $\ 1810.08$ .
Q.3. Determine the cuboid's total surface area and lateral surface area. Its dimensions are $6$ inches long, $4$ inches wide, and $3$ inches high.
Solution: As is general information, a cuboid's total surface area is equal to $2\left( {lb + bh + lh} \right)$and its lateral surface area is equal to $2h(l + b)$.
Here, the dimensions are $6$ inches long, $4$ inches wide, and $3$ inches tall.
Total surface area of the cuboid is equal to $2\left( {lb + bh + lh} \right) = 2{\text{ }}\left[ {\left( {6 \times 4} \right) + \left( {4 \times 3} \right) + \left( {6 \times 3} \right)} \right]$ sq. inches
Cuboid's total surface area $= 108$ sq. inches.
Cuboid's lateral surface area is equal to $2h\left( {l + w} \right) = 2 \times 3\left( {6 + 4} \right)$ sq. inches and is therefore equal to $60$ sq. inches.
As a result, the cuboid's lateral surface area is $60$ sq. inches and its total surface area is $108$ sq. inches.
### Practice Questions
1. Obtain the surface area of the cube where the edge is $4$ inches using the cube's surface area formula.
2. Identify the total surface area of a cylindrical container with a $28$ cm diameter and a $15$ cm height.
### Answers
1. $96$ sq. inches
2. $2552$ sq. cm
### Summary
The surface area of an object is the total area occupied by all of its surfaces. Two categories—Curved Surface Area (CSA) and Lateral Surface Area (LSA), respectively—are used to categorise the surface's area (TSA). The term "curved surface area," also referred to as "lateral surface area," refers to the surface area of a surface which is curved in shape. The bottom and top sections, as well as the lateral surface area, make up the overall surface area. The three-dimensional shapes' surface areas are included in this article. It greatly facilitates the speedy resolution of issues based on it.
Competitive Exams after 12th Science
## FAQs on Surface Area of Solids of Various Shapes (Cube Cuboid and Cylinder)
1. Define a cylinder.
One of the fundamental three-dimensional shapes in geometry is the cylinder, which has two distantly spaced parallel circular bases. A curving surface located a set distance from the centre connects the two circular bases. The axis of the cylinder is the line segment that connects the centres of two circular bases. The height of the cylinder is defined as the separation between the two circular bases. One of the actual instances of cylinders is the LPG gas cylinder. The cylinder has two main characteristics because it is a three-dimensional shape: surface area and volume.
2. Are the curved surface area and total surface area similar?
Curved Surface Area (CSA), the term refers to the entire area of all the curved surfaces. The term "lateral surface area" (LSA) refers to the total area of the surface, apart from the top and bottom areas.
Total Surface Area (TSA): This measurement considers the area of the bases as well as all of the object's surface.
3. Describe the differences between volume and surface area.
The area that an object occupies on its surface is referred to as its surface area. Geometry includes a wide range of shapes and dimensions, including spheres, cubes, cuboid shapes, cones, and cylinders, among others, whereas volume specifies how much space it has. We can only measure the area of two-dimensional shapes like squares, circles, rectangles, triangles, and so on because there is no volume present. | 2023-03-25 20:04:08 | {"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": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8832756876945496, "perplexity": 499.5345893506841}, "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/1679296945372.38/warc/CC-MAIN-20230325191930-20230325221930-00401.warc.gz"} |
http://mlwiki.org/index.php/Voting_Theory | # ML Wiki
## Voting Theory
Voting Theory studies how to take individual rankings of voters and aggregate them to form the global ranking.
Examples:
• Votes for a president of a company/country, etc. All voters communicate their results and based on that the president is chosen
• Search engines: there are many results, how to show them?
## Notation and Relations
• let $A = \{a, b, c, ...\}$ be the set of candidates
• there are $N$ voters
• each voter can express his preference on the basis of a total order
• i.e. he has to rank all the candidates
For this notation we define the following relations (Voting Theory Relations)
• Weak and Strong Preference
• Indifference
## Voting Mechanisms and Principles
A voting mechanism (or voting procedure or voting method) takes a collection of votes (individual preferences of the candidates from set $A$) and forms the global ranking. Usually it choses a single candidate from the set $A$.
There are several voting procedures:
### Criteria
How to characterize "good" voting methods?
There are several criteria
PV 2PV Borda Cond.
Monotonicity
Solution Existence
Manipulation
Separability
Condorcet Fairness
Other principles: | 2021-10-17 22:28:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6020881533622742, "perplexity": 4558.901352294538}, "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/1634323585183.47/warc/CC-MAIN-20211017210244-20211018000244-00305.warc.gz"} |
https://math.stackexchange.com/questions/949122/0-to-l-to-m-to-n-to-0-is-a-short-exact-sequence-m-is-noetherian-iff | # $0 \to L \to M \to N \to 0$ is a short exact sequence. $M$ is Noetherian $\iff$ $L$ and $N$ are Noetherian
Let $$0 \rightarrow L \xrightarrow{\alpha} M \xrightarrow{\beta} N \rightarrow 0$$ be a short exact sequence of $$A$$-modules ($$A$$ is a commutative ring).
Prove $$M \text{ is Noetherian} \iff L \text{ and } N \text{ are Noetherian.}$$
I don't understand the first part of $$\impliedby$$. It says:
Let $$M_1 \subset M_2 \subset \dots \subset M_k \subset \cdots$$ be an increasing chain of submodules of $$M$$. Then identifying $$\alpha(L)$$ with $$L$$ and taking intersection gives a chain $$L\cap M_1 \subset L \cap M_2 \subset \dots \subset L\cap M_k \subset \cdots$$ of submodules of $$L$$.
I don't get it, what does "identifying $$\alpha(L)$$ with $$L$$" even mean? I don't understand what $$L$$ is and therefore the chain of submodules of $$L$$ doesn't make sense.
Can someone bring clarity to this, what am I missing?
$\alpha(L)=\text{Im}\,\alpha$, and since the sequence is exact $\alpha$ must be mono (and injective) and thereby $\text{Im}\,\alpha$ is isomorphic to L and Noetherian.
$$\text{Im}\,\alpha\cap M_1 \subset \text{Im}\,\alpha \cap M_2 \subset...\subset \text{Im}\,\alpha\cap M_k \subset...$$ of submodules of $\text{Im}\,\alpha$, a sequence that is finite because $\text{Im}\,\alpha$ is Noetherian.
• Thank you so much, I understood it correct now. I dont get it how one can use $L$ for $Im(\alpha)$ when $L$ is the domain of the function, whatever... – user117449 Sep 28 '14 at 7:11 | 2019-09-22 19:16:13 | {"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": 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": 15, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9320542812347412, "perplexity": 117.64508105899566}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514575627.91/warc/CC-MAIN-20190922180536-20190922202536-00069.warc.gz"} |
http://physics.stackexchange.com/questions/16698/yield-strength-versus-ultimate-strength | # Yield Strength versus Ultimate Strength
What is the qualitative difference between these two:
As seen on the table Typical yield and ultimate strengths.
I am trying to resolve the meaning of the phrase "contact yield stress" from C. Thorrton 1997 to real world values.
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Could you provide a full citation for "C. Thorrton 1997" please? – deadly Feb 27 '13 at 13:36
@deadly appliedmechanics.asmedigitalcollection.asme.org/… Thornton CC. Coefficient of Restitution for Collinear Collisions of Elastic-Perfectly Plastic Spheres. J. Appl. Mech.. 1997;64(2):383-386. doi:10.1115/1.2787319. – Mikhail Feb 27 '13 at 20:08
If you look at a stress-strain diagram, the difference becomes clearer.
The initial slope is where stress is directly proportional to strain (like a spring) and the material behaves like this up to its elastic limit where it reaches its yield strength.
Beyond this the material deforms permanently (like an overstretched spring that won't return to its original shape). The material then becomes strain hardened until you reach the ultimate strength and necking starts to occur and the material becomes weaker again until it breaks apart.
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The material does not become weaker after the UTS, but due to necking the engineering strain gets lower because it is defined as $\frac{F}{A_0}$, while the true strain has a higher strain due to a smaller area, caused by the necking. – fibonatic Jul 22 '13 at 11:36
I see what you're saying in that locally (at the neck) the material is under higher stress and so might be considered stronger, but the force required to stretch the overall material (or "sample") becomes less and so in practical terms it is weaker. – deadly Jul 22 '13 at 14:10
Yield stress is the stress at which that the material deforms permanently, ultimate tensile stress is the stress at which it breaks.
There is probably some official ISO/ASME definition of how much it has to deform for it to count as having yielded.
Materials first deform elastically - when you release the stress they return to their original shape, this is what a metal spring does.
Then with more force they deform plastically - when you release the stress they have permanently been stretched into a new shape, this is yield.
Finally they break, this is ultimately tensile stress, or breaking point
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Can you expound upon the difference between breaking and deformation, for materials such as steel? (i'll give you the check :-) ) – Mikhail Nov 8 '11 at 21:08
The yield stress is often defined as the stress at a 0.2% strain. Probably because this values can be determined with an higher accuracy than the stress at which the deformation will become plastic instead of elastic. I am not 100% sure about the reason for this though. – fibonatic Jul 22 '13 at 11:42
It's more correct to say that the ultimate tensile strength (UTS) is the stress at which a materiel fails (not breaks). Only a brittle material will break at the UTS. – deadly Jul 25 '13 at 9:49
The deformation amount is 0.2% to count as yielded for steel
Hard steels and non-ferrous metals do not have defined yield limit, therefore a stress, corresponding to a definite deformation (0.1% or 0.2%) is commonly used instead of yield limit. This stress is called proof stress or offset yield limit (offset yield strength):
$\sigma t= \frac{F_S}{S_0}$
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Ultimate Tensile Strength is a common engineering parameter when designing brittle material because there is no yield point. Whereas in designing ductile material, yield stress is a common parameter.
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Acutaly ultimate stress is maximum stress above which the material will break.
Yield stress is at which it deforms permanently means plastic deformation
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## protected by Kyle KanosSep 29 '15 at 11:56
Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count). | 2016-07-02 00:17:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.6171059012413025, "perplexity": 1677.3370792678968}, "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-2016-26/segments/1466783404382.73/warc/CC-MAIN-20160624155004-00026-ip-10-164-35-72.ec2.internal.warc.gz"} |
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# xyz < 0
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xyz < 0 [#permalink] 20 Oct 2017, 02:40
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Question Stats:
75% (00:27) correct 25% (00:54) wrong based on 4 sessions
$$xyz < 0$$
Quantity A Quantity B $$x + y + z$$ $$2x + 2y + 2z$$
A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
[Reveal] Spoiler: OA
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Re: xyz < 0 [#permalink] 20 Oct 2017, 06:17
The fact that xyz < 0 does not give enough information about the unknowns to be able to compare the two quantities. We just know that two of them should be positive and the other negative but nothing more. Given that we can't compare the quantities answer is D!
Re: xyz < 0 [#permalink] 20 Oct 2017, 06:17
Display posts from previous: Sort by | 2019-02-23 08:59:23 | {"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": 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": 0.39802098274230957, "perplexity": 3705.1899747040447}, "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-2019-09/segments/1550249495888.71/warc/CC-MAIN-20190223082039-20190223104039-00173.warc.gz"} |
https://www.nature.com/articles/s41564-022-01211-y/?error=cookies_not_supported | ## Main
Chagas disease, caused by the protozoan parasite Trypanosoma cruzi, remains the highest-impact parasitic disease in Latin America and one of the major causes of infection-induced myocarditis worldwide1. For more than five decades, two nitroheterocyclic compounds, benznidazole and nifurtimox, have been available for treatment of the infection, but are relatively rarely used due to their inconsistent efficacy and high frequency of side effects. Recent trials of potential new therapies have yielded disappointing results2,3.
Among the challenges for drug development in T. cruzi infection is the parasite’s predominantly intracytoplasmic location in mammals and its ability to invade a wide variety of host cell types and tissues, although it shows a clear preference for muscle cells, including cardiac, skeletal and smooth muscle of the gut. The recent discovery of arrested ‘dormant’ intracellular forms of T. cruzi that are relatively and transiently resistant to otherwise highly effective trypanocidal compounds4 may partially explain why these therapeutics must be given for extended periods of time (60 d is common) but still have a high failure rate.
Previous work from our laboratories have identified a novel class of boron-containing molecules, the benzoxaboroles5,6 as having potent activity against protozoans including Trypanosoma brucei7, Leishmania donovani8 and Plasmodium falciparum9. Screening of the Anacor benzoxaborole compound library against T. cruzi revealed several hits, but initial assessment of structure-activity relationships (SARs) suggested limited opportunity for improvement of potency and/or selectivity, particularly in those subclasses previously found to have activity against T. brucei and L. donovani. Here we have taken advantage of this benzoxaborole scaffold and the multiple natural host species for T. cruzi to move rapidly from in vitro detection of trypanocidal activity in lead compounds into facile in vivo tests of efficacy in mice and ultimately in naturally infected non-human primates (NHPs). The result is identification of a class of benzoxaboroles that provide high rates of parasitological cure of T. cruzi infection. AN15368 from this class is the first, extensively validated and safe potential clinical candidate in over 50 yr for the prevention/treatment of Chagas disease.
## Results
### In vitro activity and SARs
The initial lead benzoxaborole 6-carboxamide AN4169 (Fig. 1) provided 100% cure of mice infected with the T. cruzi Brazil strain5; however, rodent tolerability studies suggested that an insufficient therapeutic margin existed for further progression of this compound. Profiting from a concurrent project evaluating analogues of AN4169 against Trypanosoma congolense10, several compounds with submicromolar activity against T. cruzi in vitro and good metabolic stability in an in vitro mouse S9 liver fraction assay were identified, among these an ester of the 6-valine ‘transposed’ carboxamide AN10443 (Fig. 1). Further manipulation of this compound, in particular inclusion of a methyl group at C(7) of the benzoxaborole ring as in AN11735, drastically increased in vitro activity against T. cruzi (IC50 < 10 nM), whereas substituents larger than methyl at C(7) ablated activity (Extended Data Fig. 1a).
SAR development of the ester region in the C(7)-methyl series showed that potency was not substantially impacted by substituents on the benzyl ester, with the exception of the 4-(3-pyrrolidinylethyloxy) analogue AN14502 and the 4-methylsulfonyl analogue AN14561, which were markedly less potent (Supplementary Table 1). Physicochemical properties were more affected, with most simple halogenated analogues being poorly soluble in aqueous media. Metabolic stability, as estimated by incubation with the mouse S9 liver fraction, was variable, and roughly tracked with lipophilicity (cLogD; Supplementary Table 1).
These observations prompted us to more substantially modify the ester region of the molecule through preparation of aliphatic and heterocyclic esters that would be expected to be less lipophilic, more water soluble and less susceptible to metabolism (Supplementary Table 2). Several interesting SARs emerged from this group of analogues: (1) esters containing basic amines (for example, AN15143, AN15144, AN15658, AN15678, AN15129, AN15192, AN15078, AN14504 and AN15159) were less active than neutral compounds and (2) small aliphatic esters were quite potent except for the t-butyl ester (AN15134). The relationship between lipophilicity and solubility or metabolic stability continued to exist for these compounds and provided reasonably wide latitude for modulation of such properties by choice of ester substituent.
### In vivo activities
In addition to being very potent in vitro, the valine esters were also of generally good stability, including in mouse and human S9 liver fraction assays (Supplementary Table 3). Several valine esters also exhibited low clearance (<20% hepatic blood flow) following intravenous dosing and good bioavailability following oral administration to mice, achieving good to excellent exposure (area-under-the-curve (AUC) > 10 µg h ml−1) with low mg kg−1 doses (Supplementary Table 3). Concurrent testing results of these compounds in vivo for the ability of a single oral dose to reduce an established focal infection in the footpad of mice over 3 d11,12 were very encouraging, with AN14353 emerging as the lead on the basis of activity at reduced doses (Fig. 1b). Notably, AN14353 demonstrated rapid in vivo trypanocidal activity (Fig. 1c), had high in vitro potency for a range of T. cruzi isolates for different genetic lineages (discrete typing units, DTUs; Extended Data Fig. 1b) and could consistently resolve established T. cruzi infections at a dose of 25 mg kg−1 in a standard5,13 40 d treatment protocol in wild-type mice (Fig. 1d) as well as infections in immunodeficient mice (Fig. 1e).
### Lead benzoxaboroles are prodrugs activated by a parasite serine carboxypeptidase
We next sought to understand the essentiality of the ester group for activity, as this functional group carries liabilities for hydrolytic and metabolic instability. An early indication of the importance of the ester function to anti-T. cruzi activity was evident from the already noted lack of potency of the t-butyl ester AN15134. Amide, N-methyl amide, ketone, ether and acylsulfonamide analogues of AN11735, as well the 1,2,4-oxadiazole ester bioisostere AN14562 all lacked activity (Supplementary Table 4). Furthermore, the expected carboxylic acid metabolite, AN14667, had ~1,000-fold reduced activity on both intra- and extracellular amastigotes (Figs. 1a and 2a). Thus, the ester functionality is absolutely essential for anti-T. cruzi activity.
We hypothesized that the esters might be prodrugs to the carboxylic acid and were cleaved within either the host cell or by a parasite enzyme. The high sensitivity of extracellular amastigotes to AN14353 but not to the carboxylic acid AN14667 (Fig. 2a) virtually eliminated the requirement for a host peptidase in prodrug activation. Several candidate enzymes with potential ester cleavage activity in T. cruzi were identified, including a serine carboxypeptidase (CBP; TcCLB.508671.20) present at 2–18 copies in different T. cruzi strains14 and a metallocarboxypeptidase 2 (TcCLB.504045.60). CRISPR/Cas9-driven disruption of all copies of the serine carboxypeptidases (Fig. 2b,c) but not the metallocarboxypeptidase (Supplementary Fig. 1) decreased in vitro sensitivity of T. cruzi amastigotes by up to 100-fold. Accumulation of the AN14353 analogue, AN15368, and its conversion to the cleaved product in wild-type T. cruzi epimastigotes and amastigotes but not in the TcCLB knockout (KO) line was documented by quantitative liquid chromatography tandem mass spectrometry (Fig. 2d), thus confirming the serine carboxypeptidase-dependent prodrug to drug conversion within T. cruzi.
### Mouse test of cure studies
Evaluation of dose proportionality of exposure with AN14353 and generation of the carboxylic acid metabolite AN14667 in vivo (Supplementary Fig. 2) suggested solubility-limited absorption of this compound, prompting us to attempt to further optimize aqueous solubility. Focusing on the ester region, a variety of more polar, non-basic substituents such as aliphatic and cyclic ethers as well as hydroxyvaline analogues (predicted to be less hydrophilic) of our lead compounds were evaluated for both solubility and in in vitro trypanocidal assay (Supplementary Table 5). The highest in vitro-active compounds in this set were then evaluated and found to have anti-T. cruzi activity in the 2 d in vivo assay (Fig. 3a). Extensive in vitro and in vivo pharmacokinetic analysis and metabolic stability assays in mouse and human liver S9 fraction, and both mouse and human plasma and in vivo pharmacokinetics studies (summarized in Supplementary Table 6) ultimately identified three additional compounds of particular interest—AN14817, AN15368 and AN16109.
On the basis of these aggregate data, we progressed these and related compounds into a series of ‘test of cure’ assays in mice with a terminal immunosuppression period to reveal residual infection5. An initial screening using 10 mg kg−1 for 40 d identified several compounds for which the treated animals showed no parasite recovery by haemoculture and no parasite DNA detection in skeletal muscle using PCR (Fig. 3b). A more stringent test using only 20 d of treatment (a treatment period for which benznidazole is only partially effective in generating cure5) further distinguished the highest potency compounds from less potent ones (Fig. 3c). On the basis of these results, lower doses of the candidates were evaluated in a short-term (non-cure) treatment course of intraperitoneal infection with luciferase-expressing parasites (Fig. 3d and Supplementary Fig. 3) from which a dose of 2.5 mg kg−1 was identified and shown effective in a 40 d treatment cure assay (Fig. 3e).
### Non-human primate test of cure study
AN14353, AN14817 and AN15368 were progressed to an array of preliminary safety pharmacology, genotoxicity and toxicology studies. All three compounds were found to exhibit little to no affinity for a broad array of mammalian enzymes, receptors and ion channels, were non-genotoxic in standard Ames test and in vitro micronucleus studies, and did not demonstrate marked inhibition of representative cytochrome P450 enzymes at 10 uM. High-dose 7 d toxicology studies did not distinguish the three compounds from each other, but the non-dose proportional exposure noted previously with AN14353 (Supplementary Fig. 2) was also seen with the benzylic ester AN14817, probably a consequence of solubility-limited oral absorption. In contrast, the more hydrophilic analogue AN15368 exhibited good dose proportional exposure in rats (Supplementary Fig. 4) and modest effects on haematology and clinical chemistry at 150 mg kg−1, but none at 120 mg kg−1 d−1 or lower. Total plasma exposure (AUC0–24 h) in rats at 120 mg kg−1 d−1 was approximately 30,000 ng h ml−1, with no evidence of drug accumulation between the first and seventh days of the study. On the basis of these observations, AN15368 was selected as a pre-clinical candidate for the treatment of Chagas disease and for evaluation in rhesus macaques (Macca mulatta) infected with T. cruzi via natural exposure in the United States.
We chose to treat NHP for 60 d as this is the standard length of treatment employed for human infections and in previous clinical trials2,3 (Fig. 4a). On the basis of pharmacokinetic studies in NHPs (Supplementary Fig. 5) and allometric scaling15 (see Methods for additional details), a dose of 30 mg kg−1 was chosen for NHP to provide a high possible rate of cure without compromising safety. To maximize the power of detecting an expected high rate of cure, 19 animals were enrolled in the single-arm treatment study and 3 animals served as untreated controls. Pre-treatment data on the animals are shown in Supplementary Table 8; the mean age of the animals in the treatment group was 19.4 yr and the average minimal length of infection was 5.7 yr.
All animals in the trial were positive by PCR for T. cruzi DNA in blood at one or more time points pre-treatment and were seropositive for anti-T. cruzi antibodies by standard facility screening tests and via our Luminex-based multiplex serological assay16,17 (Extended Data Figs. 2 and 3, and Fig. 4). Parasites were also isolated from haemocultures of pre-treatment blood samples from 18 of the 19 animals in the treatment group and 2 of 3 control animals. Importantly, the isolated parasite lines showed a variability in genetic types (DTU), indicating a diversity of parasite lineages in these animals, even though they acquired infection in the relatively restricted geographic footprint of the primate colony (Extended Data Fig. 2). These latter characteristics are similar to those found in a previous US-based study in cynomolgus macaques with naturally acquired T. cruzi infection17.
The primary endpoints of the trial were detection of parasite DNA in blood and culture of parasites from blood, for which all animals in the study were assayed a minimum of 7 times at 2–4.5-week intervals following the end of treatment (Fig. 4a). All AN15368-treated animals were negative by both assays at all time points (Fig. 4b and Supplementary Table 7). In contrast, 2 of the 3 untreated animals were positive by one or both haemoculture and PCR at multiple time points.
A secondary determinant of treatment efficacy was the detection of T. cruzi DNA by PCR in post-necropsy tissues. For this purpose, 9 of the 19 treated animals and 2 of the untreated controls were euthanized and tissues collected. DNA was extracted and analysed by PCR for T. cruzi kinetoplast DNA (kDNA) using both individual and pooled tissue samples (Fig. 4a,b and Supplementary Table 8). For control animal C1, 30 of 68 (39.4%) single or pooled tissue samples from multiple tissues yielded positive PCR results. In contrast, for the 9 treated animals, tissue samples assayed singly or in pools (range 84–99 total sample sites per animal) from heart, quadriceps, biceps, small and large intestine, oesophagus, tongue, liver, spleen, abdominal fat and brain were all negative. Interestingly, we also failed to detect parasite DNA in non-treated control animal C2 by blood or tissue PCR despite its being positive by blood PCR and haemoculture in pre-treatment sampling.
A third measure of treatment efficacy was declining antibody levels to a set of recombinant T. cruzi proteins in the multiplex serological assay16,17. Monitoring for decreases in anti-T. cruzi antibodies has been useful for assessing treatment efficacy in humans and conversion to seronegative is considered the standard for determining infection cure, although in many cases this can take years post-treatment to achieve18,19,20. Nine of the remaining 10 treated macaques not killed at the end of treatment were returned to the breeding colony and thus were available for continued periodic monitoring over >3 yr (Fig. 4a). All animals showed declines in antibody levels to multiple recombinant proteins from T. cruzi over the 42 months after the end of treatment (Fig. 4c and Extended Data Fig. 3). Importantly, blood PCR and haemoculture samples at these additional post-treatment time points also remained negative in all treated macaques (Extended Data Fig. 2 and Supplementary Table 7). Thus, the exhaustive examination of blood and tissues for parasite DNA and long-term monitoring of anti-T. cruzi antibodies conclusively establish a 100% efficacy of AN15368 in a population of time-variable, naturally infected NHPs harbouring genetically diverse populations of T. cruzi.
Throughout the dosing period, macaques readily accepted food treats containing compound and no post-dose nausea or other interruption of normal activity was observed. No adverse events were noted in any of the 19 treated macaques during the 60 d treatment period and repeated physical examinations revealed no clinical signs that could be associated with drug administration. According to the blood-based health screening performed throughout the study, the mean values of the liver enzymes alanine aminotransferase and alkaline phosphatase, and the levels of lymphocytes and monocytes in the blood were mildly elevated during the drug-treatment phase of the study and returned to pre-study values by the final blood draw at the end of the study (Supplementary Data 1). The 9 female animals that returned to the breeding colony have shown no abnormalities in the yearly examinations and have produced 13 healthy and T. cruzi-seronegative infants in the first 2 yr following treatment. These latter numbers are wholly consistent with the fecundity rate of the colony in general. At necropsy, 3 of 11 euthanized animals (2 treated animals and 1 control) were identified on gross examination to have pale areas in the heart that could be consistent with myocardial damage associated with Chagas disease. Histological examination revealed inflammation in several cases but did not differentiate the treated from untreated (control) study animals and no T. cruzi amastigotes were detected in any tissues from the study animals (Supplementary Data 1). Thus, AN15368 is both highly effective in curing long-standing T. cruzi infections and presents no overt safety or reproductive health concerns in a 60 d treatment course in NHPs.
### Target identification
During the course of this work, several benzoxaborole analogues of AN15368 with efficacy in the treatment of African trypanosomiasis in cattle were shown to target the Cleavage and Polyadenylation Specificity Factor (CPSF3), an important factor in messenger RNA processing21,22. Similar to findings reported in these studies, the overexpression of CPSF3 in T. cruzi (Extended Data Fig. 4) also resulted in a 3–5-fold increase in resistance to AN15368 (Fig. 5) as well as cross resistance to other benzoxaborole analogues (Fig. 5b and Extended Data Fig. 4), suggesting that CPSF3 is at least one of the targets of AN15368 in T. cruzi. This conclusion is also supported by the marked and continuing reduction in parasite mRNA as early as 6 h following addition of AN14353 to T. cruzi-infected host cells, but not in benznidazole-treated cultures (Fig. 5c). Furthermore, introduction into T. cruzi of the Asn232 mutation to His in CPSF3, reported by ref. 21 to disrupt binding of benzoxaborole compounds in T. brucei brucei, also conferred resistance to AN15368 (Fig. 5d). Lastly, and as noted above for AN14353 (Fig. 2), AN15368 and other related analogues all function as prodrugs and require activation by the T. cruzi CBP to efficiently kill intracellular T. cruzi (Fig. 5e and Extended Data Fig. 4c), as is the case for the benzoxaboroles effective against African trypanosomes23. Thus, the highly effective benzoxaborole AN15368 is a prodrug that enters host cells and then T. cruzi, wherein it is cleaved by a T. cruzi peptidase. The product of this cleavage selectively targets CPSF3-mediated mRNA maturation in intracellular amastigotes.
Although the target of these benzoxaboroles in T. cruzi and in African trypanosomes appears to be the same, a number of the compounds with previously reported potent activity in vitro to T. congolense had activity on extracellular amastigotes but not on T. cruzi amastigotes in host cells (Fig. 5e and Extended Data Fig. 4d). Reasoning that this differential effect could be due to variable or limited entry into or metabolism by T. cruzi host cells, we screened for the activity of these compounds on extracellular amastigotes of T. cruzi. With one possible exception, all of these compounds have low nanomolar activity on extracellular amastigotes, indicating an additional selectivity of benzoxaborole activity on T. cruzi due to this parasite’s intracellular lifestyle.
## Discussion
The benzoxaborale AN15368 is the first highly effective compound for the treatment of T. cruzi infection discovered in >50 yr, and the only compound so far shown to achieve unequivocal and apparently uniform cure of infection in NHP with long-term naturally acquired infections of diverse T. cruzi genetic types. AN15368 is orally bioactive and exhibited no overt toxicity in a 60 d course of treatment in NHP. Thus, AN15368 is a very strong candidate for ultimately progressing into human clinical trials.
Despite some successful vector control efforts, the risk of infection with T. cruzi remains variable but substantial for human and other animals from the southern United States to southern South America. The currently available drugs benznidazole and nifurtimox suffer from variable efficacy and high rates of adverse events. Consequently, these drugs are not routinely used despite their relatively wide availability. The absence of highly effective treatments undermines the use of widespread and routine screening that would detect the usually asymptomatic early infection before irreversible damage is done24. A relatively large number of potential candidates have been targeted for development, some for decades (reviewed in ref. 25), but those that have been progressed to human clinical trials2,3,26 have performed significantly worse than currently available drugs.
The benzoxaboroles have become a rich source of development candidates for treatment of protozoal infections, with an apparent target of all being the mRNA-processing endonuclease, CPSF310,21,22,23,27,28,29,30,31. Although it was initially hypothesized that AN15368 and the analogue AN11736 may not target CPSF3 on the basis of their limited effect on mRNA processing22, subsequent work demonstrated that overexpression of CPSF3 in T. b. brucei induced resistance to killing by AN1173621 and we arrive at a similar conclusion with respect to the activity of AN15368 in T. cruzi, indicating CPSF3 as one probable target. Also, similar to a number of benzoxaboroles highly effective against the African trypanosomes, AN15368 requires processing into its carboxylate form to achieve full potency and we show that this activation is mediated by a parasite serine carboxypeptidase. However, unlike the case in the extracellular African trypanosomes, AN15368 must traffic unprocessed through both the host and the parasite plasma membranes to reach these activating enzymes. This requirement appears to account for the differential activity of a number of highly similar benzoxaboroles on African trypanosomes and T. cruzi, and emphasizes the need to tailor drugs to match the specific biology of the pathogen, even when the processing/activation requirements and the target of the compounds are the same. Likewise, this outcome highlights the challenge of designing compounds with cross-species activity for genetically related but biologically diverse pathogens such as the kinetoplastids.
The differential dosing requirements for the benzoxaboroles in T. cruzi infection, where 20 or more days of treatment is necessary for sterile cure as compared with African trypanosomes where a single dose effects cure in cattle10, is remarkable and further underscores the difficulties of drug discovery for T. cruzi. T. cruzi can invade diverse host cell types in tissues throughout the body, presenting a challenge for any one drug to reach effective levels in all tissues. Furthermore, T. cruzi amastigotes have recently been shown to assume a non-dividing, apparently low-metabolic state that provides substantial resistance to drugs4,32. Fortunately, these properties do not prevent drug-induced sterile cure but appear to necessitate an extended treatment course, as observed herein, and previously for other anti-T. cruzi drugs32. The NHP trial conducted as part of this study was initiated before knowledge of this dormancy property in T. cruzi and thus utilized daily dosing for 60 d, as is common for the currently used benznidazole and nifurtimox. Despite these lengthy treatment periods, drug-induced resistance in T. cruzi has neither been reported with respect to benznidazole and nifurtimox nor have we observed resistance during the extended treatment courses using benzoxaboroles, even though all three drugs are produgs. Nevertheless, shortened or modified treatment regimens may be possible with the benzoxaboroles (Fig. 3c) as is the case for benznidazole5,32,33, to further reduce this possibility. As a prodrug, AN15368 has the liability of potential selection of resistance via loss of the processing carboxypeptidase23, although deletion of the CBP array in T. cruzi substantially reduces but does not totally abolish susceptibility to AN15368.
One noteworthy advantage of drug discovery in T. cruzi is the very wide natural host range of the parasite, including most wild and domesticated mammals as well as mice, canines and NHP, in addition to humans. In all these hosts, T. cruzi appears to behave similarly, infecting the same host cell types, being controlled (but rarely eliminated) by similar immune effector mechanisms, generating analogous pathologies, and being affected correspondingly by the same drugs. Animals in T. cruzi-endemic areas, including the southern United States, are at risk of acquiring T. cruzi infection and in some areas, this risk is severe, leading to 20–30% new infections per year in some populations34 as well as infections in zoo animals35,36. The commonly used indoor/outdoor housing of NHPs in T. cruzi-endemic regions also results in naturally acquired T. cruzi infection in animals in these facilities, despite vector control and other preventative measures37. Infections in these NHPs mirror that in humans, initiating at different points in life and extending for decades in some cases, involving genetically and phenotypically diverse parasite populations, and leading to a diversity of immune responses and disease outcomes17. All these characteristics make these NHPs incredibly valuable resources for trialling anti-T. cruzi drugs before human clinical trials. The observed 100% cure with AN15368 in macaques harbouring long-term infections with genetically diverse parasite lineages and without any apparent drug toxicity bodes well for the potential safety and efficacy of AN15368 in humans. It is noteworthy that the only other documented treatment trials in T. cruzi -infected NHPs recorded a high degree of failure for posaconazole (100%)38 and benznidazole (>60%; R.L.T. et al., manuscript in preparation), in agreement with the high failure rate of these drugs in human clinical trials2,3,26. We also validate the same methods used for monitoring treatment efficacy in humans, specifically serial blood PCR for T. cruzi DNA and changes in immune profiles, for use in NHPs and reinforce these metrics with extensive tissue PCR in a subset of necropsied animals. Lastly, the ability to return infection-cured macaques to the breeding colony extends the utility of these resources, providing the potential to monitor disease development and the susceptibility to reinfection, for example, in hosts previously cured of T. cruzi infections.
Surprisingly, we also observed one apparent self-cure among the three untreated NHPs. Although documentation of spontaneous cures is relatively rare39,40,41, they are not without precedent and may occur more frequently than currently appreciated42 but have not been previously observed in this NHP colony.
The success of this project provides insights into some best practices for drug discovery in T. cruzi and perhaps related organisms. In addition to taking advantage of the potency of the oxaboroles in general as anti-infectives, and the med-chem knowledgebase in this class of compounds, this project also benefitted substantially from the accessibility of infection systems for screening and testing compounds. Specifically, the power of the mouse-based assays to quickly, easily and quantitatively assess in vitro-active compounds for in vivo activity was instrumental in rapidly identifying the compounds with the highest potential. Coupled with the availability of substantial numbers of NHPs with naturally acquired T. cruzi infections for pre-clinical validation, we should be able to avoid the clinical trial failures that have accompanied previous drug discovery efforts in Chagas disease.
## Methods
### Compound synthesis
All compounds used in this study were prepared as described in US patent 10,882,272 granted on 5 January 2021. The syntheses of representative compounds AN14353 and AN15368 are described in Supplementary Text 1.
### Parasites and mice
C57BL/6J mice were purchased from the Jackson Laboratory and B6.129S7-Ifngtm1Ts/J (IFN-gamma deficient) were bred in-house at the University of Georgia Animal Facility. The SKH-1 ‘hairless’ mice backcrossed to C57BL/6J were a gift from Dr Lisa DeLouise (University of Rochester). All the animals were maintained at the University of Georgia Animal Facility under specific-pathogen-free conditions at 22 °C, 50% humidity and in a 12:12 h light:dark cycle. Male and female mice 6–9 weeks of age were used. All mouse experiments were carried out in strict accordance with the Public Health Service Policy on Humane Care and Use of Laboratory Animals and the Association for Assessment and Accreditation of Laboratory Animal Care accreditation guidelines. The protocol was approved by the University of Georgia Institutional Animal Care and Use Committee (IACUC). T. cruzi tissue culture trypomastigotes of the wild-type Brazil strain, Colombiana strain co-expressing firefly luciferase and tdTomato reporter proteins4, and CL strain expressing the fluorescent protein tdTomato11 were maintained through passage in Vero cells (American Type Culture Collection) cultured in RPMI 1640 medium with 10% fetal bovine serum at 37 °C in an atmosphere of 5% CO2. Parasite genotypes were determined as previously described17.
### In vitro amastigote growth inhibition and killing assays
The in vitro anti-T. cruzi amastiogote activity assay was performed and optimized on the basis of the protocol described previously11. The change in tdTomato fluorescence intensity was determined as a measurement of growth over 72 h of culture. For assaying drug effects on extracellular amastigotes, trypomastigotes were collected from infected Vero cell cultures and converted in acidic media as previously described43. Amastigotes (50,000 per well) were incubated with 2-fold serial dilutions of compounds for 48 h. ATP production was used as an indication of growth in this case and was measured by ATPlite Luminescence ATP detection assay system (PerkinElmer). Both fluorescence and luminescence were read using a BioTek Synergy hybrid multi-mode reader equipped with the software Gene5 v 2.0 (BioTek). The dose-response curve was generated by linear regression analysis with GraphPad Prism v9.4.0 (GraphPad software). IC50 was determined as the drug concentration that was required to inhibit 50% of growth compared to that of parasites with no drug exposure.
### In vivo compound screens
#### Rapid assays
C57BL/6J mice were injected in the hind footpads with 2.5 × 105 tdTomato-expressing T. cruzi (CL strain) and orally treated with a single dose of the compounds (50 mg kg−1) at 2 days post infection (dpi). Fluorescent intensity of the feet was measured at 2 dpi before compound administration and at 4 dpi in the Maestro in vivo imaging system equipped with the software Maestro 2.1.0 (CRi) as previously described44. The proliferation index was estimated as PI = [(T4d – T2d)/(mUnt4d − mUnt2d)] × 100; where T4d and T2d are the fluorescence intensity of the feet of the treated animals at days 4 and 2 post infection, respectively; mUnt4d and mUnt2d are the average fluorescence intensity of the feet of the untreated animals at 4 and 2 dpi, respectively.
#### Cure assays
Male or female C57BL/6J mice were intraperitoneally infected with 104 trypomastigotes of the Brazil strain. Mouse infection was confirmed at 25–30 dpi by detection of CD8+ T cells specific against the T. cruzi TSKb20 peptide in blood45. Compounds resuspended in 1% carboxymethyl cellulose and 0.1% Tween 80 were administered daily by gavage at the specified concentrations. To optimally detect persistent infection, immune responses in the mice were suppressed by intraperitoneal (i.p.) injection of four doses of cyclophosphamide every 2–3 d (200 mg kg−1 d−1), beginning at 1 week after the end of therapeutic treatment. At the end of the immunosuppression regimen, peripheral blood was checked under light microscope for parasites and cultured in LDNT media5. Mouse skeletal muscle samples were obtained at the end of the immunosuppression and processed for T. cruzi DNA detection by qPCR as previously described32.
#### In vivo killing time assays
C57BL/6J mice were infected with 2.5 × 105 luciferase-expressing T. cruzi (Brazil strain) in the footpads, and 2 d later one oral dose of AN14353 (25 mg kg−1) was administered. The bioluminescent signal in the feet after i.p. injection of d-luciferin (PerkinElmer; 250 mg kg−1) was measured in a Lumina II IVIS imager (PerkinElmer).
#### Low-dose short treatment
Hairless mice (SKH-1) were infected intraperitoneally with 5 × 104 luciferase-expressing T. cruzi (Colombiana strain) and orally treated from 12 to 22 dpi with 1, 2.5 or 5 mg kg−1 of AN16109 or AN15368. The bioluminiscence signal of the whole body was measured after d-luciferin injection in a Lumina II IVIS in vivo imager equipped with the Living Imaging 4.0 software (PerkinElmer).
### Generation of CBP knockout
The CBP knockout was produced using ribonucleoprotein (RNP) complexes as previously described46. Briefly, Brazil tdTomato strain epimastigotes were electroporated with RNP complexes containing SaCas9 and a single-guide RNA (sgRNA) targeting the CBP gene TcBrA4_0048170, plus a repair template containing stop codons in all three reading frames and the M13 sequence for use as a PCR anchor. The sgRNA targeted the ‘GATTTACGTTGACCAGCCTGC’ sequence. After 2 d of recovery, single-cell clones were derived by depositing epimastigotes into a 96-well plate at a density of 0.5 cells per well by using a MoFlo Astrios EQ cell sorter (Beckman Coulter). DNA isolation was performed for clones, and the primer pair 5’-ACGTTGACCAGCCTGCAG-3’ and 5’- TGTGTATGGGTCTGTGAG-3’ was used for amplifying the wild-type allele, while the primer pair M13-F: 5’- TGTAAAACGACGGCCAGT-3’ and 5’-TGTGTATGGGTCTGTGAG-3’ was used for amplifying the mutant allele.
### Western blot
A total of 5 × 107 epimastigotes were collected at 4 °C and washed once with cold PBS. Pellets were suspended in RIPA buffer (150 mM NaCl, 20 mM Tris HCl pH7.5, 1 mM EDTA, 1% SDS, 0.1% Triton X-100) with 1% protease inhibitor cocktail (Thermo Fisher) and incubated for 1 h on ice. Then the suspension was sonicated (Sonics & Materials, model 501) for 10 s using microtip probe at 25 amplitude, and the sonicate centrifuged at 16,000 g for 10 min to remove the pellets and obtain total protein. Western blot was performed according to the general established protocol. TcCBP-specific antibody (a gift from Drs Juan José Cazzulo and Gabriela Niemirowicz at Instituto de Investigaciones Biotecnológicas, Buenos Aires, Argentina) was diluted at 1:500 and β-tubulin antibody was diluted at 1:1,000. The IRDye 800CW donkey anti-rabbit IgG (Li-COR) was used as secondary antibody for both TcCBP and tubulin at 1:10,000 dilution. Images were taken with the Bio-Rad ChemiDoc imaging system with the software ImageLab Touch 2.4.03.
### Generation and confirmation of the CPSF3 overexpression line
The CPSF3 gene was amplified using primers ‘ATGCTCCCTGCGGCAGCAGCAGTAA’ and ‘TTACACAGCCTCCTCTGGCAAAGGCT’, and integrated into the pTrex vector47 by NEBuilder HiFi DNA assembly (New England Biolabs). The construct pTrex-CPSF was then transfected into Brazil tdTomato epimastigotes and selected by 60 ug ml−1 blasticidin.
To confirm the CPSF overexpression in the selected transfectants, RNA was extracted as previously described14 and converted to complementary DNA (cDNA) using SuperScript reverse transcriptase (Invitrogen). Quantitative PCR reactions were performed in triplicate on the C1000 Touch Bio-Rad CFX96 real-time PCR detection system for CPSF using primer sets CPSF-1 (5’-TGAAACAGCAGCATGCCAAC-3’ and 5’-CGCGTCTGTCTACCATCAGA-3’) and CPSF-2 (5’-CGGCTCATTCTGATGGTAGACA-3’ and 5’-TGTGCGTTGCACACTGAATG-3’) in both control and CPSF overexpressing parasites. The expression level was normalized to tubulin and amplified using primers ‘AAGTGCGGCATCAACTACCA’ and ‘ACCCTCCTCCATACCCTCA’.
### Generation of CPSF3 mutants
To generate CPSF3 mutants, an RNP complex was transfected into Brazil tdTomato strain epimastigotes to target the CPSF3 gene (TcBrA4_0124800), together with a repair template that contained the mutation of Asn231 to His21. The sgRNA targeted the ‘TCTGATTGCGGAAAGCACAA’ site. After 24 h of recovery, 20 uM AN15368 was added to the parasites to select CPSF3 mutants that were resistant to drug treatment. The ultimate resistant parasites were validated to have acquired the Asn231His mutation in CPSF3 via sequencing.
### RNA-seq sample preparation, sequencing and analysis
Vero cells (106) were infected with 107 CL strain trypomastigotes of T. cruzi for 2 d before treating with either 5 uM benznidazole or 30 nM AN14353. The drug concentration used for treatment was set at 5 times the IC50. Samples were collected at several time points for RNA extraction as previously described48. Ribosomal RNA (rRNA)-depleted RNA library construction and RNA sequencing using Illumina Nextseq 75PE was carried out by the Georgia Genomics and Bioinformatics Core (GGBC, University of Georgia, Georgia). Illumina reads with mean quality lower than 30 (Phred Score based) were removed from analysis, then mapped to the CL Brener genome (TritrypDB release-33) and the African green monkey genome49 using the HiSAT software package v0.1.650 with default parameters. The mapping rate was quantified by HTseq v0.6.151.
### LC–MS/MS analysis of intracellular AN15368 and AN14667
Wild-type and peptidase-knockout T. cruzi epimastigotes (5 × 108) were treated with AN15368 (10 uM) or with DMSO vehicle control for 6 h. The cells were then pelleted and resuspended in 100 µl PBS. The cell suspension was mixed with 200 µl acetonitrile and centrifuged at 735 g for 10 min at room temperature. After extraction, the supernatant was further diluted with methanol:water (30:70; v/v) containing 0.4 nM internal standard AN14817 to a concentration within the calibration range. Each sample was diluted in triplicate to provide technical replicates and the diluted sample (10 µl) was injected for subsequent LC–MS/MS analysis.
LC–MS/MS analysis was performed on a Waters ACQUITY I-Class UPLC system coupled to a Xevo TQ-S triple quadrupole mass spectrometer. An ACQUITY UPLC BEH C18 column (130 Å, 1.7 µm, 2.1 mm × 50 mm) was used for chromatographic separation, and the column temperature was 40 °C. The mobile phase consisted of water (A) and methanol (B), both containing 0.1% (v/v) formic acid. The following gradient elution was performed at a flow rate of 0.4 ml min−1: 0–0.5 min, 30% B; 0.5–3 min, 30–95% B; 3–4 min, 95% B; 4–4.1 min, 95–30% B; and 4.1–5 min, 30% B. The MS ionization was carried out in the positive electrospray ionization mode with following conditions: capillary voltage = 1.50 kV, desolvation temperature = 500 °C, desolvation gas flow = 1,000 l h−1 and nebulizer gas pressure = 7.0 bar. The MS/MS transitions used for detection and quantification were 390.1→174.9 for AN15368, 292.0→174.9 for AN14667 and 416.1→109.0 for AN14817. Data were processed using TargetLynx v4.1 software (Waters).
### NHP resource and facilities
All NHPs utilized for these studies were acquired from the approximately 1,000-animal Rhesus Macaque (Macaca mulatta) Breeding and Research Resource housed at the AAALAC accredited Michale E. Keeling Center for Comparative Medicine and Research (KCCMR) of The University of Texas MD Anderson Cancer Center in Bastrop, Texas. This is a closed colony, which is specific-pathogen-free (SPF) for Macacine herpesvirus-1 (Herpes B), Simian retroviruses (SRV-1, SRV-2, SIV and STLV-1) and Mycobacterium tuberculosis complex. All animals are socially housed in shaded, temperature-regulated indoor-outdoor enclosures with numerous barrels, perches, swings, and various feeding puzzles and substrates to mimic natural foraging and feeding behaviours. Standard monkey chow, ad libitum water and novel food enrichment items are provided daily. Study animals that were seropositive for T. cruzi had acquired the infection naturally through exposure to the insect vector of the parasite while in their indoor-outdoor housing facilities. The NHP experiments were performed at the KCCMR and all protocols were approved by the MD Anderson Cancer Center’s IACUC and followed the NIH standards established by the Guide for the Care and Use of Laboratory Animals52.
### Pharmacokinetic (PK) analysis in NHP
Pre-treatment PK analysis of AN15368 distribution and clearance was performed to assist in determining the treatment dosing regimen. While under sedation/general anaesthesia, AN15368 was administered at various dosing levels either intravenously (IV) or via oral gavage (PO) to a T. cruzi -seronegative rhesus macaque, and 500 ul blood samples were collected before dosing and at 2, 5, 15, 30 and 60 min post-dose administration at which time the animal was recovered from general anaesthesia. Additional 500 ul blood samples were collected at 3, 6, 9 and 24 h post-dose administration under light anaesthesia/sedation. After the initial IV/PO PK assessment, a pre-regimen PK assessment of oral dosing was conducted, with administration of a single dose of AN15368 in 3 animals over 3 dosing periods, with AN15368 (30 or 50 mg kg−1 dose) administered in food treats. For this pre-regimen phase, blood samples were collected at pre-dose, then at 0.25, 0.5, 1, 3, 6, 9 and 24 h post dose.
Mid- and end-regimen PK assessments were also performed. A ‘peak and trough’ mid-regimen (day 30) PK assessment was performed on 3 treated animals: (1) blood was collected before drug dosing, (2) the animals were gavage-dosed with AN15368 in pumpkin slurry and (3) a second blood sample was collected at 3 h post dosing. The end-regimen PK analysis was performed on the 60th (and final) day of AN15368 dosing, with a non-serial sparse sampling design utilized for 18 of the 19 treated animals. For this study, 3 animals had blood collected before provision of the AN15368 in food treats. The other 15 animals were provided AN15368 in food treats and then blood was collected from 3 separate animals at 0.5, 1, 3, 6 and 9 h post dosing. Only 1 blood sample was collected at a single time point from each of the 18 animals. The composite plasma PK profile on day 60 was obtained using the mean concentrations (n = 3) at each sampling time point (pre-dose, 0.5, 1, 3, 6 and 9 h). All blood samples (500 ul) for PK analysis (pre-, mid- and end-regimen) were collected into EDTA microtainers and plasma collected for the determination of AN15368. The plasma samples were provided to Pharmout Labs (Fremont, California) for analysis using LC–MS/MS. The mean pre-dose concentration was also depicted as the 24 h post-dose concentration and used for the calculation of the post-treatment AUC0–24 value on day 60.
For calculation of PK parameters, the Cmax (maximum concentrations) and tmax (time to maximum concentrations) were determined by visual inspection of the plasma concentration vs time curves from the pre-regimen and end-regimen periods. PK calculation was not performed for the mid-regimen PK samples since only two time points were collected. The AUC values for the pre- and end-regimen were calculated using the linear-trapezoidal rule with the following equation:
$${{{{{\mathrm{AUC}}}}}}_{({{{t}}}1 - {{{t}}}2)} = [({{{C}}}_{{{{t}}}2} + {{{C}}}_{{{{t}}}1}) \times \left( {{{{t}}}2-{{{t}}}1} \right)]/2$$
where t1 and t2 are consecutive sampling time points, AUC(t1-t2) is the fractional AUC over time intervals t1 and t2, Ct2 is the concentration at time t2 and Ct1 is the concentration at time t1. The total AUC (AUC0–24) over the dosing interval (24 h) was calculated by summation of all fractional AUC values over the intervals between 0 (pre-dose) and 24 h (post-dose).
When a terminal elimination phase was apparent in the plasma concentration vs time curve, the terminal half-life (t1/2) was estimated using the equation t1/2 = 0.693/ƛz, where ƛz is the elimination rate constant estimated from the slope of the terminal elimination phase. AUC0- ∞ was estimated using the following equation:
$${{{{{\mathrm{AUC}}}}}}_{0 - \infty } = {{{{{\mathrm{AUC}}}}}}_{{{{{{\mathrm{last}}}}}}} + {{{C}}}_{{{{{{\mathrm{last}}}}}}}/{\bar{\lambda}}_{{{{{\mathrm{z}}}}}}$$
where AUC0–∞ is AUC from zero to infinity, AUClast is the AUC from zero to the last measurable time point and Clast is the concentration at the last measurable time point. AUC0–∞ and t1/2 were not estimated when the terminal elimination phase was not defined.
Plasma clearance (CLp) after the IV dose was estimated using the following: CLp = Dose/AUC0–∞. Bioavailability (%F) after a single oral dose was estimated using the following:
$$\% {{{F}}} = \left( {{{{{{\mathrm{AUC}}}}}}_{0 - \infty ,\,{{{{{\mathrm{PO}}}}}}}/{{{{{\mathrm{Dose}}}}}}_{{{{{{\mathrm{PO}}}}}}}} \right)/\left( {{{{{{\mathrm{AUC}}}}}}_{0 - \infty ,\,{{{{{\mathrm{IV}}}}}}}/{{{{{\mathrm{Dose}}}}}}_{{{{{{\mathrm{IV}}}}}}}} \right)$$
Mean AN15368 plasma concentrations and PK parameters after a single IV or PO dose, and mean AN15368 (total and free) plasma concentrations and PK parameters in the pre- and end-regimen periods are depicted in Supplementary Fig. 4.
### NHP treatment study
A total of 22 rhesus macaques that had been confirmed to be serologically and PCR-positive for T. cruzi were utilized in these studies. Using 19 animals in the treatment group provided 85% power of detecting 100% efficacy. The 19 animals were treated with a 30 mg kg−1 dose of AN15368 delivered in food treats once a day for 60 d. The remaining 3 animals in the study were maintained as untreated control animals and received food treats but were not dosed with AN15368.
The selected dose of 30 mg kg−1 in NHP was determined on the basis of the following rationale. The minimal efficacious dose in mice was determined to be 2.5 mg kg−1 (Fig. 3e) and PK studies in mice at a 10 mg kg−1 dose yielded an exposure of 17.5 (AUC0–last (μg h kg−1); Supplementary Table 6). Assuming linear exposure, a minimal curative exposure in mice was estimated at 4.375 μg h kg−1 (that is, 17.5/4). PK analysis of a 30 mg kg−1 dose in NHP indicated an exposure of 3.5–4.7 μg h kg−1 (Supplementary Fig. 5). On the basis of allometric scaling15, the no-observed-adverse-effect level (NOAEL) of 120 mg kg−1 in rats translates to a NOAEL dose of 60.5 mg kg−1 in monkeys and 19.4 mg kg−1 in humans. Thus, the 30 mg kg−1 NHP dose was expected to achieve an efficacious level on the basis of PK comparisons of mouse and NHP, and be safe as it is well below the estimated NOAEL dose determined in rats.
Under light anaesthesia/sedation, peripheral blood samples were collected from each animal before treatment and at 7 time points after treatment. Blood analysis (complete blood count, CBC and serum chemistry assays) and physical exams to evaluate the health of the study animals were performed on each animal before the beginning of the study and also at 3 time points during the treatment protocol (see Supplementary Data). At the termination of the study, 9 treated and 2 control animals were euthanized, necropsied, and blood and tissues were evaluated histologically, and using PCR and haemoculture for evidence of active T. cruzi infection. The remaining 10 treated and 1 control animals from the study were returned to the breeding colony at the Keeling Center.
### NHP blood and tissue PCR for T. cruzi DNA and haemoculture
Blood samples from each macaque were collected at various time points and processed for quantification of T. cruzi DNA by real-time qPCR. Between 8 to 10 ml of whole blood collected in EDTA anticoagulant tubes was subjected to DNA extraction using the Omega E.Z.N.A. blood DNA maxi kit (Omega BioTek) following the manufacturer’s instructions for up to 10 ml whole blood and using a total of 650 µl of elution buffer. Each round of extractions included a negative (no-template) control composed of 10 ml PBS. The concentration of DNA in the eluted solution was quantified after each extraction using an Epoch microplate spectrophotometer (BioTek).
DNA from each sample was then subjected to a series of two qPCR assays for detection of T. cruzi satellite DNA. The first qPCR used the cruzi 1, 2 primer set and cruzi 3 TaqMan probe as previously described53,54, using Bio-Rad iTaq Universal Probes Supermix (Bio-Rad). This qPCR amplifies a 166 bp region of a repetitive satellite DNA sequence and is sensitive and specific for T. cruzi when compared with other PCR techniques55. To rule out false negative PCR results due to inhibition, an internal amplification control (IAC) was added to the second qPCR reaction, which was run as a multiplex as previously described, with the cruzi 1/2/3 primers and probe, and the IAC primers and probe54, except that the IAC sequence was synthesized as a gene fragment by a commercial laboratory (gBlocks Gene Fragments, Integrated DNA Technologies) and was added at the time of PCR rather than before extraction. Positive (DNA extracted from T. cruzi Sylvio X10 clone 4, American Type Culture Collection (ATCC), 50800, known concentration 1.7 × 10−3 parasite equivalents) and negative (water) controls were included in each PCR plate for both assays. A C1000 Touch Bio-Rad CFX96 real-time PCR detection system was used for both assays under the following cycling conditions: (1) initial denaturation, 95 °C, 3 min; (2) denaturation, 95 °C, 15 s; (3) annealing, 58 °C, 1 min; (4) ×50 cycles. We selected FAM and VIC channels for each read.
Frozen tissues were screened for T. cruzi DNA using 8 mm biopsies (Sklar instruments, 96-1130), collecting 3 to 10 individual ~100 µl tissue punches for each tissue type and one or more pooled samples consisting of five punches from different areas of the tissue, totalling ~500 µl per pool. The tissues sampled included liver, heart, fat, oesophagus, quadricep, bicep, large intestine, and brain, as well as tongue and spleen in a few instances. Two individual and one pooled sample of tissues from an uninfected macaque was collected for each sampling batch. DNA from macaque tissue was extracted and analysed as previously described32, with the exceptions that the purification was scaled up to accommodate the larger amount of tissue in the pooled samples and the range for the standards was 2.6 × 102 – 2.6 × 10−3 parasite equivalents using the kDNA minicircle S35 and S36 primers. The Bio-Rad CFX manager software version 3.1 was used to analyse PCR data. For samples to be considered positive, both replicates per sample must show a product (cQ value) of <40 and less than that of the included naïve sample, and melt curves had to be in the same temperature range as the standards for each plate.
For haemoculture determinations, peripheral blood from macaques was collected and cultured at 26 °C in supplemented liver digest neutralized tryptose medium as described previously17. The presence of T. cruzi parasites was assessed every week for 3 months under an inverted microscope. T. cruzi DTU of the macaque isolates was determined as previously described17.
### Multiplex serological analysis
Luminex-based multiplex serological assays were performed as previously described16,17. For a number of smaller proteins, fusions of up to 2 individual genes are employed for some target proteins to expand the array of antibodies being detected while controlling costs and complexity of the assay (TritrypDb.org identifiers: Tc1, fusion of TcBrA4_0116860 and TcYC6_0028190; Tc2, fusion of TcBrA4_0088420 and TcBrA4_0101960; Tc3, fusion of TcBrA4_0104680 and TcBrA4_0101980; Tc4, fusion of TcBrA4_0028480 and TcBrA4_0088260; Tc5, fusion of TcYC6_0100010 and TcBrA4_0074300; Tc7, fusion of TcYC6_0083710 and TcBrA4_0130080; Tc8, TcYC6_0037170; Tc11, TcYC6_0124160; Tc17, fusion of TcBrA4_0028230 and TcBrA4_0029760; Kn107, TcCLB.508355.250; G10, TcCLB.504199.20). Macaque antibody binding to individual beads in the multiplex assays was detected with donkey anti-human IgG (H+L) conjugated to phycoerythrin (Jackson ImmunoResearch) in a 1:200 dilution.
### Statistics and reproducibility
The non-parametric Mann-Whitney U test and the unpaired t-test from the software GraphPad Prism v9.4.0 were used. Values are expressed as mean ± s.e.m. Statistical significance was evaluated using *P ≤ 0.05, **P ≤ 0.01 and ***P ≤ 0.001. All mouse experiments were performed at least twice with similar results. All in vitro parasite proliferation assays were repeated at least once with similar results. PCR and western blot assays depicted as representative microphotographs in Fig. 2b were repeated three times and one time, respectively, with similar results. The quantitative liquid chromatography tandem mass spectrometry assay described in Fig. 2d was performed once. Due to cost and complexity, the NHP trial was performed once.
### Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article. | 2023-03-30 23:14:13 | {"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": 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": 0.3676662743091583, "perplexity": 10301.36562422886}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949506.62/warc/CC-MAIN-20230330225648-20230331015648-00339.warc.gz"} |
http://thaimaizeandproduce.org/hq7hpcl/combustion-of-magnesium-equation-ad85b6 | Final temperature, T, "С "С "C 21:5 3. CH4 reacts with oxygen (O2) to make carbon dioxide (CO2) and water (H2O). In Ref. How old was Ralph macchio in the first Karate Kid? Show how equations (1),(2) and (3) in the introduction can be combined to give the equation for the combustion of magnesium. Chemistry Q&A Library Write a complete balanced equation for the combustion of ethanol, the combustion of aspirin, and the reaction between magnesium oxide and hydrochloric acid. B. Age range: 13â15 . All Rights Reserved. Through experimentation it was found that the enthalpy of change for the combustion of magnesium is -593.3KJ/mol and that the thermo chemical equation (target equation) for the combustion of magnesium is (see right ) . It will not be practical to measure the reaction enthalpy in your foam cup calorimeter, but you can What is the word equation and balanced equation for combustion of Magnesium? The combustion mechanism of laser-ignited magnesium particles in the 100 ν-size range was studied by cinemicrography of burning particles and by scanning electron micrography of quenched samples. A very bright flame accompanies the production of magnesium oxide, as shown in the photograph on the right. ... the equation (1) was calculated to be about 2238 K (1965°C), which is much higher than 1800 K, indicating Demonstrations ⺠Chemical Reactions I ⺠4.1. Magnesium also reacts with carbon dioxide to form magnesium oxide and carbon: $Mg_{(s)} + CO_2 \rightarrow 2 MgO_{(s)} + C_{(s)} \tag{3}$ Hence, carbon dioxide fire extinguishers cannot be used for extinguishing magnesium fires either. Combustion of Magnesium . HEAT OF COMBUSTION: MAGNESIUM DATA AND CALCULATIONS Name: Reaction 1 (Mgo) Reaction 2 (Mg) ml 1. B. Determination of the Enthalpy of Combustion of Mg Using Hessâs Law The calibrated calorimeter will be used to determine th e enthalpy of combustion of magnesium by application of Hessâs law. What is exact weight of male Bengal tiger? magnesium + oxygen → magnesium oxide 2Mg (s) + O₂ (g) → 2MgO. What date do new members of congress take office? The magnesium ribbon has been pre-cut to the proper length by your teacher. Also, burning magnesium gives off a very bright white light, which is why there are magnesium flairs. Abstract: Two reactions, listed in Background, were carried out to determine the heat of combustion of magnesium. Combustion process: 2Mg (s) + O2 (g) --> 2MgO (s) When you burn Magnesium (Mg), it reacts with Oxygen (O2) to form magnesium oxide (MgO). Change in temperature, AT "С "C 5. What is the WPS button on a wireless router? In Experiment 18, you learned about the additivity of reaction heats as you confirmed Hess’s law. Initial temperature, T 4. This is an ionic bond so atoms are giving and accepting electrons. The oxide is in liquid state with emissivity of 0,9. surface ⦠Is there a way to search all eBay sites for different countries at once? Jack Hume 14 March 2015 Richards, HN Chemistry 8B. Heat of Combustion: Magnesium . Other names: Magnesium monoxide Permanent link for this species. ... We can go further and translate the picture equation for the reaction between magnesium and oxygen to a chemical equation: 2 Mg + O 2 â 2 MgO. should not be carried out. The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. What was lasik visons competitive priority? Where is Jesse De-Wilde son of Brandon deWilde? combustion, the particle consists of pure magnesium with spectral emissivity of about 0.25. Magnesium metal burns rapidly, releasing light and heat, as observed in photo flashbulbs or by burning magnesium ribbon. The increase in mass is due to the fact that oxygen from the air has combined with the magnesium to make magnesium oxide, MgO. WORD EQUATION: BALANCED FORMULA EQUATION: B) T combustion of sulfur produces dioxide, SO2-. in the same way as the iron wool. Timing: ... mass when sulfur, carbon, magnesium and copper are burnt. What does contingent mean in real estate? The word equation for the result of this experiment is Magnesium + Oxygen= Magnesium Oxide. Simple composition or formation Reaction between magnesium metal and oxygen gas finishing with balanced equation The oxidation or combustion of magnesium in air has long been used as a source of intense light in photography and for other photochemical reactions. CH 4 + 2O 2 â 2H 2 O + CO 2. Therefore, the mass of the product(MgO) is included mass of Magnesium and Oxygen, that is, ofcourse more than ⦠At the initial phase of During the combustion process, magnesium oxide is formed locally on the particle surface. 20.8 "С 2. Chemistry. Liquid Phase Heat Capacity (Shomate Equation) C p ° = A + B*t + C*t 2 + D*t 3 + E/t 2 hydrochloric acid to be 1 g/mL. In this work, the combustion process of a single magnesium particle in water vapor is studied both experimentally and theoretically. The magnesium was lit and then there was a white bright light as it ignited and it formed this chalky white powder. The reaction is presented by the equation: This equation can be obtained by combining equations (2), (3) and (4): By combining these three equations, you can obtain equation (1). Heat of Combustion: Magnesium Chemistry with Vernier 19 - 3 Reaction 2 9. The negative sign tells us it is exothermic (energy is released). Heat of Combustion: Magnesium In Experiment 5, you learned about the additivity of … 0.411 mol x -601.6 kJ/mol = -247 kJ. 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Issue A&A Volume 552, April 2013 A94 6 Galactic structure, stellar clusters and populations https://doi.org/10.1051/0004-6361/201220396 04 April 2013
## 1. Introduction
Owing to its nearness and extreme nature, Radcliffe 136 (Feast et al. 1960) has been not only a template for other more extreme and distant starburst clusters, but also a good example of how a typical galactic globular cluster might have looked soon after its birth. To obtain its physical characteristics in a way firmly rooted in observations has been an important goal as the many studies of this interesting object attest (Melnick 1985; Weigelt & Baier 1985; Campbell et al. 1992; Parker & Garmany 1993; Malumuth & Heap 1994; Hunter et al. 1995; Brandl et al. 1996; Hunter et al. 1996; Walborn & Blades 1997; Massey & Hunter 1998; Andersen et al. 2009; Bosch et al. 2009; Campbell et al. 2010; Crowther et al. 2010; de Marchi et al. 2011; Hénault-Brunet et al. 2012, and the whole series of VLT-FLAMES Tarantula Survey papers; Sabbi et al. 2012). Selman et al. (1999b, henceforth SMBT) studied the IMF of R136 and provided several of its physically relevant parameters, including the mass density profile of the cluster. Three very important results of that paper gave insight into the age structure of the cluster, the normal nature of its IMF, and the scale-free character of the mass density profile between 0.4 and 10 pc.
The cuspy profile and the small core radius has been used to posit a post-core-collapse (PCC) state for this cluster (Campbell et al. 1992; but see opposite views by Malumuth & Heap 1994; Brandl et al. 1996; Mackey & Gilmore 2003). PCC clusters are characterized by a high central density and a density profile that can be modeled by a normal King profile, which shows a break and turns into a power law near the center. High central density means, in this context, that the relaxation time is shorter than the age of the system. Observations of Galactic globular clusters show that they can be separated into “core-collapsed” and non-core-collapsed depending on whether their photometric profile turns into a power law near the center (Harris 1996; McLaughlin & van der Marel 2005; Chatterjee et al. 2013, and references therein). The cluster R136 has been claimed to have the characteristics of a PCC cluster: a cuspy density profile (Mackey & Gilmore 2003), and a large number of runaway stars (see Fujii et al. (2012) and references therein). Nevertheless, the time scale for core collapse is too long if we believe the value of 3 × 104 M/pc3 determined by Mackey & Gilmore (2003) for the central density. Fujii et al. (2012) invoke the fact that smaller clusters evolve faster, thereby speeding up the evolution via hierarchical merging of smaller substructures. This idea has been backed up by the recent work of Sabbi et al. (2012). Using theoretical isochrones for MS and PMS evolution, and Hubble Space Telescope (HST) optical and NIR photometry, they find that the central part of the cluster is very young with ages below 1 My, while an overdensity ~5.4 pc to the northeast of the center is closer to 2.5 My. This led them to propose that R136 is a double cluster that is currently interacting. Similar ages were found in SMBT, but with fits to the Geneva tracks in the upper part of the HR diagram. That similar age structure is found by using these two sets of tracks give credibility to both of these results.
Such a complex age structure is quite a complication when modeling to convert magnitudes and colors into masses. If the average age of the stellar population depends on radius, this will result in systematic errors in our estimates of physical parameters unless variable ages are allowed in the modeling. Most work on R136 so far assume simple stellar populations. The only work that we are aware of that does not is SMBT, which also considers variable reddening determined in a star-by-star basis, and does a full completeness analysis, which is fundamental when working in the optical bands.
Using ground-based observations in combination with the HST work of Hunter et al. (1995; 1996), SMBT determined the density profile of the cluster showing that between 0.4–10 pc it is represented well by a single power law, giving explicit expressions. Although a lower limit to the central density of the cluster can be derived from those numbers, it was not given explicitly in the paper. In this work we give the explicit results. If we use the same core parameter as used by Mackey & Gilmore (2003) we find a striking similarity between our result and that of those authors. Furthermore, using the recent estimates of the core parameter by Campbell et al. (2010), we find that the central density is considerably higher than the previous estimate, making core collapse a virtual certainty for this cluster, or an indication that the latter estimate of the core parameter is wrong.
## 2. Radial profile, central density, and total mass in R136
SMBT give the following relation for the stellar number density normalized to 1 pc, derived counting stars with masses between 10 M < M < 40 M: (1)where \begin{lxirformule}$r$\end{lxirformule} is the distance to the cluster center in parsecs. As previously noted, a strong effort was made by SMBT to correct for incompleteness, and although the work was based on optical data, Eq. (1) should not be affected by differential reddening and should be complete in the specified mass range.
A form often used to parametrize the stellar density of a star cluster was given by Elson et al. (1987): (2)where a is the core parameter1, and ρ0 is the central density. This form is suitable for clusters that are young enough not to exhibit tidal truncation. We note that the core parameter a is usually obtained by fits to the 2D profile, but if the fit is a good one, then the value thus obtained is the same as for the cluster 3D density profile. Given a core parameter, then and only then can we calculate ρ0. If we only have valid measurements in the power-law region, where r ≫ a, we can only obtain a lower limit to ρ0. Thus, the challenge in determining the central density is really a challenge in determining the core parameter. There are basically two methods of determining this parameter: fits to star counts, and fits to the integrated light profile. Brandl et al. (1996) cautions that determining the core parameter using surface brightness profiles is not reliable, since the light can be dominated by a few very bright stars (as is indeed the case near the center of R136), and mimic a power-law cusp. A similar caveat can be found in Mackey & Gilmore (2003).
We can write the results for the stellar volume density found in SMBT using the Elson et al. (1987) form for an arbitrary value of a but fitting the SMBT power-law data: (3)showing a clear power law of slope γ = 1.85, without signs of a core between 0.4–10 pc. Figure 1 reproduces Fig. 13 from SMBT, which gives a profile totally determined counting stars in mass bins. The innermost point suffers from strong incompleteness.
Fig. 1 Radial density profile for the stars with 10 M⊙ < M < 40 M⊙. For the innermost bins we have calculated the points using the Hunter et al. (1995) data following the procedure described in SMBT. The line is a power law with exponent –1.85. The point at the smallest radius is strongly affected by incompleteness. Reproduced with permission from SMBT. Open with DEXTER
Using the value a = 0.4 pc as an upper limit we can write the previous result as (4)This expression is valid for stars with M in the range 10 M < M < 40 M. To convert this number density profile to a mass profile, we need the spectrum of stellar masses to extrapolate the density from the given mass range to the full mass range. We choose to use the spectrum of masses at birth, i.e. the IMF, instead of the present-day mass function, although it does not make much of a difference given the young age of the central parts of R136. At 1.2 My the most massive stars have lost less than 4% of their mass and at 2 My this figure is still only 20%. The lowest mass for which an IMF has been given for R136 so far is 1 M (Andersen et al. 2009). These authors found consistency with a Salpeter slope down to this mass limit. We used this result as a check that the IMF does not show a break down to this mass limit but used the SMBT IMF slope extrapolated down to this mass, because it was determined in a rigorous and self-consistent way. We thus use a Kroupa IMF (Kroupa 2002) modified in the high mass range to have the slope determined in SMBT (Γ = 1.17), (5)With this IMF the fraction of stars with masses between 10 M < M < 40 M is 0.3967%. The fraction in the mass range 1 M < M < 120 M is 7.286%, and the average mass is 3.85 M. Thus, if we choose to integrate only down to 1 M, we estimate the central mass density limit, (6)The result in Eq. (6) has been derived almost directly from observations. There are modeling uncertainties, and an extrapolation from 10 M down to 1 M, but the latter supported by observations. To extend the result to even lower masses where the mass function has not been determined observationally is somewhat risky. Depending on the actual dynamical state of the cluster, we could have the effects of mass segregation invalidating the result. The mass density profile we obtain with this extrapolation of the IMF to the full 0.01 M < M < 120 M range where the average mass is 0.46 M is given by (7)Mackey & Gilmore (2003) determine an upper limit to the core parameter of a < 0.32 pc, and with this value we can estimate a lower limit to the central density of 2.8 × 104 M/pc3, in excellent agreement with the value of 3 × 104 M/pc3 given by those authors. Because of the IMF extrapolation we have not preserved the inequality sign in this equation, which was quite certain for Eq. (6).
These numbers can also be used to determine a total mass. The complication here is that the slope of the density profile results in an ever increasing total mass as one goes to larger radii. Here we choose to cut the integration at a radius of 10 pc, the distance at which the star counts become noisier due to the presence of a number of substructures. The same slope value that diverges for large radii ensures convergence when integrating from the center. Thus, we will give our estimates of the mass of the cluster as a range where the lower total mass limit is given by the upper limit of the core parameter, and the upper total mass limit is given for a power law all the way to the center. With these assumptions we can then write (8)where we have defined (9)and in what follows we integrate numerically. We note that for γ > 2 the integral converges in the limit R/a → ∞. For γ = 2 it diverges logarithmically.
With these definitions the constraint to the total mass becomes (10)and for R136 using a < 0.4 pc, (11)or in terms of total number of stars Ntot, (12)
## 3. Discussion
As noted in the previous section, the estimate of the central mass density depends very strongly on the limits put on the core parameter. Because it was determined by star counts, we feel that the limit given in SMBT is a very conservative one. Other authors have placed a much more stringent constraint by analyzing the light profile (e.g. Campbell et al. 1992; Andersen et al. 2009). One of the latest such studies, and one that reaches similar conclusions to other studies that use the surface brightness method, is that of Campbell et al. (2010). They used the multiconjugate adaptive optics instrument (MAD) at ESO’s Melipal 8-m telescope to perform star counts and surface photometry in H and K, on frames with typical Strehl ratios in K of 15–25%. Although there is little leverage for disentangling masses and ages in the infrared bands, they are less affected by reddening so with proper care a good estimate of the radial profile can be obtained in the case of a simple stellar population. These bands are also separate from the peak emission for very massive stars so the cluster’s light profile is less affected by the presence of a few very massive stars. The magnitude limit of the MAD data corresponds to approximately 5 M. Within approximately r < 2 pc they use the light profile integrated in annuli, while for r > 0.7 pc they use star counts. They merge the two profiles into a single one using the area of overlap to normalize the two sets. They find γ = −1.6 ± 0.1, and a = 0.025 pc. Because of the presence of radially dependent stellar populations with multiple ages, we only use their value for the core radius, while for the slope we use the value determined in SMBT where allowance has been made for multiple epochs of star formation.
The issues complicating the determination of the core parameter using integrated light profiles are not addressed at all by Campbell et al. (2010), but are fundamental in these kinds of studies. Their Fig. 20 shows that the power-law slope is virtually the same for the segment derived with star counts and the segment derived by the surface brightness fit. This might indicate that the surface brightness is a good proxy for mass surface density. Nevertheless, we must point out that the very bright central sources could give a similar signal due to scattering of light in the optical elements of the instrument. It has been known for a long time that the profile of a stellar image contains a kernel and a power-law part of index ~− 2 (de Vaucouleurs 1958; King 1971). The origin of this power-law halo has been ascribed to optics imperfections and dust. According to Kormendy (1973) the slope is flatter at ~− 1.54. This index is close to the –1.6 index determined with MAD so scattered light might mimic the cluster light profile. This shallow-slope power law is expected to dominate a few seeing radii away from the core of the PSF. Nevertheless, between the core of the PSF and this radius, we expect a much steeper uncompensated halo that falls off with a Moffat β ~ 11/3, approximately as r-3.7 (Moffat 1969; Roddier 1995; Racine 1999).
The results of Campbell et al. of a = 0.025 pc, imply moderately larger bounds for the total mass and number of stars given by (13)and central density of (14)which is much higher than any other estimates of the R136 central density.
Is R136 a PCC cluster? If we take the MAD data at face value, the cluster must be in a PCC state since the relaxation time for a cluster with such a central density is only ~8 × 103 y, and the time to core collapse is approximately 15 times this value (using Binney & Tremaine 1987, as presented in Eq. (8) by Mackey & Gilmore 2003), or ~1 × 105 y, considerably below the current estimate of 106 y for the age of the cluster.
Two other subtleties related to relaxation and core collapse must be pointed out. First, the identification of “core-collapsed” clusters in the galaxy rests in identifying a break in the inner part of the light profile. When properly studied, that is by star counts, R136 shows a scale-free distribution from 10 pc all the way down to at least 0.4 pc. Second, the relaxation times used in the literature assume a single mass component, while for this very young cluster we have a spectrum of masses spanning at least three orders of magnitude. In such a case the relaxation time is diminished by the ratio of the lowest mass to the highest mass star. We do not delve deeper in this subject because the physics of core-collapse is beyond the scope of this work.
Fig. 2 a) Radial profile of color excess, as determined in Selman et al. (1999a). b) Color excess versus selective extinction parameter S. Reproduced with permission from Selman et al. (1999a). Open with DEXTER
A central density as high as implied by Eq. (14) would have other consequences. Selman et al. (1999a), the first paper of the series and dedicated in part to the study of the reddening distribution in R136, found a puzzling trend in reddening increasing toward the center of the cluster. Their Fig. 6a, reproduced here as Fig. 2 shows clearly that trend. The figure shows that in the inner regions most of the stars show a somewhat limited amount of variable extinction that increases toward the center. In the outer parts more than half of the stars follow the same trend, while another group, probably associated with the nebulosity that surrounds the cluster, show a much higher variation of extinction. The amount of dust associated with the inner component was estimated at 30–60 M within 15 pc, and its origin was a mystery.
The details of the calculation were not given in SMBT, so we give them in Appendix A. From Fig. 2 we see that the extinction increases from values near 0.2–0.3 at 80′′ from the center reaching values slightly in excess of 0.5 near the center. We assume that the former values sum the contributions from the Milky Way, the LMC and the larger 30 Doradus environment, and we assume the values above 0.2–0.3 to come from a component associated to R136 itself. Thus, using EB − V ≈ 0.1−0.2 for the extinction associated with R136, we find Mdust ~ 40−80 M within 15 pc from the cluster center, or Mdust ~ 18−36 M within 10 pc from the cluster center.
In SMBT, we stated our surprise to see dust in such an extreme environment. Any dust particle near the center of the cluster would, if not destroyed, be removed very rapidly. Thus, a source of dust is needed to replenish it. A natural source of matter is the mass loss from the massive stars. It is only the stars with masses above 15 M that lose significant amounts of mass, and the total amount of mass loss depends strongly on the age of the system as the rate increases rapidly with age. According to SMBT the central cluster stars belong to the first burst with ages below 1.5 My. A more recent estimate of the age of the central stars of the cluster is given by Crowther et al. (2010). These authors used rotating models and UV and IR spectroscopic observations of the central WN stars to constraint the age to 1.7 ± 0.2 My. We use for our estimate an upper limit of 2 My with the evolutionary models without rotation of Meynet et al. (1994). These authors doubled the mass loss rates of de Jaeger et al. (1988). With this artificial increase in the mass loss rate, these authors were able to explain the number ratio of WR stars to O stars, which otherwise would have come too low. With the current state of the art rotating models they are able to produce WR stars even if there is no wind! The new models use the mass loss rates for O stars by Vink et al. (2000), and for WR stars by Nugis & Lamers (2000) that have mostly revised the rates downward (Maeder & Meynet 2012). Thus, using the Meynet et al. (1994) 2 My isochrone for Z = 0.008, weighted by the IMF in Eq. (5) we can calculate an upper limit to the total mass loss due to stellar evolution that is 0.35%. If we set the total mass of the cluster to 105 M, then the total mass loss corresponds to 350 M. Although a natural source of matter is the mass loss from the massive stars, it is not clear that they are a natural source of dust. To convert this to an amount of dust, we need the uncertain gas-to-dust ratio in the wind of such hot stars. The amount of gas from mass loss, if it were to have the gas-to-dust ratio of 347 found by Pei (1992) for the LMC would correspond to only 1.0 M, more than an order of magnitude below what we see. Using 100 for the gas-to-dust ratio results in only 3.5 M of dust. Thus, in SMBT we realized that we needed a different source of dust. Dust around massive stars, B-types, and WR had been found in the past (Geisel 1970; Allen & Swings 1972), which was later ascribed to colliding winds (Usov 1991), so we speculated that this dust could be produced in colliding winds of massive stars. Some work has been done in colliding winds since that time, and it has become clear that to produce dust we need a binary with a WC star as a member (Crowther 2003). There is only one WC star near the center of R136 in projection, Brey 83 (Breysacher 1981). This star is found at approximately 3 pc from the cluster center, and given the age structure of the cluster, it is probably farther away from the center. Even so, one star is not enough to explain the large amount of dust found. We would like to point out another possibility here.
If the IMF is normal, which it appears to be down to the lowest masses studied, we can assume that the star formation mechanism must also be normal, and we can expect the presence of circumstellar disks. The observational results are somewhat difficult to interpret. Rubio et al. (1998) used JHKs photometry to detect infrared excess sources around the 30 Doradus nebula. Most of the sources with excess are more than 5 pc away from the central cluster. These authors interpret this emission as coming from YSOs whose formation was triggered by the central cluster. Maercker & Burton (2005) use L-band observations with the 60-cm South Pole InfraRed Explorer (SPIREX) Telescope to detect L-band excess sources that they interpret as possessing circumstellar disks. They find a disk fraction of 42%. These observations are limited to the brightest sources, and the resolution does not permit studying the central regions of the cluster. It is likely that these sources are very bright YSOs that are still embedded in their dust shells. Stolte et al. (2010) studied the galactic star-burst Arches cluster near the center of the galaxy using L-band excess to find candidate disk sources. In addition they confirm the disk nature of these excess sources using SINFONI VLT observations to detect the signature of rotation in the CO bandhead emission (Bik & Thi 2004). They find that the disk fraction in Arches increases from 2.7% ± 1.8% in the cluster core (within 0.16 pc from the center), to 5.4% ± 2.6% for 0.16 pc < r < 0.3 pc, reaching 9.7% ± 3.7% outside. The overall disk fraction for the B stars in the Arches cluster is 6%, much lower than what would expect from the disk fraction age relation (Haisch et al. 2001). They explain the decrease in disk fraction toward the center by either UV radiation destruction, winds, or the tidal destruction mechanism of Olczak et al. (2012) that has the important characteristic, in the context of this work, of destroying the disk without destroying the dust.
In a recent series of papers, Olczak et al. (2012, and references therein) have proposed, based upon numerical simulations, that in an environment as dense as the core of the Arches cluster, where the density is expected to be >105 M/pc3, encounters can destroy up to one third of the circumstellar disks. Olczak et al. (2010) posit a number density of 105 pc-3 as the threshold above which disk destruction occurs rapidly. If Eq. (14) represents the physical state of the cluster, then the disk destruction process might be responsible, at least in part, for the inner extinction that we have measured. According to recent studies, disks are formed in all environments, varying only the lifetimes of the disks, which are much shorter for massive stars where they are destroyed by photo-evaporation (see Williams & Cieza 2011, and references therein). There is a loose linear relation between disk mass and stellar mass. The median disk mass is approximately 1% of the parent star mass, and the variation is approximately ±0.5 dex. For our estimate we need the arithmetic mean, which for stellar masses between 0.015–4 M has been found to vary between 2% to 5% of the parent stellar mass for different galactic star forming regions studied (Andrews et al. 2010; Scholz et al. 2006; Manning & Sargent 2000). Thus, if we assume a total mass of 105 M for the cluster, we expect the total mass in disks to be 2000–5000 M. Using a gas-to-dust ratio of 100 implies 20–50 M of dust, while using the gas-to-dust ratio of the 30 Doradus region of the LMC results in 6–15 M. Thus, to explain the inner reddening of R136 with dust coming from the destruction of circumstellar disks requires (1) that most of these disks were destroyed recently, within a few 105 y or else the gas would have been removed from the central region; and (2) the gas-to-dust ratio must be closer to 100 than to the value of 347 measured elsewhere in the 30 Doradus region. If both of these conditions are not met, then an alternative origin must be sought.
## 4. Conclusions
In this work we estimate a conservative limit to the central mass density of R136 of 1.5 × 104 M/pc3 for a < 0.4 pc. From this we estimate that the total mass of the cluster enclosed within 10 pc must be in the range or in terms of total number of stars Ntot, In this case the observed scale-free profile must be in place at the moment of formation because relaxation and core collapse have not had time to act in this young object.
On the other hand, if we use the recent estimate of the core parameter by Campbell et al. (2010), we get the moderately larger bound for the total mass and the more extreme constraint for the central density which would imply that core collapse becomes a certainty.
Such high central density can have other effects. We have shown that it is unlikely that the reddening distribution, observed by SMBT to increase toward the inner parts of the cluster, could come from normal mass loss from massive stars. It is posited in this work that a sizeable fraction of this dust could come from the recent destruction of most of the circumstellar disks in R136.
1
The core parameter in the Elson, Fall and Freeman form is related to the King profile core radius by This formula is incorrectly quoted in Campbell et al. (2010), but correctly given in Mackey & Gilmore (2003). We also note that rc is equivalent to the r0 in the King (1962) model only when it is much smaller than the tidal radius. This is not valid for low-concentration clusters, but certainly valid for R136.
2
Note a slight departure from the Pei (1992) notation. We use nX(a) for the distribution of grain sizes instead of NX(a), and use NX only as the normalization constant.
## Acknowledgments
The authors would like to thank C. Olczak for the estimate of the number of disks needed to be destroyed to explain the total mass of dust implied in Selman et al. (1999a). The speculation that the color excess seen in the inner parts of R136 is due to such a process lies only with the authors. C. Olczak also pointed out to us the effect that a spectrum of masses can have on the relaxation time of a stellar system. We would also like to thank the anonymous referee, whose constructive criticism helped improve the paper. We would also like to thank Dave Jones and the language editor for help with the English.
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## Appendix A: The total mass of dust from the observed E(B – V)
For our estimates we follow Purcell (1969), Spitzer (1978), and Pei (1992). This last author is interested in the gas-to-dust ratio, and all his derivations were done in terms of that parameter. However, the extinction can be directly converted into a mass column density of dust based solely on the assumed dust model. Spitzer (1978) gives an expression for the mass of dust as a function of E(B − V), but valid only for the Milky Way and in terms of the average extinction per kpc. We therefore redo the more modern Pei (1992) calculation to directly get the dust column density implied by the observed E(B − V).
The extinction optical depth at wavelength λ, τ(λ), is given by (A.1)where the sum is over species X, graphite, and silicate; a is the size of a grain; nX(a) is the distribution of grain sizes of type X; amin and amax are the lowest and largest grain sizes; and QX(λ,a) is the extinction efficiency factor for grains of type X. The distribution of grain sizes is that of Mathis et al. (1977)2: (A.2)with β = 3.5, amin = 0.005 μm, and amax = 0.25 μm. Then, aV is given by (A.3)which for this grain size distribution results in aV = 0.737 μm. With these definitions they write (A.4)where is the extinction efficiency factor of grains of type X given by Draine & Lee (1984). They have defined . They use this expression, together with a dust grain model, to fit the extinction data to the Milky Way, the LMC, and the SMC to find the value of rX that fits the data. With these definitions the column density of dust, Σd, is given by where ρC = 2.26 g cm-3 and ρS = 3.3 g cm-3 are the mass density of the graphite and silicate grains, and RV is the total-to-selective extinction parameter, which has the value 3.16 for the LMC. For the LMC the extinction curve is best fit with rX values given by rC = 0.018 and rS = 0.083. The values for the Milky Way and the SMC are somewhat different and can be found in Pei (1992), and can be used to finally obtain (A.8)Thus, an average value E(B − V) ~ 0.1−0.2 implies, in the LMC, a total mass of 40–80 M within 15 pc, and 18–36 M within 10 pc.
## All Figures
Fig. 1 Radial density profile for the stars with 10 M⊙ < M < 40 M⊙. For the innermost bins we have calculated the points using the Hunter et al. (1995) data following the procedure described in SMBT. The line is a power law with exponent –1.85. The point at the smallest radius is strongly affected by incompleteness. Reproduced with permission from SMBT. Open with DEXTER In the text
Fig. 2 a) Radial profile of color excess, as determined in Selman et al. (1999a). b) Color excess versus selective extinction parameter S. Reproduced with permission from Selman et al. (1999a). Open with DEXTER In the text
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Initial download of the metrics may take a while. | 2021-02-28 16:22:05 | {"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": 0.6618119478225708, "perplexity": 1885.4148531924177}, "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/1614178361510.12/warc/CC-MAIN-20210228145113-20210228175113-00460.warc.gz"} |
https://space.stackexchange.com/questions/44930/increase-in-period-and-semi-major-axis-following-near-head-on-collision-between | # Increase in period and semi-major axis following near head-on collision between Cerise and Ariane debris (1996)
This report describes the collision of the French satellite Cerise with a piece of Ariane rocket debris (object no. 18208) in 1996: http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1997ESASP.393..589A&defaultprint=YES&filetype=.pdf
The approach angle between the two objects was 158.9 degrees (nearly head-on).
Near the end of section 4.4, the report states: "It is astonishing to note that the period and hence the semi-major axis of object no. 18208 increased during the presumed collision whereas the two objects were going in opposite directions. The delta-V in the trajectory of debris no. 18208 is in the direction of its velocity and not in the opposite direction."
Has the increase in delta-V of the Ariane debris ever been satisfactorily explained? If not, what are some possible explanations for how a near head-on collision of two orbital bodies could increase the period and semi-major axis of one of the objects?
• +1 This is a really interesting question! A good answer will have several parts; 1) establish that this is really "astonishing" to begin with and not just counterintuitive, 2) find out if it has "ever been satisfactorily explained" and 3) include a summary of the explanation. – uhoh Jun 21 at 22:46
• – uhoh Jun 22 at 0:30
• @uhoh this redirection you showed in the link would not add deltaV to the objects. The object redirected "upside" would gain Apoapsis hight, but it would lower the periapsis. And another point is: you cannot apply elastic bumbs in orbital speeds. – CallMeTom Jun 22 at 10:37
• @CallMeTom what makes you think so? Can you demonstrate this rigorously using math? If so then that would address point 1) in my first comment. If not, then for now we'll just have to call it "counterintuitive". In the mean time, if a ping-pong ball and a bowling ball collide with equal and opposite velocities, does not the ping-pong ball recoil with a higher speed than before the collision? – uhoh Jun 22 at 10:42
• @uhoh assuming both objects (1 and 2) in the figure you showed are on a (near) circular orbit, this would mean: a horizontal line is parallel to earths surface up is zenit, down is nadir. after the collision just looking at the "new orbit" would come from a lower alt and go to a higher alt, so it would be elliptical with a periapsis lower, and a apoapsis higher than before the crash – CallMeTom Jun 22 at 10:57 | 2020-10-24 09:49:05 | {"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": 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": 0.5365626215934753, "perplexity": 1261.1752189782103}, "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/1603107882103.34/warc/CC-MAIN-20201024080855-20201024110855-00717.warc.gz"} |