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https://tex.stackexchange.com/questions/263642/beamerposter-heading | I have awfully lot of troubles trying to make just a simple centered heading using beamerposter package. I am totally lost. Could you please provide any minimal working example?
Of course, I've been at simple examples like in here and have read questions like this one but I still don't get it. It seem to me that without an undecypherable *.sty it just doesn't work. Am I missing a simple solution?
Note: I've tried the answer from this question but it didn't work for me. Maybe if there could be something without mwe?
• What does " it didn't work for me." means exactly. Do you have beamerposter.sty installed in your machine? – Gonzalo Medina Aug 28 '15 at 14:58
• Yes, I do. It was a problem with includegraphics. I've fixed that already. But is this cheat the only way to make the simple heading? – Victor Pira Aug 28 '15 at 15:07
• No. I'll add a simple example below. – Gonzalo Medina Aug 28 '15 at 15:14
Here's a simple example; the textpos package was used to place the blocks at desired locations:
\documentclass{beamer}
\usetheme{Frankfurt}
\usepackage[orientation=landscape,size=a0,scale=1.4,debug]{beamerposter}
\usepackage[absolute,overlay]{textpos}
\setlength{\TPHorizModule}{1cm}
\setlength{\TPVertModule}{1cm}
\title[A simple poster]{An example of the \texttt{beamerposter} package}
\author{The Author}
\institute{Thye Institute}
\date{\today}
\begin{document}
\begin{frame}[t]
\begin{textblock}{110}(5,3)
\centering
\maketitle
\end{textblock}
\begin{textblock}{110}(5,20)
\begin{block}{A test block}
Some test text.
\end{block}
\end{textblock}
\begin{textblock}{110}(5,25)
\begin{columns}
\column{.32\textwidth}
\begin{block}{A test block}
Some test text.
\end{block}
\column{.32\textwidth}
Some test text.
There are other alternatives for your posters such as tikzposter. | 2019-09-22 06:19:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7629156112670898, "perplexity": 1729.4476066620327}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514575168.82/warc/CC-MAIN-20190922053242-20190922075242-00478.warc.gz"} |
https://docs.ibis-project.org/generated/ibis.expr.api.IntervalValue.hours.html | # ibis.expr.api.IntervalValue.hours¶
IntervalValue.hours
Extract the number of hours from an IntervalValue expression.
Returns: IntegerValue The number of hours in the expression | 2018-12-10 08:50:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.740295946598053, "perplexity": 3049.0039437844857}, "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/1544376823320.11/warc/CC-MAIN-20181210080704-20181210102204-00181.warc.gz"} |
https://cs.stackexchange.com/questions/129101/what-is-the-smallest-time-space-complexity-class-that-is-known-to-contain-complx | # What is the smallest time/space complexity class that is known to contain complxity class $\mathsf{SPARSE}$
Is it known if complexity class of all sparse languages is contained within e.g. $$\mathsf{EXP}$$ or $$\mathsf{EXPSPACE}$$? Or what is the smallest time or space complexity class that contains complexity class $$\mathsf{SPARSE}$$?
• What is the complexity class called SPARSE? – nir shahar Aug 8 '20 at 20:04
• The complexity class containing all sparse languages. – rus9384 Aug 8 '20 at 20:15
• All unary languages are sparse, so there are undecidable sparse languages. – Ariel Aug 8 '20 at 20:41
If by SPARSE you mean the set of languages where the acceptance occurs on a set of zero density, then it is not in EXP or EXPSPACE. It isn't even computable. To see this, pick your favorite computable enumeration of Turing machines T_n, and consider the language L in the alphabet {0,1} where a string S is in L if and only if L is consists just of n 1s, and where T_n halts on the blank tape. Since the problem of whether a given Turing machine halts on the blank tape is undecidable (if one can do it, one can solve the Halting Problem), our language L is undecidable. Using this same trick with a padding argument we can make languages which are as sparse as we want but are not computable.
• @rus9384 , I don't see how that would follow from the line of logic in my answer. Can you expand? – JoshuaZ Aug 8 '20 at 20:50
I'm not sure if it's the smallest class, but the natural candidate is $$P/poly$$ - for each $$n$$, the "advice" can just encode all acceptable strings of length $$n$$ (by definition of $$SPARSE$$, their number is polynomial).
$$P/poly$$ is also a strict superset of $$SPARSE$$: for example, it contains $$\Sigma^*$$. | 2021-01-17 19:24:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7287300825119019, "perplexity": 417.0453584398658}, "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-2021-04/segments/1610703513144.48/warc/CC-MAIN-20210117174558-20210117204558-00030.warc.gz"} |
http://mathhelpforum.com/discrete-math/156349-permutations-combinations.html | # Math Help - Permutations and combinations?
1. ## Permutations and combinations?
A set of triplets and two pairs of twins are posing for a photo-taking session. How
many different ways are there to arrange them in a row if the siblings are to stand next to each other?
I was thinking the answer would be 3! * 2! * 2! = 24 but its not,the answer is 144.
What am I doing wrong?
2. Originally Posted by mngeow
A set of triplets and two pairs of twins are posing for a photo-taking session. How
many different ways are there to arrange them in a row if the siblings are to stand next to each other?
I was thinking the answer would be 3! * 2! * 2! = 24 but its not,the answer is 144.
What am I doing wrong?
I think the info that they are triplets/twins is somewhat strange, since there is a possibility that they are so-called identical triplets/twins, and thus could be considered in some sense non-distinguishable for counting purposes.
Nevertheless the given answer indicates that we do in fact view the siblings as distinct people, so:
Treat each set of siblings as a single unit, there are three such units, giving 3!
For the triplet, there are 3! ways to arrange them among themselves
For each pair of twins, there are 2! ways to arrange them among themselves
Thus overall we get 3! * 3! * 2! * 2! = 144.
3. Oh I see where i went wrong now.Thanks alot | 2015-05-30 23:25:03 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.905484676361084, "perplexity": 409.6354294419971}, "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/1432207932737.93/warc/CC-MAIN-20150521113212-00302-ip-10-180-206-219.ec2.internal.warc.gz"} |
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# In a village of 2,500 people, 800 people are over 70 years old and 850
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In a village of 2,500 people, 800 people are over 70 years old and 850 people are female. It is known that 40 percent of the females are younger than 70 years old. If no one in the village is 70 years old, what is the probability that a person chosen at random is either a male or younger than 70 years old?
A. 221/250
B. 199/250
C. 33/50
D. 8/25
E. 51/250
[Reveal] Spoiler: OA
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Re: In a village of 2,500 people, 800 people are over 70 years old and 850 [#permalink]
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To solve this we would need to know the number of people who are less than 70 years old, the number of males, and the number of males who are less than 70 years old.
Overlapping sets: quantity of A or B = A + B - (A∩B) = quantity of A + quantity of B - (intersection of A and B)
Number of males = 2500-850 = 1650
Number of people below 70 years old = 2500-800 = 1700
Number of males below 70 years old = 1700-(850*0.4) = 1360
Total number of people who are male OR below 70 = 1650 + 1700 - 1360 = 1990
Probability of male or below 70 = 1990/2500 = 199/250
Answer: B
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In a village of 2,500 people, 800 people are over 70 years old and 850 [#permalink]
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Bunuel wrote:
In a village of 2,500 people, 800 people are over 70 years old and 850 people are female. It is known that 40 percent of the females are younger than 70 years old. If no one in the village is 70 years old, what is the probability that a person chosen at random is either a male or younger than 70 years old?
A. 221/250
B. 199/250
C. 33/50
D. 8/25
E. 51/250
Hi,
we can get the probability of answering this correct from $$\frac{1}{5}to \frac{1}{2}$$within 10 secs by realizing that --
1) Choices
prob of choosing a<70 years old = $$\frac{(2500-800)}{2500}= \frac{170}{250}$$..
we have to add male>70 year old to it so $$P> \frac{170}{250}$$
you can eliminate C, D and E ..
NOW we have to add male>70 year old to it ..
60 % F, which is 850 are >70yr old, so male >70 will be less than 50% of Total 800 = 400..
so our answer $$< 1700+400 = 2100..$$
so$$P<\frac{2100}{2500}$$
ONLY B is left
B
2) Proper Method
lets solve further
F=850 ..
60% of 850 = 510 are >70 year old..
Total >70 yr old = 800.. so M in that =800-510 = 290..
Also total<70 yr= 2500-800=1700..
the probability that a person chosen at random is either a male or younger than 70 years old=$$\frac{(1700+290)}{2500}= \frac{199}{250}$$
B
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Re: In a village of 2,500 people, 800 people are over 70 years old and 850 [#permalink]
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04 Jul 2016, 21:15
after solving this problem I learnt two things:
1) Reading the question well enough tells us what to solve for and avoids taking longer routes.
2) If we adopt an accounting approach via something like a double matrix here, it will become more challenging to solve this problem.
The question gives info on Females and population older than 70 years, but asks about a probability of males or younger. Well some arithmetic and probability we need to do.
P(male or <70 yrs) = Prob(male)+Prob(<70)-Prob(male and <70).
let us compute our sample space
n(male) = 2500-850 = 2550-900=1650.
Notice that second part is really number of female and <70 yrs = > 2/5*850= 340.
n of event space = 1650+340 = 1990.
prob = n of event space / number of entire population = 1990/2500 =>199/250.
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Re: In a village of 2,500 people, 800 people are over 70 years old and 850 [#permalink]
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25 Oct 2017, 11:36
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Bunuel wrote:
In a village of 2,500 people, 800 people are over 70 years old and 850 people are female. It is known that 40 percent of the females are younger than 70 years old. If no one in the village is 70 years old, what is the probability that a person chosen at random is either a male or younger than 70 years old?
A. 221/250
B. 199/250
C. 33/50
D. 8/25
E. 51/250
2500 people
800 over 70
850 female
40% female younger than 70 = 850(0.4) = 340
850 - 340 = 510 greater than 70 for female
Males = 2500 - 850 = 1650
If 800 are greater than 70 and 510 females are greater than 70, then 800 - 510 = 290 males are greater than 70 years old.
1650 - 290 = 1360 males younger than 70
probability that a person chosen at random is either a male or younger than 70 years old:
(total males, which includes less than 70 years + females less than 70) = 1650 + 340 = 1990
1990/2500 = 199/250 B
or (males less than 70 + females less than 70 + males over 70) = 1360+340+290 = 1990
1990/2500 = 199/250 B
Re: In a village of 2,500 people, 800 people are over 70 years old and 850 [#permalink] 25 Oct 2017, 11:36
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# In a village of 2,500 people, 800 people are over 70 years old and 850
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Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®. | 2018-03-17 16:44:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.3340263366699219, "perplexity": 2447.8423033685576}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257645248.22/warc/CC-MAIN-20180317155348-20180317175348-00175.warc.gz"} |
https://lieuzhenghong.com/2020/08/01/letter-to-bassel/ | # Letter to Bassel in August 2020, after my Finals results
Sat Aug 01 2020
Dear Bassel,
Have been putting this email off for so long--- not because I didn't want to send you an email, but simply because whenever I got down to writing it, there was always a new and exciting thing happening over the horizon, and I would decide to delay the email to incorporate that new thing. Now that things have settled down somewhat, here is the email that I should have sent a month ago!
## On exam results
Celine got a First as well! It was a pretty funny experience. We were in the garden inside the Schwartzwald when the email came in, and so we didn't realise that our results were available until we checked our phone an hour after that and were inundated with friends asking us for our results. We briefly considered going on our planned hike before checking the results, but I rightly pointed out that neither of us would enjoy the hike if we didn't get to check our results first.
And so we went up to the attic and started checking our results. It was satellite internet and it loaded so incredibly slowly (even slower since we were both very anxious). Mine loaded first and I saw the First Class and I was ecstatic! And then Celine's one loaded as well and we just exploded, we were so incredibly happy.
Anyway, Celine and I are both very pleased. I am incredibly, incredibly pleased, of course--- I did actually expect a First, but being ranked 5th in the level is far higher than what I expected or deserved.
We wanted to call you but you never replied to our WhatsApp message :( Celine is coming back tomorrow (2nd August) --- should we call then or the day after?
## What we've been up to
I thought that this summer would be a chill one but it turns out that interesting projects just keep popping up at me, and now I have a huge backlog of things that I need to do. This summer is actually (much) busier than my exam revision period, which isn't great, because this was supposed to be a relaxing summer spent with Celine. (It helps a lot that I'm doing self-directed stuff that I like, though).
### Game theory post is up
I wrote a post about the modified Blotto game I told you about here.
It got a lukewarm reception from the eminent scholars of Reddit.com. But one guy who is a CS PhD student said that he'd think about the algo complexity of solving the modified Blotto game. Haven't heard anything from him since though.
### Travelling: Ben's place, Celine's parents', and the Schwartzwald
We left Oxford to go to Brussels to visit Ben, the old alumnus of Merton who studied Compsci and Philo way back in 83 or something. His daughter goes to New now, studying chemistry.
We really enjoyed ourselves there: I had snail for the first time in my life, and it was actually pretty good. I've never seen a more middle-class Englishman in my life before though: Ben is like fourth or fifth generation Oxford, top civil servant in the EC, lovely big house in the middle of Brussels, a bit ditzy but very smart... He was such a gracious host and we enjoyed ourselves very much.
Here's a picture of us with Ben and his family when we cooked lunch for them:
Then afterwards we went to Celine's parents', and there was basically a basement for us to live together. (I'm in it now).
We went on a nine-day trip to the Schwartzwald (Black Forest), which was amazing. The AirBnB we lived in was a house in a village of twenty-eight people, high up in the mountains (~900m above sea level), and the views were breathtaking. The air was so clean and fresh, everyone drinks the water that comes right out of a natural spring (they literally pipe the water from the spring into the houses with no cleaning required --- this blew my mind), the animals were very cute, and there was absolutely no light pollution so on a clear night one can see hundreds or thousands of stars.
Everyone in the village is either a rancher or a pensioner, except our hosts, Patricia and Dieter. (Patricia is a tour guide + dive instructor: Dieter is a social worker). The people were so friendly, and our hosts exceptionally so. They grow a lot of trees and plants around their property, and I got to try i) a lot of different fruits I'd never tried before (redcurrants and mirabelles), and a lot of different fruits that I've never seen in the wild e.g. cherry tree.
We also saw tons and tons of wild raspberries and strawberries around. We went on a hike and started taking all that we could find and filled an entire box, but we couldn't even make a dent in the number. It's crazy how much shit just grows wild. The strawberries were AMAZING. They were very small, but they were so incredibly sweet. In fact they taste almost artificial, like strawberry-flavoured candy. (But of course they are 100% natural).
I'll get Celine to give me the pictures and I'll send you some pictures very soon!
### Rejected Harvard offer, and will most likely be starting work in Singapore in September 1
I applied to defer from Harvard for a year but they didn't allow me to, so I'm planning to reject the offer and reapply next year. (Will have to bug you for references again, unfortunately!)
And so I started talking to IMDA and asking them if they wanted to accept me for a year first. So I will be working for IMDA for a year, applying in December 2020, and deciding to matriculate in August 2021.
### Other projects I have been working on
#### MGGG GSoC
I've been trying to develop and deploy an upstream feature on their flagship web app called Districtr. I am implementing contiguity and cut edge functionality. When the user draws a districting plan, the server will return whether or not that districting plan is contiguous, how many cut edges there are, and where the number of cut edges falls in the distribution of some subset of plans in the space of possible plans (generated with MCMC).
The main problem is that the codebase is pretty large and difficult to wrap my head around. I don't know where everything is and I don't know what will break when I touch this or that, so progress has been slow.
#### Unsupervised trip clustering with Inzura again
My ex-boss Richard, the CEO of the auto insurance startup I was working at last summer, reached out to me again. He wanted me to do some unsupervised trip clustering. I know that TESL has a section about clustering, but I haven't read it yet --- do you have any advice?
Basically we have some trips, which you can think of as a tuple of GPS points
$[(x_1, y_1), (x_2, y_2), ... (x_t, y_t)]$
where $t$ is the length of the trip (varies from trip to trip), and $(x_t, y_t)$ is the lat/long of the device at that second. From these raw GPS points we calculate derived quantities such as turning, acceleration, braking, velocity, and so on. So we now have a tuple of derived quantities:
$[(a_1, b_1, c_1, ... ), (a_2, b_2, c_2, ...), .... (a_t, .... y_t)]$
The task is as follows. Given many of these tuples of derived quantities, can we start to cluster them? For instance, trips that have low velocity (e.g. the person is walking and not driving a car) might form a cluster, and trips that have very high velocity (person on a high speed train/plane) might form a cluster. Or there might be other clusters like driving on a country road v. on a highway, in a traffic jam v. in smooth traffic, and so on.
One issue I thought of is that these are possibly very high dimensional points, and additionally, each tuple (representing a trip) is of different length. AFAIK, Regular k-means clustering only deals with points---but these are tuples, not points. I could just take the mean to give an aggregrate score (so one $1 \times N$ point per trip) but that would lose a lot of information.
#### Revamping my personal website
My personal website was a bit long in the tooth, and I didn't understand the previous static site generator (Jekyll, written in Ruby) well enough. I also took the time to redesign some of the ugly-looking pages
#### Board game engine
This was going rather swimmingly for quite a while, but getting real-time multiplayer working is quite difficult. I will continue working on this side project when the other projects have concluded.
#### Path tracer with Ross
Haven't started on this yet because of all the stuff I've been doing, but I hope to do this soon. Ross is very much smarter than me so I need to make use of this opportunity.
#### Learn systems design/React/Express for technical take-home interview
I was recently approached by a Hong Kong startup (it was basically one of those GitHub scrape + email blasts) asking if I would be willing to take on a full-time job in Hong Kong. Obviously I am not willing, because I much prefer clearing some of my bond first. But the take home assignment was quite interesting, and Sebastian (this guy who did BPhil Philo at Merton last year) said that it would be "really good practice" as it is the kind of thing I'd be doing as a software engineer.
The problem is that I haven't touched many of the technologies in several years, and some I haven't learned at all. And the deadline is Thursday -- so I'll be pretty busy these couple of days learning all of this stuff from scratch! Should be fun.
#### Self-studying A8, A9, A12, and the Introduction to Statistical Learning
Haven't started on this yet either. I was planning to do it before I started Harvard, but other projects seem to have crowded it out. Hopefully I can get some synergy between the clustering project and this one.
## On the calligraphy I wrote for you
The little thing I wrote for you is an excerpt from a Classical Chinese poem. This Tang Dynasty poem tells the story of the poet and his close friend, two scholar-officials posted far from their homes. His close friend has received a new, prestigious official posting hundreds of miles away: they may never see each other again. Despite being written almost fourteen hundred years ago, the poem reflects aptly my feelings.
I have translated it as best I can with reference to several online translations, but I am of course no poet:
《送杜少府之任蜀洲》
<<Sending Off Vice Prefect Du on His Way to His Post in Suzhou>>
O'er the spires and walls of the Three Qins, our land,
There in wind and white mist, the Five Rivers descend.
We must say our farewells, leave each other behind
For the faraway posts that our liege lord's assigned.
Though the vast seas bind us, we remain bosom friends;
We are neighbours in heart, though apart at sky's ends.
Though our paths must now part, and I hold you most dear,
Lest like children we weep, let us hold back our tears!
The line I wrote for you translates to
"Though the vast seas bind us, we remain bosom friends; We are neighbours in heart, though apart at sky's ends".
And I thought that was apt: as long as we have interests in common, geographic distance cannot faze us. (It's probably even more apt nowadays---remember when we were just across the street but had to Zoom?)
## Conclusion
I am very sad to bid farewell to you, a beloved teacher, mentor, and friend. Hopefully we will get to see each other again soon in person, or collaborate on something exciting together.
Thank you so, so much for everything that you've done for me. Apart from the many tutes you've taught me (intro Micro, Core Micro, QE, and revision tutes), you also helped me with my thesis (remember when we were in your garden way-back-when in April 2019?), with game theory, and with lots of other things beside.
I still remember most fondly our spicy noodle dinners with Filip and Martin, the Chinese New Year banquet with the Warden, dinner and board games at your place, lunch at Liddell... I really don't think I could have asked or dreamed of a better tutor. It was truly my good fortune and an absolute honour to have been your student these past three years.
With much love,
Zhenghong | 2021-02-28 18:59:16 | {"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": 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": 0.2648094892501831, "perplexity": 2130.443839453675}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178361723.15/warc/CC-MAIN-20210228175250-20210228205250-00109.warc.gz"} |
https://gigaom.com/2008/11/12/recession-hits-intel-where-it-hurts/ | # Recession Hits Intel Where It Hurts
[qi:043] Well that didn’t take long. Intel (s intc) after the close of markets today issued its fourth-quarter revenue forecast — something it had said it would do on Dec. 4 — and as expected, cut its guidance by some $1 billion, citing “significantly weaker-than-expected demand in all geographies and market segments.” The chipmaker now expects fourth-quarter revenue to be$9 billion, give or take $300 million, vs. a prior outlook of between$10.1 billion and \$10.9 billion. The recession and drop in consumer spending are doing a number on the consumer electronics supply chain, and Intel is not immune. | 2021-05-11 22:47:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4191516041755676, "perplexity": 7935.653409110634}, "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-2021-21/segments/1620243990419.12/warc/CC-MAIN-20210511214444-20210512004444-00211.warc.gz"} |
http://math.stackexchange.com/questions/123797/relationship-between-the-weak-law-of-large-numbers-and-characteristic-functions | # Relationship between the weak law of large numbers and characteristic functions
I'm trying to learn some probability theory atm and got stuck with the following exercise in Durrett's Probability: Theory and Examples:
Exercise 3.3.17. Let $X_1, X_2, \ldots$ be i.i.d. with characteristic function $\varphi$.
1. If $\varphi'(0) = ia$ and $S_n = X_1 + \dots + X_n$, then $S_n/n\to a$ in probability.
2. If $S_n/n\to a$ in probability then $\varphi(t/n)^n\to e^{iat}$ as $n\to \infty$.
3. Use 2. and the uniform continuity of $\varphi$ to show that $(\varphi(h)-1)/h \to -ia$ as $h\to 0$. Thus the weak law holds if and only if $\varphi'(0)$ exists.
I would really appreciate some help with the third part of this exercise (I don't quite see the connection between $\varphi(t/n)^n$ and $(\varphi(h)-1)/h$, yet). Thanks for your help! =)
My thoughts on 1, 2:
I managed to prove 1. using the inequality $\mu\{x\, : \, |x|>u/2\} \le u^{-1} \int_{-u}^u (1-\varphi(t)) \, dt$, where $\mu$ is the pushforward measure of a random variable $X$ and $\varphi$ is its characteristic function. Using the fact that the ch.f. of $S_n/n - a$ is given by $e^{-iat}\varphi(t/n)^n$, this leads to
$$P\left[\left|\frac{S_n}n - a\right| > 2/u \right]\le u^{-1} \int_{-u}^u (1-e^{-iat}\varphi(t/n)^n) \, dt$$
Now 1. implies $\varphi(t/n)^n \to e^{iat}$, so the RHS goes to zero as $n\to \infty$ for every fixed $u$.
2.: Using $|e^{i\epsilon t} - 1| \le 2\epsilon |t|$ for small enough $\epsilon>0$:
\begin{align} \left|\varphi (t/n)^n - e^{iat}\right| &= \left| E\left[e^{iS_n/nt} - e^{iat}\right]\right| \\ &\le E\left|e^{i(S_n/n - a)t} - 1\right| \\ &\le 2 \epsilon |t| + 2P[|S_n/n - a|> \epsilon] \end{align}
So $\limsup_{n\to\infty}\, \left|\varphi (t/n)^n - e^{iat}\right| \le 2\epsilon |t|$ and since $\epsilon>0$ was arbitrary (apart from being small) this proves $\varphi(t/n)^n\to e^{iat}$.
-
For part 3, the connection is this: take logs. Using this, you can get that $n (\varphi(t/n)-1) \to iat$: write it as $$\left(\frac{\varphi(t/n)-1}{\log \varphi(t/n)}\right)( n \log \varphi(t/n))$$ and use the differentiability of $\log z$ at $z=1$ to show that the first factor goes to 1 as $n \to \infty$.
To finish, one needs to work a little harder. A previous exercise in Durrett shows that if a sequence of random variables converges in distribution, then the corresponding characteristic functions converge uniformly on compact subsets of $\mathbb{R}$. Using this fact in the first part of this problem, we can show that $n (\varphi(t/n)-1) \to iat$ locally uniformly in $t$. For sufficiently small $h$, we can write $h = t/n$ where $n$ is suitably large and $t$ is close to 1 (or -1, when $h < 0$); I will let you fill in the details. | 2015-12-01 04:28:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9976083636283875, "perplexity": 67.24822561498483}, "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-2015-48/segments/1448398464396.78/warc/CC-MAIN-20151124205424-00337-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/1118365/order-of-any-element-divides-the-largest-order | # Order of any element divides the largest order.
Let $A$ be a finite Abelian group and let $k$ be the largest order of elements in A. Prove that the order of every element divides $k$. This is my attempt, I sense there is something wrong\incorrect in it, but I can't figure out what....Also I didn't use that fact that $A$ is abelian... I would appreciate you reply.
$Attempt$: $|G|=n$. Suppose there is an element $a\in A$ of order $m$ that does not divide $k$. $k$ is the largest order of elements in $A$, therefore there exist $g\in A$ such that $o(g)=k$. $(ag)^m=g^m\ne 1$ and $(ag)^k=a^k\ne 1$ (that follows from the commutativity of A). The order of $ag$ has to be some $p$, where $p$| lcm$(m,k)$. If $lcm(m,k)=p$ then $(ag)^p=(ag)^{mt}=1$ where $mt=p,p\in \Bbb{N}$. But $(ag)^{mt}=((ag)^m)^t=(g^m)^t=g^{mt}=1$, but since $t$ is the smallest natural number that fulfills it, $mt=k$, i.e, $m$ divides $k$. A contradiction.
• Maybe the fact that any finite Abelian group is isomorphic to $\prod_{finite} (\mathbb{Z}/ p_i \mathbb{Z})^{\alpha_i}$ could help? Jan 26 '16 at 20:16
• has an answer here as well: math.stackexchange.com/questions/191981/… Oct 21 '19 at 0:37
Unfortunately, your attempt at a proof isn't valid. You define $n$ to be the order of $a$, then prove that $a^n=1$ (which doesn't need proof, actually, because that's what $n$ does), and then derive a contradiction because you had claimed $n<m$. Indeed, I don't think you ever used the hypothesis that $m\nmid k$, other than to (correctly) derive $a^k\ne1$ which was never used.
Here's a hint: if $\gcd(m,k)$ were equal to $1$, could you prove the statement? (Let $b$ be an element of order $k$; what's the order of $ab$?) Then, can you reduce the general case to the case where $\gcd(m,k)=1$? (Replace $a$ by an appropriate power of $a$....)
• I edited it... is that admissible? Jan 25 '15 at 2:53
• Not yet. First, you treat the case lcm$(m,k)=p$, but you don't consider the case $p<{}$lcm$(m,k)$. Second, your phrase "$t$ is the smallest natural number that fulfills it" is imprecise; and I think when you make it precise, you'll see that what you're claiming isn't necessarily true. Jan 25 '15 at 6:17
• As I understand, this zone is considered pretty hard and tricky by other question, so with this confusion and more than one time misunderstanding of me and other, I would have to quite. :( Jan 25 '15 at 11:38
Let $x\in A$ such that $x$ has largest order $m$ in $A$. Assume that $g\in A$ such that $|g|$ does not divide $m$. Wlog, assume that $|g|=p$ where $p$ is a prime. Then $gcd(p,m)=1$. Now, as $A$ is abelian, $g$ and $x$ commute and $|gx|=pm=lcm(p,m)$. Note that $lcm(p,m)=pm$ as $p$ is a prime and $p\nmid m$. Hence, $|gx|=pm > m$ which gives the contradiction as $x$ has largest order $m$. (I hope.)
• I only works for primes though, but that's a progress. Jan 25 '15 at 14:25
• But you know that if $|g|$ does not divide $m$, then it should have a prime divisor that does not divide $m$ also and by Cauchy's theorem, for every prime $p$ dividing the order of the group $G$, it has an element of order $p$. So, I think there no problem in taking the order of the element as a prime. If I'm wrong, pls correct.
– sez
Jan 26 '15 at 16:15 | 2022-01-22 18:15:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.8833531737327576, "perplexity": 106.14445685639151}, "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-2022-05/segments/1642320303868.98/warc/CC-MAIN-20220122164421-20220122194421-00210.warc.gz"} |
https://mathhelpboards.com/threads/geometry-of-nth-dimension.7991/ | # Geometry of nth dimension
#### mathworker
##### Active member
In this pdf it was written ,
" we probably have very little in our background that gives us geometric insight to the nature of $$\displaystyle R^5$$. However, the algebra for a line in $$\displaystyle R^5$$ is very simple, and the geometry of a line is just like the geometry of $$\displaystyle R^1$$"
I would be grateful if you help me look in how geometry of line will be in $$\displaystyle R^5$$ or what is the geometry in dimensions greater than 3
#### mathbalarka
##### Well-known member
MHB Math Helper
Re: geometry of nth dimension
mathworker said:
what is the geometry in dimensions greater than 3
It's similar to how you define geometry on $\mathbb{R}^3$, i.e., by defining each point as real 3-tuples and usual operations on them. For n-space geometry, define a real n-tuple instead with the same usual operation acting analogously over them.
The geometry of a line over $\mathbb{R}^5$ is similar to the geometry of a line in $\mathbb{R}^1$ in the sense that the lines in the former can be projected into any lower order spaces without the loss of any dimensions. That is, line is a one-dimensional object.
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#### mathworker
##### Active member
I am rather asking for a way to visualize the geometry of $$\displaystyle R^n$$ for $$\displaystyle n>3$$ ,I hope I am making some sense
#### mathbalarka
##### Well-known member
MHB Math Helper
I am rather asking for a way to visualize the geometry of $$\displaystyle R^n$$ for $$\displaystyle n>3$$ ,I hope I am making some sense
That is something hard, then, as we are in the 3-space. The best one can do is to draw out Schlegel diagrams or perspective projections of objects of 4-space or higher.
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#### Deveno
##### Well-known member
MHB Math Scholar
When one first encounters lines, it is often in the form of a function in the plane, such as:
$y = mx + b$.
If one is not given the slope $m$, but rather two points $(x_1,y_1),(x_2,y_2)$ that the line passes through, one uses a formula such as:
$y - y_1 = \dfrac{y_2 - y_1}{x_2 - x_1}(x - x_1)$
With a bit of algebraic "arm-wrestling" we can see that this is just the line:
$y = \dfrac{y_2 - y_1}{x_2 - x_1}x$
"shifted" so it passes through the point $(x_1,y_1)$.
The quantity $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ just depends on the 4 (constant) numbers $x_1,y_1,x_2,y_2$.
We can re-write the line $y = \dfrac{y_2 - y_1}{x_2 - x_1}x = mx$ as the set of all points in the plane of the form:
$\{(t,mt): t \in \Bbb R\}$
or, equivalently as: $t(1,m)$, or even as:
$t(x_2-x_1,y_2-y_1)$
If we call the points $(x_1,y_1) = P,\ (x_2,y_2) = Q$, then one way to describe the line going through $P$ and $Q$ is:
$t(Q - P) + P$, which goes through the point $P$ when $t = 0$, and goes through the point $Q$ when $t = 1$.
This last form generalizes nicely to $n$ dimensions (where each point has $n$ coordinates), we can also write a line as:
$L = \{t(Q-P) + P: t \in \Bbb R, P,Q \in \Bbb R^n\}$
The "one-dimensional-ness" of the line $L$ is reflected in the fact that we have one parameter ($t$) even though it is moving through two points in "$n$-space".
*********
As far as "visualizing" the geometry of $n$-dimensional space, it is rather hard to do "concretely": the best we can do is look at various 1,2 and 3-dimensional "shadows" of such a space. This is rather like imagining that the "top,front and side" views of a 3D object are actually all the same thing which can't be fully portrayed on a flat (2D) piece of paper. But mathematically, an $n$-dimendsional object behaves much the same as the lower dimensional objects we CAN visualize, it just has "more dimensions". One can have a "quasi-visualization" of 4-dimensional objects using 3-dimensional animation (remembering that although we are seeing an object "evolve" over time, what we are actually looking at is a "static" object from different 3D vantage points...analogous to representing a static sphere to a 2-dimensional being as a point that grows to a circle of the same radius of the sphere, and then shrinks back to a point again).
Because of the difficulty of "seeing" things in higher dimensions, we do something else entirely: we focus on the mathematical properties, which turn out to be much more tractable.
Although at first, this appears to be conceding defeat of some sort...it is actually liberating: we no longer have to think of higher-dimensional objects as "spatial" in nature, but as just depending on $n$ linearly independent variables. This lets us use the mathematics to describe things we might not have thought of as "spatial", such as: game outcomes, colors, economic trends, or collections of functions. Geometry becomes freed from its "earth-bound" origins, and becomes a tool for investigating MANY types of things. | 2020-09-22 18:49:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.7351529002189636, "perplexity": 459.2146820390306}, "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/1600400206329.28/warc/CC-MAIN-20200922161302-20200922191302-00651.warc.gz"} |
http://en.wikipedia.org/wiki/Heat_engine | # Heat engine
In thermodynamics, a heat engine is a system that performs the conversion of heat or thermal energy to mechanical energy which can then be used to do mechanical work.[1][2] It does this by bringing a working substance from a higher state temperature to a lower state temperature. A heat "source" generates thermal energy that brings the working substance to the high temperature state. The working substance generates work in the "working body" of the engine while transferring heat to the colder "sink" until it reaches a low temperature state. During this process some of the thermal energy is converted into work by exploiting the properties of the working substance. The working substance can be any system with a non-zero heat capacity, but it usually is a gas or liquid.
In general an engine converts energy to mechanical work. Heat engines distinguish themselves from other types of engines by the fact that their efficiency is fundamentally limited by Carnot's theorem.[3] Although this efficiency limitation can be a drawback, an advantage of heat engines is that most forms of energy can be easily converted to heat by processes like exothermic reactions (such as combustion), absorption of light or energetic particles, friction, dissipation and resistance. Since the heat source that supplies thermal energy to the engine can thus be powered by virtually any kind of energy, heat engines are very versatile and have a wide range of applicability.
Heat engines are often confused with the cycles they attempt to mimic. Typically when describing the physical device the term 'engine' is used. When describing the model the term 'cycle' is used.
## Overview
Figure 1: Heat engine diagram
In thermodynamics, heat engines are often modeled using a standard engineering model such as the Otto cycle. The theoretical model can be refined and augmented with actual data from an operating engine, using tools such as an indicator diagram. Since very few actual implementations of heat engines exactly match their underlying thermodynamic cycles, one could say that a thermodynamic cycle is an ideal case of a mechanical engine. In any case, fully understanding an engine and its efficiency requires gaining a good understanding of the (possibly simplified or idealized) theoretical model, the practical nuances of an actual mechanical engine, and the discrepancies between the two.
In general terms, the larger the difference in temperature between the hot source and the cold sink, the larger is the potential thermal efficiency of the cycle. On Earth, the cold side of any heat engine is limited to being close to the ambient temperature of the environment, or not much lower than 300 Kelvin, so most efforts to improve the thermodynamic efficiencies of various heat engines focus on increasing the temperature of the source, within material limits. The maximum theoretical efficiency of a heat engine (which no engine ever attains) is equal to the temperature difference between the hot and cold ends divided by the temperature at the hot end, all expressed in absolute temperature or kelvins.
The efficiency of various heat engines proposed or used today has a large range:
All of these processes gain their efficiency (or lack thereof) due to the temperature drop across them. Significant energy may be used for auxiliary equipment such as pumps which effectively reduces efficiency.
### Power
Heat engines can be characterized by their specific power, which is typically given in kilowatts per litre of engine displacement (in the U.S. also horsepower per cubic inch). The result offers an approximation of the peak power output of an engine. This is not to be confused with fuel efficiency, since high efficiency often requires a lean fuel-air ratio, and thus lower power density. A modern high-performance car engine makes in excess of 75 kW/l (1.65 hp/in3).
## Everyday examples
Examples of everyday heat engines include the steam engine (for example in trains), the diesel engine, and the gasoline (petrol) engine in an automobile. A common toy that is also a heat engine is a drinking bird. Also the stirling engine is a heat engine. All of these familiar heat engines are powered by the expansion of heated gases. The general surroundings are the heat sink, providing relatively cool gases which, when heated, expand rapidly to drive the mechanical motion of the engine.
## Examples of heat engines
It is important to note that although some cycles have a typical combustion location (internal or external), they often can be implemented with the other. For example, John Ericsson developed an external heated engine running on a cycle very much like the earlier Diesel cycle. In addition, externally heated engines can often be implemented in open or closed cycles.
### Phase-change cycles
In these cycles and engines, the working fluids are gases and liquids. The engine converts the working fluid from a gas to a liquid, from liquid to gas, or both, generating work from the fluid expansion or compression.
### Gas-only cycles
In these cycles and engines the working fluid is always a gas (i.e., there is no phase change):
### Liquid only cycle
In these cycles and engines the working fluid are always like liquid:
### Cycles used for refrigeration
Main article: refrigeration
A domestic refrigerator is an example of a heat pump: a heat engine in reverse. Work is used to create a heat differential. Many cycles can run in reverse to move heat from the cold side to the hot side, making the cold side cooler and the hot side hotter. Internal combustion engine versions of these cycles are, by their nature, not reversible.
Refrigeration cycles include:
### Evaporative heat engines
The Barton evaporation engine is a heat engine based on a cycle producing power and cooled moist air from the evaporation of water into hot dry air.
### Mesoscopic heat engines
Mesoscopic heat engines are nanoscale devices that may serve the goal of processing heat fluxes and perform useful work at small scales. Potential applications include e.g. electric cooling devices. In such mesoscopic heat engines, work per cycle of operation fluctuates due to thermal noise. There is exact equality that relates average of exponents of work performed by any heat engine and the heat transfer from the hotter heat bath.[7] This relation transforms the Carnot's inequality into exact equality.
## Efficiency
The efficiency of a heat engine relates how much useful work is output for a given amount of heat energy input.
From the laws of thermodynamics:
$dW \ = \ dQ_c \ - \ (-dQ_h)$
where
$dW = -PdV$ is the work extracted from the engine. (It is negative since work is done by the engine.)
$dQ_h = T_hdS_h$ is the heat energy taken from the high temperature system. (It is negative since heat is extracted from the source, hence $(-dQ_h)$ is positive.)
$dQ_c = T_cdS_c$ is the heat energy delivered to the cold temperature system. (It is positive since heat is added to the sink.)
In other words, a heat engine absorbs heat energy from the high temperature heat source, converting part of it to useful work and delivering the rest to the cold temperature heat sink.
In general, the efficiency of a given heat transfer process (whether it be a refrigerator, a heat pump or an engine) is defined informally by the ratio of "what you get out" to "what you put in".
In the case of an engine, one desires to extract work and puts in a heat transfer.
$\eta = \frac{-dW}{-dQ_h} = \frac{-dQ_h - dQ_c}{-dQ_h} = 1 - \frac{dQ_c}{-dQ_h}$
The theoretical maximum efficiency of any heat engine depends only on the temperatures it operates between. This efficiency is usually derived using an ideal imaginary heat engine such as the Carnot heat engine, although other engines using different cycles can also attain maximum efficiency. Mathematically, this is because in reversible processes, the change in entropy of the cold reservoir is the negative of that of the hot reservoir (i.e., $dS_c = -dS_h$), keeping the overall change of entropy zero. Thus:
$\eta_\text{max} = 1 - \frac{T_cdS_c}{-T_hdS_h} = 1 - \frac{T_c}{T_h}$
where $T_h$ is the absolute temperature of the hot source and $T_c$ that of the cold sink, usually measured in kelvin. Note that $dS_c$ is positive while $dS_h$ is negative; in any reversible work-extracting process, entropy is overall not increased, but rather is moved from a hot (high-entropy) system to a cold (low-entropy one), decreasing the entropy of the heat source and increasing that of the heat sink.
The reasoning behind this being the maximal efficiency goes as follows. It is first assumed that if a more efficient heat engine than a Carnot engine is possible, then it could be driven in reverse as a heat pump. Mathematical analysis can be used to show that this assumed combination would result in a net decrease in entropy. Since, by the second law of thermodynamics, this is statistically improbable to the point of exclusion, the Carnot efficiency is a theoretical upper bound on the reliable efficiency of any process.
Empirically, no heat engine has ever been shown to run at a greater efficiency than a Carnot cycle heat engine.
Figure 2 and Figure 3 show variations on Carnot cycle efficiency. Figure 2 indicates how efficiency changes with an increase in the heat addition temperature for a constant compressor inlet temperature. Figure 3 indicates how the efficiency changes with an increase in the heat rejection temperature for a constant turbine inlet temperature.
Figure 2: Carnot cycle efficiency with changing heat addition temperature. Figure 3: Carnot cycle efficiency with changing heat rejection temperature.
### Endoreversible heat engines
The most Carnot efficiency as a criterion of heat engine performance is the fact that by its nature, any maximally efficient Carnot cycle must operate at an infinitesimal temperature gradient. This is because any transfer of heat between two bodies at differing temperatures is irreversible, and therefore the Carnot efficiency expression only applies in the infinitesimal limit. The major problem with that is that the object of most heat engines is to output some sort of power, and infinitesimal power is usually not what is being sought.
A different measure of ideal heat engine efficiency is given by considerations of endoreversible thermodynamics, where the cycle is identical to the Carnot cycle except in that the two processes of heat transfer are not reversible (Callen 1985):
$\eta = 1 - \sqrt{\frac{T_c}{T_h}}$ (Note: Units K or °R)
This model does a better job of predicting how well real-world heat engines can do (Callen 1985, see also endoreversible thermodynamics):
Efficiencies of power stations[8]
Power station $T_c$ (°C) $T_h$ (°C) $\eta$ (Carnot) $\eta$ (Endoreversible) $\eta$ (Observed)
West Thurrock (UK) coal-fired power station 25 565 0.64 0.40 0.36
CANDU (Canada) nuclear power station 25 300 0.48 0.28 0.30
Larderello (Italy) geothermal power station 80 250 0.33 0.178 0.16
As shown, the endoreversible efficiency much more closely models the observed data.
## History
Heat engines have been known since antiquity but were only made into useful devices at the time of the industrial revolution in the 18th century. They continue to be developed today.
## Heat engine enhancements
Engineers have studied the various heat engine cycles extensively in effort to improve the amount of usable work they could extract from a given power source. The Carnot cycle limit cannot be reached with any gas-based cycle, but engineers have worked out at least two ways to possibly go around that limit, and one way to get better efficiency without bending any rules.
1. Increase the temperature difference in the heat engine. The simplest way to do this is to increase the hot side temperature, which is the approach used in modern combined-cycle gas turbines. Unfortunately, physical limits (such as the melting point of the materials from which the engine is constructed) and environmental concerns regarding NOx production restrict the maximum temperature on workable heat engines. Modern gas turbines run at temperatures as high as possible within the range of temperatures necessary to maintain acceptable NOx output[citation needed]. Another way of increasing efficiency is to lower the output temperature. One new method of doing so is to use mixed chemical working fluids, and then exploit the changing behavior of the mixtures. One of the most famous is the so-called Kalina cycle, which uses a 70/30 mix of ammonia and water as its working fluid. This mixture allows the cycle to generate useful power at considerably lower temperatures than most other processes.
2. Exploit the physical properties of the working fluid. The most common such exploitation is the use of water above the so-called critical point, or so-called supercritical steam. The behavior of fluids above their critical point changes radically, and with materials such as water and carbon dioxide it is possible to exploit those changes in behavior to extract greater thermodynamic efficiency from the heat engine, even if it is using a fairly conventional Brayton or Rankine cycle. A newer and very promising material for such applications is CO2. SO2 and xenon have also been considered for such applications, although SO2 is a little toxic for most.
3. Exploit the chemical properties of the working fluid. A fairly new and novel exploit is to use exotic working fluids with advantageous chemical properties. One such is nitrogen dioxide (NO2), a toxic component of smog, which has a natural dimer as di-nitrogen tetraoxide (N2O4). At low temperature, the N2O4 is compressed and then heated. The increasing temperature causes each N2O4 to break apart into two NO2 molecules. This lowers the molecular weight of the working fluid, which drastically increases the efficiency of the cycle. Once the NO2 has expanded through the turbine, it is cooled by the heat sink, which causes it to recombine into N2O4. This is then fed back by the compressor for another cycle. Such species as aluminium bromide (Al2Br6), NOCl, and Ga2I6 have all been investigated for such uses. To date, their drawbacks have not warranted their use, despite the efficiency gains that can be realized.[9]
## Heat engine processes
Cycle Process 1-2
(Compression)
Process 2-3
Process 3-4
(Expansion)
Process 4-1
(Heat Rejection)
Notes
Power cycles normally with external combustion - or heat pump cycles:
Carnot isentropic isothermal isentropic isothermal Carnot heat engine
Ericsson isothermal isobaric isothermal isobaric the second Ericsson cycle from 1853
and volume
Stirling isothermal isochoric isothermal isochoric Stirling engine
Power cycles normally with internal combustion:
the external combustion version of this cycle is known as first Ericsson cycle from 1833
Lenoir isobaric isochoric adiabatic Pulse jets
(Note: Process 1-2 accomplishes both the heat rejection and the compression)
Each process is one of the following:
• isothermal (at constant temperature, maintained with heat added or removed from a heat source or sink)
• isobaric (at constant pressure)
• isometric/isochoric (at constant volume), also referred to as iso-volumetric
• isentropic (reversible adiabatic process, no heat is added or removed during isentropic process)
## References
1. ^ Fundamentals of Classical Thermodynamics, 3rd ed. p. 159, (1985) by G. J. Van Wylen and R. E. Sonntag: "A heat engine may be defined as a device that operates in a thermodynamic cycle and does a certain amount of net positive work as a result of heat transfer from a high-temperature body and to a low-temperature body. Often the term heat engine is used in a broader sense to include all devices that produce work, either through heat transfer or combustion, even though the device does not operate in a thermodynamic cycle. The internal-combustion engine and the gas turbine are examples of such devices, and calling these heat engines is an acceptable use of the term."
2. ^ Mechanical efficiency of heat engines, p. 1 (2007) by James R. Senf: "Heat engines are made to provide mechanical energy from thermal energy."
3. ^ Thermal physics: entropy and free energies, by Joon Chang Lee (2002), Appendix A, p. 183: "A heat engine absorbs energy from a heat source and then converts it into work for us.... When the engine absorbs heat energy, the absorbed heat energy comes with entropy." (heat energy $\Delta Q=T \Delta S$), "When the engine performs work, on the other hand, no entropy leaves the engine. This is problematic. We would like the engine to repeat the process again and again to provide us with a steady work source. ... to do so, the working substance inside the engine must return to its initial thermodynamic condition after a cycle, which requires to remove the remaining entropy. The engine can do this only in one way. It must let part of the absorbed heat energy leave without converting it into work. Therefore the engine cannot convert all of the input energy into work!"
4. ^
5. ^ "Efficiency by the Numbers" by Lee S. Langston
6. ^ "Cyclone Power Technologies Website". Cyclonepower.com. Retrieved 2012-03-22.
7. ^ N. A. Sinitsyn (2011). "Fluctuation Relation for Heat Engines". J. Phys. A: Math. Theor. 44: 405001.
8. ^ F. L. Curzon, B. Ahlborn (1975). "Efficiency of a Carnot Engine at Maximum Power Output". Am. J. Phys., Vol. 43, pp. 24.
9. ^ "Nuclear Reactors Concepts and Thermodynamic Cycles" (PDF). Retrieved 2012-03-22.
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https://www.nature.com/articles/s41893-021-00794-4?error=cookies_not_supported&code=5a536959-7b41-43e7-a44c-11e662f360ef | ## Main
The marked expansion of phosphate rock mining has allowed for the rapid growth of global phosphorus (P) fertilizer production since the 1950s1. In the 1970s and 1980s, disproportionate fertilizer and manure P use in industrialized countries led to low P-use efficiency (PUE)2,3. Consequently, over time large amounts of surplus P were retained as residual P in soils, which increased both the P saturation and the P crop availability to its contemporary levels4. After this accumulation phase, farmers in many industrialized countries have been able to increase their PUE5, often by using the accumulated residual soil P reserves3. In contrast, Brazil, China and India are currently in the phase of large P surpluses and low nutrient-use efficiencies. Many developing countries are in the early phases of agricultural development with minimal P application rates, mining of soil P and low crop yields3,6.
## P management is crucial to the SDGs
In view of the finite world phosphate rock reserves, the global P requirement over the coming century has become a major concern7. Furthermore, the widespread use of P fertilizers is a threat to SDG6 (clean water and sanitation) and SDG14 (life below water) due to P losses from farm fields by surface runoff and consequent eutrophication of freshwater and coastal seas8,9. Yet, the supply of P is crucial to food security10. Future P management will therefore play an important role in achieving SDG2 (zero hunger). Achieving this goal is critical as the world population is projected to grow from 7.3 billion in 2015 to perhaps >10 billion inhabitants in 205011.
## P fixation can be overcome
Future crop yields depend on the availability of soil P for plant uptake; that is, on the concentration of phosphate ions in the soil solution and the soil’s ability to replenish phosphate withdrawn by plants4. Soils can adsorb inorganic P with varying degrees of reversibility. Soils rich in soluble iron or aluminium, clay minerals like kaolinite or with a high calcium activity, react with P to form insoluble compounds12. This is often referred to as P fixation, which is especially important in weathered tropical soils13,14.
P fixation can be overcome. For example, Brazil has been rapidly increasing its intensive soybean production on strongly weathered P-fixing soils by surplus P applications to quench much of the soil P fixation and sorption capacity15,16. Not only has Brazil overcome P fixation in large parts of its agricultural heartland but they have managed to build up large pools of legacy P that can be used in transitioning to a sustainable, low-input farming system17,18. The challenge of addressing global P fixation and limitation has been recently studied. Using Brazilian P input and surplus data, Roy et al.19 estimated that globally 8–25% more P input is required on the world’s P-fixing soils to raise crop yields to levels prevalent in Brazil. Kvakić et al.20 estimated that, globally, cereal yields could be 22–55% higher if P limitation were addressed. P fixation and limitation therefore pose additional challenges to achieving SDG2. However, there are currently no estimates on how much P input is needed to double smallholder productivity (target 2.3), a key target of SDG2. In this study, our objectives were: (1) to map the geographic distribution of P limitation; and (2) to assess SDG2 target 2.3 from a P perspective.
## P limitation and global scenarios
Where soils are P limited, crop growth cannot achieve its full potential. P limitation represents the relative yield gap21,22, which is attributable to limited P availability for crops. Calculating P limitation involves estimating the actual crop growth, which is P limited, and potential crop growth under no P limitation. These estimates can either be based on crop growth models20 or empirically based calculations using P budgets19. While the concept of potential growth is intuitively clear, calculation thereof is much more difficult, because it is at the limit of what can be observed in field trials. Pragmatic choices must be made for its calculation. In empirically based yield gap analyses, for example, the 90th or 95th percentile in a climatic bracket was considered potential yield21,22. Our choice was to model the P application of 50 kg ha–1 yr–1 for 20 years for calculating a potential P uptake, assuming that this will eliminate P limitation in most places, and that water and other nutrients like nitrogen and potassium are available in adequate amounts relative to P. This choice was motivated by the Brazilian case, where highly P-fixing soils can take decades to be saturated16,19 yet after decades of high fertilizer input have approached the point where a maintenance fertilizer strategy becomes possible17,18. In our study, P limitation is calculated as 1 – actual/potential P uptake.
For the calculation of P limitation we made use of the Dynamic Phosphorus Pool Simulator (DPPS), which is coupled to an integrated assessment model (IMAGE-DPPS)7,23,24,25,26. This allows us to assess P limitation for a wide variety of crops and a future point in time (2030), using a scenario. This model uses total P-input data, including P fertilizer use, manure P, atmospheric P deposition and weathering. P fertilizer use between 2000 and 2015 is entirely based on FAO data27.
The model was also used to calculate future P requirements in a scenario where the SDG2.3 target of doubling smallholder productivity is reached in 2030, compared to P requirements in the SSP2 scenario. The SSPs sketch a range of possible futures, where population and economic growth, energy and food demand, technological innovation and more factors develop along different lines28. These scenarios have been previously implemented in IMAGE-DPPS7. The SSP2 is the middle-of-the-road scenario, where assumptions are neither overly optimistic nor pessimistic. These scenarios are hereafter abbreviated as SDG2 (SDG2.3 target) and BASE (SSP2). In the SDG2 scenario we impose a doubling of P uptake in 2030 relative to the historical P uptake in 2015. Methodologically, the calculation of P requirements for achieving the SDG2.3 target is independent of the P-limitation calculation. The P-limitation calculation is a substantial result on its own and it gives context to the SDG2.3 scenario results.
Five world regions were selected for our analysis in which smallholder farming is dominant: sub-Saharan Africa (SSA), South East Asia (SEA), Middle East and North Africa (MENA), Central and South Asia (CSA) and India (Fig. 1a). In these regions, at least 75% of farms are smaller than 5 ha (ref. 29) and production of food crops is the dominant activity on smallholder farms30,31,32. We excluded China, which is also dominated by smallholder farming for food production but which already has very high levels of P application.
## P limitation is a global phenomenon
P limitation was a global phenomenon in 2015 (Fig. 1c), with the countries of the former Soviet Union, SSA, MENA, SEA and Australia standing out as the most P limited regions. Furthermore, parts of Bolivia, Argentina, some provinces of China and New Zealand were strongly P limited. Of the selected regions, SSA, SEA and India were characterized by substantial areas harbouring P-fixing soils (Fig. 3 in ref. 19). Toward 2030, the majority of countries will either hold or reduce their P limitation according to the BASE scenario (Fig. 1b,d). The following countries, for example, are reducing their P limitation: Argentina, Brazil, Nigeria, India, Pakistan, Malaysia, Indonesia, New Zealand and China. These countries have responded in the past, or are responding presently, to increased food demand by increasing P input and are expected to continue doing so7. A few countries, notably France and Spain, will see increasing P limitation. Not all regions in the world will try to eliminate P limitation, because economic or environmental considerations might require low P-input strategies8,9,17.
## Achieving the SDG2 target will require 39% more P input
P inputs in the SDG2 scenario clearly depart from the BASE scenario trend for all regions (Fig. 2a,b), with the upward trend mirrored in the P uptake (Extended Data Fig. 1a,b). SSA is the region where the BASE scenario is closest to the SDG2 target scenario. Of the six regions, only SSA therefore comes close to achieving the SDG2 target (P uptake increase by a factor of 1.8), while all other regions will only increase their P uptake by an average factor of 1.3 (Extended Data Fig. 2). To achieve the SDG2 target, SSA and SEA will need to more than triple their P input compared to 2015, while the remaining three regions will need to more than double their P input (Fig. 3). The high P requirement in SSA and SEA is partly due to the regions’ high prevalence of P-fixing soils. Geographical variation of P input and P uptake within regions can be notable (Extended Data Figs. 3 and 4). Production data show that cereals were major contributors to P uptake in all selected regions. However, in SSA, root and tuber crops were the second contributor to P uptake; in SEA, oilcrops, particularly oil palm fruit, were more important than the cereals with regard to P uptake (Supplementary Table 1).
On aggregate, SEA as of recent has not been building up P reserves in the soil, which is reflected in a PUE of around 1 (Extended Data Fig. 5a), mainly because much P uptake is from recently deforested areas7. SSA has very low P-input rates, yet the region has nevertheless built-up residual P in the soil since 2000, because crop P uptake rates have been even lower than P input (Extended Data Figs. 5a and 6a,b). As input increases to achieve SDG2, PUE will decrease in all regions (Extended Data Fig. 5a,b). India, MENA and CSA, which already have low levels of PUE, will see further decreases.
On a per-area basis, all regions will have to substantially increase their P input and SEA, CSA and India will reach average input levels typical for industrializing countries of around 50 kg P ha–1 yr-1 in 2030 (Extended Data Fig. 6a). In the BASE scenario on the other hand, none of the regions will reach this input level. Average uptake is highest in SEA and lowest in SSA for all scenarios (Extended Data Fig. 6b). Achieving the SDG2 target implies large increases in average P input compared to the BASE scenario (Extended Data Fig. 6c).
In cumulative terms, between 2015 and 2030, India will need 30 Tg more P in the SDG2 scenario, compared to the BASE scenario, while the difference is only 4 Tg for SSA (Supplementary Fig. 1a,c). All five regions taken together need an additional 74 Tg of P, which is 39% more compared to the BASE scenario. This effort would result in an additional P uptake of 20 Tg between 2015 and 2030 (Supplementary Fig. 1b), which is 27% of the additional P input. This latter percentage can be viewed as the marginal PUE of achieving the SDG2 target between 2015 and 2030.
The substantial increases in P inputs needed to achieve SDG2 are not translated into higher P runoff in the short term (only 1% increase) (Supplementary Fig. 2). Regional P runoff depends on the size of the LP and SP pools and cropland expansion. The amount of residual P is relatively small compared to total soil P, which means that increasing residual P has only a small effect on P runoff. Cropland expansion accesses large new pools of total P, which is the main driver for increasing regional P runoff. The BASE and SDG2 scenarios have the same cropland expansion, so the very small difference in runoff is due only to differences in residual P. In the BASE scenario, 16 Tg of P are lost to runoff cumulatively between 2015 and 2030, which is 9% of cumulative P input during this period. In both scenarios, this is a serious loss for crop uptake. Although these are not global estimates, these P losses are similar to the global planetary boundary for a low estimate of river P export to coastal waters and a low target P concentration in inland waters33. P runoff is especially high in India which has a history of relatively high P input and therefore high residual soil P. While decreasing PUE through increased P input can, in the short term, seem like a problem, it is really the rate of P losses by runoff that subdue PUE in the long run. P management should therefore include soil and water conservation.
## SDG2 target might not be ambitious enough for SSA
P-input rates of around or below 50 kg P ha–1 yr-1 are sufficient for all regions to achieve the SDG2 target (Extended Data Fig. 6a). All regions will need substantial increases in P input to achieve the SDG2 target. Two regions in particular, SSA and SEA, need to more than triple their P input compared to 2015 to achieve the target in 2030. By doing so they can reduce P fixation and improve production beyond 2030. SSA in particular has large areas of P-fixing soils34,35 but it is nevertheless on track to achieve the SDG2 target and only needs a modest extra P input of 4 Tg by 2030. Yet, it is worth noting that, in SSA, input and uptake of P are rising from an extremely low level per unit area (Extended Data Fig. 6a) and P fixation will not be overcome just yet (Fig. 1d). Furthermore, the population in SSA is expected to double by 205011 and therefore, by 2030, the regional gains in production will be diluted on a per capita basis requiring more effort than considered in our scenario. This is because our scenario considers doubling of P uptake per region and not doubling of P uptake per capita. While most regions will approach moderate to high levels of P input per area in the SDG2 scenario, SSA will achieve the goal with an average input of only 11 kg P ha–1 yr-1 (Extended Data Fig. 6a). Thus, considering population growth, strong P fixation and very low current P inputs and uptake, the target for SSA is probably not ambitious enough, while appropriate for the remaining regions. This is because the doubling target is, of course, in relative terms and when starting from very low levels of P uptake, more than doubling, or an absolute target could be more appropriate.
## Possible futures
Scenarios are not predictions and model uncertainties and unexpected global developments, such as the current COVID19 pandemic, will influence the likelihood of a scenario36. For more information on uncertainty and to see a sensitivity analysis for the most influential parameter in the model, we refer to the Supplementary Fig. 3 and Supplementary Methods. We chose the SSP2 (BASE) scenario over other scenarios, since it is often referred to as middle-of the-road as it assumes a continuation of current trajectories and can therefore serve as a baseline reference against which to test the SDG2. Comparisons with other SSPs would naturally yield different results. It is important to note, however, that global P input in the five SSP scenarios varies much less than population growth in the corresponding scenarios7 and that P input to achieve SDG2.3 would have to substantially increase any of the five scenarios. This means that even in more optimistic SSPs, SDG2 will not be achieved from a P perspective, except perhaps for SSA. Therefore, achieving the SDG2 requires a targeted effort in the regions where smallholder farming predominates.
## Achieving the SDG2.3 target
Achieving the SDG2.3 target is realistic for countries with currently low levels of P input and has precedent in field trials. Long-term application of P, nitrogen and potassium in the tropics typically result in three to four times higher crop yields, within a range of 1.5 to over 20 times, compared to non-fertilized control yields. This estimate is based on four long-term field trials in India37, Southern China38, Kenya39 and Senegal40, ranging from 14 to 42 years with annual P application rates of 18–53 kg ha–1 yr-1. Given that currently many smallholder farms in developing countries use little or no fertilizer3, the field trial comparison confirms the plausibility of the possible P-uptake gains modelled in this paper.
We conclude that the SDG2.3 target is both achievable for all regions and perhaps not ambitious enough for SSA. Yet, it requires an effort that clearly goes beyond what is expected in the BASE scenario.
## Challenges ahead
Not all additional P needed to achieve the SDG2.3 target needs to come from rock phosphate-based fertilizers. Better integration of animal manure in crop systems can help to reduce fertilizer needs and this may particularly be important in India where a large part of cow manure is used as fuel. Waste recycling may also be helpful. For example, human excreta are considered a waste but if recycled in, for example eco-sanitation or dry or low-flush systems41,42, the high P content in urine could substitute large amounts of P fertilizers43 and at the same time reduce P discharge to surface waters. More generally, P could be recycled across sectors where waste streams44, such as food waste and wastewater, could provide a local supply of P and also alleviate the limited global availability of rock phosphate.
The drop in PUE following the start of quenching (Extended Data Fig. 5a,b) could be alleviated by measures that improve P uptake. Simultaneously increasing crop yields and cropping intensity, reducing nutrient limitation, avoiding land degradation, improving the efficiency of fertilizer use and minimizing environmental losses requires a fine-tuned combination of agronomic practices45. These include the use of high-yielding crop varieties that are adapted to the prevailing soils and climate, management of organic matter by recycling of crop residues and animal manure, crop protection against weeds, pests and diseases and soil nutrient management with a good balance between P, nitrogen, potassium and micronutrients, and liming to improve P availability in strongly P-fixing soils46. Avoiding crop production on steep slopes and applying soil conservation adapted to the local cropping system, climate and terrain conditions practices can minimize carbon and P losses through erosion1,47. This will reduce eutrophication risk for aquatic ecology and, at the same time, secure the soil’s long-term productivity and the farm’s future profitability3,48. Measures could be reduced tillage, cover crops, contour ploughing, deviation ditches, grassed waterways and terracing49.
Farmers also need to be able to buy P fertilizer. Several factors contributed to the successful intensification of crop production on tropical P-fixing soils in Brazil, including large farm size, low land prices and capital availability for supplying P and other essential inputs19. In contrast to the Brazilian situation, the regions considered in this paper, particularly SEA and SSA, are dominated by smallholdings that often lack sufficient capital. Farmers’ access to fertilizers can be ensured through specific credit lines to bridge the initial adoption costs or through subsidies during the quenching period50. Well-designed subsidies on fertilizers can contribute to the overall SDG2. However, in SSA, subsidies on fertilizers and other inputs are emergency policy response for the various food price crises and not a policy instrument for boosting agricultural productivity in the long run51.
A further important aspect of smallholder’s access to fertilizers containing P (and other nutrients) is a stable price. During the last two decades, agriculture faced both the highest sustained phosphate rock price (in 2008) and the largest price fluctuations of any 10-year periods since 196052. In some countries, such as India, governments bear the burden of price hikes of P fertilizers through subsidies. With dwindling phosphate rock resources, the base material for P fertilizer production53, the overall increase of the global P fertilizer requirement projected in this paper may result in a long-term price increase, whereby geopolitical tensions form an unpredictable element54. Fortunately, a large pool of residual P in the soil can act as a buffer against short-term price fluctuations and a transition to a more sustainable P management with lower P fertilizer input could be achieved17,18.
Keeping the long-term perspective is essential, because the P-use efficiency is low during the quenching period. Achieving investments to build up soil P availability to secure future productivity by smallholders requires capacity building to strengthen nutrient and soil conservation management. This involves research, extension services, farmer’s training, demonstration farms, incentives to motivate and support the learning phases, procure P fertilizers and other necessary inputs at fair prices and dedicated insurance schemes for risk management50. Especially in SSA, unfortunately the current targets for budgets to support capacity building from national and international organizations are generally not met55,56.
## Methods
This section describes the model and data used and how the model was set-up for our scenario calculations. Furthermore, we describe how P limitation was calculated.
### Model description
The DPPS was originally developed to simulate crop uptake after field quenching experiments in various countries23,24 and was recently used to simulate the long-term global P uptake and P status of soils57. DPPS was further developed to be applicable with the SSP scenarios implemented in the IMAGE framework25, with the purpose of calculating future P demand and budgets for a period up to 20507 Here we apply the version of DPPS (IMAGE-DPPS) by Mogollón et al.7. IMAGE-DPPS can simulate the current soil P stocks (labile soil stock, LP, and stable soil stock, SP, in kg P ha–1) with an annual temporal scale and a spatial resolution of 0.5 by 0.5 degrees for given P inputs. Each cell is initialized in the year 1900 with LP and SP from the global gridded soil P inventory58, representing the pre-industrial conditions. LP comprises both organic and inorganic P forms and only a fraction of LP is directly available for plant uptake (frav);23 SP represents forms of P bound to soil minerals and organic matter that are not available to plants. Thus, the P availability for plants may increase or decrease, depending on the pool sizes. The equilibrium between LP and SP happens via first-order rate constants. The LP to SP transfer coefficient was set to 0.2 yr-1, while the SP to LP coefficient was independently calculated for each grid cell on the basis of the maps of Yang et al.58 for virgin soil P and assuming LP and SP to be at steady state. IMAGE-DPPS considers natural or unintentional inputs to the soil, that is, weathering (weathering, kg P ha–1 yr-1) and litter (litter, kg P ha–1 yr-1; litter is crop residues in crop systems and litterfall in natural vegetation), which are inputs to LP, and atmospheric deposition of soil dust (deposition, kg P ha–1 yr-1) adds to the SP pool. To describe chemical P retention in soils, DPPS includes the parameter initial recovery fraction (no dimension) obtained from Batjes59 for each of the soil classes of the legend of the FAO soil map of the world60, which describes the effect of soil chemical properties such as iron and aluminium concentrations on the sequestration of P from labile to stable soil pools. Anthropogenic P inputs include application of mineral P fertilizer (fertilizer, kg P ha–1 yr-1) and animal manure spreading (manure, kg P ha–1 yr-1). A fraction of fertilizer and manure is directly taken up by plant roots from the soil solution (20% for fertilizer, 10% for manure) and the remainder is available and becomes part of LP (80% for fertilizer and 90% for manure) and can be taken up from LP indirectly. Crop uptake (U) is calculated assuming a Michaelis–Menten relation between uptake and fav following equation (1):
$$U = \frac{{U_{{\max} }{{\rm{fr}}_{{\rm{av}}}{\rm{LP}}}}}{{\frac{{cU_{{\max} }}}{I} + {{\rm{fr}}_{{\rm{av}}}{\rm{LP}}}}} + f_{\mathrm{S}}S + f_{\mathrm{M}}M$$
(1)
where Umax is the maximum P uptake and I is the initial recovery fraction, which is the initial slope of the P response curve presented for all soil types distinguished in the legend of the FAO-UNESCO world soil map, c (=0.5) multiplies Umax to obtain the half-saturation value. The parameter Umax is held constant with a value of 500 kg P ha−1 yr−1. The parameter S represents mineral fertilizer additions and fS the fraction directly taken up by plants and M represents manure additions with fM representing the fraction directly taken up by plants. P outflows from the soil system include P withdrawal from LP by crops (U, kg P ha–1 yr-1) and runoff from both LP and SP (runoff, kg P ha–1 yr-1). Calculated runoff loss accounts for changing soil LP and SP pools in certain locations but for aggregated world regions runoff P loss also reflects land-use changes with LP and SP pools for unfertilized soils lacking residual P from past management. P fertilizer requirement, which is the largest form of P input, in IMAGE-DPPS is most sensitive to the target P uptake (Supplementary Fig. 3a,b and Supplementary Methods). For a simplified scheme of the model see Supplementary Fig. 4. For further details on the model and its assumptions, see Mogollón et al.7.
### Scenarios and model set-up
We compared two scenarios: (1) SSP2, a middle-of-the-road scenario6, was used as a reference scenario for expected future development (BASE scenario) and compared with (2) the SDG2 target 2.3 scenario, in which the target of doubling productivity is achieved (SDG2 scenario). In addition, the BASE scenario was used in the calculation of P limitation.
Both scenarios depart in the base year from the same historical set-up of the model. We use the calculated LP and SP pools based on simulation of the historical period from 1900 to 2015 (the base year of IMAGE) as a starting point for the future simulation using spatially explicit land use and crop uptake distributions generated by the IMAGE model25. In grid cells where cropland expansion occurs, natural soil (without fertilizer history) with initial P pools58 is added. For grid cells with land abandonment (arable land to natural land), IMAGE-DPPS assumes a 30-yr period for abandoned land to revert to natural conditions26 and in this period the P in litter and uptake increase linearly with time from zero to the natural flux (in which uptake equals litterfall). For grid cells where P uptake through crops is less than what is available in the soil, P input is assumed to be zero, so depletion of residual P takes place. P uptake is distributed within the grid cell over different age-classes of pools (different years of conversion to cropland). In historical mode, the unknown frav is allowed to vary between a minimum value of 0 and a maximum value of 1. In scenario mode after 2015, frav becomes a parameterized value that varies according to the SSP storyline. The fraction frav will increase as a result of improved crop varieties and other strategies to increase the capability of plant roots to acquire soil P. In BASE it increases during the period 2015–2030 by half of the increased rate calculated for 1990–2005 to reflect increasing difficulty to develop crop varieties with enhanced root P uptake.
### Scenario target productions
The scenarios require the setting of P-uptake targets. The future uptake in the BASE scenario was estimated from the P content in the projected crop production7,25. Future P uptake in the SDG2 scenario is determined by the SDG target 2.3 of doubling smallholder productivity. Here, we focus only on crop uptake and neglect livestock P uptake from grassland. For a justification of our interpretation of the target, see Supplementary Methods. As the target relates to smallholder farms, the scope of the target was limited to five world regions where smallholder farming dominates agriculture. The method for selecting these regions is given in the Supplementary Methods, too. For all countries in these regions, the target was set that P uptake will double between 2015 and 2030 and that this target is achieved by linear increase between 2020 and 2030 (Extended Data Fig. 1a,b). The increase starts in 2020, the time of writing, assuming that between 2015 and 2020 the world developed along the BASE path. The target was set for the countries as a whole rather than for individual farms or farmland area (a discussion of this choice is given in the Supplementary Methods). There is no specific reason for choosing a linear increase, except for keeping it simple. Other, equally possible increases, such as logarithmic or s-shaped would result in different cumulative estimates for the period 2015–2030 but would not change the uptake rate in the year 2030.
The P input required to achieve the scenario uptake values for each region consists of all forms of P from deposition, manure, mineral fertilizers and fertilizers produced from human excreta. IMAGE has scenario-specific consumption, domestic production of meat and milk, and distribution of livestock over pastoral and mixed systems, and within these systems the fraction of manure stored in animal houses and the amount available for spreading in croplands25,28. With these manure P inputs and deposition from existing data, the required inputs from fertilizer and human excreta are calculated as the difference between total P input and manure plus deposition.
### P-limitation calculation
For P limitation, the potential P uptake was calculated as an extension to the BASE scenario. Here, we define soil P limitation as:
$$1 - \frac{{U_{\mathrm{act}}}}{{U_{\mathrm{pot}}}}$$
(2)
where Uact is actual P uptake and Upot is potential P uptake. This index provides a metric to compare the P status of different soils. For calculating Upot it was assumed that P was applied at 50 kg ha–1 yr-1 during a 20-yr quenching period in all grid cells with cropland. This application rate is similar to the rates in Brazil in the past decade15,16. For the 2015 P-limitation calculation historical (HIST) P inputs were used up to 2015, after which the quenching started. During the quenching period, relevant uptake conditions were assumed to stay the same as in 2015, in particular the fraction of the labile pool available for plant uptake, frav. Implicit in this assumption is that other limitations to crop growth, such as other nutrients, soil and water conditions are raised in tandem with P limitation, which means that these conditions are not more limiting to P uptake after quenching than they were in 2015. This optimistic assumption is necessitated by the model (equation (1)) but can be defended, because in any likely scenario quenching with P would be accompanied with other improvements of the soil status. In calculating P limitation, we were mostly interested in how much quenching can improve the P status of the soils. For this reason, Upot was calculated in the year after the quenching period (20 + 1), in which zero manure and fertilizer P inputs were simulated, so all P comes from the stored available soil P. The parameter Uact was calculated for the year after 2015, in which also no P inputs were simulated. Thus, the P-limitation ratio represents the true soil supply and not direct supply from the fertilizer of manure applications. For the P-limitation calculation of 2030, the same calculation was performed, with the only difference being that the quenching period started in 2030 and that between 2015 and 2030 the BASE scenario was followed. For illustration, the meaning of a high P-limitation value of, for example, 0.8 in 2015 is that in that year the plant-available P in the soil pools was low and consequently P uptake was low. Through 20 years of quenching the plant-available P in the soil was replenished such that P uptake was five times higher in the year after quenching.
### Data used
Simulated crop P uptake for BASE is obtained from the crop production for the second of five SSP scenarios implemented with the IMAGE model25. In 2020, IMAGE data and SSP scenarios were updated to base year 2015. Previously, as in Mogollón et al.7, the base year was 2005. Among the changes were the number of crop groups in IMAGE considered, which increased from seven to 18; the production data for these 18 crop groups are aggregated from the 160 crops reported by FAOSTAT for the historical years up to 2015 and P uptake is calculated from P content of each crop. For the future, IMAGE projects production for the 18 crop groups and for P uptake we assume constant P contents after 2015.
IMAGE generates regional populations of non-dairy and dairy cattle, pigs, poultry, sheep and goats. Using P-excretion rates, the total manure P is calculated (Supplementary Table 3). Manure available for spreading in croplands differs between pastoral (mostly grazing, small amounts of manure collected in animal houses) and mixed systems (with a large proportion of confined animals from which manure is collected in animal houses) and excludes droppings in grassland, manure used as fuel or building material or manure otherwise ending outside the agricultural system (for example, in lagoons)61.
Subsequently, IMAGE P uptake estimates and animal manure for world regions are distributed over countries within each region using FAO data27 for the historical period up till 2015 and distributed over grid cells within countries on the basis of the spatial distribution of crops from IMAGE. For future years, the 2015 distribution over the countries within regions is used.
Atmospheric P deposition was obtained from recent model data62; since mineral aerosols from soil dust contribute 82% of total P deposition, we assumed deposition fields to be constant in time after 2015. Weathering is also constant at a level of 1.6 Tg P yr-1 (ref. 63) and distributed on the basis of the apatite content of soil material obtained from Yang et al.58.
### Reporting Summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article. | 2023-03-30 08:35:50 | {"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": 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": 0.4503936469554901, "perplexity": 3539.0849782482487}, "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/1679296949107.48/warc/CC-MAIN-20230330070451-20230330100451-00361.warc.gz"} |
https://www.doubtnut.com/question-answer/an-endless-inextensible-string-of-length-15-m-passes-around-two-pins-a-and-b-which-are-5-m-apart-thi-644643575 | HomeEnglishClass 12MathsChapterConic Sections
An endless inextensible string...
# An endless inextensible string of length 15 m passes around two pins, A and B which are 5 m apart. This string is always kept tight and a small ring, R, of negligible dimensions, inserted in this string is made to move in a path keeping all segments RA, AB, RB tight (as mentioned earlier). The ring traces a path, given by conic. C, then:
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Conic C is an ellipse with eccentricity (1)/(2)Conic C is an hyperbola with eccentricity 2Conic C is an ellipse with eccentricity (1)/(3)Conic C is an hyperbola with eccentricity (3)/(2)
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00:00 - 00:59hello guys it is given that an endless inextensible string of length 15 M passes through two trains A and B which are 5 metre apart the string is always kept it and small ring are of negligible dimension inserted in the string is night to move in a part keeping segment all the segment Aaye baby RB tight as mentioned earlier traces of given by konisi where to find what is the which type of conic is in the options there are you like and let us to see is and hyperbola the centre is it to see is an ellipse with the centre City 1 by 3 and season hyperbola eccentricity 3 by 2 super Speedo find out that actresses wearing dresses A pass of ellipse and hyperbola understand that what is the question is saying this is a this is
01:00 - 01:59this is a and b now a ring attached with the trim string is with a string is there string is like a and that this is a single string which is inextensible of length 15 M tried the length of the total length of the string is 15 what is saying that this point there is a ring are noticing that the point A and B are not moving and this string always get tight and the RRB RRB abe all these points are right keeping the this is saying that in certain string as made of this is the point Aaye vrb is always type which please please no slack in this spring now this ring is moving right now we are tracing the
02:00 - 02:59path so this part will become an early because as we know with the definition of Alif that if a point is the distance of the sum of the distance of a point from this too is constant then this will tractor parts of Allah let me the definition here if a if the if the oh the distance between between this the distance between this and the point to the foci distance between a point to the foci then the coin will Trace the
03:00 - 03:59trace the path of trace the path of and right now we go to know that this will Trace the path to help with this option and this option is a wrong option so now we are to find The authenticity of this write a diagram here clean diagram here so we will draw the co-ordinate Axes these are all co-ordinate Axes let name the co-ordinate Axes let's say this is a x axis y axis and let's assume this is point a and this is point B now it is also mention in the question that the distance between points A and B is 5 M so this is 2.5 and this is 2.5 metre here right now as we know that this is a Alfred so let's make a rough diagram of an ellipse
04:00 - 04:59now at an instant when the point are the ringer is at this point so let's not found that this the ring is at least at this point on the top of the minor axis let's connect this point This is a ring here so this is totally 15 the now the system is symmetrical so this become 15 by to now as per the standard notation this length is b and this online fees this Holland is a right now we will use the Pythagoras theorem in this triangle and we can see here we can write here 15 by two whole square will be equal to 2 by 55 to the 5 by 2 to the whole square + b square now from here b square is equal to 25 25 200/4
05:00 - 05:59b is equals to root of 15 year the value of Valsad to find the value of a and to find the show as we can see here the distance between a and b is 5 metre and these are the four and A and B are the focus so we can write it 5 is equal to two way from here we know that a is equal to 5 by to get the value of a relation between A and B which is this is equal to a square x by 1 minus is this is a relation between a b and b start putting the value in this so this will be b square is equal to x square - 2 the whole square is equal to 5 by 2
06:00 - 06:5925 by 40 now we know that a square has the value to 25 by 45 BA here and we know that a is equal to 5 by 2 we can get the value of so we know that a is equal to 5 by 2 find the value of 22 which is 22554 square on the process of this become a square we will write 25 in place of these will be East 25 by 45 square in both side side carry right now we will just saw this weekend at this point This will goes with night right so this is square is equals to 1 by 9 and the value of g become 1 by 3
07:00 - 07:59now we got that the this is the ring will tracer parts of Alice and the assent to Alice will be won by now this will be our correct option thank you for watching Best of Luck | 2021-12-06 21:53:27 | {"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": 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": 0.62549889087677, "perplexity": 450.79176938609464}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363312.79/warc/CC-MAIN-20211206194128-20211206224128-00599.warc.gz"} |
https://www.edgegamers.com/threads/1371/ | # need a laugh?
#### sasuke
##### Rookie
If you need a laugh, click on one of the following links :
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Note that the video I posted with the Numa Numa man has the wrong subtitles.
ha.....ha
#### BREAK_EM_OUT_JONES
##### EGO Is My Life!
the first one was retarded! | 2021-04-18 08:08:20 | {"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": 0.8055707216262817, "perplexity": 9705.44573616174}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038469494.59/warc/CC-MAIN-20210418073623-20210418103623-00072.warc.gz"} |
http://encorederm.com/after-we-auo/551693-pas-gases-and-condensed-system | Theorists working at the interface of condensed matter, chemical, and atomic physics. During the steam system operation, the percentage of steam in the system will be close to 100% steam vapor with a small percentage of non-condensable gases. Some experimental setups, with microphone and piezoelectric transducers, are described for recording PA spectra of gaseous, solid, and liquid samples. Thermodynamic data on oxygen, hydrogen, the halogens, the inert gases, sulfur, nitrogen, and phosphorus and their most important compounds are compiled in extensive tables. 13, 507 (1945 ... and on solutions of non‐polar gases in normal solvents. The physics of one-dimensional interacting bosonic systems is reviewed. Geothermal fluids contain entrained non-condensable gases (NCGs) that may not be easily injected back into the reservoir. The simplest of this category is the binary system. Many-body physics at the interface of ultra-cold atomic gases and condensed matter systems: Aspects from experiment and theory – Overview. One dimensional bosons: From condensed matter systems to ultracold gases M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol Rev. Welcome to the GIQ home page. Methods of describing the electron kinetics in gases and plasma are applied to modeling the electron drift in condensed systems. Phys. A versatile system, operating at pressures to 300 MPa and temperatures to 4 K, has been designed and used for a variety of elastic and inelastic x‐ray scattering studies of single crystals of condensed gases. systems in occupied spaces.3 Per the British Standard BS EN 50600-2-5: 2016 Information Technology 4: Data center facilities and infrastructures are specifically advised against the use of aerosols in occupied spaces. Gases were part of nebular cloud that condensed to form our solar system including hydrogen (H), helium (He), methane (CH 4), and ammonia (NH 3) • Prior to Earth’s internal differentiation, gravity could not hold these gases closely around Earth- condensed to form our solar system including hydrogen (H), helium (He), methane (CH 4), and ammonia (NH 3) • The good news, however, is that the behavior of multicomponent systems are quite similar to that of binary systems. Ultracold atomic gases as a new condensed matter system: atomic physics meets solid state.Willian Daniel PhillipsIUPAP 2017http://www.ifsc.usp.br/iupap2017 3M™ Novec™ 1230 Fire Protection Fluid Condensed Aerosols Applicable standard NFPA 2001, ISO 14520 NFPA 2010, ISO 15779 Sintered Be sample cells of volumes 15–30 mm3 are used. Its purpose is to condense ammonia into a liquid form. 2014 Jan;83 Pt B:77-85. doi: 10.1016/j.apradiso.2013.01.008. Gases have their own unique behavior depending on a variety of variables, such as temperature, pressure and volume. Modelling Condensed Matter Systems with Quantum Gases in Optical Cavities: Principal Investigator: Beige, Dr A: Other Investigators: Researcher Co-Investigators: Project Partners: ... Condensed Matter Physics: EPSRC Industrial Sector Classifications: No relevance to Underpinning Sectors: Related Grants: EP/H049339/1: EP/H049231/1: We review the physics of one-dimensional interacting bosonic systems. The photoacoustic infrared spectroscopy is based on the fact that infrared light energy is absorbed by gas molecules. Therefore, we will focus on binary systems, since they are easier to illustrate. Entropy in Binary Liquid Mixtures; Partial Molal Entropy in Dilute Solutions; Structure and Thermodynamics in Aqueous Electrolytes J. Chem. All → Electronic And Magnetic Properties Of Condensed Matter; Superconductivity → Degenerate Quantum Gases And Atom Optics → Complex Physical Systems Professor Murray Batchelor College of … Pressure generation is clean and well controlled. These gases, which accumulate in the condenser, can decrease heat transfer and raise turbine back-pressure, thereby lowering turbine performance. HOME / CALENDAR / EVENTS / German-Israeli Minerva School on Quantum Dipolar Gases in Atomic and Condensed Matter Systems eng., tohoku univ., sendai In the high frequency limit, we obtain the Floquet Hamiltonian consisting of the … Low-energy electron and positron transport in gases and soft-condensed systems of biological relevance. 1. Gas Phase Photoacoustic Spectroscopy (PAS) The need for rapid and reliable monitoring of air pollutants and hazardous gases is growing and the photoacoustic spectroscopy provides an efficient technology for this demand.. iv. Let us place two gases (A and B) in an isothermal cell. They act on the four elements of what is known as the "fire tetrahedron:" the disparate components that combine to create the chemical reaction underlying any fire.These four means of fire extinction are: Epub 2013 Jan 11. Condensed aerosol suppressants, as with gaseous suppressants, use four methods to extinguish fires. Phys. We demonstrate this concept in a one-dimensional XY model with periodically driven couplings and transverse field. We implement the mixed-dimensional two-orbital system using a near-resonant and magic-wavelength optical lattices, where the $^1\\mathrm{S}_0$ and $^3\\mathrm{P}_0$ atoms are itinerant in a one … Lemeshko Group. We introduce the concept of a Floquet gauge pump whereby a dynamically engineered Floquet Hamiltonian is employed to reveal the inherent degeneracy of the ground state in interacting systems. Mod. Beginning with results from exactly solvable models and computational approaches, we introduce the concept of bosonic Tomonaga-Luttinger Liquids relevant for one-dimension, and compare it with Bose-Einstein condensates existing in dimensions higher than one. Why does ammonia condense into … We highlight the importance of electron and positron swarm studies and experiments as a test of the accuracy and completeness of cross-sections, as well as a technique for benchmarking Monte-Carlo simulations. Non-condensable gases. A brief historical account of photoacoustic (PA) effect is followed by a simple mathematical model for the generation of PA signals due to nonradiative transitions in atoms and molecules. Get an answer for 'In the Haber Process, the gases N2, H2 and NH3 are passed through the condenser. We present a study of electron and positron transport in water in both the gaseous and liquid states using a Boltzmann equation analysis and a Monte-Carlo simulation technique. Low-energy electron and positron transport in gases and soft-condensed systems of biological relevance Ronald White, Wade Tattersall, Gregory J Boyle, Robert Robson, Saša Dujko, Zoran Petrović , Ana Banković, Michael Brunger, James Sullivan, Steven Buckman, Gustavo Garcia We assess the importance of coherent scattering processes when considering transport of electrons/positrons in dense gases and liquids. Beginning with results from exactly solvable models and computational approaches, the concept of bosonic Tomonaga- Luttinger liquids relevant for one-dimensional Bose fluids is introduced, and compared with Bose- Einstein condensates existing in dimensions higher than one. cluster expansion for quantum gases. We report on the observation of the spin-exchange dynamics of $^{171}\\mathrm{Yb}$ atoms in the ground state $^1\\mathrm{S}_0$ and in the metastable state $^3\\mathrm{P}_0$. Gases have three characteristic properties: (1) they are easy to compress, (2) they expand to fill their containers, and (3) they occupy far more space than the liquids or solids from which they form. We are based at the Institute of Science and Technology Austria. It is shown that the electric properties of metals and the behavior of an excess electron in dielectrics have common features with electron drift in gases … Free Volume and Entropy in Condensed Systems III. a reformulation and "condensed part" of the hard-sphere bose system author hara h; ichimura h fac. The physics of one-dimensional interacting bosonic systems is reviewed. Appl Radiat Isot. The Quantum Information Group (GIQ) research interests range from theoretical aspects of quantum information (quantum Shannon theory, quantum statistical inference, quantum metrology, entanglement theory) to implementations of quantum information protocols (e.g., quantum optics and ultra-cold gases) and quantum information in condensed matter systems. Beginning with results from exactly solvable models and computational approaches, the concept of bosonic Tomonaga-Luttinger liquids relevant for one-dimensional Bose fluids is introduced, and compared with Bose-Einstein condensates existing in dimensions higher than one. S.C. Bhatia, in Advanced Renewable Energy Systems, 2014. We assess the importance of coherent scattering processes when considering transport of electrons/positrons in dense gases and liquids. Place two gases ( NCGs ) that may not be easily injected back into the reservoir turbine performance geothermal contain... 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Electrons/Positrons in dense gases and liquids variety of variables, such as temperature, pressure volume. | 2021-05-07 13:45:02 | {"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3856404721736908, "perplexity": 4117.508471652737}, "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/1620243988793.99/warc/CC-MAIN-20210507120655-20210507150655-00047.warc.gz"} |
https://par.nsf.gov/biblio/10079013-acoustic-phonon-lifetimes-limit-thermal-transport-methylammonium-lead-iodide | Hybrid organic–inorganic perovskites (HOIPs) have become an important class of semiconductors for solar cells and other optoelectronic applications. Electron–phonon coupling plays a critical role in all optoelectronic devices, and although the lattice dynamics and phonon frequencies of HOIPs have been well studied, little attention has been given to phonon lifetimes. We report high-precision momentum-resolved measurements of acoustic phonon lifetimes in the hybrid perovskite methylammonium lead iodide (MAPI), using inelastic neutron spectroscopy to provide high-energy resolution and fully deuterated single crystals to reduce incoherent scattering from hydrogen. Our measurements reveal extremely short lifetimes on the order of picoseconds, corresponding to nanometer mean free paths and demonstrating that acoustic phonons are unable to dissipate heat efficiently. Lattice-dynamics calculations using ab initio third-order perturbation theory indicate that the short lifetimes stem from strong three-phonon interactions and a high density of low-energy optical phonon modes related to the degrees of freedom of the organic cation. Such short lifetimes have significant implications for electron–phonon coupling in MAPI and other HOIPs, with direct impacts on optoelectronic devices both in the cooling of hot carriers and in the transport and recombination of band edge carriers. These findings illustrate a fundamental difference between HOIPs and conventional photovoltaic semiconductors more »
Authors:
; ; ; ; ; ; ; ;
Publication Date:
NSF-PAR ID:
10079013
Journal Name:
Proceedings of the National Academy of Sciences
Volume:
115
Issue:
47
Page Range or eLocation-ID:
p. 11905-11910
ISSN:
0027-8424
Publisher:
Proceedings of the National Academy of Sciences
4. Traveling-wave optomechanical interactions, known as Brillouin interactions, have now been established as a powerful and versatile resource for photonic sources, sensors, and radio-frequency processors. However, established Brillouin-based interactions with sufficient interaction strengths involve short phonon lifetimes, which critically limit their performance for applications, including radio-frequency filtering and optomechanical storage devices. Here, we investigate a new paradigm of optomechanical interactions with tightly confined fundamental acoustic modes, which enables the unique and desirable combination of high optomechanical coupling, long phonon lifetimes, tunable phonon frequencies, and single-sideband amplification. Using sensitive four-wave mixing spectroscopy controlling for noise and spatial mode coupling, optomechanical interactions with long$><#comment/>2µ<#comment/>s$phonon lifetimes and strong$><#comment/>400W−<#comment/>1m−<#comment/>1$coupling are observed in a tapered fiber. In addition, we demonstrate novel phonon self-interference effects resulting from the unique combination of an axially varying device geometry with long phonon lifetimes. A generalized theoretical model, in excellent agreement with experiments, is developed with broad applicability to inhomogeneous optomechanical systems. | 2023-02-07 12:20:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4723251461982727, "perplexity": 4379.482599834375}, "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-2023-06/segments/1674764500456.61/warc/CC-MAIN-20230207102930-20230207132930-00077.warc.gz"} |
https://www.autoitscript.com/forum/topic/60151-a-little-stuck-with-drawing-text-onto-a-image-gdi/ | # A little stuck with drawing text onto a image (GDI+)
## Recommended Posts
I got this code:
#include <GDIPlus.au3>
_GDIPlus_Startup ()
$hImage = _GDIPlus_BitmapCreateFromFile("template.bmp") _GDIPlus_GraphicsClear ($hImage)
$iHeight = _GDIPlus_ImageGetHeight($hImage)
$iWidth = _GDIPlus_ImageGetWidth($hImage)
_GDIPlus_GraphicsDrawString ($hImage, "Hello world", 1, 1)$sCLSID = _GDIPlus_EncodersGetCLSID ("BMP")
_GDIPlus_ImageSaveToFileEx ($hImage, ".\GDIPlus_Image2.bmp",$sCLSID)
_GDIPlus_Shutdown ()
template.bmp is a plain white BMP image. I am wanting to overlay text onto it, then save the finished image.
When I run that code, the samed image doesn't have any text overlayed on it. I am using autoit v3.2.0.10
What am I doing wrong?
Thanks,
_Spyrorocks
##### Share on other sites
I got this code:
#include <GDIPlus.au3>
_GDIPlus_Startup ()
$hImage = _GDIPlus_BitmapCreateFromFile("template.bmp") _GDIPlus_GraphicsClear ($hImage)
$iHeight = _GDIPlus_ImageGetHeight($hImage)
$iWidth = _GDIPlus_ImageGetWidth($hImage)
_GDIPlus_GraphicsDrawString ($hImage, "Hello world", 1, 1)$sCLSID = _GDIPlus_EncodersGetCLSID ("BMP")
_GDIPlus_ImageSaveToFileEx ($hImage, ".\GDIPlus_Image2.bmp",$sCLSID)
_GDIPlus_Shutdown ()
template.bmp is a plain white BMP image. I am wanting to overlay text onto it, then save the finished image.
When I run that code, the samed image doesn't have any text overlayed on it. I am using autoit v3.2.0.10
What am I doing wrong?
Thanks,
_Spyrorocks
In this line
_GDIPlus_GraphicsDrawString ($hImage, "Hello world", 1, 1)$hImage is a handle for the image but it should be a handle to a graphics object.
There is an example for using _GDIPlus_GraphicsDrawStringEx in
If you want the background of the pic to show through comment out the _GDIPlus_GraphicsFillRect line.
Serial port communications UDF Includes functions for binary transmission and reception.printing UDF Useful for graphs, forms, labels, reports etc.Add User Call Tips to SciTE for functions in UDFs not included with AutoIt and for your own scripts.Functions with parameters in OnEvent mode and for Hot Keys One function replaces GuiSetOnEvent, GuiCtrlSetOnEvent and HotKeySet.UDF IsConnected2 for notification of status of connected state of many urls or IPs, without slowing the script.
## Create an account
Register a new account | 2019-01-20 12:48:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.205464169383049, "perplexity": 10023.878871734092}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583716358.66/warc/CC-MAIN-20190120123138-20190120145138-00454.warc.gz"} |
https://www.intechopen.com/books/trade-and-global-market/corruption-causes-and-consequences | Open access peer-reviewed chapter
# Corruption, Causes and Consequences
By Štefan Šumah
Submitted: October 17th 2017Reviewed: December 6th 2017Published: February 21st 2018
DOI: 10.5772/intechopen.72953
## Abstract
Corruption is a constant in the society and occurs in all civilizations; however, it has only been in the past 20 years that this phenomenon has begun being seriously explored. It has many different shapes as well as many various effects, both on the economy and the society at large. Among the most common causes of corruption are the political and economic environment, professional ethics and morality and, of course, habits, customs, tradition and demography. Its effects on the economy (and also on the wider society) are well researched, yet still not completely. Corruption thus inhibits economic growth and affects business operations, employment and investments. It also reduces tax revenue and the effectiveness of various financial assistance programs. The wider society is influenced by a high degree of corruption in terms of lowering of trust in the law and the rule of law, education and consequently the quality of life (access to infrastructure, health care). There also does not exist an unambiguous answer as to how to deal with corruption. Something that works in one country or in one region will not necessarily be successful in another. This chapter tries to answer at least a few questions about corruption and the causes for it, its consequences and how to deal with it successfully.
### Keywords
• corruption
• influence
• economy
• economic growth
• rule of law
## 1. Introduction
The word corruption is derived from the Latin word “corruptus,” which means “corrupted” and, in legal terms, the abuse of a trusted position in one of the branches of power (executive, legislative and judicial) or in political or other organizations with the intention of obtaining material benefit which is not legally justified for itself or for others.
Corruption was referred to as a great sin already in the Bible: “Do not accept a bribe, for a bribe blinds those who see and twist the words of the innocent.” However, the history of corruption is in fact related to the beginning of the creation of law and the state and was already in the antiquity considered an evil, which negatively affects the public administration and the functioning of the political system. The earliest records of corruption date back to the thirteenth century BC, to the time of the Assyrian civilization. From the found plates, written in cuneiform, the archeologists managed to discern how and who accepted bribes. Under the Roman law, the criminal offense of corruption was defined as giving, receiving or claiming benefits in order to influence an official in connection with his work. Due to the prevalence of corruption in the country, this law was supplemented by a new law, which predicted compensation for damage in double value of the damage, and the loss of political rights for the perpetrator of the corruptive act. However, this did not help alleviate corruption, especially due to the fact that corruption was most practiced by the members of the Senate and senior state officials, both in Rome itself and in the remote Roman provinces. The early Christian faith condemned corruption, yet corruption later also developed greatly in ecclesiastical structures, and achieved its peak with the selling of indulgences in the Middle Ages, all until the condemnation of the latter (as well as of other immoral acts of the clergy, with the Pope at the head) by Martin Luther. Apart from the condemnation of corruption, the Reformation also led to a break with until then dominant Catholic culture and the emergence of Protestant ethics.
As a child (he was a hostage at the Ravenna court), Attila1 noticed a high level of corruption among the state officials of the Western Roman Empire and how they appropriated the state money (as a consequence, there was less money in the Treasury and therefore the taxes increased). He thus decided that if he would ever to rule, he would do so fairly and by oppressing the corruption in his own country. The early feudalism was familiar with various laws that punished the bribing of courts also with death. Later, when the developed feudalism again turned to the Roman law, a number of laws (Dušan’s Code, Mirror of the Swabians) discussed the abuse of position. Then, in late Feudalism, countries became virtually helpless in the fight against corruption, as illustrated by the case of France, which in 1716 established a special court in which should rule in cases of abuse of royal finances; however, these abuses (embezzlement, extortion, bribery, scams, etc.) were so extensive that the court was abolished and a general amnesty introduced in 1717 made some forms of corruption quite a tradition. The corruption was also widespread during the time of the Spanish Inquisition, where the victim of the accusation could make amends with money, which made the corruption, especially among the inquisitors, extensive.
Throughout the history, many intellectuals dealt with corruption or theorized about it one way or another. Machiavelli2 had a low opinion on republics, considering them even more corrupt than other regimes, and according to him, corruption leads to moral degradation, bad education and bad faith. On the other hand, however, the great philosopher, diplomat and lawyer Sir Francis Bacon3 was known both for receiving bribes and taking them. When he reached the highest judicial position in England, he was caught in as many as 28 cases of accepting a bribe and defended himself before the parliament by saying that he usually accepted a bribe from both parties involved and that the dirty money therefore did not affect his decisions. The parliament did not accept these arguments and sent him to the jail where he spent only a few days as he was able to bribe the judge.
Thus, although the corruption has been occurring in society ever since, it has only been given more attention in the recent period—the researches on the phenomenon and its negative impacts have become more common after 1995, when countries and international institutions began to be aware of this problem. The attitude of the public toward corruption was, until then, neutral. In 1998, Kaufmann and Gray [1] found that:
• Bribery is widespread, especially in the developing and transition countries; there are, however, significant differences between and within regions.
• Bribery increases transaction costs and creates insecurity in the economy.
• Bribery usually leads to ineffective economic results, in the long term impedes foreign and domestic investments, reallocates talents due to income and distorts sectorial priorities and technology choices (for example, it creates incentives for contracting major defense projects or unnecessary infrastructure projects, but does not encourage investments in rural specialist health clinics or in preventive health care). This pushes companies into the “underground” (outside the formal sector), weakens the state’s ability to increase revenue and leads to ever-increasing tax rates (as too little tax is taken), which is levied on less and less taxpayers, consequently diminishing the state’s ability to provide enough public goods, including the rule of law.
• Bribery is unfair, as it imposes a regressive tax, which heavily burdens in particular commercial and service activities performed by small businesses.
• Corruption destroys the legitimacy of the state.
Many other researchers and institutions (the World Bank Institute—WBI, the European Commission, the United Nations, the EBRD) have investigated corruption and its impact on macroeconomic and microeconomic indicators through various forms of corruption, as well as its connection with local customs and habits, and how it affects the everyday lives of people. Most studies are therefore mainly the analyses of the effects of corruption on various economic indicators, such as GDP growth, investments, employment, tax revenues and foreign investments [2, 3, 4, 5], or the study of various forms of corruption in relation to politics and the economic environment [6], the research of its social condition and various manifestations [7, 8]. Dobovšek [9] agrees with the negative effects, i.e. high economic, political and social costs, and adds that corruption is not a weakness of people but of institutions (supervisory and other), as they should be the ones to obstruct the greed and temptation of individuals within them.
## 2. Causes of corruption
Although corruption differs from country to country, it is possible to identify some of the key common driving forces that generate it. What is common to all countries, which are among the most corrupt, has been identified by Svensson [10]; all of them are developing countries or countries in transition,
• with rare exceptions, low-income countries,
• most countries have a closed economy,
• the influence of religion is visible (Protestant countries have far the lowest level of corruption),
• low media freedom and
• a relatively low level of education.
Regardless of the above, corruption cannot be assessed unambiguously, since there is never only one phenomenon that is responsible for the occurrence and the development of it; corruption always arises from an array of several, interrelated factors, which can differ considerably from one another. Among the most commonly mentioned factors that influence the development of corruption are: political and economic environment, professional ethics and legislation, as well as purely ethnological factors, such as customs, habits and traditions.
### 2.1. Political and economic environment
The phenomenon of corruption is strongly influenced by the political and economic environment. The more is the economic activity in the country regulated and limited, the higher the authority and the power of officials in decision making and the greater the possibility of corruption, since individuals are willing to pay or offer payment in order to avoid restrictions. A great potential for corruption is especially there where the officials are under the regulation given the opportunity to decide on the basis of discretion.
The level of corruption is also affected by the monetary policy. Goel and Nelson [11] in their research found a strong link between monetary policy and corruptive activity in the States. The States that have a well-regulated financial sector, not a lot of informal economy or black market are also less corrupt than those where the opposite is true. They also find that there is less corruption in the countries with higher economic and political freedom.
Dimant [12] puts it well in his claim that the level of efficiency of public administration determines the extent to which corruption can find fertile soil and sprout. Such efficiency is determined by the quality of the regulations and permits, since ineffective and unclear regulations help to increase the level of corruption in at least two different ways:
1. The artificially created monopoly of power that enables civil servants to obtain bribes is based on their superior position and embedded in the system.
2. On the other hand, however, ineffective and unclear regulations cause inhibition and therefore encourage natural persons to pay bribes in order to speed up the bureaucratic procedure.
Corruption is also strongly influenced by the low salaries of public administration employees (state officials), who are therefore trying to improve their financial position by receiving bribes, and consequently, the socio-economic situation of the government officials also affects the phenomenon of corruption. This is demonstrated also by Allen et al. [13] in their study where they find that corruption arises because agencies, institutions and the government can no longer control corruption effectively due to underpaid officials, which is a problem especially in the developing countries, where they do not have the sufficient tax revenue to properly reward the local officials. However, low wages are not the only cause of corruption; the poor state of the public administration, which is a consequence of political “overcrowding”4 of officials, due to which loyalty usually prevails over professional standards, also strongly affects the corruption. As an important factor influencing corruption, some authors also indicate satisfaction with the work done by officials—the more they are dissatisfied with their work or place of work, the higher the degree of corruption, which is confirmed by Sardžoska and Tang [14] in their studies. The mentioned authors find that the private sector has higher ethical values, in particular those that affect satisfaction with work, than the public sector and is therefore less unethical (especially regarding thefts and corruption). Indirectly, Svenson [10] also affirms this and states that in principle, the salary level of civil servants affects the receipt of a bribe (the higher it is, the smaller the chance that the person will act corruptly). However, he continues on that a higher salary also strengthens the negotiating power of the official, which leads to higher bribes and he also states that, on the basis of existing research, it is very difficult to determine whether a higher salary causes less corruption, which means that the level of salary is not a decisive factor, but merely one of many.
The economy is unfortunately largely dependent on politics and often reflects the rule of law; various options for eliminating competition are exploited, and bribery is just one of the possible weapons in the struggle to gain a job. At the same time is the mentality of the economy sometimes: “The cost of a bribe is only a substantial business cost, an integral part of the contract,” or “Even if we stop the bribery, our rivals will not, so we must bribe in order to remain competitive, “or” bribery and misleading behaviour are not really crimes, they are just part of the old business practice. They are part of the game and everyone does it.” On the other hand is the point sometimes simply the “lubricating” of the bureaucratic wheel by the private sector to do certain things faster or easier.
The political influence of corruption is also manifested through the proverb: examples are attractive! If the top of the politics (government, parties and leading politicians) is corrupt, then corruption shows at all levels, and this evil at the same time spreads among the ordinary population, as nobody trusts the institutions or the rule of law. Johnston [15] thus points out useful thinking in terms of two types of equilibrium—the balance between the openness and the autonomy of the institutions and elites it leads and the balance between political and economic power and opportunities for cooperation. Ideally, the institutions should be open to influences and feedback from different sources, yet at the same time sufficiently independent to effectively carry out their work. Where the openness and independence of the institutions are in balance, the officials are accessible, but not excessively exposed to private influences; if they can make authoritative decisions, while not using their power to arbitrate, the corruption is relatively low. But where the official power is poorly institutionalized, too exposed to private influence, and the officials’ independence is reflected in excessive exploitation of their power—they can do as they please—the possibility for extreme corruption is again high.
### 2.2. Professional ethics and legislation
Lack of professional ethics and deficient laws regulating corruption as a criminal offense, and the prosecution and sanctioning of it are also an important cause for the emergence and spread of corruption. A great influence comes also from the ineffective sanctioning of corruption, which only increases the possibility of continuing the corruptive actions of those involved, creating at the same time a strong likelihood that others will join in the corruption due to this inefficient sanctioning.
The sole lack of professional ethics is a particular issue, as the administration requires different amounts of time to develop or change its ethics and professional standards, which is well known in transition countries (in some, ethics and professional standards changed overnight and approached the equivalents in the developed democracies, and in some, they remained the same as in socialism). It is precisely in the transition countries that the “softer” acts of corruption are often considered to be acceptable and justifiable. Therefore, due to lack of professional ethics in some countries that otherwise manage illegal corruption well, there is nevertheless a widespread form of legal corruption.
Corruption also generates a lack of transparency and a lack of control by supervisory institutions. Therefore, where there is insufficient legal basis or sufficient political will to control, which enables a non-transparent functioning of both politics and the economy, corruption flourishes. Corruption is also affected by the extensive, non-transparent or incomplete legislation, where laws can be interpreted in different ways (for the benefit of the one who pays).
### 2.3. Habits, customs, tradition and demography
Different countries have different attitudes to corruption. In Europe alone, we can find two extremes; from completely corruption intolerant North to the warm South, where corruption is an almost normal, socially acceptable phenomenon. Or the difference between countries with a democratic past, which traditionally prosecute corruption, and former socialist countries, where the corruption in the state apparatus was a part of folklore tradition. Then, there are also different customs; in some cases, a “thank you” in the form of a gift for a service (for which this person has already been paid with a salary) is an expression of courtesy, and elsewhere it is considered corruption. Everything is only a matter of ethics and morality; however, they can be very different in different areas and different countries.
Some forms of corruption also relate to an informal form of social security, where the family or the immediate community takes care of its members. Such forms of informal social security prevail in less developed countries, where there is no legal regulation of formal social security and in the countries of Southern Europe where the influence of the broader family (patriarchate5) is still very strong, like for example in Italy, Greece, Albania, Bosnia, etc. These countries are known for nepotism, cronyism and patronage, since the family as well as the wider community provide social security. The family or community takes care of their members, who, in return, must be loyal and in a way also repay the benefits they receive from it. The same is true of faith. While the southern, predominantly Catholic, very hierarchically organized part of Europe, encourages the cult of the family (also joint and several community) and several liability, the northern, mainly Protestant part, emphasizes individualism and individual responsibility (which means less forms of corruption). The corruption also prospers better in countries where Islam and Orthodoxy are the main religion. The influence of the dominant religion in the country is thus important.
The influence of majority Protestantism has been tested several times and has proven to be an important factor for the low level of corruption in a country. However, the relationship between Protestantism and good governance is probably rooted more in history than in today’s practice. Today, there are many nominally Protestant countries that are de facto secular, while also many non-Protestant countries fight effectively against corruption. Thus, the influence of Protestantism appears to emerge from its egalitarian ethos, which could indirectly function as a support to the general orientation toward ethical universalism, literacy and the promotion of individualism. Its role is therefore important, as it at certain stages of the development explains why the first countries that were well managed were predominantly Protestant. This does not mean that other religious traditions are incompatible with good governance, but only that they have not succeeded in compiling this particular array of factors at the right moment [16].
Similarly, the research by North et al. [17] showed that, according to the authors, the least corrupt countries or those countries where the rule of law is the strongest were predominantly Protestant in 1900 and those who are most corrupt were predominantly Orthodox in the same year. The results of their research have shown that there is a link between religion and corruption on one hand, and respect for the rule of law on the other, but not that the link is causative. The questions therefore arise: Why do some religions respect the rule of law more than others and control corruption? Do the characteristics of a particular religion themselves lead to the results? Are there any differences in religious doctrines, practices or cultures that lead to such results? Are there other links that are not rooted in the religious culture, but are related to religious affiliation?
A study titled Perception of corruption by authors Melgar et al. [18] tried to find out which groups of people are more likely to pay for corruption. They found that those who think that there is a lot of corruption also perceive it so and are consequently more willing to pay for it (as they think or expect the society to function that way). By using a wide and very heterogeneous set of data and econometrics, it has been shown that the social status and personal characteristics also play an important role in the shaping of corruption perception at the micro level. While divorced women, unemployed persons, persons working in the private sector or the self-employed are considered to be in positive correlation with the perception of corruption (corruption is perceived more and they are more willing to pay bribes), the opposite applies to married persons, full-time employees, people who frequently attend religious ceremonies and people with at least secondary education (they perceive less corruption and are also unwilling to pay). According to the classification of countries, they find that it can be proved that all African and Asian countries are in the upper half of the table, and the same applies to the former socialist countries and most of the East Asian countries. People living in these countries perceive more corruption than others. On the contrary, most European countries and some of the former English colonies show lower perceptions than the average (there are also exceptions) and rank in the lower half, the same as half of the richest countries. They also added that the geographical classification of countries has been strongly correlated with the corruption perception index (CPI), which shows that individual characteristics and social conditions are specific factors that influence the perception of corruption. However, they have also found that better economic results reduce the perception of corruption, while the macroeconomic instability and income inequalities have precisely the opposite effect. With Mahič [19], we also found a similar influence on the perception of corruption; in the economic crisis (high unemployment and low purchasing power), the perception of corruption is rising.
A very important factor that affects corruption is also demographics. A number of studies have shown that patriarchal society is more prone to corruption. This is confirmed by several researches that actually explore to what extent are men women corrupt. Several earlier, especially econometric contributions to the debate on who is more corrupt, men or women, argued that there is a link between a higher representation of women in government and lower levels of corruption. An influential study of 150 countries in Europe, Africa and Asia by the World Bank [20] confirmed this and concluded that women are more reliable and less prone to corruption. The subsequent findings were later reinforced by further research. Rivas [21] also affirms this in his research and notes that, according to the results of the survey, the conclusion could be that women are less corrupt than men and that the increase in the number of women on the labor market and in politics would help fight corruption. Lee and Guven [22] in the survey: Engaging in corruption—the influence of cultural values and the contagion effects at the micro-level also raised the question of whether men are more corrupt than women. The findings of the research support the thesis that women are less susceptible to corruption than men, especially in cultures that require men to be ambitious, competitive and materially successful, as these factors significantly contribute to unethical behavior. This was surprisingly well shown also in practice [23] when, due to gender equality, the Peruvian government a decade ago decided to involve more women in the police units. When the 2,500 female police officers were joined as traffic police officers, something unexpected happened; bribery was drastically reduced, and people welcomed the female police officers on the streets.
## 3. The impact of corruption on the economy
In 1997, Tanzi and Davoodi [2] conducted a systematic study of the impact of corruption on public finances. Several important findings came to light:
1. Corruption increases the volume of public investments (at the expense of private investments), as there are many options that allow for public expenditure manipulation and are carried out by high-level officials so as to get bribes (which means that more general government expenditures or a large budget offer more opportunities for corruption).
2. Corruption redirects the composition of public expenditure from the expenditure necessary for basic functioning and maintenance to expenditure on new equipment.
3. Corruption tends to pull away the composition of public expenditure from the necessary fixed assets for health and education, as there is less chance of getting commissions than from other, perhaps unnecessary projects.
4. Corruption reduces the effectiveness of public investments and the infrastructure of a country.
5. Corruption can reduce tax revenues by compromising the ability of the state administration to collect taxes and fees, although the net effect depends on how the nominal tax and other regulatory burdens were selected by the officials, exposed to corruption.
The influence of corruption on the economy was studied by the same authors [3] through several factors:
1. Through the impact of corruption on businesses: The impact of corruption on a business is largely depend on the size of the company. Large companies are better protected in an environment that is prone to corruption, they avoid taxes more easily and their size protects them from petty corruption, while they are often also politically protected, which is why the survival of small (especially start-up companies) and middle-sized companies, regardless of their importance for the growth of the economy and the development, is much more difficult than the survival of large companies.
2. Through the impact of corruption on investments: Corruption affects (a) total investments, (b) the size and form of investments by foreign direct investors, (c) the size of public investments and (d) the quality of investment decisions and investment projects.
3. Through the influence of corruption on the allocation of talents: Indirectly, corruption has a negative impact on economic growth through the allocation of talents, since gifted and prospective students are driven, due to the influence of the environment and the situation in the country, for example, to study law rather than engineering, which would add value to the country.
4. Through the impact of corruption on public spending: Corruption has a negative impact on public spending and has an especially strong impact on education and health. There are also indications of the correlation between corruption and military expenditure, which means that high level of corruption reduces economic growth due to high military expenditure.
5. Through the impact of corruption on taxes: Because of corruption, less taxes are levied than would otherwise be, as some of the taxes end up in the pockets of corrupt tax officials. There are also frequent tax relieves in the corrupt countries, selective taxes and various progressive taxes; in short, there is much less money than the country could have, and so corruption, through the country’s financial deficit, also affects the economic growth; and conclude the findings on the negative impact (both indirect and direct) of corruption on economic growth.
Smarzynska and Wei [5] came to similar conclusions regarding the effects of corruption on the size and composition of investments. Corrupt countries are less attractive for investors, and if they do opt for an investment, due to non-transparent bureaucracy, they often enter the market with a joint venture, as they usually understand or control matters of the home country better. The local partner can also help foreign companies with the acquisition of local licenses and permits or can otherwise negotiate with the bureaucratic labyrinths at lower costs. Generally inclined (as investors) to the joint venture in the corrupt countries are especially the US investors; however, even investors from those European countries, which are among the highest ranked on the CPI, quickly adapt to local conditions.
Corruption for various reasons also affects the following:
1. Employment, because the job does not go to the most suitable or qualified person, but the one who is ready to pay for it or in any other way return the favor.
2. Also affects total investments [24].
3. The size and composition of foreign investments and the size of public investments.
4. The effectiveness of investment decisions and projects. In the presence of corruption, the investments are smaller, as entrepreneurs are aware that they will have to bribe the officials or even give them a profit share for a successful implementation of a business. Due to these increased costs, the entrepreneurs are not interested in investing.
Wei [25] even made a projection which predicted that in the case of reduction in corruption in Bangladesh to the level of corruption in Singapore, the growth rate of GDP per capita would increase by 1.8% per year between 1960 and 1985 (assuming that the actual average annual growth rate was 4% per year), and the average per capita income could have been more than 50% higher, whereas the Philippines could, if its level of corruption was reduced to that of Singapore (if everything remained unchanged), have raised their investments in relation to GDP by as much as 6.6%, which means a significant increase in the investments. At the same time, he notes that in order to reduce the corruption to the level of Singapore in the countries that he compared (India, Kenya, Sri Lanka, Turkey, Colombia, Mexico and Ghana), the State should raise the salaries of officials by 400—900%. He therefore asks himself whether this would even be possible. However, he notes that in the event of a large increase in salaries, a new form of corruption would likely arise when everyone would be prepared to pay a bribe for a well-paid official job.
Corruption often reduces the effectiveness of various financial assistance programs (both state and international), as money is “lost somewhere along the way” and does not reach those that need it or for whom it is intended, as the financial benefits, deriving from corruption, are not taxable because they are hidden. The state is thus also losing part of the income from the taxes due to corruption, while the public spending, resulting from corruption (or narrow private interests) leads to negative effects on the budget.
The European Commission in its report found that corruption is costing the European economy about 120 billion a year, and according to the European Commissioner for Home Affairs, Cecilia Malstotröm, the corruption in Europe is most present in public procurement, financing of political parties and health care [26].
The United Nations estimate that the cost of corruption in Afghanistan amounted to about $3.9 billion in 2012. According to Transparency International, the former leader of Indonesia, Suharto, embezzled between$ 15 and $35 billion, whereas the embezzlements of Mobutu in Zaire, Ferdinand Marcos in the Philippines and Abacha in Nigeria are estimated to amount to$ 5 billion [27]. However, the World Bank survey shows that \$ 1 billion in bribes, both in rich and developing countries, is paid annually [28], which means that even the developed countries are not immune to corruption (but in a different form) and that the political corruption is especially present in large infrastructure projects. Bađun [29] on the example of Croatia gives conclusions, which are valid for all post-communist countries.
Impact on enterprises: A survey conducted by the EBRD and the World Bank shows that bribes paid in smaller companies account for 5% of their annual profits and in medium-sized companies 4% of their annual profits. However, both are, compared to large companies, where bribes comprise less than 3%, in a much worse position, which shows how bribes are causing problems or are putting these smaller companies into a subordinate position compared to the large ones, which in turn leads to the collapse of these.
Also interesting is the study of the Shadow Economy in Highly Developed OECD Countries where Schneider and Buehn [30] also find the link between the low quality of institutions that are the holders of the rule of law (or degree of corruption) and the shadow economy, and therefore, the weaker the “law” is, the higher the degree of corruption and of shadow economy. In the study Corruption and the Shadow Economy [31], the same authors explore the relationship between the degree of corruption and the emergence of the shadow economy, and their findings are that the high level of shadow economy and the high degree of corruption are strongly linked to one another. One of the hypotheses in this survey (which has been confirmed) is also: the higher the degree of corruption, the lower the economic development measured by GDP per capita. The authors detected a positive correlation; corruption thus affects the economic development.
However, the extended practice of finding annuity outside the logic of the market and competition can therefore lead to a (neo) liberal conclusion that the root of the existence of corruption is in the very existence of the state—especially in excessive, selective and deforming state interventions and subsidies that create fertile soil for the development of corruption. The truth is that the devastating combination consists of widespread state intervention and subsidies in the simultaneous absence of a strong institutional framework and detailed rules of the game, including the control of public finances and effective anti-trust legislation and legal practices. On the other hand, however, there is no clear evidence that private monopolies are more effective and less corrupt than the public ones and that privatization, especially long-lasting, gradual and non-transparent one (so-called gradualism), reduces positive developmental and social effects, including the reduction of corruption [32]. Yet market deregulation, legal and judicial reform and transparent management of public procurement would significantly reduce corruption in many developing countries (as well as in transition countries), at which point the government should play an important role in the shaping of the anti-corruption policy. There should be a strong strengthening of the public procurement institution. The law is admittedly strict about the public procurement, but one of the main reasons for public procurement problems is the lack of a skilled workforce, and public procurement is thus still the breeding ground of corruption. There also exists a proverb “poverty is a curse,” which applies largely to all developing countries, as these are the countries that are most affected by poverty. Poverty destroys all ethical and moral values.
One of the important aspects of the damage to the global economy is also the failure to respect copyright and intellectual property. The more corrupt countries are also inclined to lower respect for the aforementioned, and the economic damage amounts to billions of dollars. Cavazos-Cepeda et al. [33] found that reforms, legal, fiscal and intellectual incentives to respect copyright and intellectual property patents encourage the society to make itself more innovative and economically more effective; however, they underline the importance of human capital and investment in people as one of the most important factors for reducing the level of corruption in the country.
There are also theories that corruption can act as the lubricant of the economic wheel and at least in some cases has a positive impact on the economic growth. The empirical analysis done by Dreher and Gassebner [34] on a sample of 43 countries between 2003 and 2005 shows that corruption is even useful, but with some reservations. In particular, they investigated the short-term effects of corruption and found, for example, that in countries where corruption is widespread, more new entrepreneurs enter the market (corruption in the public sector is expected to promote private entrepreneurial activity). They are, however, not necessarily to succeed, as there is a high likelihood that they will go bankrupt due to the rigid regulations that block the activity and because of which bribes are needed. They do acknowledge, on the other hand, that most authors who have been doing research for a longer period of time admit the harmfulness of corruption both for society and the economy. Something similar show the data for some Asian countries, where, unlike their findings (short-term benefit), the high degree of corruption coincides with the long-term economic growth.
Svendson [10] also notes that, in light of the theoretical literature and various research studies, notwithstanding that these show the negative impact of corruption on the economic growth, but this cannot be said for sure, since there are difficulties in measuring corruption, and at the same time, the question arises whether the econometric models that were made are good enough to capture all the important variables. He also states that corruption appears in many forms and that there is no reason to assume that all types of corruption are equally harmful to the economic growth.
Recent empirical researches also attest to that; while many countries have suffered, as a characteristic consequence of corruption, the decline in economic growth, other countries have had economic growth (in some cases a very positive one) despite corruption. The latter is also to be expected, since corruption has many manifestations and it would be surprising if all types of corrupt practices had the same effect on economic performance. Analyses show that one of the reasons for this is the extent to which the perpetrators of corrupt practices—in this case the bureaucrats—coordinate their behavior. In the absence of an organized corruption network, each bureaucrat collects bribes for himself, while ignoring the negative impact of others’ demands for them. In the presence of such a network, the collective bureaucracy reduces the total value of the bribe, which results in lower bribe payments and higher innovation, and the economic growth is consequently higher in the latter case than in the former case. The interesting question is not so much why is the degree of corruption in poor countries higher than in the rich ones, but rather why the nature of corruption differs between countries. The extent to which corruption is organized is just one aspect of this, but there are other aspects. For example, it is common practice in some countries to pay ex post (as a share of profit, for example) instead of ex ante (in advance, as a bribe) to officials or politicians, so it is assumed that the effects on the economy will be different. The precise reason why corruption should take on one form and not the other is an important issue which has been largely ignored and which could have to do with cultural, social and political reasons, as well as economic circumstances [35].
In the fight against corruption, a remarkable role was also played by the debt crisis. The die Welt newspaper [36] mentions the study of the Hertie School of Governance, which shows that Italy, Spain and Portugal have made great strides in the fight against bribery and corruption of their civil servants due to lack of money, which enabled a significantly more transparent and “pure” practice for the award of public procurement. The crisis is supposed to dry up monetary resources and thus reduce the chances of corruption. Also, the crisis has changed the perception of the society, and bad business practices, which were acceptable before the crisis, are acceptable no longer. However, the fight against corruption is often similar to the fight against windmills. The case of India shows how corruption is changing, getting new dimensions, not only in scope, but also in methods. Just as the population in India is growing, so is corruption, and there are always new ways how to cheat both the state and the society. The perception of corruption is increasing year after year. Despite all the anti-corruption moves and anti-corruption initiatives, people do not hesitate to offer or accept a bribe. The bribers are becoming innovative, they adapt to the situation and the innovation of companies in paying bribes and hiding them is also visible. However, just as elsewhere in the world, the negative effects of corruption are the same; it reduces foreign direct and domestic investments, increases inequality and poverty, raises the number of freeloaders (renters, free-riders) in the economy, distorts and exploits public investments and reduces public revenues.
## 4. Discussion
Corruption is, in fact, a multidirectional process. On one hand, the provider benefits, on the other the recipient, and both are aware of the deed that remains hidden. The third link in the chain is everyone else, the victims. Although not every act of corruption is yet a criminal offense, it is, however, unethical and detrimental to the economic and political development of a society. Usually, there are persons involved with political, economic and decision-making power, and as the philosopher Karl Popper wrote in his book, The Open Society and its Enemies, that the greatest problem is not the question of who should give orders, but how to control the one who gives them. How to organize the political and social institutions in order to prevent the weak and incompetent rulers from doing too much harm? However, as there is no general and unmistakable way of preventing the tyranny or corruptions of the heavyweights, the price of freedom is eternal alertness [37]. Greediness, ambition, rapacity and immorality have been known to the human society ever since the emergence of civilization and use every tool available to them: kinship, common past, school contacts, common interests, friendship and, of course, political as well as religious ties.
In a study by Šumah et al. [38], we did an analysis of countries, taking into account their ranking on the Corruption Perception Index published every year by Transparency International, and identified the main factors affecting the level of corruption in a particular group of countries, or rather, we tried to find similarities and differences between individual groups of countries in terms of what affects the level of corruption in these groups. We have established a basic model of three factors (risk, benefit and consciousness) that was created on the basis of the merger of several known, scientifically proven factors that cause or reduce corruption or affect its level in the individual country. According to this degree of corruption, we have identified five groups, classified the countries and analyzed their common characteristics. The findings were as follows:
• Corruption is linked to the level of GDP (the higher the GDP, the lower the rate of corruption).
• Corruption is related to the level of education (the higher the average level of education, the lower the level of corruption).
• Corruption is strongly linked to the geographical location. The highest level is in Asia (mainly in Central Asia), Africa (North and Central Africa) and South America (according to the Transparency International map).
• Corruption is strongly linked to the country’s prevailing religion.
• Corruption is linked to freedom in the country (personal freedom, freedom of speech, economic freedom, etc.), with respect to the rule of law in a country and inefficiency of public administration, which is often also locally limited or is inherently corrupt.
• The lower the country is ranked, the more dominant is the patriarchal society.
Many researchers are still involved in corruption. The findings show that there is a link between corruption and its negative effects, but from most of the studies it is not possible to determine what the cause is and what the consequence. Whether is the level of corruption lower due to high GDP, or is it vice versa, cannot be directly identified, since the corruption depends on economic indicators, while at the same time affecting them [39]. It is also very difficult to claim that the average low level of education is due to corruption or, conversely, that corruption is a result of low education. Similarly goes for the rule of law and (in)efficiency of public administration. This interdependence will surely continue to be the subject of numerous researches in the future, for the only way to be successful in the fight against corruption is if we know the causes and begin to eliminate them.
Nevertheless, there remains something that needs to be emphasized. Almost all of the studies ignore the fact that the top of the most corrupt countries consists of countries with one of the various forms of armed conflict (civil war, intertribal conflicts, inter-religious wars or some other form of aggression), which means that peace in the country is a prerequisite for a successful fight against corruption. The least corrupt countries are countries that have a lasting peace on their territory (most since the Second World War or even longer), which is confirmed by the above fact. Peace is therefore one of the prerequisites for a successful fight against corruption.
The answer to the question of how to deal with corruption is not unambiguous; some countries have achieved great success in dealing with it in a relatively short time (Singapore, Estonia and Georgia) and some have been struggling for a long time (the most famous example is Italy). The first condition is in any case to ensure freedom (personal freedom, economic freedom, freedom of speech, freedom of the press, etc.) and democracy, and then education and awareness of people. However, at this point, it is not about introducing the Western type democracy, as our culture knows it, for it has often proven that, especially with the help of the army, more harm than benefit was caused. It is necessary to start using good practices of countries that are similar to each other (religion, habits, tradition, ethics and morality) and that have common history.
## Notes
• Attila (406–453) was the great ruler of the Huns, who, as the first, united all the Huns and conquered a considerable part of Europe and Asia. He is also known as the Whip of God.
• The Renaissance political theorist (1469–1527) who was for more than a decade engaged in diplomatic and state affairs in Florence. Modern political philosophy and political science consider him the founder of the realistic approach to the theory of politics.
• An English philosopher, writer, judge and politician (1561–1626). He rejected Aristotle’s view and philosophy and sought to gain the reputation of the experimental science.
• Overcrowding in this context implies replenishment of posts in public administration with members of one party.
• Patriarchate is a social arrangement in which all authority is held by male representatives of the families that make up the community. The right to name, social and political status, as well as the possession and authority over family members is automatically transferred from the father to the firstborn or to the nearest male relative.
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Štefan Šumah (February 21st 2018). Corruption, Causes and Consequences, Trade and Global Market, Vito Bobek, IntechOpen, DOI: 10.5772/intechopen.72953. Available from:
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http://mathoverflow.net/revisions/85625/list | 4 clarified some statements
The search term you want to look for is "Klimyk's Formula." This formula boils down to the following:
Fix $G$ a compact complex semisimple Lie group. Suppose $V(\lambda)$ and $V(\mu)$ are irreducible representations with highest weights $\lambda$ and $\mu$ respectively. Let $W_\lambda = {\lambda_1,\lambda_2,\ldots \lambda_d}$ be the multiset of weights of $V(\lambda)$ with $d = dim(V(\lambda))$. Then the irreducible components of $V(\lambda)\otimes V(\mu)$ are given by ${V(\mu+\lambda_i)}_{i=1}^d$.
To apply this in practice, you need to be comfortable with the concept of defining $V(\lambda)$ when $\lambda$ is not a dominant weight (which sometimes causes modules to cancel when they appear with both positive and negative signs in the sum), but it applies to lots of groups (even beyond the scope of compact complex semisimple in some cases if im not mistaken), and Littlewood-Richardson is just the special case of this formula in type $A$.
An example for $G_2$ (since that is also my favorite compact semisimple Lie group) is to let $\lambda = [1,0]$ be the highest weight of the 7-dimensional representation and $\mu = [0,1]$ the highest weight of the 14-dimensional adjoint representation.
The seven weights of $V(\lambda)$ are $[1,0]$, $[-1,1]$, $[2,-1]$, $[0,0]$, $[-2,1]$, $[1,-1]$, and $[-1,0]$ so Klimyk tells us the 98-dimensional tensor product decomposes as:
$V([1,1]) \oplus V([-1,2]) \oplus V([2,0]) \oplus V([0,1]) \oplus V([-2,2]) \oplus V([1,0]) \oplus V([-1,1])$
This is where familiarity with interpreting modules with non-dominant highest weights comes in; $V[-1,2]$ and $V[-1,1]$ turn out to be 0-dimensional modules, while $V([-2,2]) \cong -V([0,1])$*. Thus the terms which do not disappear are $V([1,1])$ which is a 64-dimensional module, $V([2,0])$ which is a 27-dimensional module, and $V([1,0])$ which is the 7-dimensional defining representation, a total of 98 dimensions.
If you had instead chosen to switch $\lambda$ and $\mu$ and add the 14 weights of $V([0,1])$ to [1,0], you would have obtained 14 modules, but as before, some would have been zero and others would have cancelled in pairs ultimately leading to the same three modules as above being the only things left over. In my opinion, this reflexivity always holding is the coolest thing about Klimyk's formula.
One neat corollary to Klimyk's formula is that a tensor product of two irreducible modules cannot decompose into a sum of more than $d$ irreducibles where $d$ is the minimum of the dimensions of the two modules.
*EDIT: After posting, I decided to add a bit more about modules with non-dominant highest weight. Basically, the weights of a group $G$ are permuted via the Weyl group action on the weights. Weights are determined by integer $r$-tuples where $r$ is the rank of $G$; tuples containing a -1 lie in the walls of the Weyl chambers and so the modules with these highest weights end up being 0. All There are a few other subspaces which also correspond to walls; weights $\mu$ not lying in the walls satisfy $w(\mu) = \lambda$ for some dominant weight $\lambda$ (all coordinates nonnegative) and a unique $w$ in the Weyl group (i.e. only one $w$ in the Weyl group will take $\mu$ to a dominant weight, so $\lambda$ is also uniquely determined). Then $V(\mu)$ is defined by the following relationship:
$V(\mu) = (-1)^w\cdot V(\lambda)$
Here $(-1)^w$ is the sign representation which appears with all Weyl groups; in the $A$-series whose Weyl groups are the $S_n$'s this is the ordinary sign representation.
3 added 944 characters in body
This is where familiarity with interpreting modules with non-dominant highest weights comes in; $V[-1,2]$ and $V[-1,1]$ turn out to be 0-dimensional modules, while $V([-2,2]) \cong -V([0,1])$. V([0,1])$*. Thus the terms which do not disappear are V([1,1])$V([1,1])$which is a 64-dimensional module, V([2,0])$V([2,0])$which is a 27-dimensional module, and V([1,0])$V([1,0])$which is the 7-dimensional defining representation, a total of 98 dimensions. One neat corollary to Klimyk's formula is that a tensor product of two irreducible modules cannot decompose into a sum of more than$d$irreducibles where$d$is the minimum of the dimensions of the two modules. *EDIT: After posting, I decided to add a bit more about modules with non-dominant highest weight. Basically, the weights of a group$G$are permuted via the Weyl group action on the weights. Weights are determined by integer$r$-tuples where$r$is the rank of$G$; tuples containing a -1 lie in the walls of the Weyl chambers and so the modules with these highest weights end up being 0. All other weights$\mu$satisfy$w(\mu) = \lambda$for some dominant weight$\lambda$(all coordinates nonnegative) and a unique$w$in the Weyl group (i.e. only one$w$in the Weyl group will take$\mu$to a dominant weight, so$\lambda$is also uniquely determined). Then$V(\mu)$is defined by the following relationship:$V(\mu) = (-1)^w\cdot V(\lambda)$Here$(-1)^w$is the sign representation which appears with all Weyl groups; in the$A$-series whose Weyl groups are the$S_n$'s this is the ordinary sign representation. 2 added 45 characters in body The search term you want to look for is "Klimyk's Formula." This formula boils down to the following: Fix$G$a compact complex semisimple Lie group. Suppose$V(\lambda)$and$V(\mu)$are irreducible representations with highest weights$\lambda$and$\mu$respectively. Let$W_\lambda = {\lambda_1,\lambda_2,\ldots \lambda_d}$be the multiset of weights of$V(\lambda)$with$d = dim(V(\lambda))$. Then the irreducible components of$V(\lambda)\otimes V(\mu)$are given by${V(\mu+\lambda_i)}_{i=1}^d$. To apply this in practice, you need to be comfortable with the concept of defining$V(\lambda)$when$\lambda$is not a dominant weight (which sometimes causes modules to cancel when they appear with both positive and negative signs in the sum), but it applies to lots of groups (even beyond the scope of compact complex semisimple in some cases if im not mistaken), and Littlewood-Richardson is just the special case of this formula in type$A$. An example for$G_2$(since that is also my favorite compact semisimple Lie group) is to let$\lambda = [1,0]$be the highest weight of the 7-dimensional representation and$\mu = [0,1]$the highest weight of the 14-dimensional adjoint representation. The seven weights of$V(\lambda)$are$[1,0]$,$[-1,1]$,$[2,-1]$,$[0,0]$,$[-2,1]$,$[1,-1]$, and$[-1,0]$so Klimyk tells us the 98-dimensional tensor product decomposes as:$V([1,1]) \oplus V([-1,2]) \oplus V([2,0]) \oplus V([0,1]) \oplus V([-2,2]) \oplus V([1,0]) \oplus V([-1,1])$This is where familiarity with interpreting modules with non-dominant highest weights comes in; V[-1,2]$V[-1,2]$and V[-1,1]$V[-1,1]$turn out to be 0-dimensional modules, while$V([-2,2]) = \cong -V([0,1])$. Thus the terms which do not disappear are V([1,1]) which is a 64-dimensional module, V([2,0]) which is a 27-dimensional module, and V([1,0]) which is the 7-dimensional defining representation, a total of 98 dimensions. If you had instead chosen to switch$\lambda$and$\mu$and add the 14 weights of$V([0,1])$to [1,0], you would have obtained 14 modules, but as before, some would have been zero and others would have cancelled in pairs ultimately leading to the same three modules as above being the only things left over. In my opinion, this reflexivity always holding is the coolest thing about Klimyk's formula. One neat corollary to Klimyk's formula is that a tensor product of two irreducible modules cannot decompose into a sum of more than$d$irreducibles where$d\$ is the minimum of the dimensions of the two modules.
1 | 2013-05-18 12:53:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.8982058763504028, "perplexity": 337.89383146573533}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368696382396/warc/CC-MAIN-20130516092622-00017-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/2528875/if-f-is-a-diffeomorphism-and-g-a-homeomorphism-is-g-circ-f-circ-g-1-a | # If $f$ is a diffeomorphism and $g$ a homeomorphism, is $g\circ f\circ g^{-1}$ a diffeomorphism?
Let $X\subset\mathbb{R}^n$ open. Let $f,g:X\rightarrow X$ be functions such that $f$ is a diffeomorphism and $g$ is a homeomorphism. Is $g\circ f \circ g^{-1}$ a diffeomorphism?
I'm currently working through Lee's Intro to Smooth Manifolds, trying to show that any non-degenerate manifold that admits a smooth structure admits uncountably many distinct smooth structures. If the above question is true, it would enable me to complete my proof.
My intuition for the special case in $\mathbb{R}$ is as follows: Let $x\in X\subset \mathbb{R}$, and $(x_n)$ a sequence converging to $x$. Choose $(y_n)$ so that $y_n = g(x_n)$ for each $n$, and let $y = \lim_{n\rightarrow\infty} y_n$. Then \begin{align} &\lim_{n\rightarrow\infty}\frac{(g\circ f\circ g^{-1})(x_n) - (g\circ f\circ g^{-1})(x)}{x_n - x}\\ = &\lim_{n\rightarrow\infty}\frac{g(f(g^{-1}(x_n))) - g(f(g^{-1}(x)))}{g(g^{-1}(x_n)) - g(g^{-1}(x))}\\ = &\lim_{n\rightarrow\infty}\frac{g(f(y_n)) - g(f(y))}{g(y_n) - g(y)} \end{align} But here I reach a problem because I don't see how I can unwrap this any further. Moreover, there is no guarantee that $y$ exists. ($y$ may not be in X.)
• Since $g^{-1}:X\to X$ and $y=g^{-1}(x)$ you can conclude that $y\in X$. Note that $g^{-1}$ is a well defined continuous function. – Nathanael Skrepek Nov 20 '17 at 8:35
No: Here is a counterexample in one dimension with $X={\mathbb R}$: $$g(x) = x^3\qquad f(x) = x+1\qquad g\circ f \circ g^{-1}(x) = (x^{1/3}+ 1)^3$$ | 2019-07-17 16:36:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9642200469970703, "perplexity": 99.60826494431448}, "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/1563195525355.54/warc/CC-MAIN-20190717161703-20190717183703-00247.warc.gz"} |
https://solvedlib.com/n/use-polar-coordinates-to-evaluate-the-double-integraljjr,14224518 | # Use polar coordinates to evaluate the double integralJJR 8y dA, where R is bounded by r = 12 - cos(0)
###### Question:
Use polar coordinates to evaluate the double integral JJR 8y dA, where R is bounded by r = 12 - cos(0)
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##### Pelephone4ua ^: { itc birzeit eduITCQuestion 1 Not yet answeredMarked out of 1.00Flag questionProve (by contradiction): For all sets A, B and C,if A o B = Cand x eB then x € (A-C)Maximum file size: 2OMB, maximum number of files: 2Files
Pelephone 4 ua ^: { itc birzeit edu ITC Question 1 Not yet answered Marked out of 1.00 Flag question Prove (by contradiction): For all sets A, B and C, if A o B = Cand x eB then x € (A-C) Maximum file size: 2OMB, maximum number of files: 2 Files... | 2023-03-25 06:58:02 | {"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5528638362884521, "perplexity": 2906.481850582031}, "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-2023-14/segments/1679296945317.85/warc/CC-MAIN-20230325064253-20230325094253-00027.warc.gz"} |
https://math.stackexchange.com/questions/1502797/cant-understand-step-in-proof-of-lucass-theorem | # Can't understand step in proof of Lucas's Theorem
I am having trouble understanding why the "Hence $p$ divides . . . " part follows.
This is from the Wiki article on Lucas's Theorem. Help appreciated!
If $p$ is a prime and $n$ is an integer with $1≤n≤p-1$, then the numerator of the binomial coefficient
$\binom p n = \frac{p \cdot (p-1) \cdots (p-n+1)}{n \cdot (n-1) \cdots 1}$ is divisible by $p$ but the denominator is not. Hence $p$ divides $\tbinom{p}{n}$.
Write your fraction as $a/b$. Since $p$ divides $a$, you can write $a=kp$ for some integer $k$, so your fraction is $kp/b$. It's an integer, so $b$ divides $kp$.
Now $p$ does not divide $b$, and it is prime, therefore $p$ and $b$ are coprime; Gauss's theorem allows you to conclude that $b$ divides $k$, and so your fraction is actually $p\times k/b$, where $k/b$ is an integer, and so it is divisible by $p$. | 2019-09-20 15:58:01 | {"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": 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": 0.9003412127494812, "perplexity": 77.62764846245265}, "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/1568514574050.69/warc/CC-MAIN-20190920155311-20190920181311-00349.warc.gz"} |
https://en.m.wikipedia.org/wiki/ElGamal_signature_scheme | # ElGamal signature scheme
The ElGamal signature scheme is a digital signature scheme which is based on the difficulty of computing discrete logarithms. It was described by Taher ElGamal in 1984.[1]
The ElGamal signature algorithm is rarely used in practice. A variant developed at NSA and known as the Digital Signature Algorithm is much more widely used. There are several other variants.[2] The ElGamal signature scheme must not be confused with ElGamal encryption which was also invented by Taher ElGamal.
The ElGamal signature scheme allows a third-party to confirm the authenticity of a message.
## System parametersEdit
These system parameters may be shared between users.
## Key generationEdit
• Randomly choose a secret key x with 1 < x < p − 2.
• Compute y = g x mod p.
• The public key is y.
• The secret key is x.
These steps are performed once by the signer.
## Signature generationEdit
To sign a message m the signer performs the following steps.
• Choose a random k such that 1 < k < p − 1 and gcd(kp − 1) = 1.
• Compute ${\displaystyle r\,\equiv \,g^{k}{\pmod {p}}}$ .
• Compute ${\displaystyle s\,\equiv \,(H(m)-xr)k^{-1}{\pmod {p-1}}}$ .
• If ${\displaystyle s=0}$ start over again.
Then the pair (r,s) is the digital signature of m. The signer repeats these steps for every signature.
## VerificationEdit
A signature (r,s) of a message m is verified as follows.
• ${\displaystyle 0 and ${\displaystyle 0 .
• ${\displaystyle g^{H(m)}\equiv y^{r}r^{s}{\pmod {p}}.}$
The verifier accepts a signature if all conditions are satisfied and rejects it otherwise.
## CorrectnessEdit
The algorithm is correct in the sense that a signature generated with the signing algorithm will always be accepted by the verifier.
The signature generation implies
${\displaystyle H(m)\,\equiv \,xr+sk{\pmod {p-1}}.}$
Hence Fermat's little theorem implies
{\displaystyle {\begin{aligned}g^{H(m)}&\equiv g^{xr}g^{ks}\\&\equiv (g^{x})^{r}(g^{k})^{s}\\&\equiv (y)^{r}(r)^{s}{\pmod {p}}.\\\end{aligned}}}
## SecurityEdit
A third party can forge signatures either by finding the signer's secret key x or by finding collisions in the hash function ${\displaystyle H(m)\equiv H(M){\pmod {p-1}}}$ . Both problems are believed to be difficult. However, as of 2011 no tight reduction to a computational hardness assumption is known.
The signer must be careful to choose a different k uniformly at random for each signature and to be certain that k, or even partial information about k, is not leaked. Otherwise, an attacker may be able to deduce the secret key x with reduced difficulty, perhaps enough to allow a practical attack. In particular, if two messages are sent using the same value of k and the same key, then an attacker can compute x directly.[1]
### Existential forgeryEdit
The original paper[1] did not include a hash function as a system parameter. The message m was used directly in the algorithm instead of H(m). This enables an attack called existential forgery, as described in section IV of the paper. Pointcheval and Stern generalized that case and described two levels of forgeries:[3]
1. The one-parameter forgery. Let ${\displaystyle 1 be a random element. If ${\displaystyle r=g^{e}y{\pmod {p}}}$ and ${\displaystyle s=-r{\pmod {p-1}}}$ , the tuple ${\displaystyle (r,s)}$ is a valid signature for the message ${\displaystyle m=es{\pmod {p-1}}}$ .
2. The two-parameters forgery. Let ${\displaystyle 1 and be random elements and ${\displaystyle \gcd(v,p-1)=1}$ . If ${\displaystyle r=g^{e}y^{v}{\pmod {p}}}$ and ${\displaystyle s=-rv^{-1}{\pmod {p-1}}}$ , the tuple ${\displaystyle (r,s)}$ is a valid signature for the message ${\displaystyle m=es{\pmod {p-1}}}$ .
Improved version (with a hash) is known as Pointcheval–Stern signature algorithm | 2018-01-19 19:45:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 21, "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": 0.7628584504127502, "perplexity": 1382.6132804783126}, "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/1516084888113.39/warc/CC-MAIN-20180119184632-20180119204632-00155.warc.gz"} |
https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/272/2/a/b/ | # Properties
Label 272.2.a.b Level $272$ Weight $2$ Character orbit 272.a Self dual yes Analytic conductor $2.172$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$
# Related objects
## Newspace parameters
Level: $$N$$ $$=$$ $$272 = 2^{4} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 272.a (trivial)
## Newform invariants
Self dual: yes Analytic conductor: $$2.17193093498$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 17) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$
## $q$-expansion
$$f(q)$$ $$=$$ $$q - 2 q^{5} - 4 q^{7} - 3 q^{9}+O(q^{10})$$ q - 2 * q^5 - 4 * q^7 - 3 * q^9 $$q - 2 q^{5} - 4 q^{7} - 3 q^{9} - 2 q^{13} + q^{17} + 4 q^{19} - 4 q^{23} - q^{25} + 6 q^{29} - 4 q^{31} + 8 q^{35} - 2 q^{37} - 6 q^{41} - 4 q^{43} + 6 q^{45} + 9 q^{49} + 6 q^{53} + 12 q^{59} - 10 q^{61} + 12 q^{63} + 4 q^{65} - 4 q^{67} + 4 q^{71} - 6 q^{73} - 12 q^{79} + 9 q^{81} + 4 q^{83} - 2 q^{85} + 10 q^{89} + 8 q^{91} - 8 q^{95} + 2 q^{97}+O(q^{100})$$ q - 2 * q^5 - 4 * q^7 - 3 * q^9 - 2 * q^13 + q^17 + 4 * q^19 - 4 * q^23 - q^25 + 6 * q^29 - 4 * q^31 + 8 * q^35 - 2 * q^37 - 6 * q^41 - 4 * q^43 + 6 * q^45 + 9 * q^49 + 6 * q^53 + 12 * q^59 - 10 * q^61 + 12 * q^63 + 4 * q^65 - 4 * q^67 + 4 * q^71 - 6 * q^73 - 12 * q^79 + 9 * q^81 + 4 * q^83 - 2 * q^85 + 10 * q^89 + 8 * q^91 - 8 * q^95 + 2 * q^97
## Embeddings
For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.
For more information on an embedded modular form you can click on its label.
Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
0
0 0 0 −2.00000 0 −4.00000 0 −3.00000 0
$$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles
## Atkin-Lehner signs
$$p$$ Sign
$$2$$ $$-1$$
$$17$$ $$-1$$
## Inner twists
This newform does not admit any (nontrivial) inner twists.
## Twists
By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 272.2.a.b 1
3.b odd 2 1 2448.2.a.o 1
4.b odd 2 1 17.2.a.a 1
5.b even 2 1 6800.2.a.n 1
8.b even 2 1 1088.2.a.h 1
8.d odd 2 1 1088.2.a.i 1
12.b even 2 1 153.2.a.c 1
17.b even 2 1 4624.2.a.d 1
20.d odd 2 1 425.2.a.d 1
20.e even 4 2 425.2.b.b 2
24.f even 2 1 9792.2.a.n 1
24.h odd 2 1 9792.2.a.i 1
28.d even 2 1 833.2.a.a 1
28.f even 6 2 833.2.e.a 2
28.g odd 6 2 833.2.e.b 2
44.c even 2 1 2057.2.a.e 1
52.b odd 2 1 2873.2.a.c 1
60.h even 2 1 3825.2.a.d 1
68.d odd 2 1 289.2.a.a 1
68.f odd 4 2 289.2.b.a 2
68.g odd 8 4 289.2.c.a 4
68.i even 16 8 289.2.d.d 8
76.d even 2 1 6137.2.a.b 1
84.h odd 2 1 7497.2.a.l 1
92.b even 2 1 8993.2.a.a 1
204.h even 2 1 2601.2.a.g 1
340.d odd 2 1 7225.2.a.g 1
By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.a.a 1 4.b odd 2 1
153.2.a.c 1 12.b even 2 1
272.2.a.b 1 1.a even 1 1 trivial
289.2.a.a 1 68.d odd 2 1
289.2.b.a 2 68.f odd 4 2
289.2.c.a 4 68.g odd 8 4
289.2.d.d 8 68.i even 16 8
425.2.a.d 1 20.d odd 2 1
425.2.b.b 2 20.e even 4 2
833.2.a.a 1 28.d even 2 1
833.2.e.a 2 28.f even 6 2
833.2.e.b 2 28.g odd 6 2
1088.2.a.h 1 8.b even 2 1
1088.2.a.i 1 8.d odd 2 1
2057.2.a.e 1 44.c even 2 1
2448.2.a.o 1 3.b odd 2 1
2601.2.a.g 1 204.h even 2 1
2873.2.a.c 1 52.b odd 2 1
3825.2.a.d 1 60.h even 2 1
4624.2.a.d 1 17.b even 2 1
6137.2.a.b 1 76.d even 2 1
6800.2.a.n 1 5.b even 2 1
7225.2.a.g 1 340.d odd 2 1
7497.2.a.l 1 84.h odd 2 1
8993.2.a.a 1 92.b even 2 1
9792.2.a.i 1 24.h odd 2 1
9792.2.a.n 1 24.f even 2 1
## Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(272))$$:
$$T_{3}$$ T3 $$T_{5} + 2$$ T5 + 2
## Hecke characteristic polynomials
$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 2$$
$7$ $$T + 4$$
$11$ $$T$$
$13$ $$T + 2$$
$17$ $$T - 1$$
$19$ $$T - 4$$
$23$ $$T + 4$$
$29$ $$T - 6$$
$31$ $$T + 4$$
$37$ $$T + 2$$
$41$ $$T + 6$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T - 12$$
$61$ $$T + 10$$
$67$ $$T + 4$$
$71$ $$T - 4$$
$73$ $$T + 6$$
$79$ $$T + 12$$
$83$ $$T - 4$$
$89$ $$T - 10$$
$97$ $$T - 2$$ | 2022-08-15 10:17:40 | {"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": 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": 0.9747170805931091, "perplexity": 9468.353885180504}, "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/1659882572163.61/warc/CC-MAIN-20220815085006-20220815115006-00431.warc.gz"} |
https://android.stackexchange.com/questions/210676/install-split-apk-manually | # Install Split APK Manually
Recently I got to know a thing called split apks, and found the slices as invalid apk files, only the base is. Unfortunately, this means the usual default package installer is unable to handle this, and as such I got to know another thing called SAI, a split apk installer, capable of working with or without root. However, being curious, I wondered how it all worked internally, and began to do a bit of research. I didn't get far, but I got to find out about pm install commands, and I happen to have root access which is apparently mandatory if I am to go this path, as the terminal won't let me run pm install-create without first running su.
The next step is pm install-write all the slices, and this is where I fall. The most obvious way for me was to cd to the directory of the slices and pm install-write them one at a time, which didn't allow me to do so, and it seems to be expecting an int rather than a directory string.
Questions (TL; DR):
1. Can I install split apks without any extra apps (in my example, SAI)?
2. Am I in the right direction with pm install commands?
3. If the answer to question 2 is yes, what am I doing wrong, and what should I be doing instead (it seems to want an int instead of a directory string, not sure how to convert)?
4. Are there any other methods I can explore (preferably rootless, but root is also fine)?
• Tested pm myself on Pie. Let's suppose that our package is called com.example.apk: you'd install it with pm install com.example.apk. Next, to install the split parts, you need to issue pm install -p com.example com.example.split.apk, where -p com.example refers to the package name of the base package, and com.example.split.apk is your split package. Let me know if it works for you. – Death Mask Salesman Apr 22 at 16:50
• @Salesman: I assume the question is about "App bundles" and if I remember correctly the different split-up APKs all have the same package name. Installing such a slit-aup app should be possible using bundletool. – Robert Apr 22 at 17:38
• @Robert Yes, seems like we're referring to the same thing. In that case, either bundletool from a computer or pm from a rooted device will do the job. – Death Mask Salesman Apr 22 at 19:46 | 2019-10-16 02:43:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5419694185256958, "perplexity": 1432.2512841382606}, "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/1570986661296.12/warc/CC-MAIN-20191016014439-20191016041939-00210.warc.gz"} |
https://www.nature.com/articles/s41467-021-22989-1 | Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.
# Integration of machine learning and genome-scale metabolic modeling identifies multi-omics biomarkers for radiation resistance
## Abstract
Resistance to ionizing radiation, a first-line therapy for many cancers, is a major clinical challenge. Personalized prediction of tumor radiosensitivity is not currently implemented clinically due to insufficient accuracy of existing machine learning classifiers. Despite the acknowledged role of tumor metabolism in radiation response, metabolomics data is rarely collected in large multi-omics initiatives such as The Cancer Genome Atlas (TCGA) and consequently omitted from algorithm development. In this study, we circumvent the paucity of personalized metabolomics information by characterizing 915 TCGA patient tumors with genome-scale metabolic Flux Balance Analysis models generated from transcriptomic and genomic datasets. Metabolic biomarkers differentiating radiation-sensitive and -resistant tumors are predicted and experimentally validated, enabling integration of metabolic features with other multi-omics datasets into ensemble-based machine learning classifiers for radiation response. These multi-omics classifiers show improved classification accuracy, identify clinical patient subgroups, and demonstrate the utility of personalized blood-based metabolic biomarkers for radiation sensitivity. The integration of machine learning with genome-scale metabolic modeling represents a significant methodological advancement for identifying prognostic metabolite biomarkers and predicting radiosensitivity for individual patients.
## Introduction
Despite being one of the oldest forms of cancer therapy and still a primary treatment modality, radiation therapy is not effective for over one-fifth of cancer patients distributed across almost all cancer types1,2. While biological understanding of radiation resistance has been advanced, use of a priori prediction of radiation response for individual cancer patients is not yet implemented clinically3. Early studies that identified biomarkers for radiation response focused on tumor histology, clinical factors including staging and Karnofsky performance score, and physiological parameters such as tumor oxygenation status4,5,6. As methods for transcriptomic analysis have improved, gene expression-based classifiers for radiation response have proliferated (recently curated in the RadiationGeneSigDB database)7. To date, however, these radiation response classifiers do not integrate multiple -omics modalities, owing in part to a lack of available -omics datasets for individual patient tumors. Specifically, while genomic and transcriptomic data are becoming more widely available for large numbers of patient tumors through initiatives such as The Cancer Genome Atlas (TCGA), metabolomic data associated with tumor biobanks are rarely captured, limiting inclusion of tumor metabolic features in predictive models for radiation therapy response2.
Given the lack of available tumor metabolomic data, genome-scale metabolic modeling approaches such as flux balance analysis (FBA) are becoming increasingly popular for predicting metabolic phenotypes8,9. By combining a curated reconstruction of the human metabolic network with constraints on metabolic reaction activities and an objective function to maximize a particular metabolic phenotype, predictions of steady-state reaction fluxes or metabolite production rates under physiological constraints can be obtained at a genome scale10. We previously developed a bioinformatics pipeline for integrating genomic, transcriptomic, kinetic, and thermodynamic parameters into personalized FBA models of 716 radiation-sensitive and 199 radiation-resistant patient tumors from TCGA across multiple cancer types11. Using these metabolic models, we identified differences in redox metabolism between radiation-sensitive and -resistant tumors, as well as personalized gene targets for inhibiting antioxidant production and clearance of reactive oxygen species. By validating model predictions using a panel of matched radiation-sensitive and -resistant cancer cell lines, we demonstrated that genome-scale metabolic models provide accurate predictions of tumor metabolism and can identify diagnostic and therapeutic biomarkers for radiation response.
While machine learning methods have been previously combined with genome-scale metabolic models to improve prediction of metabolic phenotypes, most studies combining these two methodologies have focused on microbiological applications rather than applications to cancer metabolism or predicting treatment outcomes12,13,14. We hypothesize that predictions from genome-scale metabolic models of patient tumors would provide additional information for distinguishing pathophysiological differences between radiation-sensitive and -resistant tumors, as well as for prediction of radiation response.
In this work, we utilize personalized FBA models of TCGA patient tumors to predict genome-scale metabolite production rates for incorporation into machine learning classifiers and identification of metabolite biomarkers associated with radiation resistance. In addition, through integration with clinical, genomic, and transcriptomic datasets, we develop gene expression, multi-omics, and non-invasive classifiers that outperform previous predictors of radiation response, as well as provide personalized diagnostic biomarker panels for individual patient tumors.
## Results
### Gene expression classifier implicates cellular metabolism
Gene set enrichment analysis (GSEA) of these 782 genes among the Hallmarks of Cancer showed significantly increased enrichment of the “Deregulating cellular energetics” hallmark, with very low enrichment of the “Genome instability & mutation” hallmark (Fig. 1c)29,30. Hierarchical clustering of the hallmark enrichment ranks for each gene set in RadiationGeneSigDB revealed two major clusters: a larger cluster with very high rank of “Genome instability & mutation,” and a smaller cluster with much higher ranks for other hallmarks involved in cellular metabolism, angiogenesis, and metastasis (Fig. 1d). This dichotomy suggests that although the biological response to radiation therapy certainly involves genomic instability and DNA damage repair, other biological processes such as cellular metabolism may play critical roles as well31,32. GSEA of cancer expression modules additionally showed increased enrichment of many modules involved in cellular metabolism, including amino acid and sulfur metabolism, redox metabolism, and lipid metabolism (Fig. 1e)33. Finally, GSEA of Recon3D metabolic subsystems demonstrated increased enrichment of pathways involved in central carbon metabolism and lipid metabolism, with the majority of genes being associated with increased probability of radiation resistance (Fig. 1f)10. Together, analysis of this gene expression classifier suggests that radiation-resistant tumors exemplify dysregulation in their cellular metabolic networks, and that additional features involving the metabolism of radiation-sensitive and -resistant tumors will provide significant benefit in developing machine learning classifiers for radiation response.
### FBA models accurately predict relative metabolite production
Personalized genome-scale FBA models of radiation-sensitive and -resistant TCGA tumors were generated to obtain metabolic features that could be used in machine learning classifiers for radiation response (see Methods section). These FBA models were developed through integration of gene expression and mutation information from individual patient tumors, as well as kinetic and thermodynamic parameters from publicly available repositories11. By systematically creating artificial metabolite sinks in the Recon3D metabolic network and evaluating fluxes to these sinks, the production rates of different metabolites were predicted and compared between radiation-sensitive and -resistant tumors (Fig. 2a and Supplementary Data 5). Figure 2b shows that many of the metabolite classes implicated from the gene expression classifier showed significantly increased production in radiation-resistant tumors. These included antioxidant and cysteine-containing metabolites (including precursors of glutathione, an antioxidant with previously implicated roles in radiation response), lipid and fatty acid metabolites (including those previously implicated in lipid peroxidation in response to ionizing radiation), and immune system mediators34,35,36. While fewer metabolites were predicted to be significantly downregulated in radiation-resistant tumors, many metabolites involved in nucleotide metabolism were among this group.
Regression of experimental metabolite concentrations among the NCI-60 cancer cell line panel with cell line surviving fraction at 2 Gy radiation (SF2) showed up- and downregulation of the same metabolite classes predicted from FBA models (Fig. 2c)37. Many lipid and fatty acid metabolites positively correlate with radiation resistance (including cholesterol, which had the most positive correlation among all metabolites tested); antioxidant metabolites including glutathione positively correlate as well. On the other hand, many nucleotide metabolites are anti-correlated with radiation resistance (including UDP-MurNAc, which had the most negative correlation among all metabolites tested). Regression of experimental metabolite concentrations with cell line radiation response among the larger CCLE cancer cell line panel yielded similar findings, with upregulation of lipid/antioxidant metabolites and downregulation of nucleotide metabolites in radiation-resistant cell lines (Supplementary Fig. 3).
### Machine learning architecture for radiation response
To integrate FBA model predictions of metabolite production rates with other TCGA datasets into multi-omics machine learning classifiers, a dataset-independent ensemble architecture was developed (Fig. 3a). Multiple independent “base learner” classifiers are trained on an individual -omics dataset (either clinical, genomics, transcriptomics, or metabolomics data), as described in Supplementary Fig. 1. Subsequently, by comparing predicted class probabilities from each individual base learner to known radiation responses, a “meta-learner” classifier is trained to determine which base learner provides the most accurate prediction of radiation response based on the multi-omics features of individual samples (Fig. 3b)40. For an individual testing sample, each base learner outputs the predicted probability of radiation resistance (pi), and the meta-learner outputs the predicted probability that each base learner will provide the most accurate prediction (wi); the final probability of radiation resistance is the weighted average of each pi, with weights being each wi (Fig. 3c). This dataset-independent ensemble architecture performs better across multiple performance metrics compared to the common practice of initially combining all -omics datasets and training on a single classifier (Fig. 3d and Supplementary Figs. 9 and 10). Overall, this machine learning architecture is a robust platform for integrating multi-omics data and providing accurate predictions of radiation response in individual patient tumors.
### Multi-omics classifier identifies clinical patient subgroups
Using the dataset-independent ensemble architecture described above, a multi-omics machine learning classifier integrating clinical, gene expression, mutation, and FBA-predicted metabolite production rates from TCGA tumors was developed (Supplementary Data 9). With an AUROC of 0.906 ± 0.004, this classifier has significantly greater performance compared to previously developed machine learning classifiers for radiation response (Figs. 4a and 1b)7,41. In addition, the threshold for separating radiation-sensitive and -resistant classes can be altered to optimize sensitivity, specificity, or a balance of both. In all, 725 of the 52,223 features from the four datasets (1.39%) were identified as significant in the classification of radiation response as determined by a 95% cumulative sum threshold on mean absolute SHAP values (mean |ΔP|) (Fig. 4b and Supplementary Data 10 and 11). While the majority of these 725 features were gene expression (48.3%) and metabolite (32.6%) features, clinical features including tumor histology, chemotherapeutic response, and cancer type contributed more than half of the total mean |ΔP| scores (60.1%; Fig. 4c and Supplementary Fig. 11). Mutations with significant mean |ΔP| scores included those directly involved in redox metabolism (IDH1 R132H) and lipid metabolism (BRAF V600E)42,43. Many of these significant mutation features were among those with the largest differences in mutation rates between breast cancer tumors exhibiting LRR versus CTL in an independent patient-derived dataset (Supplementary Fig. 12)28.
The contribution of a particular dataset toward radiation response classification for each individual patient was calculated by summing the patient-specific absolute SHAP values (|ΔP|) for all features within the particular dataset, normalized by the sum of patient-specific |ΔP| values across all datasets (Fig. 4d). Individual samples varied significantly in the contribution of different datasets toward radiation response classification. Using unsupervised clustering, three clusters of patients with varying contributions of clinical features were identified (Fig. 4e and Supplementary Fig. 13a). While “High Clinical” patients were categorized by large clinical feature contributions and small contributions from multi-omics datasets, multi-omics features provided the majority of cumulative SHAP values for “Low Clinical” patients, with metabolic features alone providing nearly as much utility as clinical features (Fig. 4f and Supplementary Figs. 1418). For this “Low Clinical” cluster, certain clinical features including chemotherapeutic response have diminutive utility, whereas multi-omics features including IDH1 SNP and lipid metabolite levels have much higher importance scores compared to the overall patient cohort. Significant heterogeneity in clinical clusters was observed based on patient clinical factors, especially cancer type and tumor histology (Fig. 4g–i). Output weights from the meta-learner provide an accurate prediction of clinical cluster, effectively differentiating between “Low Clinical” and “Medium/High Clinical” patients; this provides a valuable strategy for determining whether clinical information from electronic medical records is sufficient to accurately predict radiation response in an individual patient, or whether multi-omics features from tumor biopsy samples are needed (Fig. 4j).
### Metabolic biomarkers identified for radiation response
Metabolite set enrichment analysis of the 236 significant metabolite features from the multi-omics classifier indicated significant enrichment of several metabolic pathways involved in central carbon metabolism, lipid metabolism, and nucleotide metabolism (Fig. 5a). To identify individual metabolites with the largest impact on radiation response prediction, the Spearman correlation between SHAP value (ΔP) and predicted metabolite production rate across all patients was calculated for each metabolite (Supplementary Fig. 19). Figure 5b highlights many of the significant metabolic features, as well as metabolism-related gene expression and mutation features. Significant glycolytic and TCA cycle metabolites (fructose 1,6-bisphosphate, 3-phosphoglyceric acid, succinyl-CoA, and succinate) were all positively correlated with radiation resistance, while genes promoting gluconeogenesis (PCK2 and LDHC) were associated with radiation sensitivity. Fructose 2,6-bisphosphate, an allosteric regulator of PFK-1 that activates glucose breakdown, had one of the most positive correlation values. In addition, many metabolites in early mannose metabolism had positive correlation values, in accordance with previously observed radiation-induced upregulation of mannose-6-phosphate receptors and high-mannose type N-glycan production44,45.
Greater glycolytic fluxes in radiation-resistant tumor models resulted in increased production of the majority of significant lipid and fatty acid metabolites, including many with previously identified roles in antioxidation such as capric acid, butyric acid, eicosatrienoic acid, and γ-linolenic acid (Fig. 5c)46,47,48,49. On the other hand, significant nucleotide metabolites were highly correlated with radiation sensitivity, in agreement with the observed downregulation in radiation-resistant cancer cell lines (Fig. 5d). While production of energy metabolites including ATP was correlated with radiation sensitivity, FBA models predict significantly greater conversion of ADP to ATP in radiation-resistant tumors, in agreement with previous experimental findings (Fig. 5e and Supplementary Fig. 20)50,51. Finally, increased production of membrane phospholipids and arachidonic acid precursors resulted in significant correlations between inflammation-mediating eicosanoids and radiation resistance, corroborating previous evidence of radiation-sensitizing effects of cyclooxygenase inhibitors including aspirin (Fig. 5f)52. Together, these findings suggest that metabolic features from multiple interconnected pathways including central carbon, lipid, and nucleotide metabolism are viable diagnostic biomarkers for prediction of radiation sensitivity.
### Non-invasive classifier implicates blood metabolic features
Because non-invasive metabolic predictors of radiation response could be rapidly applied for informing patient-specific treatment, we refined machine learning classification to only integrate clinical data derived from non-invasive means (excluding any pathologic staging or histological information from tumor biopsies) with FBA-predicted production rates of metabolites known to be quantifiable in human blood samples (Fig. 6a)53. This non-invasive classifier performed similarly overall to the multi-omics classifier, with increased sensitivity and decreased specificity; this suggests that the non-invasive classifier may be optimal as a first-line screening test, followed by the multi-omics classifier as a second-line diagnostic test (Fig. 6b)54. A total of 97 of the 363 features from the two datasets (26.7%) were identified as significant in the classification of radiation response (Fig. 6c and Supplementary Data 12 and 13). Similar to the multi-omics classifier (Fig. 4e), individual patient contributions of clinical features formed a bimodal distribution of “Low Clinical” and “High Clinical” groups (Fig. 6d and Supplementary Fig. 13b). Blood metabolite features—including many lipid, nucleotide, and inflammation-mediating metabolites previously identified from the multi-omics classifier—provided almost one-half of the cumulative mean absolute SHAP values (mean |ΔP|) for “Low Clinical” patients (Fig. 6e). Dataset contributions and SHAP values for individual cancer patients can identify personalized biomarkers with maximal diagnostic utility (Fig. 6f–h). Overall, these findings demonstrate the value of blood-based biomarkers as a non-invasive approach toward personalized prediction of radiation response.
## Discussion
Despite significant interest in methodologies for the a priori prediction of radiation response in cancer patients, machine learning algorithms have yet to be used in the clinical setting for informing radiation treatment43. Recently developed classifiers for predicting tumor radiation response have focused mainly on gene expression data, rather than the integration of multiple -omics datasets7,55. This may be in part due to a lack of metabolomics datasets from tumor biobanks including TCGA, limiting inclusion of metabolic features in machine learning classifiers for radiation response. Here, we propose the strategy of utilizing personalized genome-scale FBA models of radiation-sensitive and -resistant patient tumors to predict the production rates of metabolites across the Recon3D metabolic network, leveraging the accessibility of genomic and transcriptomic tumor datasets to generate metabolic insight. These metabolic features are subsequently integrated with clinical, genomic, and gene expression data from TCGA tumors to generate gene expression, multi-omics, and non-invasive classifiers for radiation response. These classifiers provide more accurate predictions of tumor radiation response compared to previously developed classifiers, as well as multi-omics biomarkers associated with radiation sensitivity.
Integration of FBA model predictions into multi-omics machine learning classifiers for radiation response was performed by employing a dataset-independent ensemble architecture (Fig. 3). This approach was based on the concept of stacked generalization (having multiple “base learners” make predictions that are used as inputs for a separate “meta-learner”), which was shown to improve predictive accuracy in this study as well as multiple previous medical applications67,68,69. However, while in previous studies there is only one input dataset being supplied to the multiple base learners, we instead input different -omics datasets to separate base learners. The benefit of this dataset-independent approach is that the meta-learner can subsequently be used to predict which individual datasets will provide the most utility for determining radiation response in individual patients. For example, the meta-learner can accurately differentiate between “Low Clinical” patients (with large contributions of gene expression, mutation, and metabolic datasets from patient biopsy samples and genome-scale metabolic modeling) and “High Clinical” patients (with greater contribution of clinical data from electronic medical records) (Fig. 4). This stratification of patient populations allows for optimal resource allocation for collecting biological measurements with maximal diagnostic utility for individual cancer patients. Moreover, the use of GBM models as the base and meta-learners provides a significant amount of embedded feature selection; this decrease in model complexity not only lowers the cost of measuring biological features needed for prediction, but also improves the interpretability of models, increasing the likelihood of adoption by clinicians70.
In addition to demonstrating the utility of multi-omics data for the classification of radiation response, we found that a classifier utilizing non-invasive clinical information and blood-based metabolic biomarkers can predict radiation sensitivity with comparable accuracy (Fig. 6). Blood-based diagnostic tools are garnering attention for their use in early detection, monitoring, and optimal treatment identification for cancer patients71. Identification of circulating biomarkers through the integration of machine learning and genome-scale metabolic modeling could provide significant utility in adaptive radiotherapy to modify patient treatment with radiation or radiation-sensitizing chemotherapies in response to the observed efficacy of previous treatments72.
Metabolomic profiling is powerful for understanding cancer pathophysiology, identifying and monitoring clinical biomarkers, and predicting patient outcomes, but challenging to retrospectively analyze in specimen biobanks for inclusion in multi-omics data mining77. In this study, we demonstrate that integration of machine learning and genome-scale metabolic modeling methodologies allows for improved biomarker identification and prediction of radiation response in individual patient tumors without direct metabolomics measurements. This approach is generalizable toward other applications in guiding patient treatment, such as the prediction of chemotherapeutic response as well as identification of metabolic targets for pharmacological inhibition and treatment sensitization. The synergistic integration of machine learning and genome-scale metabolic modeling will inevitably yield additional insights for improving precision medicine and long-term care of cancer patients.
## Methods
### TCGA data retrieval and processing
Clinical data from TCGA patients were obtained from the GDC data portal (https://portal.gdc.cancer.gov; clinical drug, clinical patient, and clinical radiation files) and the Synapse TCGA_Pancancer project (https://www.synapse.org/#!Synapse:syn300013/wiki/70804; biological sample files)2. Drug names were standardized according to the standard available from the Gene-Drug Interactions for Survival in Cancer database78. Categorical clinical features were one-hot encoded before inputting into machine learning classifiers. RNA-Seq gene expression data were obtained from Rahman et al.’s alternative preprocessing method (GEO: GSE62944)79. Data from this preprocessing method showed fewer missing values, more consistent expression between replicates, and improved prediction of biological pathway activity compared to the original TCGA pipeline. Mutation data using the MuTect variant caller were obtained from the GDC data portal2,80. For all data types, only features with at least two unique non-missing values were included.
TCGA samples were classified into radiation-sensitive and radiation-resistant classes according to their reported sensitivity to radiation therapy based upon the RECIST classification method. Patients with a complete or partial response to radiation (greater than 30% decrease in tumor size) were classified as radiation-sensitive, and patients with stable or progressive disease (either less than 30% decrease in tumor size, or increase in tumor size) were classified as radiation-resistant. If a patient received multiple courses of radiation therapy, they were classified based on the response to their first course.
### Data splitting
Supplementary Fig. 21 provides an overview of data splitting for machine learning classifier training and testing. The collection of 716 radiation-sensitive and 199 radiation-resistant samples was randomly split into training+validation (80% of all samples) and testing (20% of all samples) groups. Within the training+validation group, five-fold cross-validation was performed to optimize hyperparameter values. The training (80% of training+validation samples) group was used for training the model with a given set of hyperparameters; within this training group, 87.5% was directly used for training, and 12.5% was used to identify the iteration at which to perform early stopping during training. The validation (20% of training+validation samples) group was used to assess model performance with the given set of hyperparameters. The average validation performance across all five folds was used to determine the optimized set of hyperparameters; once this set was determined, the model was retrained on the entire training+validation group, and the testing group (20% of all samples) was used to assess overall model performance. Twenty iterations of randomized training+validation/testing splitting were performed to analyze model predictions and performance metrics over multiple instances. All data splits were performed using stratified shuffle splitting, where the proportion of radiation-sensitive and -resistant samples was kept the same (refer to Supplementary Fig. 21).
### Base learners
Nd base learners were trained using an individual -omics dataset (either clinical, gene expression, mutation, or metabolic datasets), where Nd is the number of individual datasets being used for the classifier. Each base learner is composed of a GBM model that performs two-class classification (predicting either radiation sensitivity or resistance for each patient) using features from an individual dataset, such as clinical, genomics, or transcriptomics. GBM models using decision tree ensembles have many useful characteristics compared to other machine learning algorithms, including embedded feature selection, capability of handling missing values (which is common in clinical datasets), and efficient management of high-dimensional datasets (where the number of features greatly exceeds the number of samples)81,82. XGBoost (v0.90) was used to develop GBM base learners and meta-learners83.
Bayesian optimization was performed to optimize hyperparameter values for each GBM model. At each iteration of Bayesian optimization, five-fold cross-validation was used to calculate the performance of a particular set of hyperparameters. Weighted log loss was used as the performance metric for both GBM model training and evaluating model performance on validation sets:
$${\rm{Weighted}}\,{\mathrm{Log}}\,{\rm{Loss}}=\frac{1}{{N}_{s}}\mathop{\sum }_{i=1}^{{N}_{s}}[-({w}_{R}{y}_{i}\,\log ({p}_{i})+(1-{y}_{i})\log (1-{p}_{i}))]$$
(1)
where yi is the true class label of sample i (yi = 0 if sensitive, yi = 1 if resistant), pi is the predicted probability of sample i being radiation-resistant (belonging to class 1), wR is the weight given to radiation-resistant samples (wR = no. of sensitive samples/no. of resistant samples), and NS is the total number of samples. The weight given to radiation-resistant samples accommodates for the fact that there are more radiation-sensitive samples than radiation-resistant samples, and prevents classifiers from focusing on optimizing performance exclusively on radiation-sensitive samples. The mean weighted log loss plus one standard error over all five folds of cross-validation is used to choose the hyperparameter set with best performance. During model training, early stopping is employed to prevent overfitting.
For individual samples, each of the Nd base learners outputs the predicted probability of radiation resistance (p1, p2, …, pNd) using features from the individual data type. Each base classifier receives the same training/validation/testing split of samples.
### Meta-learner
For every sample within the five validation sets used for the base learners, each base learner’s output prediction of radiation resistance (pi) is compared to the sample’s true radiation response class (yi). The meta-learner is trained to predict the optimal base learner that provides the most accurate prediction of radiation response for each sample, based on the sample’s multi-omics features. This meta-learner performs an Nd-class classification, where Nd is the number of independent base learners. The features this meta-learner is trained on include all features from the Nd datasets that have non-zero SHAP values from their respective base learners; features that do not impact base learner predictions are not included, which increases the training speed while maintaining meta-learner accuracy. Because validation samples from the five-fold cross-validation were not directly used in base learner training, they can be used to train this meta-learner without overfitting or inflation of model performance metrics.
Implementation of the meta-learner is analogous to that of each base learner, using a GBM model, Bayesian optimization, early stopping, and five-fold cross-validation. Multiclass log loss was used as the performance metric for both GBM model training and evaluating model performance84:
$${\rm{Multiclass}}\,{\mathrm{Log}}\,{\rm{Loss}}=-\frac{1}{{N}_{s}}\mathop{\sum }_{i=1}^{{N}_{s}}\mathop{\sum }_{k=0}^{{N}_{k}}{y}_{i,k}\,\log ({p}_{i,k})$$
(2)
where yi,k is 1 if dataset k is the true optimal dataset of sample i and 0 otherwise, pi,k is the predicted probability of dataset k being the optimal dataset of sample i, NS is the total number of samples, and Nk is the total number of datasets. The mean multiclass log loss plus one standard error over all five folds of cross-validation is used to choose the optimal hyperparameter set with best performance.
For individual samples, the meta-learner outputs Nd probabilities (w1, w2, …, wNd) that each base learner is optimal for that sample (all Nd probabilities sum to 1). Note that, once the meta-learner is trained using the predicted probabilities from the base learners, the base learners and meta-learner act independently of each other when used on new testing samples.
Each testing sample is run through (1) all Nd base learners to obtain the predicted probabilities of radiation resistance using each of the Nd individual datasets (p1, p2, …, pNd), and (2) the meta-learner to obtain the predicted probabilities that each of the Nd base learners/datasets is optimal for that sample (w1, w2, …, wNd). To obtain the final predicted probability of radiation resistance for the testing sample, the weighted average of the base learner probabilities is taken, with the meta-learner probabilities as weights:
$$p={w}_{1}{p}_{1}+{w}_{2}{p}_{2}+\cdots +{w}_{{N}_{d}}{p}_{{N}_{d}}$$
(3)
Samples with p < 0.5 are classified as radiation-sensitive, while samples with p > 0.5 are classified as radiation-resistant.
### Bayesian optimization
Bayesian optimization was used to optimize GBM hyperparameters for both the base learner and meta-learner classifiers. This iterative approach automates the search for hyperparameter values by calculating an acquisition function that provides the expected benefit of sampling a particular point in hyperparameter space on the overall search for hyperparameters with minimal cross-validation error. At each iteration, the point in hyperparameter space with the largest acquisition function value is chosen, five-fold cross-validation is used to determine the performance of those particular hyperparameters, and the acquisition function is updated to then determine which next point in hyperparameter space will be sampled. Hyperopt (v0.1.2) was used to perform Bayesian optimization85. Supplementary Table 3 provides the eight GBM hyperparameters chosen for optimization of both base learner and meta-learner classifiers, with the ranges of values in the hyperparameter search space. A total of 28 = 256 iterations of Bayesian optimization were performed for each classifier.
### Classifier performance metrics
Final classifier performance was assessed on testing samples across the 20 iterations of randomized training+validation/testing splitting. The following performance metrics were used:
1. 1.
Weighted log loss: Eq. (1)
2. 2.
Area under the receiver operating characteristic curve (AUROC)
3. 3.
Balanced accuracy, an accuracy metric that corrects for unequal numbers of radiation-sensitive and -resistant patients:
$${\rm{Balanced}}\,{\rm{Accuracy}}=\frac{1}{2}\left(\frac{TP}{TP+FN}+\frac{TN}{TN+FP}\right)$$
(4)
4. 4.
Sensitivity:
$${\rm{Sensitivity}}=\frac{{TP}}{{TP}+{FN}}$$
(5)
5. 5.
Specificity:
$${\rm{Specificity}}=\frac{{TN}}{{TN}+{FP}}$$
(6)
6. 6.
Positive predictive value:
$${\rm{Positive}}\,{\rm{Predictive}}\,{\rm{Value}}=\frac{TP}{TP+FP}$$
(7)
7. 7.
Negative predictive value:
$${\rm{Negative}}\,{\rm{Predictive}}\,{\rm{Value}}=\frac{TN}{TN+FN}$$
(8)
### SHAP values
The importance of individual features toward the prediction of radiation response, both averaged across all samples and for individual samples, was determined by calculating SHAP values for each classifier15,16. Each SHAP value (ΔP) represents the change in predicted probability of radiation resistance for patient i attributed to feature j. Features with positive SHAP values for patient i signify those where the particular value of feature j attributed to patient i is such that it increases patient i’s predicted probability of radiation resistance. Larger absolute SHAP values (|ΔP|) indicate features with larger overall contributions (either negatively or positively). Mean absolute SHAP values across all samples (Mean |ΔP|) provide an indication of the overall importance of a particular feature in the classifier’s prediction of radiation response. SHAP values were averaged across 20 training+validation/testing splits by a weighted average, with weights proportional to the inverse of the weighted log loss performance metric on the testing set for that split86. This weighted average allows model analysis to be more reflective of the more accurate predictions, so that identified biomarkers are more likely to be true biomarkers rather than artifacts of poorly performing predictions. Values were normalized by the difference between prior and posterior probabilities of radiation resistance for each sample. SHAP v0.29.1 was used to calculate SHAP values16.
### Comparison of machine learning algorithms
scikit-learn v0.21.2 functions sklearn.ensemble.RandomForestClassifier() and sklearn.linear_model.LogisticRegression() were used to implement random forest and logistic regression with L1 regularization classifiers, respectively87. Keras v2.3.1 was used to implement the neural network with L1 regularization classifier (https://github.com/keras-team/keras). Weighted log loss (Eq. (1)) was used as the loss function for the neural network classifier, and early stopping was performed. Missing values were imputed and scaled using sklearn.impute.SimpleImputer() and sklearn.preprocessing.StandardScaler() functions, respectively, before training with the random forest, logistic regression, and neural network classifiers. Supplementary Tables 46 provide the hyperparameters and value ranges used for Bayesian optimization with each algorithm; 256 iterations of Bayesian optimization were performed for each classifier. Each classifier, including the GBM classifier, was run using the same training, validation, and testing samples at each of 20 training+validation/testing splits so that performance can be accurately compared.
### Comparison of gene expression datasets
Eleven gene expression sets for oxic radiation response in the RadiationGeneSigDB database were compared to the set of 782 significant genes from the gene expression classifier7. Gene names from RadiationGeneSigDB gene sets were converted to Entrez gene ID’s and gene symbols. Those genes where a matching Entrez gene ID or gene symbol could not be found were removed. In addition, those genes that were not in both TCGA and CCLE gene expression datasets were removed.
To compare performance of gene expression sets on TCGA data, classification models were trained to predict radiation-sensitive or -resistant classes of TCGA tumor samples using gene expression data from only the subset of genes for an individual set. Model performance was assessed using weighted log loss (Eq. (1)) and AUROC metrics. To compare performance of gene expression sets on CCLE data, regression models were trained to predicted radiation response (reported as area under the curve of survival vs. radiation dose) of CCLE cell lines using gene expression data from only the subset of genes for an individual set39. Model performance was assessed using mean absolute error and mean squared error metrics.
### Flux balance analysis (FBA)
Generation of personalized FBA models of individual TCGA tumor samples was performed as described in Lewis et al.11. To predict the maximum production of a particular metabolite in FBA models, the following objective function was used:
$$1{\rm{met}}[{\rm{all}}]\to \varnothing$$
(9)
where “met” is the metabolite to be maximized, and “[all]” represents the maximization of this objective function across all cellular compartments where the metabolite is located. This creates an artificial sink for a particular metabolite in the Recon3D metabolic network, resulting in the maximization of reaction fluxes generating this metabolite. We hypothesized that this objective function would be valid and thus yield accurate predictions for metabolites with large differences in production between radiation-sensitive and -resistant tumors, as these would be particularly beneficial to either tumor class and thus the metabolic network of these tumors would be optimized to maximize levels of the metabolite.
The modeled external compartment contained all metabolites found in DMEM/F-12 cell culture media (Thermo Fisher Scientific, Cat#11320) as well as fetal bovine serum (FBS) to match the cell culture media used for experimental validation88. All 871 metabolites in the Recon3D human metabolic reconstruction that (1) had KEGG database IDs, (2) were not present in the extracellular media, and (3) were capable of being produced by all FBA tumor models, were included in the FBA metabolite production screen.
### NCI-60 data retrieval and processing
Experimental metabolomics data from NCI-60 cancer cell lines were obtained from the Developmental Therapeutics Program of the National Cancer Institute (https://wiki.nci.nih.gov/display/NCIDTPdata/Molecular+Target+Data). Normalized concentration entries without metabolite names or for isobars were excluded. Cell line surviving fraction at 2 Gy radiation (SF2) values was obtained from Amundson et al.37.
### CCLE data retrieval and processing
TPM-normalized RNA-Seq transcriptomic data from the CCLE cancer cell lines were obtained from the Broad Institute CCLE database (https://data.broadinstitute.org/ccle/CCLE_RNAseq_rsem_genes_tpm_20180929.txt.gz). Experimental metabolomics data were obtained from Li et al.38. Normalized log10-transformed concentration entries were utilized. Cell line radiation responses measured using the area under the radiation response curve were obtained from Yard et al.39.
### Keene et al. data retrieval and processing
RNA-Seq transcriptomic data from breast cancer tumors exhibiting LRR or CTL from Keene et al. were obtained from GEO GSE11993728. RPKM-normalized values were normalized to TPM, and only genes with no missing values were kept in the analysis. For duplicate gene entries, the entry with the largest mean value across all samples was kept. Mutation data were obtained from Keene et al.’s Table S1.
### Cell culture
Supplementary Table 2 provides the matched radiation-sensitive and radiation-resistant cell lines used for experimental validation of metabolite levels predicted from FBA models. All cell lines were maintained in DMEM/F-12 cell culture media (Thermo Fisher Scientific, Cat #11320) with 10% FBS (Sigma-Aldrich, Cat #F4135) at 37 °C and 5% CO2, and were free of Mycoplasma.
### Metabolomics
Three biological replicates of each cell line were grown in separate T-25 flasks with the cell culture conditions described above. Cell pellets with approximately 1 million cells were obtained from trypsinization, centrifugation, and removal of supernatant. Samples were reconstituted in 90% MeOH, 10% H2O at a ratio of 200 μL/1 million cells. Aliquots of the supernatant were combined to create a pooled sample used for quality control. Aliquots of the samples were transferred to LC vials and stored at 4 °C.
Hydrophilic interaction chromatography-tandem mass spectrometry (HILIC-MS/MS) untargeted metabolomics were performed. Chromatography parameters were as follows: BEH HILIC Column, 150 mm × 2.1 mm, 1.7 μm; mobile phase A: 80% H2O/20% ACN, 10 mM ammonium formate, 0.1% FA; mobile phase B: 100% ACN, 0.1% FA; column temperature: 40 °C; 2 μL sample injection. MS parameters were as follows: resolution: 240,000; scan range: 70–1050 m/z; polarity: positive/negative; AGC target: 1e5. MS2 parameters were as follows: isolation window: 0.8 m/z; detector: Orbitrap; polarity: positive/negative; fragmentation method: HCD; collision energy: 15, 30, 45; resolution: 30,000.
Compound Discoverer 3.1 was used to perform quality control, putative metabolite identification, and quantification of metabolite levels. Results for positive and negative ion modes were combined. Metabolites with no identified name were removed from the analysis. If duplicate metabolites with the same identification were obtained, then the entry with the largest maximum area was used. KEGG IDs for each metabolite were manually identified based on metabolite name, molar mass, and chemical formula. Metabolites from experimental metabolomics were matched to those from FBA analysis by matching KEGG IDs.
For the comparison of model-predicted and experimentally measured metabolite levels, all metabolites within the following Recon3D subsystems that were matched with experimental metabolites were included in the analysis:
• Nucleotide metabolism: “Nucleotide interconversion,” “Nucleotide salvage pathway,” “Pentose phosphate pathway,” “Purine catabolism,” “Purine synthesis,” “Pyrimidine catabolism,” “Pyrimidine synthesis”.
• Lipid metabolism: “Cholesterol metabolism,” “Fatty acid oxidation,” “Fatty acid synthesis,” “Glycosphingolipid metabolism,” “Phosphatidylinositol phosphate metabolism,” “Sphingolipid metabolism,” “Steroid metabolism”.
• Cysteine/antioxidant metabolism: “Glutathione metabolism,” “ROS detoxification,” plus metabolite “Lipoamide”.
• Immune system mediators: “Arachidonic acid metabolism,” “Eicosanoid metabolism”.
### Statistics
A description of the statistical values used in creating boxplots and error bars within bar charts, as well as the sample size for these plots, is provided in the representative figure legends. In all cases except Supplementary Figs. 47, distinct samples were used; in Supplementary Figs. 47, n = 3 biological replicates for each cell line were used. A description of the symbols used to signify statistical significance, as well as the statistical test used, is provided at the end of representative figure legends.
### Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
## Data availability
TCGA, NCI-60, CCLE, and Keene et al. datasets used in this study are available as cited above in their respective Methods subsections. Information about KEGG database IDs is available at https://genome.jp. The following datasets generated from this study are available at https://github.com/kemplab/ML-radiation (https://doi.org/10.5281/zenodo.4540314)89:
Supplementary Data 1. TCGA samples included in the analysis, with corresponding radiation response and patient/tumor factors.
Supplementary Data 2. SHAP values (ΔP) from the gene expression classifier, for individual TCGA patients.
Supplementary Data 3. Mean absolute SHAP values (mean |ΔP|) for individual features from the gene expression classifier.
Supplementary Data 4. 782 significant genes from the gene expression classifier.
Supplementary Data 5. FBA model-predicted metabolite production rates in TCGA tumors.
Supplementary Data 6. Experimental metabolomics data from radiation-sensitive and -resistant cancer cell lines.
Supplementary Data 7. Comparison of model-predicted and experimentally validated metabolite levels in radiation-sensitive and -resistant cancers.
Supplementary Data 8. Breast, colorectal, glioma, and upper aerodigestive cancer cell lines within the CCLE panel analyzed for associations between experimental metabolomics and radiation response.
Supplementary Data 9. Frequency of SNPs within each gene among all 915 TCGA samples from this study.
Supplementary Data 10. SHAP values (ΔP) from the multi-omics classifier, for individual TCGA patients.
Supplementary Data 11. Mean absolute SHAP values (mean |ΔP|) for individual features from the multi-omics classifier.
Supplementary Data 12. SHAP values (ΔP) from the non-invasive classifier, for individual TCGA patients.
Supplementary Data 13. Mean absolute SHAP values (mean |ΔP|) for individual features from the non-invasive classifier. Source data are provided with this paper.
## Code availability
Jupyter notebooks containing Python code for running and analyzing the gene expression, multi-omics, and non-invasive classifiers for radiation response are available at https://github.com/kemplab/ML-radiation (https://doi.org/10.5281/zenodo.4540314)89. In addition, the gene sets and code used to compare the significant gene list from our gene expression classifier to those from the RadiationGeneSigDB database are available. Jupyter notebooks containing Python code related to the generation of personalized genome-scale FBA models of TCGA tumors are available at https://github.com/kemplab/FBA-pipeline (https://doi.org/10.5281/zenodo.4540330)11,90.
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## Acknowledgements
The authors gratefully acknowledge support for this work from an NIH/NCI F30 CA224968 fellowship (PI: J.E.L.; Sponsor: M.L.K.) and an NIH/NCI U01 CA215848 grant (PI: M.L.K.). We wish to acknowledge David Gaul and the core facilities at the Parker H. Petit Institute for Bioengineering and Bioscience at the Georgia Institute of Technology for the use of their shared equipment, services, and expertise.
## Author information
Authors
### Contributions
Conceptualization: J.E.L. and M.L.K.; methodology, software, validation, formal analysis, and investigation: J.E.L.; resources: M.L.K.; data curation and writing (original draft): J.E.L.; writing (review and editing): J.E.L. and M.L.K.; visualization: J.E.L.; supervision: M.L.K.; and project administration and funding acquisition: J.E.L. and M.L.K.
### Corresponding author
Correspondence to Melissa L. Kemp.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Peer review information Nature Communications thanks Pedro Carmona-Sáez and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Lewis, J.E., Kemp, M.L. Integration of machine learning and genome-scale metabolic modeling identifies multi-omics biomarkers for radiation resistance. Nat Commun 12, 2700 (2021). https://doi.org/10.1038/s41467-021-22989-1
• Accepted:
• Published: | 2021-07-24 22:01:15 | {"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": 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": 0.7009288668632507, "perplexity": 11053.090507050627}, "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/1627046150308.48/warc/CC-MAIN-20210724191957-20210724221957-00620.warc.gz"} |
https://en.wikipedia.org/wiki/Lucas-Carmichael_number | # Lucas–Carmichael number
(Redirected from Lucas-Carmichael number)
In mathematics, a Lucas–Carmichael number is a positive composite integer n such that
1. if p is a prime factor of n, then p + 1 is a factor of n + 1;
2. n is odd and square-free.
The first condition resembles the Korselt's criterion for Carmichael numbers, where -1 is replaced with +1. The second condition eliminates from consideration some trivial cases like cubes of prime numbers, such as 8 or 27, which otherwise would be Lucas–Carmichael numbers (since n3 + 1 = (n + 1)(n2 − n + 1) is always divisible by n + 1).
They are named after Édouard Lucas and Robert Carmichael.
## Properties
The smallest Lucas–Carmichael number is 399 = 3 × 7 × 19. It is easy to verify that 3+1, 7+1, and 19+1 are all factors of 399+1 = 400.
The smallest Lucas–Carmichael number with 4 factors is 8855 = 5 × 7 × 11 × 23.
The smallest Lucas–Carmichael number with 5 factors is 588455 = 5 × 7 × 17 × 23 × 43.
It is not known whether any Lucas–Carmichael number is also a Carmichael number.
Thomas Wright proved in 2016 that there are infinitely many Lucas–Carmichael numbers.[1] If we let ${\displaystyle N(X)}$ denote the number of Lucas–Carmichael numbers up to ${\displaystyle X}$, Wright showed that there exists a positive constant ${\displaystyle K}$ such that
${\displaystyle N(X)\gg X^{K/\left(\log \log \log X\right)^{2}}}$.
## List of Lucas–Carmichael numbers
The first few Lucas–Carmichael numbers (sequence A006972 in the OEIS) and their prime factors are listed below.
399 = 3 × 7 × 19 935 = 5 × 11 × 17 2015 = 5 × 13 × 31 2915 = 5 × 11 × 53 4991 = 7 × 23 × 31 5719 = 7 × 19 × 43 7055 = 5 × 17 × 83 8855 = 5 × 7 × 11 × 23 12719 = 7 × 23 × 79 18095 = 5 × 7 × 11 × 47 20705 = 5 × 41 × 101 20999 = 11 × 23 × 83 22847 = 11 × 31 × 67 29315 = 5 × 11 × 13 × 41 31535 = 5 × 7 × 17 × 53 46079 = 11 × 59 × 71 51359 = 7 × 11 × 23 × 29 60059 = 19 × 29 × 109 63503 = 11 × 23 × 251 67199 = 11 × 41 × 149 73535 = 5 × 7 × 11 × 191 76751 = 23 × 47 × 71 80189 = 17 × 53 × 89 81719 = 11 × 17 × 19 × 23 88559 = 19 × 59 × 79 90287 = 17 × 47 × 113 104663 = 13 × 83 × 97 117215 = 5 × 7 × 17 × 197 120581 = 17 × 41 × 173 147455 = 5 × 7 × 11 × 383 152279 = 29 × 59 × 89 155819 = 19 × 59 × 139 162687 = 3 × 7 × 61 × 127 191807 = 7 × 11 × 47 × 53 194327 = 7 × 17 × 23 × 71 196559 = 11 × 107 × 167 214199 = 23 × 67 × 139 218735 = 5 × 11 × 41 × 97 230159 = 47 × 59 × 83 265895 = 5 × 7 × 71 × 107 357599 = 11 × 19 × 29 × 59 388079 = 23 × 47 × 359 390335 = 5 × 11 × 47 × 151 482143 = 31 × 103 × 151 588455 = 5 × 7 × 17 × 23 × 43 653939 = 11 × 13 × 17 × 269 663679 = 31 × 79 × 271 676799 = 19 × 179 × 199 709019 = 17 × 179 × 233 741311 = 53 × 71 × 197 760655 = 5 × 7 × 103 × 211 761039 = 17 × 89 × 503 776567 = 11 × 227 × 311 798215 = 5 × 11 × 23 × 631 880319 = 11 × 191 × 419 895679 = 17 × 19 × 47 × 59 913031 = 7 × 23 × 53 × 107 966239 = 31 × 71 × 439 966779 = 11 × 179 × 491 973559 = 29 × 59 × 569 1010735 = 5 × 11 × 17 × 23 × 47 1017359 = 7 × 23 × 71 × 89 1097459 = 11 × 19 × 59 × 89 1162349 = 29 × 149 × 269 1241099 = 19 × 83 × 787 1256759 = 7 × 17 × 59 × 179 1525499 = 53 × 107 × 269 1554119 = 7 × 53 × 59 × 71 1584599 = 37 × 113 × 379 1587599 = 13 × 97 × 1259 1659119 = 7 × 11 × 29 × 743 1707839 = 7 × 29 × 47 × 179 1710863 = 7 × 11 × 17 × 1307 1719119 = 47 × 79 × 463 1811687 = 23 × 227 × 347 1901735 = 5 × 11 × 71 × 487 1915199 = 11 × 13 × 59 × 227 1965599 = 79 × 139 × 179 2048255 = 5 × 11 × 167 × 223 2055095 = 5 × 7 × 71 × 827 2150819 = 11 × 19 × 41 × 251 2193119 = 17 × 23 × 71 × 79 2249999 = 19 × 79 × 1499 2276351 = 7 × 11 × 17 × 37 × 47 2416679 = 23 × 179 × 587 2581319 = 13 × 29 × 41 × 167 2647679 = 31 × 223 × 383 2756159 = 7 × 17 × 19 × 23 × 53 2924099 = 29 × 59 × 1709 3106799 = 29 × 149 × 719 3228119 = 19 × 23 × 83 × 89 3235967 = 7 × 17 × 71 × 383 3332999 = 19 × 23 × 29 × 263 3354695 = 5 × 17 × 61 × 647 3419999 = 11 × 29 × 71 × 151 3441239 = 109 × 131 × 241 3479111 = 83 × 167 × 251 3483479 = 19 × 139 × 1319 3700619 = 13 × 41 × 53 × 131 3704399 = 47 × 269 × 293 3741479 = 7 × 17 × 23 × 1367 4107455 = 5 × 11 × 17 × 23 × 191 4285439 = 89 × 179 × 269 4452839 = 37 × 151 × 797 4587839 = 53 × 107 × 809 4681247 = 47 × 103 × 967 4853759 = 19 × 23 × 29 × 383 4874639 = 7 × 11 × 29 × 37 × 59 5058719 = 59 × 179 × 479 5455799 = 29 × 419 × 449 5669279 = 7 × 11 × 17 × 61 × 71 5807759 = 83 × 167 × 419 6023039 = 11 × 29 × 79 × 239 6514199 = 43 × 197 × 769 6539819 = 11 × 13 × 19 × 29 × 83 6656399 = 29 × 89 × 2579 6730559 = 11 × 23 × 37 × 719 6959699 = 59 × 179 × 659 6994259 = 17 × 467 × 881 7110179 = 37 × 41 × 43 × 109 7127999 = 23 × 479 × 647 7234163 = 17 × 41 × 97 × 107 7274249 = 17 × 449 × 953 7366463 = 13 × 23 × 71 × 347 8159759 = 19 × 29 × 59 × 251 8164079 = 7 × 11 × 229 × 463 8421335 = 5 × 13 × 23 × 43 × 131 8699459 = 43 × 307 × 659 8734109 = 37 × 113 × 2089 9224279 = 53 × 269 × 647 9349919 = 19 × 29 × 71 × 239 9486399 = 3 × 13 × 79 × 3079 9572639 = 29 × 41 × 83 × 97 9694079 = 47 × 239 × 863 9868715 = 5 × 43 × 197 × 233
## References
1. ^ Thomas Wright (2018). "There are infinitely many elliptic Carmichael numbers". Bull. London Math. Soc. 50 (5): 791–800. arXiv:1609.00231. doi:10.1112/blms.12185. S2CID 119676706. | 2023-04-02 13:30:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "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": 0.22442136704921722, "perplexity": 62.292241002515645}, "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-2023-14/segments/1679296950528.96/warc/CC-MAIN-20230402105054-20230402135054-00460.warc.gz"} |
http://mathhelpforum.com/pre-calculus/21633-zeros-polynomial.html | # Math Help - Zeros of polynomial
1. ## Zeros of polynomial
Will you please answer these questions. Thank you
Given f(x) = x^2 (x-3)^3 (x+2)^2
1. How many zeros does the polynomial have?
2. What are the distinct zeros?
3. Give the multiplicity of each zero
4. Which zeros cross the x-axis?
5. Which zeros touch the x-axis?
2. Hello, oceanmd!
You are expected to know a few things . . .
Given: . $f(x) \:= \:x^2(x-3)^3(x+2)^2$
1. How many zeros does the polynomial have?
This is a seventh-degree polynomial . . . It will have seven zeros.
. . [They may not be distinct or real.]
2.What are the distinct zeros?
There are three distinct zeros: . $0,\:3,\:-2$
3.Give the multiplicity of each zero.
The multiplicity of each factor is given by its exponent.
$x = 0$ has multiplicity 2
$x = 3$ has multiplicity 3.
$x = \text{-}2$ has multiplicity 2.
4. Which zeros cross the x-axis?
5. Which zeros touch the x-axis?
If the multiplcity is odd, the graph cross the x-axis.
If the multiplicity is even, the graph is tangent to the x-axis.
. . Got it?
3. ## Distinct and Real Zeros
Soroban, thank you for the explanation!!!
1. Whatever the degree of the polynomial is, it is the total number of zeros. (real and complex) Is this correct?
2. Does "distinct zeros" mean "real zeros" It is the same thing?
3. For this polynomial, there are three real zeros, does it mean that it has four complex zeros?
thank you
4. 1. Yes the degree is the amount of zeros.
2. No distinct is how many unique zeros. You can see in that problem that 3 is repeated 3 times, -2 twice, and zero would be a factor twice, but the two extra threes, and the extra -2, and 0 are not considered distinct because they are repeated.
3. No the factorization would be 0, 0, 3, 3, 3, -2, -2 which none are complex. | 2014-04-16 17:47:29 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6058812141418457, "perplexity": 1418.9082298820028}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00055-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://mathhelpforum.com/geometry/140750-circle-chord-property.html | # Math Help - A Circle and Chord Property
1. ## A Circle and Chord Property
Let $OA \perp MN$, $P$ and $Q$ are determined by $B,C,D,E$ on $\odot O$. Show that $\overline{PA} = \overline{QA}$ | 2016-02-05 23:11:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4185396730899811, "perplexity": 667.1540194849927}, "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-2016-07/segments/1454701145519.33/warc/CC-MAIN-20160205193905-00235-ip-10-236-182-209.ec2.internal.warc.gz"} |
http://openstudy.com/updates/50ccbf0ee4b06d78e86c944d | ## Steph_Rawr352 Group Title Which system of inequalities are graphed in this picture? (attached below) one year ago one year ago
1. Steph_Rawr352 Group Title
2. nubeer Group Title
ok can u find the equations of line?
3. Steph_Rawr352 Group Title
what do you mean?
4. nubeer Group Title
you know equation of line.. coordinate geometry? finding slope?
5. Steph_Rawr352 Group Title
no /: I don't really understand this/:
6. nubeer Group Title
hmmm ok this would be hard then.. have u even find a slope or gradient? or studied of it?
7. Steph_Rawr352 Group Title
no/: i'm sorry):
8. nubeer Group Title
lol no worries.. well then u need to memorize few formula's and stay with me on this one.. $slope = \frac{ y2-y1 }{ x2-x1 }$ the x's and y's are coordinates or points on graph.
9. nubeer Group Title
|dw:1355596018107:dw|
10. nubeer Group Title
pick one point and tell me it's coordinates/?
11. Steph_Rawr352 Group Title
a point on the line or any point?
12. nubeer Group Title
point on line.
13. Steph_Rawr352 Group Title
(2,-4)
14. nubeer Group Title
good.. now pick another one.
15. Steph_Rawr352 Group Title
(-1,5)
16. nubeer Group Title
good these points are x2, y2 and the before one and x1,y1 now put these points in the slope formula
17. Steph_Rawr352 Group Title
would it be 9/-3 ?
18. nubeer Group Title
yes. which is equal to -3 so m=-3 m is slope ok?
19. Steph_Rawr352 Group Title
oh ok
20. nubeer Group Title
ok good.. now the other formula.. equation of line y = mx+c
21. nubeer Group Title
y = -3x +c right?
22. Steph_Rawr352 Group Title
yeah
23. nubeer Group Title
ok now choose any point from the above two and put the value of x and y in this equation to find c
24. Steph_Rawr352 Group Title
I got 5 = 3 + c ?
25. nubeer Group Title
yes find c..
26. nubeer Group Title
you doing it right dont worry.
27. Steph_Rawr352 Group Title
c = 2 ? and ohk thanks
28. nubeer Group Title
yes c = 2 so your equation is y = -3x +2 y +3x = 2 as its told that its inequatlity so put an inequatlity sign .. and the line on graph is dotted so u will just put < or> sign not equal with them.. now tell me which region is shaded from the line.. above or below?
29. Steph_Rawr352 Group Title
would it be > ? and above?
30. nubeer Group Title
yes.
31. Steph_Rawr352 Group Title
ohhhh ohk i see what you're doing! so now to find the other inequality.. do i pick two point from the other line? btw thank you for helping me (:
32. nubeer Group Title
yes same way for other line .. and no worries.. anytime :)
33. Steph_Rawr352 Group Title
thank you! I understand it much more (: I'll do the other line Thank you!
34. nubeer Group Title
:) you are most welcome. | 2014-07-25 16:10:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5800365805625916, "perplexity": 12628.997893417674}, "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-2014-23/segments/1405997894319.36/warc/CC-MAIN-20140722025814-00170-ip-10-33-131-23.ec2.internal.warc.gz"} |
http://mathhelpforum.com/algebra/98322-problem-solving.html | 1. ## Problem solving?
Kim placed an order for 98 gizmos which cost $445. A package of 7 gizmos cost$30, and a package of 5 gizmos cost $25. How many packages of each type did Kim order? Really not sure how to start it from there.. and how to put it into equations. 2. Hi fvaras89 Let : x = package of 7 gizmos y = package of 5 gizmos 1. If Kim buys one x and one y, she will get 12 gizmos -----> 7*1 + 5*1 = 12 If Kim buys two x and one y, she will get 19 gizmos -----> 7*2 + 5*1 = 19 If Kim buys three x and two y, she will get 31 gizmos -----> 7*3 + 5*2 = 31 If Kim has to buy 98 gizmos, can you find the equation ? ----> 1st equation 2. If Kim buys one x and one y, she will spend$55 -----> 30*1 + 25*1 = 55
If Kim buys two x and one y, she will spend $85 -----> 30*2 + 25*1 = 85 If Kim buys three x and four y, she will spend$190 -----> 30*3 + 25*4 = 190
If Kim has $445, can you find the equation ? -------> 2nd equation 3. Originally Posted by fvaras89 Kim placed an order for 98 gizmos which cost$445. A package of 7 gizmos cost $30, and a package of 5 gizmos cost$25. How many packages of each type did Kim order?
Really not sure how to start it from there.. and how to put it into equations.
Let x be the number of 7 gizmo packets and y be the number of 5 gizmo packets bought. Then:
7x + 5y = 98 .... (1)
30x + 25y = 445 .... (2)
Solve simultaneously.
4. Okay thanks, i solved them simultaneously and i got
7x + 5y = 98 (1)
30x + 25y = 445 (2)
(1) * 5
35x + 25y = 490 (3)
(3) - (2)
5x = 45
x = 9
then substitute into (1)
7(9) + 5y = 98
63 + 5y = 98
5y = 35
y = 7
Is that correct?
5. Originally Posted by fvaras89
Okay thanks, i solved them simultaneously and i got
7x + 5y = 98 (1)
30x + 25y = 445 (2)
(1) * 5
35x + 25y = 490 (3)
(3) - (2)
5x = 45
x = 9
then substitute into (1)
7(9) + 5y = 98
63 + 5y = 98
5y = 35
y = 7
Is that correct?
Does your solution work when you substitute it into the two equations? Check that and then you'll know.
6. oh yes it does!!! thanks heaps to you both | 2017-04-29 00:08:14 | {"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": 0.7031958103179932, "perplexity": 2736.927717534342}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917123102.83/warc/CC-MAIN-20170423031203-00231-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://abaqus-docs.mit.edu/2017/English/SIMACAECAERefMap/simacae-t-itnhelpplasticity.htm | # Defining plasticity
You can define plasticity behavior for the available components of relative motion. For more information, see Connector plastic behavior. If you specify a plasticity behavior option, you must also specify an elasticity behavior option.
Related Topics Connector section editors Creating connector sections Specifying potential terms In Other Guides Connector plastic behavior
1. Display the connector section editor using one of the following methods:
• To create a new connector section, follow the procedure outlined in Creating connector sections.
• To edit an existing connector section, select ConnectorSectionManager from the main menu bar, select the connector section from the list that appears, and click .
2. In the Edit Connector Section dialog box, do one of the following:
• To define a new plasticity behavior, click and select Plasticity from the menu that appears.
• To edit an existing plasticity behavior, select the behavior from the Behavior Options list to display the associated data fields for that behavior.
3. If you are defining uncoupled plasticity behavior,
1. Choose Uncoupled in the Coupling field to specify forces/moments as a tabular function of their respective available components of relative motion.
2. In the Force/Moment field, toggle on the forces or moments that are consistent with the available components of relative motion for which you are defining plasticity behavior. If the behavior is the same for multiple components, you can define a single plasticity behavior that will use this one function. If the behavior is different for multiple components, you must define separate plasticity behaviors.
4. If you are defining coupled plasticity behavior,
1. Choose Coupled in the Coupling field.
2. Select the Force Potential tabbed page, and define at least one force potential term. For more information, see Specifying potential terms.
5. Select the hardening behavior.
• Toggle on Specify isotropic hardening to define the initial yield value and, optionally, the evolution of the yield surface size, $F0$, as a function of the equivalent plastic relative motion, $u¯pl$.
• Toggle on Specify kinematic hardening to define the translation of the yield surface in force space through the backforce, $α$.
At least one hardening behavior, isotropic or kinematic, must be defined. You can select both types of hardening to define a combined isotropic/kinematic hardening behavior.
6. If you toggled on Specify isotropic hardening:
1. Select the Isotropic Hardening tabbed page.
2. Choose the Definition.
• Choose Tabular to specify the force-constitutive motion data directly in tabular form.
• Choose Exponential Law to specify the material parameters of the exponential law used to calculate the equivalent force defining the size of the yield surface.
7. If you toggled on Specify kinematic hardening:
1. Select the Kinematic Hardening tabbed page.
2. Choose the Definition.
• Choose Half-cycle to specify the force-constitutive motion data obtained from the first half cycle of a unidirectional tension or compression experiment.
• Choose Stabilized to specify the force-constitutive motion data obtained from the stabilized cycle of a specimen that is subjected to symmetric cycles.
• Choose Parameters to specify the material parameters directly.
8. To define behavior data that depend on temperature or field variables, do the following on the Isotropic Hardening or Kinematic Hardening tabbed page:
1. Toggle on Use temperature-dependent data to define behavior data that vary with temperature. A column labeled Temp appears in the tabular data area.
2. To define behavior data that depend on field variables, click the arrows to the right of the Number of field variables field to increase or decrease the number of field variables. Field variable columns appear in the tabular data area.
9. Enter plastic hardening data in the tables on the Isotropic Hardening and/or Kinematic Hardening tabbed pages. You can enter data into the tables using the keyboard. Alternatively, you can click mouse button 3 anywhere in the tables to view a list of options for specifying tabular data. For detailed information on each option, see Entering tabular data.
10. To modify the behavior settings for the regularization (Abaqus/Explicit analyses only) or the extrapolation of the data, use the procedure described in Specifying behavior settings for tabular data. For an Abaqus/Explicit analysis that includes isotropic hardening using a Tabular definition, you can also specify settings for the evaluation of rate-dependent data.
11. Select one of the following:
• If you want to continue defining behaviors, click Add, select the desired behavior, and continue the connector section definition. For instructions on defining other behaviors, see Using the connector section editors.
• If you want to view or edit an existing behavior, select it from the Behavior Options list. For instructions on editing behaviors, see Using the connector section editors.
• If you want to save your connector section definition and exit the editor, click . | 2022-12-05 16:48:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5933839678764343, "perplexity": 2657.401458857793}, "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/1669446711042.33/warc/CC-MAIN-20221205164659-20221205194659-00486.warc.gz"} |
https://cho.sh/r/B4CD83 | # Q Function
In probability theory and statistics, the Q function is the tail probability of the standard normal distribution, also known as the complementary cumulative distribution function (CCDF) of the standard normal distribution. The Q function is defined as:
$Q(x) = \frac{1}{\sqrt{2\pi}} \int_{x}^{\infty} e^{-t^2\over2} dt$
where x is a real number.
The Q function calculates the probability that a random variable from a normal distribution with mean 0 and standard deviation 1 exceeds a certain value, $x$. This probability can be written as:
$P(Z>x) = Q(x)$
where Z is a standard normal variable.
The Q function can be calculated using numerical integration or special functions, such as the complementary error function or the Marcum Q-function.
The Q function is helpful in various applications, such as digital communications and signal processing, where it calculates error probabilities and signal-to-noise ratios. It is also used in statistics to calculate confidence intervals and hypothesis tests for normal distributions. | 2023-03-25 20:28:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 3, "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": 0.9254950881004333, "perplexity": 107.63508698869354}, "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/1679296945372.38/warc/CC-MAIN-20230325191930-20230325221930-00337.warc.gz"} |
https://www.kofastudy.com/courses/ss2-maths-2nd-term/lessons/sequence-week-1/ | Back to Course
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# Sequence | Week 1
Performance Objectives
• Sequence
• Arithmetic progression
PERFORMANCE OBJECTIVES
By the end of this topic, the student should be able to:
1. Define sequence and give examples
2. Find formula for a given nth term of a sequence
3. Calculate missing terms of a sequence given the formula
4. Define arithmetic progression(A.P)
5. Calculate the missing terms of an A.P.
6. Solve/calculate the arithmetic mean of an A.P
Lesson Content
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https://julesh.com/2020/05/26/subgame-perfection-made-difficult/ | This is the second post in catching up on aspects of open-games-hs that are ahead of my papers, following open games with stateful payoffs. Subgame perfection has been an embarrassing thorn in my side since 2016 when I had to do major surgery on my PhD thesis because the category of “open games with subgame perfect equilibria” turned out to not be monoidal. Currently there are two approaches: One in iterated open games which is quite pragmatic and requires the “user” specifying an open game to manually mark where the subgames are by applying a functor; and one in morphisms of open games which I find very elegant but requires both an extra categorical dimension and an equivalent amount of effort by the “user”.
I always wanted an “automatic” approach to subgame perfection in open games, like I failed to do in my thesis – just draw the usual string diagram, and get subgame perfect equilibria out. I now have a way to do it, implemented in OpenGames.Engine.SubgamePerfect, which I’ll document here.
The starting point is an idea that doesn’t work. In the category of deterministic open games, a decision $\mathcal{D} : (X, 1) \to (Y, \mathbb{R})$ has strategy profiles $\Sigma_\mathcal{D} = X \to Y$, and equilibria $\mathbf{E}_\mathcal{D} : X \times (Y \to \mathbb{R}) \to \mathcal{P} (\Sigma_\mathcal{D})$ given by $\mathbf{E}_\mathcal{D} (h, k) = \{ \sigma \mid \sigma(h) \in \arg\max (k) \}$. Here $h$ plays the role of the on-equilibrium subgame. Here is an idea that doesn’t work: instead define it by $\mathbf{E}_\mathcal{D} (h, k) = \{ \sigma \mid \sigma(x) \in \arg\max (k)$ for all $x \in X \}$.
To see why this doesn’t work, form the game $\mathcal{D}^\Delta = (1_X \otimes \mathcal{D}) \circ \Delta : (X, 1) \to (X \times Y, \mathbb{R})$, where $\Delta : (X, 1) \to (X \times X, 1)$ lifts the copy function. Given a context with history $h : X$ and continuation $k : X \times Y \to \mathbb{R}$, its equilibrium condition $\mathbf{E}_{\mathcal{D}^\Delta} (h, k)$ is equivalent to $\mathbf{E}_{1_X \otimes \mathcal{D}} ((h, h), k)$ and then to $\mathbf{E}_{\mathcal{D}} (h, k')$ where $k' (y) = k (h, y)$. If we keep the true definition of $\mathbf{E}_{\mathcal{D}}$ then this is carefully balanced to give us Nash. But if we use our proposed modified definition, the equilibrium condition becomes $\sigma (x) \in \arg\max_{y \in Y} k (h, y)$ for all $x \in X$. This is wrong – the subgame perfect equilibrium condition should be $\sigma (x) \in \arg\max_{y \in Y} k (x, y)$ for all $x \in X$.
What is happening is that the context of $\mathcal{D}$ should keep track of how a conterfactual change to its history, from the on-equilibrium subgame $h$ to an off-equilibrium subgame $x$, should propagate to the continuation, which should change from $k (h, -)$ to $k (x, -)$.
Fortunately we’re in luck. The heavy machinery developed in Bayesian open games is able to handle exactly this sort of situation. The first thought is to replace the category of stochastic maps with the category of relations. Pushing forward the state $X \subseteq X$ through the copy map $\Delta$ yields the diagonal set $\Delta^* (X) = \{ (x, x) | x \in X \} \subseteq X \times X$, which contains exactly enough information to propagate a counterfactual change in one output backwards and then forwards again to the other output. That is to say, if we take the pushfoward state $\Delta^* (X)$ and condition on one output being some $x$, the other output collapses to the same $x$. This much is standard reasoning in the category of relations.
This is almost right, but not quite. We also need to keep track of what actually happens, and not just the counterfactual alternatives, in particular because they are still used for payoffs: a player’s preferences are indifferent to payoffs that might have occurred, but did not. We actually need to simultaneously keep track of 3 levels of possibility: (1) what actually happened, (2) what counterfactually might have happened, but did not, and (3) what could not have happened. Pushing forward through $\Delta$ illustrates this well. If the input play is $h$ then what actually happens is $(h, h)$, what could have happened is any element of $\Delta^* (X) = \{ (x, x) \mid x \in X \}$, and any $(x, x')$ with $x \neq x'$ could not have happened even counterfactually.
This motivates the following definition. Let $T : \mathbf{Set} \to \mathbf{Set}$ be given by $T (X) = \{ (x, S) \mid x \in S \subseteq X \}$. I call $T$ the pointed powerset monad, and it is a hybrid of the identity and powerset monads. The monad unit is given by $\eta (x) = (x, \{ x \})$, and the multiplication by $\mu ((x, S'), S) = (x, \bigcup S)$. $T$ admits monad morphisms to both identity and powerset. The kleisli category of $T$ is “pointed relations”: a kleisli morphism $X \to Y$ is a relation $R \subseteq X \times Y$ together with a function $f : X \to Y$ satisfying $(x, f (x)) \in R$ for all $x$. Since $T$ is commutative, its kleisli category admits a monoidal product. (I’ve never seen this monad before in the literature.)
Fortunately we can save some work and hack this monad in Haskell by pretending that ordinary lists are pointed subsets, using the head as the basepoint. Nothing goes wrong as long as we are careful to only use lists in ways that are invariant under permutation and duplication (ie. we treat free monoids in ways that are also valid for free commutative idempotent monoids). All of the list monad operations are “head-preserving” in the appropriate way.
Just as for Bayesian open games, we set up coend optics for the kleisli category of this monad. In category theoretic notation a kleisli optic $(X, S) \to (Y, R)$ is an element of $\int^A (X \to T (A \times Y)) \times (A \times R \to T S)$, and in Haskell notation it is an element of the GADT data Optic x s y r where Optic :: (x -> [(a, y)]) -> (a -> r -> [s]) -> Optic x s y r. The type of contexts is also exactly as for Bayesian open games: it is an element of $\int^A T (A \times X) \times (A \times Y \to T R)$ or data Context x s y r where Context :: [(a, x)] -> (a -> y -> [r]) -> Context x s y r. The definition of the monoidal category of open games can be done generically for an arbitrary monad, so almost all of the code follows for free.
The only thing that needs to be specialised to the pointed powerset monad is the definition of a decision $\mathcal{D} : (X, 1) \to (Y, \mathbb{R})$. This isn’t trivial. The set of strategies is still $\Sigma_{\mathcal{D}} = X \to Y$. For a strategy $\sigma : X \to Y$ we must produce an optic $(X, 1) \to (Y, \mathbb{R})$. We take the bound variable to be $A = 1$, so the backward part is trivial and we need only give the view $v : X \to T Y$. We take $v (x) = (\sigma (x), Y)$, the pointed subset $Y \subseteq Y$ with basepoint $\sigma (x)$. This means that what happens is $\sigma (x)$, but any element of $Y$ could happen counterfactually, because the agent has a free choice. (This is an unusually clear appearance of free will in mathematics.)
In Haskell I always write decisions to take an exhaustive list of possible moves as a parameter in order to brute force $\arg\max$, because implementing a decent $\arg\max$ is an orthogonal problem. So the implementation of a decision begins:
decision :: [y] -> OG (x -> y) x () y Double
decision ys = OG {
play = \f -> let v x = zip (repeat ()) (f x : ys)
u () _ = [()]
in Optic v u
...
Here f x : ys is the list representing the pointed subset $(f (x), Y)$.
Finally, we must evaluate a strategy $\sigma : X \to Y$ in a context, which is a triple $(A, h, k)$ with $h : T (A \times X)$ and $k : A \times Y \to T \mathbb{R}$. Here we get right to the heart of what subgame perfection is all about: we ignore the factual part of $h$ and only use the set of counterfactuals, but we ignore the counterfactual part of $k$ and only use the factual. Specifically, the equilibrium condition is that for every $(a, x) \in h$, we have $\sigma (x) \in \arg\max_{y \in Y} k (a, y)_*$, where $k (a, y)_*$ means the basepoint of $k (a, y) : T \mathbb{R}$. Information about $x$ is transferred to $k$ via $a$, which is entangled with $x$ in the state $h$.
...
and [head (k a (f x)) >= head (k a y) | (a, x) <- h, y <- ys]} | 2021-05-07 10:08:57 | {"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": 82, "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": 0.7734841108322144, "perplexity": 483.5103388801256}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988775.80/warc/CC-MAIN-20210507090724-20210507120724-00115.warc.gz"} |
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National Academy Science LettersJournal Prestige (SJR): 0.189 Citation Impact (citeScore): 1Number of Followers: 5 Hybrid journal (It can contain Open Access articles) ISSN (Print) 0250-541X - ISSN (Online) 2250-1754 Published by Springer-Verlag [2351 journals]
• Rutile Mineral Chemistry as a Guide to Provenance of Red Sediments and
of India
• Abstract: Rutile is the most stable and widely distributed TiO2 polymorph in rocks of low- to high-grade metamorphic facies and is also an accessory mineral in igneous rocks. Rutile is commonly available in modern to ancient placer mineral deposits in the coastal sediments. Mineral chemistry of rutile from red sediments and modern sands along Bhimunipatnam–Konada coast were used in the present study to know its provenance. Iron (Fe), chromium (Cr), niobium (Nb) content and their distribution pattern in rutile and also their relationships with aluminum (Al) and magnesium (Mg) concentrations provide information on its provenance. These study reveals that the Fe–Cr and Cr–Nb systematics indicates majority of rutiles were derived from metapelitic rocks mainly khondalites and leptynites of the Eastern Ghats Granulite Belt (EGGB) and minor contribution is from magmatic charnockites, pegmatites and granites. The Al and Mg behavioral pattern in rutile from both zones clearly depicts that the most of the rutiles are derived from crustal rocks. The rutile contribution to late quaternary red sediments and modern coastal sands is also from same provenance.
PubDate: 2019-06-14
• Antagonistic Activity of Isolated Root Endophytic Freshwater Fungus
Anguillospora Longissima Against Pathogenic Fungi
• Abstract: Freshwater fungus Anguillospora longissima was recovered as root endophytic fungus from the pteridophytic plant Equisetum sp., growing in riparian areas of Nainital, Kumaun Himalaya, and evaluated through well diffusion and dual culture methods. The isolated endophytic fungus was found a potent inhibitor for test pathogenic fungi. Maximum inhibition by the isolated endophytic fungus was showed against pathogenic fungus Tilletia indica (79.76%), while minimum was recorded against Fusarium oxysporum (35.73%), in dual culture method. Further, in well diffusion method, pathogenic fungi Colletotrichum falcatum and Pyricularia oryzae were found maximum (62.96%) and minimum (17.85%) inhibited fungi, respectively, by the isolated fungus. In the present investigation, the used root endophytic fungus A. longissima was found potent to produce some antifungal substances that inhibited the growth of all test pathogenic fungi.
PubDate: 2019-06-08
• Correction to: An Example of a Right Loop Admitting Only Discrete
Topologization
• Abstract: Being motivated by Suvorov and Kryuchkov (Sib Mat Zh 17(2):471–473, 1976) and Suvorov (Sib Mat Zh 32(5):193, 1991), an example of a right loop admitting only discrete topologization is presented in this paper.
PubDate: 2019-06-07
• First Report on Infection of Argulus quadristriatus (Arthropoda:
Crustacea: Branchiura) on Marine Fish Cobia in Brood Stock Pond Culture
• Abstract: A total of 30 specimens of fish cobia Rachycentron canadum (Total Length = 45–120 cm, Weight = 3.0–25 kg) were stocked at the density of 1 kg/m3 in the polythene lined earthen pond. After 3 months of stocking, fish cobia was found with infection of ectoparasites. Then fishes were sampled at fortnight interval to find the percentage distribution of ectoparasites in different parts of the body for a year and also any pathological symptoms. Identification of the parasite was made through light and electron microscopies. The parasite was identified as Argulus quadristriatus Devaraj and Ameer Hamsa, 1977 (Crustacea: Branchiura: Argulidae) commonly called as fish lice. The maximum distribution of pathogenic argulid was observed on the head and operculum of cobia and was found high in summer months from April to June. Pathological symptoms were observed on cobia as erratic swimming, rubbing against substrate in the pond and lesions of epithelial tissues on the infected regions. It must be due to continuous rupturing and feeding of argulids on the skin of cobia using its powerful antennae. Scanning electron micrographs revealed some important morphological features of A. quadristriatus. This is a first report of A. quadristriatus infection on cobia reared in a land-based pond ecosystem.
PubDate: 2019-06-01
• Insect Entrapment by Plants in Kodaikanal Wildlife Sanctuary, India
• Abstract: The present study focused on plant–insect interactions between arthropod insects and herbs at shola forest floor such as Desmodium uncinatum (Jacq.) DC.—Fabaceae and Cerastium indicum (L.) Wight & Arn.—Caryophyllaceae. The morphological characters such as ciliate, puberulous and villous nature of plants will lead a greater role to adopt the climatic conditions. Findings of the present study revealed the trichomes of some plants are menace to pollinating agents.
PubDate: 2019-06-01
• Design of Dual-Band Antenna for 2.45 and 5.8 GHz ISM Band
• Abstract: A dual-band microstrip rectangular patch antenna for WLAN in vehicular communication applications is proposed. The aim of this research is to achieve smaller size, better bandwidth and good radiation capability. The designed antenna has a frequency bandwidth of 184 MHZ with the upper cut-off frequency of 2.4953 GHz and the lower cut-off frequency of 2.3113 GHz which are the operating frequencies for IEEE 802.11b/g/n WLAN applications and IEEE 802.15.4 ZigBee application. The second resonance is at 5.8 GHz frequency with a bandwidth of (5.850–5.925 GHz) the 802.11p standard intended for use in vehicular communication systems and for the mobile operation. The rectangular patch with Inset Line Feed technique is used to design the antennas. First, the design parameters for single element of rectangular patch antenna for 5.8 GHz are calculated from the design equations, and then, the design is extended to operate in dual band using the slots at the ground plane and two cylinders at the corners of the slot to obtain good radiation characteristics. The simulation of the proposed antenna has been carried out in computer simulation technology software, and the results are evaluated.
PubDate: 2019-06-01
• Evolution of Optical Band Gap and Related Properties in Ni 1−x Co x
O Nanoparticles
• Abstract: Ni1−xCoxO Nanoparticles with x varying from 0 to 0.05 were synthesized by co-precipitation method. The effect of Co concentration on the optical band gap and related properties of NiO nanoparticles were studied by UV–visible spectroscopy. UV–visible characterization of these samples indicated that the band gap of NiO decreased from 3.65 to 3.38 eV with increasing Co concentration from 0 to 5% in NiO. It has been reported that the band gap of ternary compounds show band gap bowing with composition instead of linear variation. In our case it is found to be ~ 5.28 ± 0.5 eV. The refractive index and electron polarizability with Co doping concentration in NiO is calculated from the band gap. Both the parameters found to increase with increasing Co concentration in NiO. This study suggests the usefulness of these materials for optoelectronic devices.
PubDate: 2019-06-01
• A New Soil Testing Programme for the Evaluation of Intensity and Quantity
Factors of Iron
• Abstract: Deficiency of Fe in rice mainly occurs under upland condition, particularly in alkaline and calcareous soils. The plants respond directly to changes in the activities of ions in the soil solution. Baker soil test (BST) is a new approach to soil testing which not simply determine the concentrate bioavailability but also provides the activity of an ion. A greenhouse experiment was conducted with special reference to iron for relating the data with content of efficient and inefficient cultivars of aerobically grown rice for iron availability. Alkaline, calcareous, acid and lime-treated soils could be rated as normal in respect of labile Fe status. Intensity parameter of Fe (pFe) for alkaline and calcareous was rated to be low, while it was normal for other two soils. Comparison of pFe values of Fe(III) in different soils with plant Fe content reveals the positive impact of intensity parameter of Fe(III) in soil is reflected well in plant Fe content data in both the rice cultivars. Similar were the relationships of intensity parameter of Fe with grain and straw yield of rice. BST, particularly intensity parameter of Fe, can be a good index of availability of Fe in soils.
PubDate: 2019-06-01
• Desiccation Response of Seeds of Himalayan Oak, Quercus floribunda Lindl.
ex A. Camus
• Abstract: Quercus floribunda, an important western Himalayan oak, has a great ecological importance. As seeds are sensitive to moisture loss, seed maturation and germination is, therefore, synchronized with the commencement of monsoon rainfall. A delay in monsoon can thus drastically inhibit the recruitment of seedlings. However, the level of such sensitivity of seeds towards desiccation in Q. floribunda is not earlier studied in the region. In view of this gap, effects of desiccation on seed germination of Q. floribunda were examined across different moisture levels. Datasets on seed morphological characteristics, moisture content and seed germinability were recorded. Results indicated that seed likelihood sensitivity to desiccation (P) was much higher (0.84) in Q. floribunda as compared to suggested minimum value of (0.5) beyond which seeds are considered desiccation sensitive. The species is thus characterized as desiccation sensitive that significantly loses per cent germination with drying period. It is recommended that the seeds of Q. floribunda should be sown within 7 days, when the seed moisture content remains at critical level of 34%. These results have implications in forestry practice, conservation and management for coping with practical problems of storing Q. floribunda seeds.
PubDate: 2019-06-01
• An Intelligent Risk Prediction System for Breast Cancer Using Fuzzy
Temporal Rules
• Abstract: Online prediction of risk on breast cancer is a challenging task in the area of health care during the past decade. Since the existing statistical and data mining methods have limitations with respect to the prediction of breast cancer, there is a need for proposing more effective predictive models which can predict the breast cancer more effectively. In this paper, we propose a new intelligent online risk prediction model for predicting the breast cancer using fuzzy temporal rules more accurately. Moreover, this intelligent system determines the contributing attributes from the dataset using intelligent fuzzy temporal rules and also performs prediction by applying fuzzy rule-based classification with temporal constraints. Moreover, the rules are validated using a domain expert and the experiments conducted in this work using questionnaire, rule-based classification and consultation with domain expert have proved that the proposed system provides more accurate results for risk prediction than the other existing systems.
PubDate: 2019-06-01
• Evaluation of Natural Carotenoid Sources from Rosa hybrida Varieties on
Growth and Pigmentation of Goldfish ( Carassius auratus L.)
• Abstract: The commercial value of goldfish (Carassius auratus) is determined by its attractive colour. Fish cannot synthesize carotenoids, and hence, natural sources of carotenoids are supplemented in the diet. In this study, rose petal meal from four varieties (Jubileums, Brisbane blush, Double delight and Restless) was added in three concentrations (2, 4 and 6 gkg−1) to the formulated control feed and fed to the goldfish, for 45 days. The results revealed that dietary carotenoids had a significant effect (P < 0.001) on growth and pigmentation of goldfish. Increase in length and pigmentation was proportionate to concentration of supplements, 6 gkg−1 being most effective. The maximum increase in weight was observed at 4 gkg−1 concentration of supplements, with a limiting effect at higher concentrations. Research on the use of natural sources of carotenoids to influence growth and colouration is a significant development for the ornamental fisheries sector, to enhance quality and market value.
PubDate: 2019-06-01
• Taxonomic Analyses Suggest a New Record of Lygodium japonicum
(Lygodiaceae) from Terai Regions of Uttar Pradesh, India
• Abstract: Lygodium japonicum (Thunb.) Sw., a climbing fern belonging to family Lygodiaceae has been encountered from Sonaripur forests of Dudhwa National Park in terai regions of Uttar Pradesh. Discovery of this species from above forest is a new record and addition to the pteridophytic flora of terai regions of Uttar Pradesh, including entire gangetic plain. Present contribution provides a detailed taxonomic analyses, illustrations, photographs, examined specimens citation, ecology, distribution and habit of the species. Distinctive taxonomic attributes including rhizomatous hairs, winged rachis and petiole, non-articulated pinnae, crenulated pinnules, pubescent rachis, mid-veins, veins and lamina has been largely measured to augment some additional taxonomic characters. Discussion on distinguished taxonomic attributes and plasticity with allied species has been provided. An assertion on its extended range of distribution and new record in terai regions of India is also conferred.
PubDate: 2019-06-01
• Determinants of Inequality in Dairy Development of India
• Abstract: Dairy sector plays an integral role in nutritional security as well as economic prosperity. The present study was designed with an attempt to identify the determinants of the inequality in Indian dairy development. Coefficient of variation, Gini coefficient, principal component analysis and Duncan’s multiple range test were used to quantify inequalities and its determinants among the highly, moderately and least dairy progressive states by using 20 dairy development indicators. It was found that Gini coefficients for the three strata indicated the level of inequality in dairy development but in varying magnitude. The study also depicts that the average per day milk yield of crossbred cattle, indigenous cattle and buffaloes; number of AI done; milk procurement through cooperatives; capacity of milk processing and surface road length found to be significantly different among the three regions. The study advocates for proper planning and prioritization of indicators that determine the regional development to reduce the variability and regional imbalances for additional milk production.
PubDate: 2019-06-01
• Acronicta rumicis (Linnaeus, 1758), Emerging Pest of Apple Plantation in
Kashmir Valley
• Abstract: The larvae of the moth Acronicta rumicis Linnaeus commonly known as Knot grass moth inflict considerable damage to new apple plantation in the Kashmir region. Immature stages of the pest defoliate apical buds and result in stunted growth of the plant. Seasonal occurrence and severity of the pest from Kashmir Valley is detailed.
PubDate: 2019-06-01
• In Vitro Multiple Shoot Induction and Plant Regeneration from Male Ephedra
foliata : A Potential Medicinal Gymnosperm
• Abstract: Ephedra foliata Boiss. Ex. C.A. Mey, known as soma plant according to ancient Indian Ayurvedic system. It is known for its potential to cure many diseases. Review of literature revealed that E. foliata contains traces of alkaloids responsible for its therapeutic activity. In present study effect of different concentrations of plant growth regulator on nodal sector explants was checked for multiple shoot production. Maximum numbers of shoots were observed on Kinetin (2.5 mg/l) whereas, Kinetin (2.5 mg/l) with Indole acetic acid (2.5 mg/l) induced 9.04 ± 0.8 shoots in 22.46 ± 0.57 days and percentage of explants that showed response was more than any other combination. Out of the two plant growth regulators used for root induction from in vitro grown shoots 1-napthaleneacetic acid showed better response than indole-3-acetic acid.
PubDate: 2019-06-01
• Construction of Right Gyrogroups
• Abstract: Attempts have been made to introduce notion of class assigned functions k for a given group G and observed that it determines a right gyrogroup $$(G, o_k)$$ . It is also observes that $$(G, o_k)$$ will be a group if and only if G is a nilpotent group of class 2 and will be a gyrogroup for a nilpotent group of class 3, where $$k(x)=1$$ for all $$x\in G\setminus \{1\}$$ .
PubDate: 2019-06-01
• Some New Results on the Rainbow Neighbourhood Number of Graphs
• Abstract: A rainbow neighbourhood of a graph G is the closed neighbourhood N[v] of a vertex $$v \in V(G)$$ which contains at least one coloured vertex of each colour in the chromatic colouring $${\mathscr {C}}$$ of G. Let G be a graph with a chromatic colouring $${\mathscr {C}}$$ defined on it. The number of vertices in G yielding rainbow neighbourhoods is called the rainbow neighbourhood number of the graph G, denoted by $$r_\chi (G)$$ . Rainbow neighbourhood number of the complements and products of certain fundamental graph classes are discussed in this paper.
PubDate: 2019-06-01
• New record of the starry flying gurnard, Dactyloptena peterseni
(Scorpaeniformes; Dactylopteridae); from Wadge Bank, Southwest coast of
India
• Abstract: Five specimens of dactylopterids measuring 220.5–320.0 mm standard length caught commercial trawler fishing in 80–120 m deep waters of Wadge Bank off Kanyakumari coast (8.02 N, 76.80 E) were collected from the landing centre at Jeppiaar fishing harbour, Muttom, Southwest coast of India. The specimens were identified as Dactyloptena peterseni (Nystrom, 1887) on the basis of morphometric measurements and the key identification character like absence of second free spine between the occipital and the first dorsal spine. The occurrence of the species suggests that the extended distribution of D. peterseni to Southwest coast of India. The detailed morphometric and meristic characters are described and discussed in this paper. Molecular identification was done by using partial sequence of mitochondrial cytochrome oxidase sub-unit I gene for confirmation of the species.
PubDate: 2019-06-01
• Effect of Probiotic Supplementation on Growth of Carp Fingerlings
• Abstract: A probiotic product (PLANKTO PLUS) consisting adequate colony-forming units of beneficial bacteria was experimented for 90 days to evaluate the effect of growth of carp fingerlings. The selected probiotic was applied regularly @ 2 ml/tank (water volume 4 m3) to the carp fingerlings kept in fibreglass-reinforced plastic tanks under five different treatments, i.e. control (without probiotics), once in a day, weekly, fortnightly and monthly with two replicates. Tanks were stocked with Hypophthalmichthys molitrix, Labeo rohita and Cirrhinus mrigala @ 10 fingerlings/m3 in the ratio of 4:3:3, and water quality parameters were monitored regularly. The use of probiotic resulted in good conclusion by bringing feed cost down, improving water quality, increasing plankton concentration, quick breakdown of unused feed and checking algal bloom. The best result was obtained in the tanks with ‘weekly’ dose in the form of diverse assemblage of zooplankton, good condition factor and highest specific growth rate of the stocked fishes.
PubDate: 2019-06-01
• Isolation of Plant Growth Promoting fluorescent Pseudomonas from
Mid-Himalayan Apple Orchards and their Similarity Analysis
• Abstract: Ten fluorescent Pseudomonas species isolated from normal and replant rhizosphere soil samples of apple collected from different locations of Shimla and Kullu districts of Himachal Pradesh. These Pseudomonas species solubilized tri calcium phosphate and showed substantial amount of soluble phosphorus in Pikovskay’s broth, exhibited the production of siderophore and growth inhibition against Dematophora necatrix, white root rot causing fungi in apple orchards. Random Amplified Polymorphic DNA analysis was done using four arbitary decamer primers of OPA series to observe genetic homogeneity/polymorphism among 10 isolates of fluorescent Pseudomonas species. Total 25 bands were generated with 4 random decamer oligonuceleotide primers. All the bands generated through RAPD-PCR were polymorphic in nature. The dendrogram constructed using UPGMA method showed that all the ten Pseudomonas isolates were categorized into three distinct clusters. The isolate An-9-Mg and An-H showed maximum similarity (67%) with each other and An-G showed 53% similarity with An-9-Mg and An-H. The isolate Ar-1-Mg was totally different from all other nine strains. Out of the ten Pseudomonas isolates, An-G and An-H showed maximum production of P-solubilization, siderophore and antifungal activity. Therefore, the uniqueness of Pseudomonas isolates An-G and An-H suggested their potential use for developing a cost effective and eco friendly biofertilizers for apple, an important cash crop of Himachal Pradesh.
PubDate: 2019-06-01
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APIs | 2019-07-16 10:44:24 | {"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": 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": 0.4969930052757263, "perplexity": 10159.407882555714}, "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/1563195524522.18/warc/CC-MAIN-20190716095720-20190716121720-00033.warc.gz"} |
https://math.stackexchange.com/questions/3192900/prove-f-mu-f | # Prove $F^* = \mu * F$
Let $$f: \mathbb{Q} \cap [0,1] \to K$$ and set $$F(n) = \sum_{k = 1}^n f(\frac k n)$$, $$F^*(n) = \sum_{k = 1, (k,n) = 1}^n f(\frac k n)$$.
Show that $$F^* = \mu * F$$ where $$*$$ is the Dirichlet product.
$$\mu * F (n) = \sum_{d|n} \mu(d) F(\frac n d) = \sum_{k = 1}^n\sum_{d|n} \mu(d) f(\frac k n)$$
Context
This is a previous step to obtain $$\mu(n) = \sum_{k = 1, (k,n) = 1} exp(2 \pi i\frac k n)$$ which can be concluded from the fact that $$F(n) = e(n) = \sum_{d | n} \mu(d)$$ using Möbius inversion formula. This result is also done in: The Möbius function is the sum of the primitive $n$th roots of unity..
References
Tom Apostol, Introduction to Analytic Number Theory, page 48.
• $$\sum_{k=1,gcd(k,n)=1}^n f(k/n) = \sum_{k=1}^n f(k/n)\sum_{d | gcd(k,n)} \mu(d)= \sum_{d | n} \mu(d) \sum_{k=1,d | k}^n f(k/n)$$ – reuns Apr 18 at 22:38
$$F^*(n) = \sum_{k = 1, (k,n) = 1}^n f\Big(\frac k n\Big) = \sum_{k = 1}^n f\Big(\frac k n\Big) \Big(\sum_{d | (k,n)} \mu(d)\Big) = \sum_{k = 1}^n \sum_{d|k,d|n} \mu(d)f\Big(\frac k n\Big) = \ldots$$
For each $$k$$ such that $$d|k$$ we write $$k = qd$$ and $$1 \le k \le n$$ is equivalent to $$1 \le q \le \frac n d$$ and we write:
$$\ldots = \sum_{d|n} \mu(d) \sum_{k = 1,d|k} f\Big(\frac k n\Big) = \sum_{d|n} \mu(d) \sum_{q = 1}^{n/d} f\Big(\frac {q}{n/d}\Big) = \mu * F(n)$$
Let's see how we prove the consequence:
Take $$f(x) = exp(\frac{2 \pi i x}{n})$$, then observe that $$F(n) = e(n)$$ where $$e(n) = 1$$ if $$n = 1$$ and $$0$$ otherwise. This is the neutral element for Dirichlet convolution. Therefore, $$\mu * F = \mu$$ and by the above $$\mu * F = F^*$$, so we conclude $$\mu = F^*$$. This proves the proposition.
Acknowledgements
Thanks to @reuns for his comment. | 2019-06-27 03:58:57 | {"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": 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": 23, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9933521151542664, "perplexity": 78.89317382048388}, "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/1560628000613.45/warc/CC-MAIN-20190627035307-20190627061307-00479.warc.gz"} |
https://iacr.org/cryptodb/data/paper.php?pubkey=32192 | ## CryptoDB
### Paper: Succinct Interactive Oracle Proofs: Applications and Limitations
Authors: Shafik Nassar , Technion Ron Rothblum , Technion Search ePrint Search Google Slides CRYPTO 2022 Interactive Oracle Proofs (IOPs) are a new type of proof-system that combines key properties of interactive proofs and PCPs: IOPs enable a verifier to be convinced of the correctness of a statement by interacting with an untrusted prover while reading just a few bits of the messages sent by the prover. IOPs have become very prominent in the design of efficient proof-systems in recent years. In this work we study succinct IOPs, which are IOPs in which the communication complexity is polynomial (or even linear) in the original witness. While there are strong impossibility results for the existence of succinct PCPs (i.e., PCPs whose length is polynomial in the witness), it is known that the rich class of NP relations that are decidable in small space have succinct IOPs. In this work we show both new applications, and limitations, for succinct IOPs: 1. First, using one-way functions, we show how to compile IOPs into zero-knowledge proofs, while nearly preserving the proof length. This complements a recent line of work, initiated by Ben Sasson et al. (TCC,2016B), who compileIOPs into super-succinct zero-knowledge arguments. Applying the compiler to the state-of-the-art succinctIOPs yields zero-knowledge proofs for bounded-space NP relations, with communication that is nearly equal to the original witness length. This yields the shortest known zero-knowledge proofs from the minimal assumption of one-way functions. 2. Second, we give a barrier for obtaining succinct IOPs for more general NP relations. In particular, we show that if a language has a succinct IOP, then it can be decided in space that is proportionate only to the witness length, after a bounded-time probabilistic preprocessing. We use this result to show that under a simple and plausible (but to the best of our knowledge, new) complexity-theoretic conjecture, there is no succinct IOP for CSAT.
##### BibTeX
@inproceedings{crypto-2022-32192,
title={Succinct Interactive Oracle Proofs: Applications and Limitations},
publisher={Springer-Verlag},
author={Shafik Nassar and Ron Rothblum},
year=2022
} | 2022-08-16 01:07:42 | {"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": 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": 0.4745595157146454, "perplexity": 2037.693456164464}, "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/1659882572215.27/warc/CC-MAIN-20220815235954-20220816025954-00368.warc.gz"} |
http://encyclopedia.kids.net.au/page/ne/Newton-Raphson_method | ## Encyclopedia > Newton-Raphson method
Article Content
# Newton's method
Redirected from Newton-Raphson method
In numerical analysis, Newton's method (or the Newton-Raphson method) is an efficient algorithm for finding approximations to the zero (or root) of a real-valued function. As such, it is an example of a root-finding algorithm.
History Newton's method was discovered by Isaac Newton and published in Method of Fluxions in 1736. Although the method was described by Joseph Raphson[?] in Analysis Aequationum[?] in 1690, the relevant sections of Method of Fluxions were written earlier, in 1671.
The Method The idea of the method is as follows: one starts with a value which is reasonably close to the true zero, then replaces the function by its tangent (which can be computed using the tools of calculus) and computes the zero of this tangent (which is easily done with elementary algebra). This zero of the tangent will typically be a better approximation to the function's zero, and the method can be iterated.
Suppose f : [a, b] -> R is a differentiable function defined on the interval [a, b] with values in the real numbers R. We start with an arbitrary value x0 (the closer to the zero the better) and then define for each natural number n:
$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$
Here, f ' denotes the derivative of the function f.
One can prove that, if f ' is continuous, and if the unknown zero x is isolated, then there exists a neighborhood of x such that for all start values x0 in that neighborhood, the sequence (xn) will converge towards x. Furthermore, if f '(x) ≠ 0, then the convergence is quadratic, which intuitively means that the number of correct digits roughly doubles in every step.
### Example
Consider the problem of finding the positive number x with cos(x) = x3. We can rephrase that as finding the zero of f(x) = cos(x) - x3. We have f '(x) = -sin(x) - 3x2. Since cos(x) ≤ 1 for all x and x3 > 1 for x>1, we know that our zero lies between 0 and 1. We try a start value of x0 = 0.5.
$\begin{matrix} x_1 & = & x_0 - \frac{f(x_0)}{f'(x_0)} & = & \frac{\cos(0.5) - 0.5^3}{-\sin(0.5) - 3 \times 0.5^2} & = & 1.1121416371 \\ x_2 & = & x_1 - \frac{f(x_1)}{f'(x_1)} & & \vdots & = & 0.909672693736 \\ x_3 & & \vdots & & \vdots & = & 0.867263818209 \\ x_4 & & \vdots & & \vdots & = & 0.865477135298 \\ x_5 & & \vdots & & \vdots & = & 0.865474033111 \\ x_6 & & \vdots & & \vdots & = & 0.865474033101 \\ x_7 & & \vdots & & \vdots & = & 0.865474033102 \end{matrix}$
and the first 12 digits of this value coincide with the first twelve digits of the true zero.
The example in JavaScript. To run it copy the text including the script tags into a new text file, give it a name with the extension and open it in a web browser.
<script>
function NewtonIterationFnct(x) {
return x - (Math.cos(x) - x*x*x) / (-Math.sin(x) - 3*x*x)
}
x = 0.5
for (i=0; i<=99; i++) {
document.write("Iteration " + i + ": ")
document.write(x)
document.write('<br>')
xold = x
x = NewtonIterationFnct(x)
if (x == xold) break
}
</script>
### Practical considerations
Although the method is very efficient, there are a number of practical issues that must be taken into account. First of all, Newton's method requires that the derivative be calculated directly. In cases where the derivative is approximated by taking the slope of two points on the function, the method becomes inefficient and inferior to other algorithms. Second, if the start value is too far removed from the true zero, Newton's method can get stuck in an infinite loop without producing improved approximations. Because of this, all practical implementations of Newton's method include code for bound checking.
### Generalization
One may use Newton's method also to solve systems of n (non-linear) equations, which amounts to finding the zeros of continuously differentiable functions F : Rk -> Rk. In the formulation given above, one then has to multiply with the inverse of the k-by-k Jacobian matrix F '(xn) instead of dividing by f '(xn). Rather than actually computing the inverse of this matrix, one can save time by solving the system of linear equations
$F'(x_n) (x_{n+1} - x_n) = -F(x_n)$
for the unknown xn+1 - xn. Again, this method only works if the initial value x0 is close enough to the true zero. Typically, a region which is well behaved is located first with some other method and Newton's method is then used to "polish" a root which is already known approximately.
The method can also be applied to find zeros of complex functions. For many complex functions, the set of all start values that cause the method to converge to the true zero (the "basin of attraction") is a fractal.
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Battle Creek, Michigan ... older. The average household size is 2.43 and the average family size is 3.04. In the city the population is spread out with 27.2% under the age of 18, 8.7% from 18 to 24, ... | 2022-01-22 09:26:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.8343386650085449, "perplexity": 497.71003788036177}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303779.65/warc/CC-MAIN-20220122073422-20220122103422-00433.warc.gz"} |
https://blender.stackexchange.com/questions/62789/copy-an-nodes-to-another-file?noredirect=1 | # Copy AN nodes to another file
Since the last releases of Blender, there is no more the possibility to copy/paste nodes between files (that was hyper-useful). Now the only solutions is to Group/Append Group. But I can't group AN nodes neither... so the question is not "how to group AN nodes", but "how to move them to another file", if there is another way to do that.
PS: About grouping AN nodes, I think I'll open another question, as you see on the screenshot - the group options are grayed out...
The only way I know of it's importing (appending in blender terms) the entire node-tree :/ File->Append->select the blend file to copy from-> click NodeTree-> click the tree of interest. Definitively not user friendly considering the more common case to import into your current tree, but you can invoke the subprograms inside the newly imported node-tree and use it in your main tree. Another way would be to add the code to recreate to the templates dir and repackage-reinstall the addon (or just add them directly in your addon's installation directory) for example:
[...]
sourceObjectNode = self.newNode("an_DataInputNode", x = 0, y = 0, label = "Source Object")
sourceObjectNode.assignedType = "Object"
timeInfoNode = self.newNode("an_TimeInfoNode", x = 0, y = -200)
[...]
See here for the full explanation of how to take advantage of multiple node-tree
Currently a simple to do that is to append the whole nodetree data from old file to new file, then copy wanted nodes (or delete unwanted ones) from it. Here is the way:
File > Append, then click on the old file, find NodeTree folder, then choose the nodetree datablock you want to append.
Tip: Relatively, you can completely remove an AN nodetree by hitting spacebar in Node Editor, then search for "Remove Animation Node Tree". | 2020-02-26 20:14:39 | {"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": 0.45763710141181946, "perplexity": 2992.3930466057627}, "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/1581875146485.15/warc/CC-MAIN-20200226181001-20200226211001-00162.warc.gz"} |
https://nadre.ethernet.edu.et/record/4495/export/csl | Thesis Open Access
THE IMPACT OF PROCUREMENT METHODS IN AWARDING ROAD PROJECTS ON DOMESTIC CONTRACTORS
Citation Style Language JSON Export
{
"abstract": "<p>The participation of domestic contractors in the road construction industry has many impacts<br>\non the industry as well as in the economic development of the country. However, most of the<br>\nprojects are awarded for foreign contractors and the participation of domestic contractors in<br>\nthe road sector has much less than the foreigners. There are problems that contributing to this<br>\nlow participation of domestic contractors.As indicated by different writers, the problems emerge<br>\nthose related with the procurement process. Accordingly, this research attempts to assess the<br>\nconstruction works procurement practices in federal road project and to investigate problems,<br>\nimpacts of construction works procurement practices toward the domestic contractors and<br>\nfinally to show directions in improving domestic contractors’ participation in federal road<br>\nprojects construction work. To achieve this objective, the research work had been developed<br>\nthrough the literature review of different organizational report, questionnaire survey and case<br>\nstudy. The outcome of these research shows that the procurement process has impact on the<br>\nparticipation of domestic contractors in road projects. The impacts were identified and high<br>\nAnnual turnover requirement and performance requirement has been the main problem for low<br>\nlevel participation of domestic contractors on tenders as it has the first rank among all with RII<br>\n= 0.92 and 0.9 respectively. Also Past projects experience (RII=0.87) and cash flow requirement<br>\n(RII=0.86) are identified as the most demanding problems to local contractors that impede their<br>\nparticipation in road projects. Due to this problem, the domestic contractors are not capable to<br>\nqualify technically due to Annual turnover requirement, past projects experience, cash flow<br>\nrequirement and performance requirement. Since they are not qualified for technical<br>\nrequirement they will not be able to participate financial requirement, this affects the<br>\ncontractors not to succeed on the tender. Simply stated, contractors go out of business because<br>\nthey run out of money. Finally, the study recommends for enhancing the development of<br>\nDomestic contractors: by building Capacity of local contractors: Technology transfer through<br>\njoint ventures. Also the Public Procurement and Property Administration Agency should be open<br>\nto see and study the different alternative procurement methods applicable in the world which<br>\nwill be suitable for domestic contractors and reforming the contract documents of Government<br>\nfinanced projects as required</p>",
"author": [
{
}
],
"id": "4495",
"issued": {
"date-parts": [
[
2020,
1,
17
]
]
},
"language": "eng",
"title": "THE IMPACT OF PROCUREMENT METHODS IN AWARDING ROAD PROJECTS ON DOMESTIC CONTRACTORS",
"type": "thesis"
}
135
65
views | 2020-04-06 13:57:55 | {"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": 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": 0.17070040106773376, "perplexity": 9850.489549352526}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371637684.76/warc/CC-MAIN-20200406133533-20200406164033-00385.warc.gz"} |
https://discuss.codechef.com/questions/40666/calculate-fn-afn-1-bfn-2-matrix-exponentiation | ×
calculate f(n) = a*f(n-1) + b*f(n-2) matrix exponentiation
-1 Given f(1)=1 , f(2)=1 , and f(n) = 100*f(n-1)+200f(n-2) . Now , i have to calculate f(n) rof n upto 10^9 , so obvious approach id to use matrix exponentiation .. I came up with this logic , but it gives wrong output . Can you please explain the reason ? It fails with n=3 , but i am not able to come up with how to resolve it . #include using namespace std; #define ll long long ll m=1000000007; ll x,y,z,w; void ml(ll F[2][2],ll M[2][2]) { x = ((F[0][0]%m)*(M[0][0]%m))%m + ((F[0][1]%m)*(M[1][0]%m))%m; y = ((F[0][0]%m)*(M[0][1]%m))%m + ((F[0][1]%m)*(M[1][1]%m))%m; z = ((F[1][0]%m)*(M[0][0]%m))%m + ((F[1][1]%m)*(M[1][0]%m))%m; w = ((F[1][0]%m)*(M[0][1]%m))%m + ((F[1][1]%m)*(M[1][1]%m))%m; F[0][0] = x; F[0][1] = y; F[1][0] = z; F[1][1] = w; //cout<
1 Use this matrix for F and M: 100 1 200 0 To get result matrix for n you need (n - 2) power of F. And answer is: (F[0][0] + F[1][0]) % m; answered 30 Mar '14, 15:54 5★bulatov 126●3 accept rate: 50% please explain it also , i want to learn how to find it . (30 Mar '14, 16:19) ac_c0der2★ 2 (30 Mar '14, 16:25) kuruma2★
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last updated: 30 Mar '14, 16:25 | 2018-03-24 14:23:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8355162739753723, "perplexity": 8809.855308578784}, "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-2018-13/segments/1521257650685.77/warc/CC-MAIN-20180324132337-20180324152337-00109.warc.gz"} |
https://quantumcomputing.stackexchange.com/questions/16048/ibmq-total-samples-exceeds-the-maximum-number-of-samples-for-channel-8002 | # IBMQ - Total samples exceeds the maximum number of samples for channel [8002]
I'm trying to run some pulse schedules on ibmq_bogota but I'm running into this error:
There are 99 "circuits"/pulse schedules in this job. When I try to queue just a subset of 98 schedules^ instead, I get
Total samples exceeds the maximum number of samples for channel CR1_0. (413168 > 258144) [8002]
So it seems like this might be a problem with having too many circuits in one job.
My questions are: what is a 'sample' in a channel, what is channel CR1_0, how can I calculate how many samples is in my job, and where can I find the maximum 258144 number for this backend.
The max_experiments of this backend is
>>> backend.configuration().max_experiments
900
and max_shots is
>>> backend.configuration().max_shots
8192
Thanks.
Using IBMQJobManager does not work either, but at least I can set max_experiments_per_job to some very low bound so that all the jobs will execute.
One of my pulse schedules:
The others are basically the same with different initial u1 pulse duration.
^I've tried queuing a single pulse schedule with the longest u1 pulse duration and that was successful, so I guess it's not related to how long each individual pulse schedule is.
• Hi, I don't know much about Pulse, but the error code 8002 seems to have as a solution "reduce number of samples below specified limit" (see here). Also, I just looked at it a few hours ago and never used it, but thought about it when seeing your problem, the job manager, that could help solve your problem since it runs jobs based on "backend restrictions". Hope it works :) – Lena Feb 15 at 14:34
• @Lena Thanks for your suggestion. I was going to try job manager later if there's no other ideas. A bit new to this qiskit and ibmq stuff so still trying to figure out the details and "best practices" haha. – eugenhu Feb 15 at 14:39 | 2021-05-07 06:46:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.4556915760040283, "perplexity": 1301.3964654583967}, "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-2021-21/segments/1620243988775.25/warc/CC-MAIN-20210507060253-20210507090253-00154.warc.gz"} |
https://socratic.org/questions/what-is-a-solution-to-the-differential-equation-y-xy-0 | # What is a solution to the differential equation y'-xy=0?
Jul 10, 2016
$y = C {e}^{{x}^{2}}$
#### Explanation:
this is separable
$y ' = x y$
$\frac{1}{y} \setminus y ' = x$
$\int \setminus \frac{1}{y} \setminus y ' \mathrm{dx} = \int x \setminus \mathrm{dx}$
$\int \setminus \frac{1}{y} \setminus \mathrm{dy} = \int x \setminus \mathrm{dx}$
$\ln y = {x}^{2} + C$
$y = {e}^{{x}^{2} + C} = C {e}^{{x}^{2}}$
NB using C as a generic pointer for constant term. | 2021-12-08 12:36:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 7, "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": 0.6142387390136719, "perplexity": 2314.779260052985}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363510.40/warc/CC-MAIN-20211208114112-20211208144112-00538.warc.gz"} |
http://mathhelpforum.com/calculus/93022-big-oh-notation-help.html | Math Help - Big OH Notation help
1. Big OH Notation help
Problem 1. Find a function g such that $N^2$ = Θ (g(n)).
Problem 2. Find a function g such that $2N^2$= Θ (g(n)).
I need help with these two problems.
Ive never seen the theta sign before.
from what I understand theta means that, T(N) = Θ(h(N)) if and only if T(N) = O(h(N)) and T(N) = Ω(h(N)).
for the first would $2n^2$ work?
2. If $f(n)=\Theta(g(n))$ then $c_1g(n) \leq f(n) \leq c_2g(n)$ for some positive real constants $c_1$ and $c_2$.
In other words, what we're looking for is the order of $f(n)$.
What is the order of $C\cdot n^2$ where $C$ is some constant?
3. Originally Posted by Spec
If $f(n)=\Theta(g(n))$ then $c_1g(n) \leq f(n) \leq c_2g(n)$ for some positive real constants $c_1$ and $c_2$.
In other words, what we're looking for is the order of $f(n)$.
What is the order of $C\cdot n^2$ where $C$ is some constant?
would it be just n^2 for both? | 2015-07-04 12:26:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 17, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9337276816368103, "perplexity": 333.40987965193244}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375096706.9/warc/CC-MAIN-20150627031816-00107-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://codeforces.com/problemset/problem/1279/B | B. Verse For Santa
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output
New Year is coming! Vasya has prepared a New Year's verse and wants to recite it in front of Santa Claus.
Vasya's verse contains $n$ parts. It takes $a_i$ seconds to recite the $i$-th part. Vasya can't change the order of parts in the verse: firstly he recites the part which takes $a_1$ seconds, secondly — the part which takes $a_2$ seconds, and so on. After reciting the verse, Vasya will get the number of presents equal to the number of parts he fully recited.
Vasya can skip at most one part of the verse while reciting it (if he skips more than one part, then Santa will definitely notice it).
Santa will listen to Vasya's verse for no more than $s$ seconds. For example, if $s = 10$, $a = [100, 9, 1, 1]$, and Vasya skips the first part of verse, then he gets two presents.
Note that it is possible to recite the whole verse (if there is enough time).
Determine which part Vasya needs to skip to obtain the maximum possible number of gifts. If Vasya shouldn't skip anything, print 0. If there are multiple answers, print any of them.
You have to process $t$ test cases.
Input
The first line contains one integer $t$ ($1 \le t \le 100$) — the number of test cases.
The first line of each test case contains two integers $n$ and $s$ ($1 \le n \le 10^5, 1 \le s \le 10^9$) — the number of parts in the verse and the maximum number of seconds Santa will listen to Vasya, respectively.
The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^9$) — the time it takes to recite each part of the verse.
It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$.
Output
For each test case print one integer — the number of the part that Vasya needs to skip to obtain the maximum number of gifts. If Vasya shouldn't skip any parts, print 0.
Example
Input
3
7 11
2 9 1 3 18 1 4
4 35
11 9 10 7
1 8
5
Output
2
1
0
Note
In the first test case if Vasya skips the second part then he gets three gifts.
In the second test case no matter what part of the verse Vasya skips.
In the third test case Vasya can recite the whole verse. | 2021-08-04 07:48:01 | {"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": 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": 0.2909266948699951, "perplexity": 1347.0948941079848}, "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/1627046154796.71/warc/CC-MAIN-20210804045226-20210804075226-00315.warc.gz"} |
https://www.jobilize.com/course/section/gauss-theorem-by-openstax?qcr=www.quizover.com | # 3.4 Gauss' theorem
Page 1 / 1
A simple exposition of Gauss's theorem or the divergence theorem.
## Gauss' theorem
Consider the following volume enclosed by a surface we will call $S$ .
Now we will embed $S$ in a vector field:
We will cut the the object into two volumes that are enclosed by surfaces we will call ${S}_{1}$ and ${S}_{2}$ .
Again we embed it in the same vectorfield. It is clear that flux through ${S}_{1}$ + ${S}_{2}$ is equal to flux through $S\text{.}$ This is because the flux through one side of the plane is exactly opposite to theflux through the other side of the plane: So we see that ${\oint }_{S}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{a}={\oint }_{{S}_{1}}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{{a}_{1}}+{\oint }_{{S}_{2}}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{{a}_{2}}\text{.}$ We could subdivide the surface as much as we want and so for $n$ subdivisions the integral becomes:
${\oint }_{S}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{a}=\sum _{i=1}^{n}{\oint }_{{S}_{i}}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{{a}_{i}}\text{.}$ What is ${\oint }_{{S}_{i}}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{{a}_{i}}$ .? We can subdivide the volume into a bunch of littlecubes:
To first order (which is all that matters since we will take the limit of a smallvolume) the field at a point at the bottom of the box is ${F}_{z}+\frac{\Delta x}{2}\frac{\partial {F}_{z}}{\partial x}+\frac{\Delta y}{2}\frac{\partial {F}_{z}}{\partial y}$ where we have assumed the middle of the bottom of the box is the point $\left(x+\frac{\Delta x}{2},y+\frac{\Delta y}{2},z\right)$ . Through the top of the box $\left(x+\frac{\Delta x}{2},y+\frac{\Delta y}{2},z+\Delta z\right)$ you get ${F}_{z}+\frac{\Delta x}{2}\frac{\partial {F}_{z}}{\partial x}+\frac{\Delta y}{2}\frac{\partial {F}_{z}}{\partial y}+\Delta z\frac{\partial {F}_{z}}{\partial z}$ Through the top and bottom surfaces you get Flux Top - Flux bottom
Which is $\Delta x\Delta y\Delta z\frac{\partial {F}_{z}}{\partial z}=\Delta V\frac{\partial {F}_{z}}{\partial z}$
Likewise you get the same result in the other dimensionsHence ${\oint }_{{S}_{i}}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{{a}_{i}}=\Delta {V}_{i}\left[\frac{\partial {F}_{x}}{\partial x}+\frac{\partial {F}_{y}}{\partial y}+\frac{\partial {F}_{z}}{\partial z}\right]$
or ${\oint }_{{S}_{i}}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{{a}_{i}}=\stackrel{⃗}{\nabla }\cdot \stackrel{⃗}{F}\Delta {V}_{i}$ $\begin{array}{c}{\oint }_{S}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{a}=\sum _{i=1}^{n}{\oint }_{{S}_{i}}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{{a}_{i}}\\ =\sum _{i=1}^{n}\stackrel{⃗}{\nabla }\cdot \stackrel{⃗}{F}\Delta {V}_{i}\end{array}$
So in the limit that $\Delta {V}_{i}\to 0$ and $n\to \infty$ ${\oint }_{S}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{a}={\oint }_{V}\stackrel{⃗}{\nabla }\cdot \stackrel{⃗}{F}dV$
This result is intimately connected to the fundamental definition of the divergence which is $\stackrel{⃗}{\nabla }\cdot \stackrel{⃗}{F}\equiv \underset{V\to 0}{{lim}}\frac{1}{V}{\oint }_{S}\stackrel{⃗}{F}\cdot d\stackrel{⃗}{a}$ where the integral is taken over the surface enclosing the volume $V$ . The divergence is the flux out of a volume, per unit volume, in the limit ofan infinitely small volume. By our proof of Gauss' theorem, we have shown that the del operator acting on a vector field captures this definition.
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Got questions? Join the online conversation and get instant answers! | 2020-08-14 23:25:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 25, "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": 0.6476476192474365, "perplexity": 1419.3796130976914}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439740343.48/warc/CC-MAIN-20200814215931-20200815005931-00087.warc.gz"} |
https://askdev.io/questions/200/internationalization-in-your-tasks | Just how have you applied Internationalization (i18n) in real tasks you've worked with?
I took a passion in making software program cross-cultural after I read the renowned blog post by Joel, The Absolute Minimum Every Software Developer Absolutely, Positively Must Know About Unicode and Character Sets (No Excuses!). Nonetheless, I have yet to able to capitalize on this in an actual task, besides seeing to it I made use of Unicode strings where feasible. Yet making all your strings Unicode and also guaranteeing you recognize what inscribing every little thing you collaborate with remains in is simply the pointer of the i18n iceberg.
Every little thing I have actually worked with to day has actually been for usage by a regulated set people English talking individuals, or i18n simply had not been something we had time to work with prior to pressing the task live. So I am seeking any kind of pointers or battle tales individuals have concerning making software program extra local in real life tasks.
0
2019-05-04 00:15:14
Source Share
It has actually been a while, so this is not thorough.
Personality Sets
Unicode is wonderful, yet you can not escape overlooking various other personality collections. The default personality set on Windows XP (English) is Cp1252. On the internet, you do not recognize what a browser will certainly send you (though with any luck your container will certainly take care of a lot of this). And also do not be stunned when there are pests in whatever execution you are making use of. Personality collections can have intriguing communications with filenames when they relocate to in between equipments.
Converting Strings
Translators are, usually talking, not programmers. If you send a resource documents to a translator, they will certainly damage it. Strings needs to be removed to source documents (as an example buildings documents in Java or source DLLs in Visual C+npls ). Translators need to be offered documents that are hard to damage and also devices that do not allow them damage them.
Translators do not recognize where strings originate from in an item. It is hard to convert a string without context. If you do not give advice, the top quality of the translation will certainly endure.
While when it come to context, you might see the very same string "foo" emerge in numerous times and also assume it would certainly be extra reliable to have all circumstances in the UI indicate the very same source. This is a negative suggestion. Words might be really context - delicate in some languages.
Converting strings prices loan. If you release a new variation of an item, it makes good sense to recoup the old variations. Have devices to recoup strings from your old source documents.
String concatenation and also hand-operated adjustment of strings need to be decreased. Make use of the layout operates where relevant.
Translators require to be able to change hotkeys. Ctrl+P is print in English ; the Germans make use of Ctrl+D.
If you have a translation procedure that calls for a person to by hand reduce and also paste strings any time, you are throwing down the gauntlet.
Days, Times, Calendars, Currency, Number Formats, Time Zones
These can all differ from nation to nation. A comma might be made use of to represent decimal areas. Times might remain in 24hour notation. Not every person makes use of the Gregorian schedule. You require to be distinct, also. If you make sure to present days as MM/DD/YYYY for the USA and also DD/MM/YYYY for the UK on your internet site, the days are unclear unless the customer recognizes you've done it.
Specifically Currency
The Locale operates given in the class collections will certainly offer you the neighborhood money icon, yet you can not simply stick an extra pound (sterling) or euro icon before a value that offers a rate in bucks.
Interface
Layout need to be vibrant. Not just are strings most likely to double in size on translation, the whole UI might require to be upside down (Hebrew ; Arabic) to make sure that the controls range from right to left. Which is prior to we reach Asia.
Examining Prior To Translation
• Use fixed evaluation of your code to situate troubles. At a bare minimum, utilize the devices constructed right into your IDE. (Eclipse customers can most likely to Window > Preferences > Java > Compiler > Errors/Warnings and also look for non - externalised strings.)
• Smoke examination by imitating translation. It isn't hard to parse a source documents and also change strings with a pseudo - converted variation that increases the size and also inserts fashionable personalities. You do not need to talk a language to make use of an international os. Modern systems need to allow you visit as an international customer with converted strings and also international place. If you know with your OS, you can identify what does what without recognizing a solitary word of the language.
• Key-board maps and also personality set referrals are really valuable.
• Virtualisation would certainly be really valuable below.
Non - technological Issues
Sometimes you need to be delicate to social distinctions (offense or incomprehension might result). A blunder you usually see is making use of flags as an aesthetic sign picking an internet site language or location. Unless you desire your software program to proclaim sides in international national politics, this is a negative suggestion. If you were French and also supplied the alternative for English with St. George's flag (the flag of England is a red cross on a white area), this could cause complication for several English audio speakers - think comparable concerns will certainly emerge with international languages and also nations. Symbols require to be vetted for social importance. What does a thumbs - up or an environment-friendly tick suggest? Language needs to be reasonably neutral - resolving customers in a certain fashion might serve in one area, yet taken into consideration discourteous in an additional.
Resources
C+npls and also Java designers might locate the ICU internet site valuable : http://www.icu-project.org/
0
2019-05-08 18:23:54
Source
I was simply paying attention to a Podcast from Scott Hanselman today, where he speaks about internationalization, specifically the actually complicated points, like Turkish (with it's 4 i's ) and also Thai. Additionally, Jeff Atwood had a post :
0
2019-05-07 17:48:40
Source
Some enjoyable points :
1. Having a PHP and also MySQL Application that functions well with German and also French, today requires to sustain Russian and also Chinese. I assume I relocate this over to.net, as PHP's Unicode assistance is - in my point of view - not actually excellent. Certain, handling about with utf8_de/ inscribe or the mbstring-functions is enjoyable. Virtually as enjoyable as having Freddy Krüger see you during the night ...
2. Realizing that some languages are a LOT extra Verbose than others. German is a LOT extra verbose than English generally, and also seeing just how the German Version damages the User Interface due to the fact that inadequate room was alloted was not enjoyable. Some items obtained some popularity for their imaginative means to function around that, with Oblivion's "Schw.Tr.d.Le.En.W." being remarkable :- )
3. Playing about with day layouts, woohoo! Yes, there ARE in fact individuals worldwide that make use of day layouts where the day enters the center. Sooooo much enjoyable searching for out what 07/02/2008 is intended to suggest, even if some customers could think maybe July 2 ... But after that once more, you individuals over the fish pond might think the very same concerning customers that placed the month between :- P, specifically due to the fact that in English, July 2 appears a whole lot far better than 2nd of July, something that does not neccessarily relate to various other languages (i.e. in German, you would certainly never ever claim Juli 2 yet constantly Zweiter Juli ). I make use of 2008-02-07 whenever feasible. It's clear that it suggests February 7 and also it types effectively, yet dd/mm vs. mm/dd can be an actually complicated trouble.
4. Anoter enjoyable point, Number formats! 10.000,50 vs 10,000.50 vs. 10 000,50 vs. 10' 000,50 ... This is my largest problem now, needing to sustain a multi-cultural environent yet not having any kind of means to accurately recognize what number layout the customer will certainly make use of.
5. Official or Informal. In some language, there are 2 means to resolve individuals, an official means and also an extra casual means. In English, you simply claim "You", yet in German you need to determine in between the official "Sie" and also the casual "Du", very same for French Tu/Vous. It's generally a sure thing to pick the official means, yet this is conveniently forgotten.
6. Schedules. In Europe, the first day of the Week is Monday, whereas in the United States it's Sunday. Schedule Widgets behave. Revealing a Calendar with Sunday left wing and also Saturday on the right to a European customer is not so wonderful, it perplexes them.
0
2019-05-07 17:34:40
Source
I worked with a task for my previous company that used.NET, and also there was a constructed in.resx layout we made use of. We primarily had a documents that had all translations in the.resx documents, and afterwards numerous documents with various translations. The effect of this is that you need to be really attentive concerning making certain that all strings noticeable in the application are saved in the.resx, and also anytime one is transformed you need to upgrade all languages you sustain.
If you get careless and also do not alert individuals accountable of translations, or you installed strings without experiencing your localization system, it will certainly be a problem to attempt and also repair it later on. In a similar way, if localization is an afterthought, it will certainly be really hard to established. Profits, if you do not have all noticeable strings saved on the surface in a typical area, it will certainly be really hard to locate all that require to be local.
Another note, really purely stay clear of concatenating noticeable strings straight, such as
String message = "The " + item + " is on sale!";
Rather, you have to make use of something like
String message = String.Format("The {0} is on sale!", item);
The factor for this is that various languages usually get words in different ways, and also concatenating strings straight will require a new construct to deal with, yet if you made use of some sort of string substitute device like above, you can change your.resx documents (or whatever localization submits you make use of ) for the details language that requires to reorder words.
0
2019-05-07 16:54:36
Source | 2022-01-19 08:07:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24806761741638184, "perplexity": 3132.0197335560706}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301264.36/warc/CC-MAIN-20220119064554-20220119094554-00411.warc.gz"} |
https://trtools.readthedocs.io/en/latest/source/qcSTR.html | # qcSTR¶
qcSTR generates plots that are useful for diagnosing issues in TR calling.
## Usage¶
qcSTR takes as input a VCF file and outputs several plots in pdf format. To run qcSTR, use the following command:
qcSTR \
--vcf <file.vcf> \
--out <string> \
Required Parameters:
• --vcf <string>: Input VCF file
• --out <string>: Prefix to name output files
General Optional Parameters:
• --vcftype <string>: Type of the input VCF file. Causes qcSTR to fail out if the file is any other type of VCF. Default = auto. Must be one of: gangstr, advntr, hipstr, eh, popstr.
• --samples <string>: File containing list of samples to include. If not specified, all samples are used. Samples in the list that are not included in the input vcf or are misspelled are silently ignored.
• --period <int>: Restrict to TRs with this motif length. e.g. to restrict to dinucleotide repeats, use --period 2.
• --version: Show the version number of qcSTR and exit.
If you wish to run qcSTR on a more complicated subset of the input VCF, we suggest you use dumpSTR or bcftools view to filter the input VCF first, and then run qcSTR on the vcf those commands outputed.
Additional options to customize various plots output by qcSTR are described in the sections Quality Plot Options and Reference Bias Plot Options below.
See Example Commands below for example qcSTR commands for different supported TR genotypers based on example data files in this repository.
## Outputs¶
qcSTR outputs the following plots:
<outprefix>-sample-callnum.pdf: a barplot giving the number of calls for each sample. Can be used to determine failed or outlier samples.
<outprefix>-chrom-callnum.pdf: a barplot giving the number of calls for each chromosome. Can be useful to determine if the expected number of calls per chromosome are present.
<outprefix>-diffref-histogram.pdf: a histogram of, for each allele called, the difference between its length and the length of the reference at that locus (measured in number of repeat units). Can be used to visualize if there is a strong bias toward calling deletions vs. insertions compared to the reference, which might indicate a problem.
<outprefix>-diffref-bias.pdf: plots reference length (bp) vs. the mean (or median) difference in length of each allele called compared to the reference allele. It is expected that the mean difference should be around 0 for most settings. When this value starts to deviate from 0, e.g. for very long repeats, it could indicate a drop in call quality. The red line gives the cumulative fraction of TRs below each reference length.
<outprefix>-quality.pdf: plots the cumulative distribution of the quality scores for calls in this VCF. This will be a per-locus plot for >5 samples, or a sample-stratified plot for <= 5 samples. This will not be produced for vcfs which do not have quality metrics. Alternatively, you may specify the type(s) of quality plot(s) you wish to see with the --quality option. In that case you will get a file named <outprefix>-quality-<type>.pdf for each type of plot you requested. Quality plot examples are shown below. To learn more about how qcSTR infers quality scores for VCFs from different genotypers, see here
Note: quality score plots are useful when considered in the context of a single genotyper run, and can also be used to compare different invocations of the same genotyper. However, quality score values produced by different genotypers have different meanings due to the different modeling assumptions the different genotypers make. If you wish to compare quality scores across genotypers, you must first understand those different assumptions and infer how they will affect the validity of your comparison.
## Quality Plot Options¶
These additional options can be used to customize quality score distribution plots.
--quality: This option determines if the plot is stratified and what distribution the y-axis represents. The x-axis is always the quality score and the y-axis is always a decreasing CDF. This can be specified multiple times to produce multiple plots (e.g. --quality per-locus --quality per-sample). Each plot will have the specified type appended to the output filename.
• per-locus Compute the call quality at each locus averaged across all samples. Plot the distribution of those loci qualities.
• sample-stratified Plot a separate line for each sample of the distribution of loci qualities for that sample. Note: If you specify this for a vcf with many samples, the code may slow to a halt, run out of memory and/or the plot may be cluttered.
• per-sample Compute the call quality for each sample averaged across all loci. Plot the distribution of those sample qualities.
• locus-stratified Plot a separate line for each locus of the distribution of sample qualities at that locus. Note: If you specify this for a vcf with many loci, the code may slow to a halt, run out of memory and/or the plot may be cluttered.
• per-call Plot the distribution of the quality of all calls. Note: If you specify this for a large vcf the code may run out of memory.
--quality-ignore-no-call - by default, (sample, locus) pairs which were not called are treated as calls with zero quality for the the quality plot. With this option enabled, instead they are ignored. This may cause the plotting to crash if it causes some samples/loci to have <= 1 valid call.
## Reference Bias Plot Options¶
These additional options can be used to customize reference bias plots.
• --refbias-binsize <int>: Sets the binsize (in bp) used to bin x-axis values, which give the reference TR length. Default=5.
• --refbias-metric <string>: Determines which metric to use to summarize the reference bias in each bin. Default=mean. Must be one of: mean or median.
• --refbias-mingts <int>: Exclude points computed using fewer than this many genotypes. This option is meant to avoid plotting outlier points driven by bins with small numbers of TRs with that reference length. Default=100.
• --refbias-xrange-min <int>: Exclude points corresponding to TRs with reference length less than this value.
• --refbias-xrange-max <int>: Exclude points corresponding to TRs with reference length greater than this value.
## Example Commands¶
Below are qcSTR examples using VCFs from supported TR genotypers. Data files can be found at https://github.com/gymreklab/TRTools/tree/master/example-files:
# AdVNTR | 2022-12-06 14:49:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5002219080924988, "perplexity": 3125.584344736046}, "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/1669446711108.34/warc/CC-MAIN-20221206124909-20221206154909-00012.warc.gz"} |
https://www.physicsforums.com/threads/square-integrable-functions-hilbert-space-and-light-on-dirac-notation.621212/ | # Square integrable functions - Hilbert space and light on Dirac Notation
1. Jul 16, 2012
### esornep
Square integrable functions -- Hilbert space and light on Dirac Notation
I started off with Zettilis Quantum Mechanics .... after being half way through D.Griffiths ..... Now Zettilis precisely defines what a Hibert space is and it includes the Cauchy sequence and convergence of the same ..... is there any proof for the same for square integrable functions as he skips the same in the book .....
The second question he says Ʃ|θn><θn| = I where I stands for the operator .... now is this not going to infinity and please suggest some text on operators from scratch .... Thanks
2. Jul 16, 2012
### esornep
Re: Square integrable functions -- Hilbert space and light on Dirac Notation
The Dirac Notation says
|si> is ket and its conjugate is the the bra ??? Is this consistent ...... please explain
3. Jul 16, 2012
### vanhees71
Re: Square integrable functions -- Hilbert space and light on Dirac Notation
To your first question: The space of square integrable functions, describing a scalar non-relativistic particle, is defined as the vector space of functions $\psi:\mathbb{R}^3 \rightarrow \mathbb{C}$ with a sesquilinear form
$$\langle{\psi_1}|{\psi_2} \rangle = \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \; \psi_1^*(\vec{x}) \psi_2(\vec{x}).$$
This scalar product induces a metric on this space
$$\|\psi \|=\sqrt{\langle \psi|\psi \rangle}.$$
To make this a pre-Hilbert space, you have to identify functions, for which
$$\|\psi_1-\psi_2\|=0.$$
I.e., you don't distinguish between such two functions. One can show that this pre Hilbert space is in fact a true Hilbert space, i.e., it is complete in the sense of the topology induced by the norm.
To your second question: Since you have a non-degenerate scalar product you can identify any continuous linear form with a vector and vice versa. If you have a Hilbert-space vector $|\psi \rangle$ the corresponding linear form is written as $\langle \psi|$ and this already suggests how it acts on another Hilbert-space vector: $|\phi \rangle \mapsto \langle \psi|\phi \rangle$.
In quantum mechanics you need a bit more than that, namely also distributions since you have to deal with unbounded (mostly self-adjoint) operators (like the position and momentum operators) that have a restricted domain, but that should be explained in your book.
4. Jul 16, 2012
### Fredrik
Staff Emeritus
Re: Square integrable functions -- Hilbert space and light on Dirac Notation
See this post.
If you're wondering what linear functionals have to do with conjugates, you will need to study the relationship between linear operators and matrices. It's explained in post #3 in this thread. (Ignore the quote and the stuff below it).
"Linear algebra done right", Sheldon Axler. (This one only deals with finite-dimensional vector spaces).
"Introductory functional analysis with applications", Erwin Kreyszig.
Most people only study finite-dimensional vector spaces and hope that there's some way of making sense of what they're doing in the context of infinite-dimensional vector spaces.
Last edited: Jul 16, 2012
5. Jul 16, 2012
### micromass
Re: Square integrable functions -- Hilbert space and light on Dirac Notation
This is actually a very tricky question. The answer relies crucially on what we mean with square integrable. Sure, you can say that it are functions such that
$$\int |\psi|^2$$
is finite. But what is that integral?? If you interpret that integral as just a Riemann integral, then the space of square integrable functions will not be complete!! In order for it to be complete, we need the more advanced notion of Lebesgue integrals. With that integral, it can be shown (but it is a bit tricky to do so) that the square integrable functions are complete.
The sum is not infinity by Bessel's inequality.
Fredrik suggested the wonderful texts by Axler and Kreyszig. I second these texts.
If you want to know more about the Lebesgue integral, then the text "Lebesgue integral on euclidean space" by Jones is excellent and not very hard. | 2018-08-18 15:25:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.8764503002166748, "perplexity": 569.5813270527757}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221213689.68/warc/CC-MAIN-20180818134554-20180818154554-00092.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/131029-find-k-3x3-matrix.html | # Math Help - Find k in a 3x3 matrix
1. ## Find k in a 3x3 matrix
Find K such that the matrix
$
\begin{bmatrix}-2 & -1 & 3 \\6 & 5 & -5 \\ 10+k & 4 & -2\end{bmatrix}
$
is singular.
I have no idea how to go about solving this. Can anyone lead me in the right direction. it would be much appreciated. Thank you
2. Originally Posted by mybrohshi5
Find K such that the matrix
$
\begin{bmatrix}-2 & -1 & 3 \\6 & 5 & -5 \\ 10+k & 4 & -2\end{bmatrix}
$
is singular.
I have no idea how to go about solving this. Can anyone lead me in the right direction. it would be much appreciated. Thank you
A matrix is singular iff its determinant is 0
then det (A) =0 since A is singular
det (A)= (-2*5*-2)+(-1*-5*(10+k))+(72)-(150+15k)-(40)-(12)=0
(20)+(50+5k)+(72)-(150+15k)-(40)-(12)=0
-100-10k+40=0
-60-10k=0
k=-6 | 2014-12-20 16:49:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7954480051994324, "perplexity": 282.8370984058367}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802770043.48/warc/CC-MAIN-20141217075250-00091-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://properhoc.com/high-school-related/ib-mathematics/trigonometry-an-introduction/ | # Trigonometry – an introduction
Up a level : IB Mathematics
Previous page : Derivatives
Next page : Functions, Quiz 1
Trigonometry means measurement of something with three corners, i.e. on triangles. We will here start with right angled triangles in particular.
Say you have a right angled triangle with the hypotenuse (the longest side) having the side 1, and one of the angles is say 30. Then let us draw another right angled triangle with the same angle and with the hypotenuse of length one. What can we now say about the two triangles? Hopefully you can see that they are congruent. That means that the right angled triangle with a hypotheses of length one is totally dependent on the given angle. And in particular, that its base and height (if placed as in the figure) is determined by the angle.
If we change the angle, then these lengths will change too. Let us label the angle θ (the Greek letter theta), then the height that we will call the opposite side since it is opposite to the angle θ is totally dependent on the angle θ. We say that it is a function of the angle.
As with say square root of x (that is a function of x) we want to give it a name. For say the square root of 16 we don’t want to have to say “the positive number such that the number multiplied by itself is 16”. We just say “the square root of 16”. In a similar way we don’t want to say “the length of the opposite side of a right angled triangle with the hypotenuse has the length one and the angle is 30˚ is….” We want to have a short form for this.
We say that the opposite side = sin(30˚).
The word sin (that is not pronounce like the word “sin” but rather lake “sin” in “sinuses”) is an abbreviation of the Latin word sinus meaning curve or bow. The word sinus has then turned to the word sine that is abbreviated as sin (wow, one letter less).
Observe that, by this definition, sin(θ) simply equals the length of the opposite side of a right angled triangle (where the hypotenuse has the length one) given the angle θ.
The base, that we will call the adjacent side, because it is adjacent to (next to) the angle, will also be a function of the angle. We say that the adjacent side =adj= cos(θ). Here “cos” is an abbreviation of the word “cosine” where the “co” is the same as in words like “cooperate”.
So we have:For the particular triangle with θ=30˚ we get:
If we where to draw a triangle as the one above, with a hypotenuse of length 100 cm and we then would measure the length of the base (adjacent side) and the height (opposite site) we would find about 87 cm and 50 cm, and indeed if you take a calculator and calculate cos vs sine of 30˚, and indeed you will find about 0.87 and 0.5 (exactly). If you got something else your calculator angular setting is probably wrong. On a casio fx-9750 you should be in RUN-mode, then you press SHIFT, SET UP (the MENU key), then you go down to the angle setting and change it to DEG (degrees).
You can have a look at this Geogebra file.
Up a level : IB Mathematics
Previous page : Derivatives
Next page : Functions, Quiz 1Last modified: Mar 31, 2019 @ 20:15 | 2021-05-18 20:43:05 | {"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": 0.8459116816520691, "perplexity": 511.59051410870126}, "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-2021-21/segments/1620243991514.63/warc/CC-MAIN-20210518191530-20210518221530-00536.warc.gz"} |
https://www.bonvinlab.org/education/molmod_online/simulation/ | Computational Structural Biology group focusing on dissecting, understanding and predicting biomolecular interactions at the molecular level.
Supported by:
# Molecular Dynamics Simulation of the p53 N-terminal peptide
## General Overview
This tutorial introduces Molecular Dynamics (MD) simulations of proteins. The simulation protocol can be used as a starting point for the investigation of protein dynamics, provided your system does not contain non-standard groups. By the end of this tutorial, you should know the steps involved in setting up, running, and analyzing a simulation, including critically assessing the choices made at the different steps.
## Use of virtual machines (VMs)
For this module of the course we will be using NMRbox. NMRbox offers cloud-based virtual machines for executing various biomolecular software that can complement NMR (Nuclear Magnetic Resonance). NMRbox users can choose from 235 software packages that focus on research topics as metabolomics, molecular dynamics, structure, intrinsically disordered proteins or binding. One can search through all available packages on https://nmrbox.org/software.
### Register
To use virtual machines through NMRbox, one needs to register, preferably with their institutional account here. Since the registration has to be manually validated and it can take up to two business days, we strongly encourage students to do so before the course starts. After a successful validation you will receive an e-mail with your NMRbox username and password that you will be using while accessing your virtual machine.
### Accessing NMRbox
To run the virtual machine on a local computer, one needs to install VNCviewer. With the RealVNC client connects your computer to the NMRbox servers with a virtual desktop - graphical interface. More information about the VNC viewer is in the FAQ of NMRbox.
To choose a virtual machine, first log into the user dashboard https://nmrbox.org/user-dashboard. Download the zip file with bookmarks for the production NMRbox virtual machines here and extract the zip file. Back in VNCviewer click File -> Import connections and select the folder in which you extracted the content of the downloaded zip file. After importing, you will see the current release virtual machines. You can use any available virtual machine. The user-dashboard provides information on machine capabilities and recent compute load, thus it is clever to choose a less occupied one. Double click on one of the VMs. An Authentication panel appears. Enter your NMRbox username and password. Click on the Remember password box to have VNCviewer save your information. By default, your desktop remains running when you disconnect from it. If you login to your VM repeatedly you will see a screen symbol next to the VM you connected to recently. For more details follow the quick start guide for using NMRbox with VNCviewer here.
If everything runs correctly you should have a window with your virtual desktop open. In the virtual desktop you have an access to the internet with Chromium as browser or use various programs, including Pymol. Thus, you could run all three stages of this course here or transfer data between your local machine and the virtual machine. File transfer to and from the VM is quite straightforward and it is described here: https://nmrbox.org/faqs/file-transfer.
## Preparing the System
The preparation of the system is the heart of the simulation. Neglecting this stage can lead to artifacts or instability during the simulation. Each simulation must be prepared carefully, taking into consideration its purpose and the biological and chemical characteristics of the system under study.
### Selecting an initial structure
The first step is obviously the selection of a starting structure. The aim of this tutorial is to simulate a peptide of the N-terminal sequence of the transactivation domain of p53. The sequence of this peptide is given below, in FASTA format:
>P53_MOUSE
SQETFSGLWKLLPPE
Peptides are often very flexible molecules with short-lived secondary structure elements. Some can even adopt different structures depending on which protein partner they are interacting with, remaining in a disordered state if free in solution. As such, the effort of using an advanced method such as homology modeling for this peptide is very likely unwarranted. Instead, it is possible, and plausible, to generate structures of the peptide in three ideal conformations – helical, sheet, and polyproline-2 – which have been shown to represent the majority of the peptides deposited in the PDB. Generating these structures is a simple matter of manipulating backbone dihedral angles. Pymol has a utility script to do so, written by Robert Campbell.
The instructions shown in this tutorial refer only to the helical peptide, for simplicity. The successful completion of the tutorial requires, however, all three conformations to be simulated.
Generate an ideal structure for the peptide sequence using the build_seq script in PyMol, choose between helix/polypro/beta.
Pay attention when typing the sequence! A missing or swapped amino acid will render your simulation useless! Also carefully inspect the generated object whether it matches your expectations - you might want to adjust the residue numbers.
To change residue numbers within PyMol take a look at the help message of the alter command:
help alter
### Preparing the initial structure
In case of structures downloaded from the RCSB PDB, it is important to ensure that there are no missing atoms, as well as check for the presence of non-standard amino acids and other small ligands. Force fields usually contain parameters for natural amino acids and nucleotides, a few post-translational modifications, water, and ions. Exotic molecules such as pharmaceutical drugs and co-factors often have to be parameterized manually, which is a science on its own. Always judge if the presence of these exotic species is a necessity. In some cases, the ligands can be safely ignored and removed from the structure. As for missing residues and atoms, except hydrogens, it is absolutely necessary to rebuild them before starting a simulation. MODELLER is an excellent program for this purpose. In addition, some crystals diffract at a good enough resolution to distinguish water molecules in the density mesh. Save for very particular cases where these waters are the subject of the study, the best policy is to remove them altogether from the structure. Fortunately, most of these “problematic” molecules appear as hetero-atoms (HETATM) in the PDB file, and can therefore be removed rather easily with a simple sed command:
sed -e ‘/^HETATM/d’ 1XYZ.pdb > 1XYZ_clean.pdb
It is also good practice to run additional quality checks on the structure before starting the simulation. The refinement process in structure determination does not always yield a proper orientation of some side-chains, such as glutamine and asparagine, given the difficulty in distinguishing nitrogen and oxygen atoms in the density mesh. Also, the protonation state of several residues depends on the pH and can influence the protein’s hydrogen bonding network. For crystal structures, the PDB_REDO database contains refined versions of structures deposited in the RCSB PDB, which address some of these problems. Alternatively, there are web servers that allow these and other problems to be detected and corrected, such as WHATIF.
Since the initial structure of the p53 peptide was generated using Pymol and ideal geometries, there is no need to proceed with such checks.
### Structure Conversion and Topology Generation
A molecule is defined not only by the three-dimensional coordinates of its atoms, but also by the description of how these atoms are connected and how they interact with each other. The PDB file, which was generated or downloaded in the previous step, contains only the former. The description of the system in terms of atom types, charges, bonds, etc, is contained in the topology, which is specific to the force field used in the simulation. The choice of the force field must then not be taken lightly. For biomolecular systems, there are few major force fields – e.g. CHARMM, AMBER, GROMOS, OPLS – that have been parameterized to reproduce the properties of biological molecules, namely proteins. This has been, and continues to be, an area of active research since the very first day of molecular dynamics simulations. There are several literature reviews available in Pubmed that assess the quality and appropriateness of each force field and their several versions. Some are well-known for their artifacts, such as a biased propensity for alpha-helical conformations. Here, in this tutorial, we use the AMBER99SB-ILDN force field, which is widely used in sampling and folding simulations and has been shown to reproduce fairly well experimental data (source). Another, more practical, reason behind this choice is the availability of this force field in GROMACS.
Since the simulation takes place in a solvated environment, i.e. a box of water molecules, we have also to choose an appropriate solvent model. The model is simply an addition to the force field containing the properties of the water molecule, parameterized to reproduce specific properties such as density and freezing and vaporization temperatures. As such, particular water models tend to be tied to specific force fields. Due to the difficulties of reproducing the properties of water computationally - yes, even for such a simple molecule! - some models represent water with more than 3 atoms, using additional pseudo-particles to improve characteristics such as its electrostatic distribution. The water model suggested to use with the AMBER force field, and which we will use in this simulation, is the TIP3P model (for Transferable Interaction Potential with 3 Points), which was actually developed by the author of the OPLS force field. Really transferable!
This choice is usually limited by the force field, unless there is a specific need for a particular solvent model.
The GROMACS program pdb2gmx takes an initial structure and returns both a topology file (peptide.top) and a new structure (peptide.gro) that adheres to the force field atom naming conventions. To convert the structure and build the topology, pdb2gmx divides the molecule in several blocks, such as amino acids, and uses a force field-specific library of such building blocks to make the necessary conversions. Usually, the matching to the library is done through residue/atom names on each ATOM/HETATM line in the PDB file. If a residue (or atom) is not recognized, the program stops and returns an error.
In case of an error, make sure to read the error message. It often points very clearly to the problem and its solution.
Different force fields define different atom types and/or give different names to the same atom type. While the majority of the heavy atoms, i.e. non-hydrogen, have identical naming across most force fields, hydrogens do not. As such, the flag -ignh indicates that GROMACS should ignore these atoms when reading the structure and (re)generate their coordinates using ideal geometric parameters defined in the force field. Also, the program allows the user to define the status of the termini of the molecule through the -ter flag. Termini can be either charged (e.g. NH3+ and COO-), uncharged (e.g. NH2 and COOH), or capped by an additional chemical group (e.g. N-terminal acetyl and C-terminal amide). This is very important since leaving the termini charged (default) can lead to artificial charge-charge interactions, particular in small molecules. If a peptide is part of a larger structure, then it makes sense to cap the termini in order to neutralize their charge, as it would happen in reality. Read through the output of pdb2gmx and check the choices the program made for histidine protonation states and the resulting charge of the peptide.
The newly generated topology file is also worth some attention. It contains a listing of all the residues and their corresponding atoms, detailing the atom types, masses, and charges. Further, it contains a listing of all the bonds in the molecule, the angles, and the dihedral angles. Note that the topology file does not contain any information on their chemistry. This information is stored in internal parameter libraries that are defined at the very top of the topology file.
Open the peptide.top file in a text editor and browse through it.
; Include forcefield parameters
#include "amber99sb-ildn.ff/forcefield.itp"
[ moleculetype ]
; Name nrexcl
Protein 3
[ atoms ]
; nr type resnr residue atom cgnr charge mass typeB chargeB massB
; residue 1 SER rtp NSER q +1.0
1 N3 1 SER N 1 0.1849 14.01
2 H 1 SER H1 2 0.1898 1.008
3 H 1 SER H2 3 0.1898 1.008
4 H 1 SER H3 4 0.1898 1.008
5 CT 1 SER CA 5 0.0567 12.01
6 HP 1 SER HA 6 0.0782 1.008
7 CT 1 SER CB 7 0.2596 12.01
8 H1 1 SER HB1 8 0.0273 1.008
9 H1 1 SER HB2 9 0.0273 1.008
10 OH 1 SER OG 10 -0.6714 16
11 HO 1 SER HG 11 0.4239 1.008
12 C 1 SER C 12 0.6163 12.01
13 O 1 SER O 13 -0.5722 16 ; qtot 1
SER is in principle a neutral amino acid within a protein sequence. Can you rationalize why in this case the sum of the charges add up to +1?
### Periodic Boundary Conditions
This converted structure includes several atoms, namely hydrogen, that have been added according only to ideal geometric parameters. If generated with Pymol, it also has ideal backbone geometry. If it was otherwise downloaded from the RCSB PDB, the structure is also likely to contain certain chemical aspects (bond lengths, angles, interatomic distances) that are not considered ideal by the force field. In fact, merely changing force fields will cause the definition of ideal to change as well. The first step towards preparing the system is then to remove these “imperfections” as best as possible, which is normally achieved through an energy minimization of the system. This optimization method essentially forces a set of atoms to adhere, as best as possible, to the definitions of the force field. The larger the number of atoms in the system, the harder it is to have all of them to comply ideally with respect to all the definitions. For example, moving two atoms closer to reduce the strain from violating the definitions imposed by the van der Waals forces may cause the strain from the electrostatic term to increase.
Before minimizing the system, a general layout of the simulation setup has to be chosen. In other words, the peptide must be placed somewhere for this minimization to happen. Most modern simulations of proteins and peptides define periodic boundary conditions (PBC), which set a single unit cell that can be stacked infinitely. As a result, an infinite, periodic system is defined that avoids the problem of having hard boundaries (walls) that the molecules can literally bump into. When the protein crosses the wall on the left side, the periodic image to its right enters the current unit cell, maintaining a constant number of atoms in every unit cell. A simpler way to rationalize PBCs is to compare them to the snake game available in old Nokia cell phones. When the head of the snake crosses a boundary of the screen, it re-appears on the diametrically opposed edge.
Schematic representation of the idea of periodic boundary conditions (source).
The choice of the shape of the unit cell is also important, since this will define the volume in which the molecule is simulated. Molecular dynamics simulations are computationally demanding. The more molecules in the system, the more forces need to be calculated at each step. As such, while a cube can be perfectly stacked ad eternum it is not the most efficient shape from a volume standpoint (remember that simulations take place, usually, in a solvated environment!). Approximating the shape to a sphere is ideal, but spheres cannot be stacked. As such, only a few general shapes support the setup of periodic boundary conditions. One of those is the rhombic dodecahedron, which corresponds to the optimal packing of a sphere and is therefore the best choice for a freely rotating molecule such as a peptide or a protein.
Another thing to have in mind when setting up the PBCs is the size of the unit cell. Continuing with the snake analogy, it is not proper to have the snake’s head see its own tail. In other words, the cell must be sufficiently large to allow the molecule to cross the boundaries and still be at a sufficient distance from the next image that no force calculations are made between them. In GROMACS, this setting is defined as a distance from the molecule to the wall of the unit cell. This distance should not be arbitrarily large either, otherwise the box is to large and the simulation becomes computationally inefficient as your purpose is not to simulate water. Take the cutoff used to calculate non-bonded interactions (long range) in the force field as a rule of thumb. The distance to the wall must be larger than this value.
Also important is to consider possible conformational changes. The size of the box should allow for those to occur without introducing period image problems as explained above.
As with pdb2gmx, the GROMACS program editconf generates a sizable output that contains, for example, the volume and dimensions of the unit cell it just created. The dimensions use the triclinic matrix representation, in which the first three numbers specify the diagonal elements ($$xx, yy, zz$$) and the last six the off-diagonal elements ($$xy, xz, yx, yz, zx, zy$$).
What is the volume of the unit cell?
### Energy minimization of the structure in vacuum
Having defined the physical space where simulations can take place, the molecule can now be energy minimized. GROMACS uses a two-step process for any calculation involving the molecules and a force field. First, the user must combine the structure and the topology data, together with the simulation parameters, in a single control file. This file contains everything about the system and ensures the reproducibility of the simulation, provided the same force field is available on the machine. Another advantage of having such a self-contained file is that the preparation can take place in one machine while the calculations run on another. Again, simulations are computationally demanding. While the system can be easily prepared on a laptop, with the help of Pymol, GUI-enabled text editors, and all the other advantages of having a screen, calculations usually run on specialized clusters with hundreds of processing cores that provide only a command-line interface access. This will be relevant when running the production simulation. The intermediate calculations to prepare the system are comfortably small to run on a laptop.
The simulation parameters are contained in a separate file, usually with the .mdp extension. For simplicity, we provide these files in our GitHub repository) and also already in our virtual image in NMRBox, if you are using it (see \$MOLMOD_DATA/mdp/). These parameters specify, for example, the cutoffs used to calculate non-bonded interactions, the algorithm used to calculate the neighbors of each atom, the type of periodic boundary conditions (e.g. three-dimensional, bi-dimensional), and the algorithms to calculate non-bonded interactions. They also specify the type of simulation, for example energy minimization or molecular dynamics, and its length and time step if appropriate. Finally, they describe also the frequency with which GROMACS should write to disk the coordinates and energy values. Depending on the aim of the simulation, this writing frequency can be increased to have a higher temporal resolution at a cost of some computational efficiency (writing takes time). MDP files support hundreds of parameter settings, all of which are detailed in the GROMACS manual.
Although GROMACS is made of several utilities, its heart is the mdrun program. It is this code that runs all the simulations. The -deffnm flag is a very convenient option that sets the default file name for all file options, both input and output, avoiding multiple individual definitions. The -v flag tells mdrun to be verbose and in this case, print the potential energy of the system and the maximum force at each step of the minimization.
Steepest Descents:
Tolerance (Fmax) = 1.00000e+01
Number of steps = 5000
Step= 0, Dmax= 1.0e-02 nm, Epot= 4.80138e+03 Fmax= 1.83867e+04, atom= 57
Step= 1, Dmax= 1.0e-02 nm, Epot= 2.69755e+03 Fmax= 6.83824e+03, atom= 56
Step= 2, Dmax= 1.2e-02 nm, Epot= 1.46780e+03 Fmax= 4.07714e+03, atom= 57
Step= 3, Dmax= 1.4e-02 nm, Epot= 1.69036e+03 Fmax= 1.47891e+04, atom= 56
Step= 4, Dmax= 7.2e-03 nm, Epot= 1.19448e+03 Fmax= 6.00160e+03, atom= 56
Step= 5, Dmax= 8.6e-03 nm, Epot= 1.03838e+03 Fmax= 4.85784e+03, atom= 56
Step= 6, Dmax= 1.0e-02 nm, Epot= 1.11613e+03 Fmax= 1.01399e+04, atom= 56
Step= 7, Dmax= 5.2e-03 nm, Epot= 8.98891e+02 Fmax= 2.73856e+03, atom= 56
Step= 8, Dmax= 6.2e-03 nm, Epot= 8.39895e+02 Fmax= 4.76931e+03, atom= 56
Step= 9, Dmax= 7.5e-03 nm, Epot= 8.05094e+02 Fmax= 5.81049e+03, atom= 56
Step= 10, Dmax= 9.0e-03 nm, Epot= 7.77891e+02 Fmax= 5.97918e+03, atom= 56
...
The steepest descent algorithm used in this minimization calculates the gradient of the energy of the system at each step and extracts forces that push the system towards an energy minimum. As such, the potential energy must decrease. This is not the case for molecular dynamics and other minimization algorithms. The minimization ends when one of two conditions is met: either the maximum force is small than the provided threshold ($$10 kJ.mol^-1$$), and the minimization converged, or the algorithm reached the maximum number of steps defined in the parameter file (5000). Ideally, a minimization should run until convergence, but except for very specific scenarios such as normal mode analysis, this is not a strict requirement.
### Solvating the simulation box
The next step is to add solvent to the simulation box. The first molecular dynamics simulations of proteins were done in vacuum, but researchers quickly realized this was a major limitation. Water molecules interact with the protein, mediating interactions between residues. In addition, water as a solvent has a screening effect for long-range interactions, such as electrostatics. In vacuum, there is nothing to prevent two opposite charge atoms to feel each other even at a very long distance, as long as they are within the cutoff used for the simulation of course. With the addition of water, this interaction is dampened significantly. The effect of water-mediated interactions are also important when choosing the size of the box. The presence of a solute, the peptide, induces a particular ordering of the water molecules in its vicinity. This might have a ripple effect that propagates the effect of the solute and causes artifacts well beyond the theoretical non-bonded cutoff.
You should have already chosen the appropriate water model – TIP3P – when running pdb2gmx. The topology file is not required for the solvation. Essentially, this operation is just a matter of placing pre-calculated chunks of water molecules inside the box and remove those overlapping with protein atoms. No chemistry involved. However, the topology must be updated to reflect the addition of the solvent.
GROMACS backs up the previous topology file before updating it. Generally, GROMACS never overwrites files, instead copying the previous one and renaming it with # symbols. At the end of the new topology file, there is an additional entry listing the number of water molecules that are now in the structure. It also added a definition that loads the water model parameters.
Open the solvated structure in Pymol.
show cell
Why is it not a problem to have water and/or protein atoms sticking out of the box?
### Addition of ions: counter charge and physiological concentration
Besides water, the cellular environment contains a number of ions that maintain a certain chemical neutrality of the system. Adding some of these to the simulation box also increases the realism of the simulation. The GROMACS program genion performs this task, but requires as input a .tpr file. The addition of ions is done by replacing certain atoms that are already in the simulation box. Since removing atoms of the peptide is not quite desired, pay attention to the group you select when running genion. The -neutral flag indicates that an excess of one ionic species is allowed to neutralize the charge of the system, if there is any.
### Energy minimization of the solvated system
The addition of ions was the final step in setting up the system (chemically) for the simulation. From here on, all that is necessary is to relax the system in a controlled manner. Adding the solvent and the ions might have caused some unfavorable interactions, such as overlapping atoms and equal charges placed too close together.
### Restrained MD – relaxation of solvent and hydrogen atoms
Despite dissipating most of the strain in the system, energy minimization does not consider temperature, and therefore velocities and kinetic energy. When first running molecular dynamics, the algorithm assigns velocities to the atoms, which again stresses the system and might cause the simulation to become unstable. To avoid possible instabilities, the preparation setup here described includes several stages of molecular dynamics that progressively remove constraints on the system and as such, let it slowly adapt to the conditions in which the production simulation will run.
The .mdp file for this simulation is substantially different from those used for the minimization runs. First, the integrator is now md, which instructs mdrun to actually run molecular dynamics. Then, there are several new options that relate specifically to this algorithm: dt, t_coupl, ref_t, and gen_vel. At the top of the file, there is a preprocessing option that defines a particular flag -DPOSRES. In the topology file, there is a specific statement that is activated only when this flag is set, which relates to a file created by pdb2gmxposre.itp. This file contains position restraints for certain atoms of the system, which prevent them from moving freely during the simulations.
Which atoms are restrained during the simulation? What could be the purpose of using such restraints?
What is the length of the simulation in picoseconds?
The inclusion of velocity in this system caused the particles and the system to gain kinetic energy. This information is stored in an binary file format with extension .edr, which can be read using the GROMACS utility energy. This utility extracts the information from the energy file into tabular files that can then be turned into plots. Select the terms of interest by typing their numbers sequentially followed by Enter. To quit, type 0 and Enter. Use the xvg_plot.py utility to plot the resulting .xvg file, passing the -i flag to have an interactive session open. If you want to change the colors of the plot, run the script with the -h flag and refer to this page for the available color maps.
Extract and plot the temperature, potential, kinetic, and total energy of the system.
### Coupling the barostat – simulating in NPT conditions
Equilibration is often conducted in two stages: first, the system is simulated under a canonical ensemble (NVT) in which the number of molecules, volume, and temperature are kept constant. The goal is to let the system reach and stabilize at the desired temperature. The second step is to couple a barostat to the simulation and maintain a constant pressure, which resembles more closely the experimental conditions. While the temperature is controlled by adjusting the velocity of the particles, the pressure is kept constant by varying the volume of the simulation box ($$PV = NRT$$).
Equilibrate the system under NPT conditions and re-analyze the several thermodynamical variables, there is no need to edit this file.
Inside 04_npt_pr_PME.mdp we define the Berendsen barostat to be used, although this weak-coupling algorithm is not rigorously compatible with a full isothermal-isobaric (NPT) ensemble. Gromacs correctly complains about this by means of a warning message. In our case, we are just equilibrating the system, and using the Berendsen barostat is perfectly fine. Therefore the warning can be safely ignored by adding --maxwarn 1 at the end of the previous command.
What is better controlled: temperature or pressure? Why?
### Releasing the position restraints
By now, the system had time to adjust to the injection of velocities and the introduction of both temperature and pressure. The heavy atoms of the peptide are, however, still restrained to their initial positions. The next and final steps of the simulation setup release these restraints, progressively, until the system is completely unrestrained and fully equilibrated at the desired temperature and pressure, thus ready for the production simulation.
The strength of the restraints is defined in the posre.itp file, created by pdb2gmx. The value of the force constant defines how strictly the atom is restrained. As such, releasing the restraints is as simple as modifying the numbers on the file.
[ position_restraints ]
; atom type fx fy fz
1 1 1000 1000 1000
4 1 1000 1000 1000
6 1 1000 1000 1000
9 1 1000 1000 1000
12 1 1000 1000 1000
13 1 1000 1000 1000
14 1 1000 1000 1000
17 1 1000 1000 1000
Decrease the strength of the force constant of the position restraints and re-run the system under NPT.
The final equilibration step is to completely remove the position restraints. This is done by removing the -DPOSRES definition at the beginning of the .mdp file, while maintaining all other parameters. For simplicity, we provide a further .mdp file without this definition (you don’t need to edit this file either).
## Production Simulation
Despite all these efforts, the system is unlikely to be in equilibrium already. The first few nanoseconds of a simulation, depending on the system, are in fact an equilibration period that should be discarded when performing any analysis on the properties of interest. To setup the simulation for production, all it takes it to generate a new .tpr file that contains the desired parameters, namely the number of steps that defines the simulation length. At this stage, there are plenty of questions to address that have varying degrees of influence in the performance of the calculations:
• At what time scale do the processes under study occur? How long should the simulation run for?
• What is the temporal resolution necessary to answer the research questions?
• Is there a need to store velocities and energies frequently?
• How often should the simulation information be written to the log file?
The simulation will run for 50 nanoseconds, which is sufficient to derive some insights on the conformational dynamics of such a small peptide. Bear in mind that a proper simulation to fully and exhaustively sample the entire landscape should last much longer, and probably make use of more advance molecular dynamics protocols such as replica exchange. In this case, since several students are expected to work on the same peptide, using different random seeds and starting from different initial conformations, we assume that individual simulations of 50 nanoseconds are informative enough.
The production run can be done in NMRBox, and we will run on the cluster over the next couple of days.
To run it in NMRBox, the only step missing is to generate a .tpr file containing the information for this simulation. Give this input file a clear name, combining the protein identifier (p53_helix, p53_extended, or p53_polypro) with your name or initials.
Run the production MD! This will take a few hours to complete.
gmx mdrun -v -deffnm p53_helix_CAH
If you wish to inspect the contents of the .tpr file, use the dump utility of GROMACS, which, as the name indicates, outputs the entire contents of the file to the screen. Pipe the output of the command to a text processor such as less or more (Unix joke) to paginate the output. Press q to quit the program.
gmx dump -s p53_helix_CAH.tpr | more
## Analysis of the Molecular Dynamics Simulation
The production run is only the beginning of the real work behind molecular dynamics simulations. The analysis of a simulation can be divided in several parts and varies substantially depending on the goal of the simulation and the research questions being asked. Generally, the first part of the analysis is to assess the quality and stability of the simulation in its entirety. If these indicate the simulation suffered from any problem, namely periodic image interactions, unstable temperature or pressure, or uncontrolled dynamics of the solute (i.e. unexpected unfolding of a protein), the simulation might have to be repeated. If otherwise the simulation is stable, then the analysis progresses to extract data that might help answer the research questions.
The production simulation produces a number of files, each containing different information. Depending on the options provided to mdrun, the names may vary. The extensions, however, remain the same. For most of the analysis, the only requirements are the compressed trajectory (.xtc) and energy (.edr) files.
• topol.tpr: Run input file, contains a complete description of the system at the start of the simulation.
• confout.gro: Structure file, contains the coordinates and velocities of the last step of the simulation.
• traj.trr: Full precision trajectory, contains the positions, velocities and forces over time.
• traj.xtc: Compressed trajectory, contains only coordinates (low precision: 0.001 nm)
• ener.edr: Energy file, contains energy, temperature, pressure and other related parameters over time
• md.log: Log file containing information about the simulation, namely performance, warnings, and errors.
### Quality Assurance
Before all else, it must be assured that the simulation finished properly. Many variables can cause a simulation to crash, especially problems related to the force field (if you use custom parameters) or insufficient or deficient equilibration of the system.
Another important source of information about the simulation and its successful conclusion is the log file. Most of this file contains information on the energies at each step of the simulation. At the end, there are several tables with detailed information about the performance of the simulation.
Writing checkpoint, step 25000000 at Thu Jul 16 21:58:00 2015
Energies (kJ/mol)
U-B Proper Dih. Improper Dih. CMAP Dih. LJ-14
5.44222e+02 4.34283e+02 1.81035e+01 -2.63369e+01 1.25435e+02
Coulomb-14 LJ (SR) Coulomb (SR) Coul. recip. Potential
2.90324e+03 7.71580e+03 -9.23198e+04 2.82229e+02 -8.03229e+04
Kinetic En. Total Energy Temperature Pressure (bar) Constr. rmsd
1.50841e+04 -6.52387e+04 3.09273e+02 -3.82873e+02 2.53229e-05
<====== ############### ==>
<==== A V E R A G E S ====>
<== ############### ======>
Statistics over 25000001 steps using 250002 frames
Energies (kJ/mol)
U-B Proper Dih. Improper Dih. CMAP Dih. LJ-14
5.20505e+02 4.57178e+02 3.03600e+01 -1.14010e+01 1.26916e+02
Coulomb-14 LJ (SR) Coulomb (SR) Coul. recip. Potential
2.89500e+03 8.09479e+03 -9.32391e+04 2.85641e+02 -8.08401e+04
Kinetic En. Total Energy Temperature Pressure (bar) Constr. rmsd
1.51323e+04 -6.57078e+04 3.10260e+02 1.51651e+00 0.00000e+00
Box-X Box-Y Box-Z
4.35387e+00 4.35387e+00 3.07865e+00
Total Virial (kJ/mol)
5.04350e+03 -1.47436e-02 1.02119e+00
-1.12579e-02 5.04338e+03 -2.17245e+00
1.02255e+00 -2.17242e+00 5.04266e+03
Pressure (bar)
1.89170e+00 2.46813e-01 -2.53503e-01
2.44845e-01 9.49682e-01 8.72892e-01
-2.54285e-01 8.72874e-01 1.70815e+00
Epot (kJ/mol) Coul-SR LJ-SR Coul-14 LJ-14
Protein-Protein -3.84871e+03 -2.18949e+02 2.89500e+03 1.26916e+02
Protein-non-Protein -1.39745e+03 -2.24747e+02 0.00000e+00 0.00000e+00
non-Protein-non-Protein -8.79929e+04 8.53849e+03 0.00000e+00 0.00000e+00
T-Protein T-non-Protein
3.13254e+02 3.10150e+02
M E G A - F L O P S A C C O U N T I N G
NB=Group-cutoff nonbonded kernels NxN=N-by-N cluster Verlet kernels
RF=Reaction-Field VdW=Van der Waals QSTab=quadratic-spline table
W3=SPC/TIP3p W4=TIP4p (single or pairs)
V&F=Potential and force V=Potential only F=Force only
Computing: M-Number M-Flops % Flops
-----------------------------------------------------------------------------
NB VdW [V&F] 1606.118220 1606.118 0.0
Pair Search distance check 725959.726480 6533637.538 0.4
NxN Ewald Elec. + LJ [F] 18070249.146840 1409479433.454 84.7
NxN Ewald Elec. + LJ [V&F] 182525.653360 23545809.283 1.4
NxN Ewald Elec. [F] 2138562.944360 130452339.606 7.8
NxN Ewald Elec. [V&F] 21600.501696 1814442.142 0.1
1,4 nonbonded interactions 2328.446520 209560.187 0.0
Calc Weights 74722.738140 2690018.573 0.2
Spread Q Bspline 1594085.080320 3188170.161 0.2
Gather F Bspline 1594085.080320 9564510.482 0.6
3D-FFT 8683320.943800 69466567.550 4.2
Solve PME 40790.304000 2610579.456 0.2
Reset In Box 1245.381900 3736.146 0.0
CG-CoM 1245.387762 3736.163 0.0
Propers 2230.719750 510834.823 0.0
Impropers 169.959600 35351.597 0.0
Virial 2834.932800 51028.790 0.0
Stop-CM 249.082242 2490.822 0.0
Calc-Ekin 4981.521738 134501.087 0.0
Lincs 2283.236373 136994.182 0.0
Lincs-Mat 49011.663888 196046.656 0.0
Constraint-V 33260.265960 266082.128 0.0
Constraint-Vir 3097.710213 74345.045 0.0
Settle 9564.597738 3089365.069 0.2
(null) 42.489900 0.000 0.0
-----------------------------------------------------------------------------
Total 1664061187.059 100.0
-----------------------------------------------------------------------------
D O M A I N D E C O M P O S I T I O N S T A T I S T I C S
av. #atoms communicated per step for force: 2 x 32013.5
av. #atoms communicated per step for LINCS: 2 x 1537.9
Part of the total run time spent waiting due to load imbalance: 1.7 %
Steps where the load balancing was limited by -rdd, -rcon and/or -dds: X 0 % Y 0 % Z 0 %
R E A L C Y C L E A N D T I M E A C C O U N T I N G
On 18 MPI ranks
Computing: Num Num Call Wall time Giga-Cycles
Ranks Threads Count (s) total sum %
-----------------------------------------------------------------------------
Domain decomp. 18 1 212450 667.934 31260.638 1.9
DD comm. load 18 1 212450 39.920 1868.341 0.1
DD comm. bounds 18 1 212450 106.924 5004.253 0.3
Neighbor search 18 1 212451 647.826 30319.561 1.9
Comm. coord. 18 1 4036539 3017.213 141211.619 8.7
Force 18 1 4248990 11911.658 557489.413 34.4
Wait + Comm. F 18 1 4248990 1200.017 56163.217 3.5
PME mesh 18 1 4248990 14556.938 681293.782 42.1
NB X/F buffer ops. 18 1 12322068 153.829 7199.495 0.4
Write traj. 18 1 888 0.358 16.767 0.0
Update 18 1 4248990 139.904 6547.776 0.4
Constraints 18 1 4248990 1876.305 87814.821 5.4
Comm. energies 18 1 424900 217.910 10198.637 0.6
Rest 57.370 2685.017 0.2
-----------------------------------------------------------------------------
Total 34594.107 1619073.337 100.0
-----------------------------------------------------------------------------
Breakdown of PME mesh computation
-----------------------------------------------------------------------------
PME redist. X/F 18 1 8497980 10470.597 490044.880 30.3
PME spread/gather 18 1 8497980 1694.125 79288.445 4.9
PME 3D-FFT 18 1 8497980 740.960 34678.396 2.1
PME 3D-FFT Comm. 18 1 16995960 1516.815 70989.972 4.4
PME solve Elec 18 1 4248990 125.464 5871.977 0.4
-----------------------------------------------------------------------------
Core t (s) Wall t (s) (%)
Time: 318506.837 34594.107 920.7
9h36:34
(ns/day) (hour/ns)
Performance: 21.224 1.131
Finished mdrun on rank 0 Thu Jul 16 21:58:00 2015
## Visually inspecting the simulation
Although most of the analysis comes down to extracting data and plotting them, molecular dynamics is first and foremost about dynamical. As such, it is possible to extract the frames from the trajectory and combine them into a movie. This alone can inform substantially about the integrity of the peptide throughout the simulation. The following Pymol commands show the peptide in a sausage-like representation colored sequentially from N- to C-terminal. To manipulate the trajectory file, use trjconv, the GROMACS swiss-knife utility. When asked to select a group to output, choose Protein only, otherwise you will end up with a box of slushy water molecules obscuring the real action!
The peptide moves all around the box, wiggling as it diffuses through the water molecules and beyond the boundaries of the box. When the movie is over, use the intra_fit command to align all the frames, so that you can better observe peptide motions. Then replay the trajectory.
How does the peptide behave during the simulation? Did it unfold completely? Are there parts that remained stable while others didn’t?
Feel free to play around with Pymol. Zoom in on specific regions, such as where the peptide is most rigid or most flexible, and check the side chain conformations (show sticks). Feel free to waste some (CPU) time on making an nice image, using ray and png. Do mind that scenes that are too complex may cause the built-in ray-tracer of Pymol to crash, so in that case you can only get the image as you have it on screen using png directly. Check out the Pymol Gallery for inspiration, or ask your instructors for tips. If you have really a lot of time to waste, you can also make a movie of the trajectory, although this is probably best done outside the virtual machine of the course, for performance reasons. You might need to extract more frames from the simulation to make a sizable movie, depending on the frame rate you choose.
Then, in the command-line interface, assuming you are in the directory where Pymol stored all the .png files:
convert -delay 1 -loop 0 -dispose Background frame_*.png dynamics.gif
## Quantitative Quality Assurance
After a first visual inspection of the trajectory, assuming the simulation went smoothly, it is time to perform additional and more thorough checks regarding the quality of the simulation. This analysis involves testing for the convergence of the thermodynamic parameters, such as temperature, pressure, and the potential and kinetic energies. Sometimes, the convergence of a simulation is also checked in terms of the root mean square deviation (RMSD) of the atomic coordinates of each frame against the initial structure and/or the average structure. Since this simulation is of a very small and flexible peptide, it is expected that it does not converge, although there might be surprises! Finally, the occurrence of interactions between periodic images must be checked as well since, if these did occur, they might lead to artifacts in the simulation.
### Convergence of the thermodynamical parameters
Start off by extracting the thermodynamic parameters from the energy file, as done previously. Of interest are the temperature, pressure, potential energy, kinetic energy, unit cell volume, density, and the box dimensions. The energy file of the simulation contains several dozen terms. Some of the energetic terms are split in groups. These groups were defined in the .mdp file and can be used to isolate specific parts of the system for future analysis, for example, looking at the interaction between specific residues.
Have a look at the plot and see how the temperature fluctuates around the specified value ($$310 K$$). The Heat Capacity of the system can also be calculated from these fluctuations. The system temperature must be extracted from the .edr energy file together with the enthalpy (for NPT) or Etot (for NVT) values. Furthermore, we have to explicitly state how many molecules we have in the system with the -nmol option (you can refer to the end of the topology file to get the total number of molecules in your system). This will allow gmx energy to automatically calculate the heat capacity and show at the end of its output. Check the GROMACS manual for more details.
The equilibration of some terms takes longer than that of others. In particular, the temperature quickly converges to its equilibrium value, whereas for example the interaction between different parts of the system might take much longer.
### Calculation of the minimum distance between periodic images
A key point of any molecular dynamics simulation analysis where periodic boundary conditions were used is to check if there have been any direct interactions between neighboring images. Since the periodic images are just a trick to avoid having hard boundaries, such interactions are unphysical self-interactions and invalidate the results of the simulation.
The occurrence of a periodic image sighting can be overlooked if it is very transient and infrequent. If it does occur frequently or consistently over a stretch of the simulation, time to go back and re-do the whole setup. Also, not only direct interactions are of concern. As mentioned before, the water around the solute has a different structure than the bulk water. To be on the safe side, add an extra nanometer when calculating the allowed minimal distance.
### Conformational dynamics and stability I – Radius of Gyration
Before analyzing any structural parameter, the trajectory has to be massaged to avoid artifacts because of the periodic boundary conditions. In addition, all the analysis tools work faster if the trajectory contains only the necessary (protein) atoms and their information.
Perhaps not entirely relevant for this particular simulation, since the goal is to sample many conformations, but another part of the quality assurance of a simulation is checking the convergence of the structure itself. This can be done either by calculating the root mean square deviation (RMSD) of the atomic coordinates of each frame with respect to the initial structure or the average structure, but also by calculating the radius of gyration of the structure over the trajectory. The radius of gyration gives an indication of the shape of the molecule and compares to the experimentally obtainable hydrodynamic radius.
### Conformational dynamics and stability II – Root Mean Square Fluctuation (RMSF)
The structure of the peptide changes throughout the simulation, but not equally. Some regions are more flexible than others, usually due to differences in the amino acid sequence. The root mean square fluctuations capture, for each atom, the fluctuation about its average position and often correspond to the crystallographic temperature (or b) factors. Comparing this experimental measure with the RMSF profile can serve as an additional quality check for a simulation. The higher the temperature factor, the more mobile the atom. An interesting collateral of this analysis is the calculation of an average structure, which can be used for future analyses.
### Conformational dynamics and stability III – Root Mean Square Deviation (RMSD)
As the calculation of the RMSF also produced an average structure, it is now possible to calculate the root mean square deviation of the entire trajectory. This metric is commonly used as an indicator of convergence of the structure towards an equilibrium state. The RMSD is a distance measure, and as such is mostly meaningful for low values. Two frames that differ by $$10Å$$ from the average structure may well be entirely different conformations. The GROMACS tools rms allows such calculations, and in particular selecting only specific groups of atoms of the molecule, such as the backbone.
While the RMSD with respect to the initial structure is relevant, if it plateaus at a relatively high value, it does not inform on the stability of the conformation. As mentioned above, two structures at $$10Å$$ can be very different. For this reason, the RMSD with respect to the average structure is likely to offer a better perspective of the evolution of structural changes throughout the simulation.
## Structural Analysis
When asked for a selection choose “Protein” if no selection is specifically stated or does not follow logically from the text.
Having assured that the simulation has converged to an equilibrium state, and that its results are likely to be valid, it is time for some real analysis that provides answers to a research question. Analysis of simulation data can be divided in several categories. One comprises the interpretation of single conformations according to some functions to obtain a value, or a number of values, for each time point. Example of these are the previously calculated RMSD and radius of gyration metrics. Next to that, the analysis can be done in the time domain, e.g. through averaging, such as (auto)correlations or fluctuations. In the next section, several different types of analyses will be performed, each providing a different but complementary view into the trajectory. Some might not be strictly necessary for this simulation, but are included as an example of what can be done elsewhere.
When running the GROMACS programs to perform the analyses, pay attention to their output as well as the plots they generate.
### Hydrogen Bonds
Secondary structures (of proteins) are maintained by specific hydrogen bonding networks. Thus, the number of hydrogen bonds, both internal and between the peptide and the solvent. The presence or absence of a hydrogen bond is inferred from the distance between a donor/acceptor pair and the H-donor-acceptor angle. OH and NH groups are regarded as donors, while O is always classified as an acceptor. N is an acceptor by default as well, unless specifically disabled. GROMACS can calculate hydrogen bonds over full trajectories with the hbond program. The program output informs on the number of hydrogen bonds, distance and angle distributions, and an existence matrix of all internal hydrogen bonds over all frames. The number of hydrogen bonds alone is a proxy for the existence of secondary structures.
In addition to global analyses, many GROMACS programs support index files, which are created with the make_ndx program. These index files allow the creation of user-specified groups, such as single residues or stretches of residues. For example, it is possible to evaluate the creation of β-hairpins by checking the existence of hydrogen bonds between the two halves of the peptide. Assume you are working on a 14-residue long peptide. The syntax within make_ndx to create an index file to check for hydrogen bonds between the two halves is as follows:
r 1-7
name 17 half_1
r 8-14
name 18 half_2
q
On the basis of this analysis, is your peptide adopting a β-hairpin structure during the simulation?
### Secondary Structure
Among the most common parameters to analyse protein structure is the assignment of secondary structure elements, such as α-helices and β-sheets. One of the most popular tools for this purpose is the dssp software. Although not part of the GROMACS distribution, dssp can be freely obtained online at the PDB REDO github page, and integrated in many of its analysis tools. Specifically, the do_dssp tool produces a plot of the different secondary structure elements of each residue in the peptide as a function of time. This matrix, in .xpm format, can be converted into a Postscript file using the gmx xpm2ps tool, and then into a PDF file using ps2pdf. The xpm2ps utility allows a scaling flag, -by, that is useful for very short sequences, as well as a -rainbow flag that controls the coloring of the output.
## Analysis of time-averaged properties
This simulation considers only one conformation. To obtain proper sampling of the peptide conformational landscape, 50 nanoseconds do not suffice. However, trajectories starting from different initial structures or starting from the same structure with a different initial random seed explore different regions of the conformational landscape. It is then desirable to combine different trajectories together and therefore obtain a much larger body of data.
Obtain different (full) trajectories from 2 of your colleagues. If possible, try to be as diverse as possible regarding initial structures.
### Preparation of a concatenated trajectory
The first step is to trim the trajectories in order to remove the first 10 nanoseconds, which can be conservatively considered as equilibration. This operation is possible through trjconv and its -b flag, which allows the user to specify an offset previous to which the trajectory data is ignored. To be able to extract only the peptide atoms, trjconv requires an dummy index file.
Why doesn’t it matter which topology file is used to process the different trajectory files?
After all three trajectories are trimmed, they can be concatenated using the GROMACS program trjcat. Make sure to note down the order in which the trajectories are provided to trjcat. The concatenation requires two particular flags to be provided as input to the program: -cat, which avoids discarding double time frames, and -settime, which changes the starting time of the different trajectories interactively. Effectively, the second trajectory will start at 40 ns and the third at 80 ns. The program will prompt for an action during the concatenation: press c, which tells trjcat to append the next trajectory right after the last frame of the previous one.
### Root Mean Square Deviations – Part II
Although the root mean square deviation (RMSD) was already calculated to check for the convergence of the simulation, it can be used for a more advanced and in-depth analysis of conformational diversity. After all, the RMSD is a metric that compares structures. By performing an all-vs-all comparison with all frames in the concatenated trajectory, it is possible to identify groups of frames that share similar structures. This also quantifies the conformational diversity of a particular trajectory (or trajectories). The matrix allows also to detect and quantify the number of transitions between different conformations during the simulations. It is as relevant to have 10 different conformations or 2 that interconvert quickly. Since an all-vs-all RMSD matrix entails a very large number of pairwise comparisons, and the peptide conformations are different enough, use only backbone atoms to fit and calculate the RMSD.
### Cluster Analysis
Using the all-vs-all RMSD matrix calculated in the previous step, it is possible to quantitatively establish the number of groups of similar structures that a trajectory (or concatenated trajectories) samples. Using an unsupervised classification algorithm, clustering, structures that are similar to each other within a certain RMSD threshold are grouped together. The size of a cluster, the number of structures that belong to it, is also an indication of how favourable that particular region of the conformational landscape is in terms of free energy. GROMACS implements several clustering algorithms in the cluster program. Here, we will use the gromos clustering algorithm with a cutoff of $$2Å$$. Briefly, the algorithm first calculates how many frames are within $$2Å$$ of each particular frame, based on the RMSD matrix, and then selects the frame with the largest number of neighbors to form the first cluster. These structures are removed from the pool of available frames, and the calculation proceeds iteratively, until the next largest group is smaller than a pre-defined number. The cluster program produces a very large number of output files that inform on several different properties of the clusters. Importantly, it also produces a PDB file with the centroids, or representatives, of each cluster.
Cluster the RMSD matrix using the GROMOS method to quantitatively extract representative structures of the simulation. Choose peptide backbone for fitting and all-atoms of peptide as output. This is important, since we have will use the output structures for docking.
gmx cluster -f p53_concatenated.xtc -s p53_helix_CAH.tpr -dm p53_concatenated_RMSD-matrix.xpm -dist p53_concatenated_rmsd-distribution.xvg -o p53_concatenated_clusters.xpm -sz p53_concatenated_cluster-sizes.xvg -tr p53_concatenated_transitions.xpm -ntr p53_concatenated_transitions.xvg -clid p53_concatenated_cluster-id-over-time.xvg -cl p53_concatenated_clusters.pdb -cutoff 0.2 -method gromos
How many clusters did the algorithm find? Tune the cutoff to obtain a reasonable number of clusters (e.g. 10-15).
What is the clustering cutoff that allows the definition of that number of clusters? Do you think these clusters are meaningful, i.e. contain only similar structures?
Open the resulting PDB file in Pymol and compare the centroids of each cluster with the others.
Are there any meaningful differences between the largest clusters?
## Picking representatives of the simulation
The aim of this simulation exercise was the sample the conformational landscape of the p53 N-terminal transactivation peptide, in order to extract representatives that could be used to generate models of its interaction with the MDM2 protein. The last step of clustering provides an unbiased method to select structures that were sampled throughout most of the trajectory (large clusters) and are likely good candidates for seeding the docking calculations.
Select 5 representatives of the clusters you obtained in the previous step and create individual PDB files using Pymol.
## Congratulations!
By the end of this tutorial, you have (we hope!) learned how to setup a molecular dynamics simulation of a small peptide and how to critically interpret and validate your results. This is no small feat. The analyses we show here are just the tip of the iceberg of what you can extract from your trajectory. If you are serious about MD simulations, be sure to read the GROMACS documentation and get acquainted with the tools it offers.
You might want to use the representatives you just selected in the tutorial for data-driven docking calculations! | 2023-03-22 08:39:15 | {"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": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.5920575857162476, "perplexity": 1609.8596978430955}, "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/1679296943809.22/warc/CC-MAIN-20230322082826-20230322112826-00543.warc.gz"} |
http://ncatlab.org/nlab/show/holonomy+groupoid | # nLab holonomy groupoid
### Context
#### Topology
topology
algebraic topology
## Examples
#### Differential geometry
differential geometry
synthetic differential geometry
∞-Lie theory
# Contents
## Idea
A holonomy groupoid is a (topological/Lie-) groupoid naturally associated with a foliation $ℱ$ of a manifold $X$. It is, in some sense, the smallest de-singularization of the leaf-space quotient $X/ℱ$ of the foliation, which is, in general, not a manifold itself. Every foliation groupoid of $ℱ$ has this de-singularization property, but the holonomy groupoid is, in some sense, minimal with respect to this property.
Explicitly, given a foliation $ℱ$ on a manifold $X$, the holonomy groupoid of $ℱ$ has as objects the points of $X$. Given points $x,y$ in the same leaf, a morphism between them is the equivalence class of a path in the leaf from $x$ to $y$, where two paths are identified if they induce the same germ of a holonomy transformation between small transversal sections through $x$ and $y$. If $x$ and $y$ are not in the same leaf, then there is no morphism between them.
This is naturally a topological groupoid and a Lie groupoid if done right.
The monodromy groupoid of the foliation is obtained from this by further dividing out the homotopy between paths in a leaf.
## Definition
### Foliation Holonomy
Let $\left(X,ℱ\right)$ be a foliated manifold with $q=\mathrm{codim}\left(ℱ\right)$. Let $L$ be a leaf of $ℱ$; let $x,y\in L$ be two points; and let $S$ and $T$ be transversal sections through $x$ and $y$ respectively (i.e., $S$ and $T$ are sub-manifolds transversal to the leaves of $ℱ$).
To a path $\gamma :\left[0,1\right]\to L$ from $x$ to $y$, we assign the germ of a (partially defined) diffeomorphism
${\mathrm{hol}}^{S,T}\left(\gamma \right):\left(S,x\right)\to \left(T,y\right),${hol^{S,T}}(\gamma): (S,x) \to (T,y),
called the holonomy transformation of the path $\gamma$ with respect to $S$ and $T$, as follows:
If there exists a single foliation chart $U$ of $ℱ$ that contains the image of $\gamma$, then there exists a sufficiently small open neighborhood $A$ of $x$ in the space $S\cap U$ for which there exists a unique smooth map $f:A\to T$ satisfying the following conditions:
• $f\left(x\right)=y$;
• For every $a\in A$, the point $f\left(a\right)$ lies in the same plaque in $U$ as $a$. Observe that $f$ is a diffeomorphism onto its image.
Then define ${\mathrm{hol}}^{S,T}\left(\gamma \right)$ to be the germ of this diffeomorphism at $x$:
${\mathrm{hol}}^{S,T}\left(\gamma \right)\stackrel{\mathrm{def}}{=}{\mathrm{germ}}_{x}\left(f\right).${hol^{S,T}}(\gamma) \stackrel{def}{=} {germ_{x}}(f).
In general, the image of $\gamma$ is not contained inside any single foliation chart $U$, but as it is a compact subspace of $X$, there exist finitely many foliation charts ${U}_{1},\dots ,{U}_{n+1}$ and numbers ${t}_{0},\dots ,{t}_{n+1}$ such that
• $0={t}_{0}<{t}_{1}<\cdots <{t}_{n}<{t}_{n+1}=1$ and
• $\gamma \left(\left[{t}_{i-1},{t}_{i}\right]\right)$ is contained in ${U}_{i}$ for $i=1,\dots ,n+1$.
Then arbitrarily choose transversal sections ${T}_{i}$ through $\gamma \left({t}_{i}\right)$ for $i=1,\dots ,n$ and define
${\mathrm{hol}}^{S,T}\left(\gamma \right)\stackrel{\mathrm{def}}{=}{\mathrm{hol}}^{{T}_{n},T}\left(\gamma {\mid }_{\left[{t}_{n},{t}_{n+1}\right]}\right)\circ {\mathrm{hol}}^{{T}_{n-1},{T}_{n}}\left(\gamma {\mid }_{\left[{t}_{n-1},{t}_{n}\right]}\right)\circ \cdots \circ {\mathrm{hol}}^{{T}_{1},{T}_{2}}\left(\gamma {\mid }_{\left[{t}_{1},{t}_{2}\right]}\right)\circ {\mathrm{hol}}^{S,{T}_{1}}\left(\gamma {\mid }_{\left[0,{t}_{1}\right]}\right).${hol^{S,T}}(\gamma) \stackrel{def}{=} {hol^{T_{n},T}} \left( \gamma|_{[t_{n},t_{n+1}]} \right) \circ {hol^{T_{n-1},T_{n}}} \left( \gamma|_{[t_{n-1},t_{n}]} \right) \circ \cdots \circ {hol^{T_{1},T_{2}}} \left( \gamma|_{[t_{1},t_{2}]} \right) \circ {hol^{S,T_{1}}} \left( \gamma|_{[0,t_{1}]} \right).
This definition is independent of the choice of ${U}_{i}$’s, ${t}_{i}$’s and ${T}_{i}$’s. It only depends on the initial and final transversal sections $S$ and $T$.
Proposition
• Two homotopic paths with the same endpoints induce the same holonomy. (Note, however, that the converse is not true. Two paths with the same endpoints inducing the same holonomy may not be homotopic.)
• If $S,S\prime$ are two transversal sections through $x$ and $T,T\prime$ two transversal sections through $y$, then
${\mathrm{hol}}^{S\prime ,T\prime }\left(\gamma \right)={\mathrm{hol}}^{T,T\prime }\left({\mathrm{const}}_{y}\right)\circ {\mathrm{hol}}^{S,T}\left(\gamma \right)\circ {\mathrm{hol}}^{S\prime ,S}\left({\mathrm{const}}_{x}\right).$hol^{S',T'}(\gamma) = {hol^{T,T'}}(const_{y}) \circ {hol^{S,T}}(\gamma) \circ {hol^{S',S}}(const_{x}).
### Holonomy Groupoid
Given a foliated manifold $\left(X,ℱ\right)$, the monodromy groupoid is the disjoint union of the fundamental groupoids of the leaves of $ℱ$, which is the groupoid having the following properties:
1. Its objects are the points of $X$.
2. There are no morphisms between two points on different leaves.
3. The morphisms between two points on the same leaf are homotopy-classes of paths lying in the leaf joining those points.
The holonomy groupoid is defined analogously, where instead of identifying two paths if they are homotopic, they are identified if they induce the same holonomy as described above.
## References
The holonomy groupoid appears in
• Charles Ehresmann, Structures Feuilletées , Proc. 5th Canadian Math. Congress, Univ. of Toronto Press 1963, 1961, pp. 109–172.
and was studied extensively in
• H. E. Winkelnkemper, The graph of a foliation , Ann. Global Anal. Geom. 1 (1983), no. 3, 51–75. | 2013-12-07 09:17:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 81, "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": 0.9853403568267822, "perplexity": 355.14575408352044}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386163053894/warc/CC-MAIN-20131204131733-00028-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://idocall.com/standard-error/difference-between-standard-error-standard-deviation-confidence-interval.html | Home > Standard Error > Difference Between Standard Error Standard Deviation Confidence Interval
# Difference Between Standard Error Standard Deviation Confidence Interval
## Contents
be higher or lower after another $100*n$ samples, say. Asked 4 years ago viewed 52528 times votes · comment · stats Related 4Excel's confidence interval function throws #NUM! Two data sets will be helpful to illustrate the concept of purpose of object scale? this contact form of variation in the population from which they are drawn.
in Behavioral Science. [PMC free article] [PubMed]3. for 20,000 samples, where each sample is of size n=16. When to weblink of 10, and so on.
## Relationship Between Standard Deviation And Confidence Interval
standard-deviation or ask your own question. We usually collect data in order to generalise from them and so use of the SD and the sample size. The next graph shows the sampling distribution of the mean (the distribution of farm workers and calculated the mean and standard deviations, as shown in table 1. National Center for deviations does this represent?
To understand it, we have to is somewhat greater than the true population standard deviation σ = 9.27 years. Imagine taking repeated samples of the that takes into account that spread of possible σ's. Br J Anaesthesiol Standard Error And Standard Deviation Difference See also unbiased estimation of standard deviation for more discussion.
the mean is a non-zero value. means is equal to the population mean. It remains that standard deviation can still be used https://www.r-bloggers.com/standard-deviation-vs-standard-error/ was found to be 88 mmHg and the standard deviation 4.5 mmHg. can say there is a 95% chance of including the parameter in our interval.
When To Use Standard Deviation Vs Standard Error Why didn't Monero article] [PubMed] [Cross Ref]3. How do I an exponent work when it's less than one?
## Difference Between Standard Error And Standard Deviation Pdf
When you gather a sample and calculate the standard deviation of that sample, as the http://stats.stackexchange.com/questions/151541/confidence-intervals-vs-standard-deviation 3 (3): 113–116. For an upcoming national election, 2000 voters are chosen at random For an upcoming national election, 2000 voters are chosen at random Relationship Between Standard Deviation And Confidence Interval Difference Between Standard Error And Variance obtained a totally different estimate if you had drawn another sample. The
By using this site, you agree to weblink our privacy policy. The standard error is the that they will vote for candidate A. BMJ 1995;310: 298. s, is an estimate of σ. Standard Error Vs Standard Deviation regarded as the normal (meaning standard or typical) range.
Rgreq-3023cd91de17e0ba4ec75e99e518afdf false Sign In to gain 187 ^ Zwillinger D. (1995), Standard Mathematical Tables and Formulae, Chapman&Hall/CRC. This is expressed How are they different and why do navigate here error" is a bit ambiguous.
Standard Error In R 25 (4): 30–32. The 99.73% limits lie three standard 80 or 120 (symmetrical).
## the population, this is how we would expect the mean to vary, purely by chance.
true population mean is the standard deviation of the distribution of the sample means. as batteries; how useful would they be? Review of the use of Standard Error In Excel than the true population standard deviation σ = 9.27 years. Isn't that more expensive standard-error basic-concepts or ask your own question.
To do this, you have available to you a sample of observations \$\mathbf{x} = to exclude the three highest and three lowest values. character in a file? http://idocall.com/standard-error/difference-between-standard-error-and-sample-standard-deviation.html
With small samples - say under 30 observations - larger sample mean is the standard error divided by the mean and expressed as a percentage. If you are interested in the precision of the means or in and other quantiles. The ages in that sample were 23, 27, 28, 29, 31, 10, but will be 8.94 or 10.95.
our privacy policy. ISBN 0-7167-1254-7 , p 53 ^ Barde, M. (2012). "What to use predictably as you acquire more data. What is this red active 4 months ago Get the weekly newsletter! Anaesth. 2003;90:514–516.
The sample SD ought to be controlling many, many undead in this converted adventure? This can be proven mathematically and each of these observations occurring is 5%. Subscribe/Recommend Click here to subscribe sampling distribution of a statistic,[1] most commonly of the mean. sample grows in size the estimate of the standard deviation gets more and more accurate.
We will discuss confidence intervals in is investigating acute appendicitis in people aged 65 and over. Doi: 10.1007/s11999-011-1908-9PMCID: PMC3148365In Brief: Standard A critical evaluation 31, 32, 33, 34, 38, 40, 40, 48, 53, 54, and 55. are ambiguous and should not be used.
The standard deviation of The sample mean will very rarely the age was 3.56 years. As a result, we need to use a distribution be expected, larger sample sizes give smaller standard errors. This formula is only approximate, and works best if 2011 May 10.
These means generally follow a normal distribution, and they often do How could MACUSA exist in 1693 The smaller standard deviation for age at first marriage with a lot of precision even if the data are very scattered. | 2017-09-20 23:45:00 | {"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": 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": 0.6449062824249268, "perplexity": 1052.41219139311}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818687582.7/warc/CC-MAIN-20170920232245-20170921012245-00433.warc.gz"} |
http://www.ams.org/mathscinet-getitem?mr=2964361 | MathSciNet bibliographic data MR2964361 15A80 Shitov, Ya. N. An example of a $(6\times6)$$(6\times6)$-matrix with different tropical and Kapranov ranks. (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2011, no. 5, 58--61; translation in Moscow Univ. Math. Bull. 66 (2011), no. 5, 227–229 Article
For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews. | 2015-08-28 07:47:07 | {"extraction_info": {"found_math": true, "script_math_tex": 1, "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": 0.9647911787033081, "perplexity": 4515.531355222251}, "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-2015-35/segments/1440644062327.4/warc/CC-MAIN-20150827025422-00280-ip-10-171-96-226.ec2.internal.warc.gz"} |
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### Author Topic: Sum of torque (Read 124810 times)
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##### Re: Sum of torque
« Reply #195 on: August 02, 2015, 04:11:13 PM »
The goal is always the same: have a torque on the red object and no torque on the support. The support turns around C1. C1 is fixed to the ground. The red object turns around C2. C2 is fixed to the support. I attrack balls from C2, all balls are attracked from C2. I want to have F1=F2 and the sum of forces of attraction from part1 = the sum of forces of attraction from part2. For that I use the left part with a higher radius than the right part (ratio 9/7). But I need to have F1=F2, so it's important to change the pressure in the same time. When the radius at left increases, the volume of balls increases too, so I can cancel the forces of attraction left/right. If I take the good value of the pressure I can cancel in the same time F1+F2 on C2. One end of each spring is attached on C2, the other end of the spring is attached on a ball. There is one spring for each ball.
Like that the red object has a torque on it. It's unstable like before, but I accelerate more and more the support in the same time. I win the energy of the rotation of the red object. The center of gravity of the red object is on C2.
« Last Edit: August 03, 2015, 01:00:00 AM by EOW »
#### Free Energy | searching for free energy and discussing free energy
##### Re: Sum of torque
« Reply #195 on: August 02, 2015, 04:11:13 PM »
#### EOW
• Sr. Member
• Posts: 383
##### Re: Sum of torque
« Reply #196 on: August 03, 2015, 09:53:59 AM »
I changed the radius of the left part for have a counterclockwise torque on the support.
With the radius at 5.5 at left and the radius 3.5 at right I have :
The vector at left :
x=-6.27
y=1.49
The vector at right:
x=5.21
y=-1.9
The sum is :
x=-1.06
y=-0.41
So the force give a counterclockwise torque on the support.
The force on C2 from F1 and F2 is 0 because I take the pressure at 3.5 for the right and 2.22 for the left.
« Last Edit: August 03, 2015, 07:39:19 PM by EOW »
#### EOW
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• Posts: 383
##### Re: Sum of torque
« Reply #197 on: August 04, 2015, 01:16:46 AM »
If I attrack balls at outer circle I can have F1=F2 and a counterclockwise torque on the support, no ?
#### Free Energy | searching for free energy and discussing free energy
##### Re: Sum of torque
« Reply #197 on: August 04, 2015, 01:16:46 AM »
#### EOW
• Sr. Member
• Posts: 383
##### Re: Sum of torque
« Reply #198 on: August 04, 2015, 06:21:04 PM »
Here I have a counterclockwise torque on the support and each red object receives a counterclockwise torque too.
#### EOW
• Sr. Member
• Posts: 383
##### Re: Sum of torque
« Reply #199 on: August 11, 2015, 02:13:53 PM »
Black lines = to the center C1. Red line = to the center C2.
Attraction: height = pressure.
Torque1=2.66
Torque2=-4
Torque3=-1
Torque4=+1
Torque5=+0.166
Torque6=-0.833
Sum of torques=-2
Force3+Force4=4
Force5+Force6=2
Sum of F3+F4+F5+F6 (vectors)=sqrt(4²+2²)=4.47
equation of the line of the sum of forces F3+F4+F5+F6 on C2: y=-2x+1; Equation of perpendicular: y=0.5x; Intersection: x=0.4,y=0.2; Distance =sqrt(0.4²+0.2²)=0.447
Torque from forces on C2 from F3+F4+F5+F6 = 4.47 * 0.447 =2
« Last Edit: August 11, 2015, 07:12:22 PM by EOW »
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##### Re: Sum of torque
« Reply #199 on: August 11, 2015, 02:13:53 PM »
#### EOW
• Sr. Member
• Posts: 383
##### Re: Sum of torque
« Reply #200 on: August 12, 2015, 11:52:58 AM »
Hello,
A wheel is turning and is moving in translation exactly like a wheel of a bike on the road. I put a gas under pressure P inside the black shape. The black shape is moving in translation (no rotation). Outside the black shape there is no pressure (a theoretical problem to simplify the problem). The angle $$\alpha$$ is very small, it can be at 0.00000001 rd for example.
I drawn several positions of the wheel+black_shape. A point "w" fixed on the wheel turns and moves in translation:
All the volumes are constant. I don't drawn the gaskets between the wheel and the black shape but there are (no gas escapes). I suppose the friction at 0 to study the sum of energy.
I don't need an energy to move in translation the black shape. So, the sum of energy must be at 0 too for the wheel, but:
If I'm looking at points A and B, one is moving more at right than one another. I calculated this with a big precision, like that it's possible to decrease the angle $$\alpha$$:
The radius of the wheel is at 1.
Quote
from mpmath import *
mp.dps=100; mp.pretty=True
da=0.000000001
a=pi/2
x1=a-sin(a)
y1=1-cos(a)
x2=a+da-sin(a+da)
y2=1-cos(a+da)
x3=a+pi-sin(a+pi)
y3=1-cos(a+pi)
x4=a+pi+da-sin(a+pi+da)
y4=1-cos(a+pi+da)
print "dx=", x2-x1-(x4-x3)
print "dy=", y2-y1+(y4-y3)
print "dx1=", x2-x1
print "dx2=", x4-x3
print "dy1=", y2-y1
print "dy2=", y4-y3
The result is that the point B moves more at right than the point A, so with the pressure of the gas this would say the point B gives more and energy than the point A need.
dx= 0.000000000000000001000000000000000124479849582226383362699685857886561868869889953568598575440719416999541559130733522
dy= -7.143671195514218638848647176908380996574437467642491632594536520172703974417113722971550136626075185 e-102
I calculated for A and B like I drawn but it's possible to change the angle of the black shape.
#### EOW
• Sr. Member
• Posts: 383
##### Re: Sum of torque
« Reply #201 on: August 13, 2015, 10:04:56 PM »
The numerical solution:
from mpmath import *
mp.dps=150; mp.pretty=True
D=0.1
S1=pi/2
S2=pi
w=0.0000000000000001
def rectangles(f,a,b,n) :
h=(b-a)/n/1.0
z=0.0
for i in range(n) :
z=z+f(a+i*h)
return h*z
def sx1(t):
y1= -1/( sin(t) / (1-cos(t)) )
if t<=pi/2.0:
a1=atan(y1)+pi
a2=-t-pi/2.0
a3=abs(abs(a1)-abs(a2))
if t>pi/2.0 and t<=pi:
a1=atan(y1)+pi
a2=-t+pi/2.0
a3=abs(abs(a1)-abs(a2))
if t>pi and t<=3.0*pi/2.0:
a1=atan(y1)+pi
a2=-t+pi/2.0
a3=abs(abs(a1)-abs(a2))
if t>3.0*pi/2.0 and t<=2.0*pi:
a1=atan(y1)+pi
a2=-t+pi/2.0
a3=abs(abs(a1)-abs(a2))
return sqrt( sin(t)*sin(t)+(1-cos(t))*(1-cos(t)) ) * cos(a3)
m1=rectangles(sx1,S1+w,S1+D+w,40000)
m2=rectangles(sx1,S2+w,S2+D+w,40000)
print m1
print m2
print abs(m1)-abs(m2)
#### Free Energy | searching for free energy and discussing free energy
##### Re: Sum of torque
« Reply #201 on: August 13, 2015, 10:04:56 PM »
#### EOW
• Sr. Member
• Posts: 383
##### Re: Sum of torque
« Reply #202 on: August 13, 2015, 10:06:16 PM »
The maths solution
All the volumes are constant. I didn't draw the gaskets between the wheel and the black shape but there are (no gas escapes). I suppose the friction at 0 to study the sum of energy. The radius of the wheel is at 1.
If I'm looking at points A and B, one is moving more at right than one another. I calculated this with a big precision, like that it's possible to decrease the angle $$\alpha$$
I don't need an energy to move in translation the black shape. So, the sum of energy must be at 0 too for the wheel, but for find the work of the circle I must integrate
$$\int_{pi/2}^{pi/2+da}(x-sin(x)) sin(x) dx$$
$$\int_{pi/2}^{pi/2+da} (1-cos(x)) cos(x) dx$$
$$\int_{3pi/2}^{3pi/2+da} (x-sin(x)) sin(x) dx$$
$$\int_{3pi/2}^{3pi/2+da} (1 - cos(x)) cos(x) dx$$
$$\int_{0}^{x}(x-sin(x))sin(x)dx=1/4.0*(-2.0*x+4.0*sin(x)+sin(2.0*x)-4.0*x*cos(x))$$
$$\int_{0}^{x}(1-cos(x))cos(x)dx=-x/2.0+sin(x)-1/4.0*sin(2.0*x)$$
The result is 0.5 for an angle of 0.1
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxNUMERICAL SOLUTIONxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
[quote]from mpmath import *
mp.dps=200; mp.pretty=True
da=0.000000001
a=pi/2
x1=a-sin(a)
y1=1-cos(a)
x2=a+da-sin(a+da)
y2=1-cos(a+da)
x3=a+pi-sin(a+pi)
y3=1-cos(a+pi)
x4=a+pi+da-sin(a+pi+da)
y4=1-cos(a+pi+da)
w1=(x2-x1-(x4-x3))*sin(a+da)
w2=(y2-y1-(y4-y3))*cos(a+da)/2
print w1
print w2
print "dx=", x2-x1-(x4-x3)
print "dy=", y2-y1-(y4-y3)
There is a difference of distance:
dx=1.00000000000000012447984e-18
dy=2.00000000000000012422984e-9
But, I calculated the mean works with the worst case:
w1=(x2-x1-(x4-x3))*sin(a+da)
w2=(y2-y1+(y4-y3))*cos(a+da)/2 ***** I need to calculate the integrale
print w1
print w2
And the results of the works are:
w1= 0.00000000000000000100000000000000012397984
w2=-0.00000000000000000100000000000000012422984
I can reduce the angle there is always a difference in the axis x.
I calculated for A and B like I drawn but it's possible to change the angle of the black shape, this could change the sign of the result.
from mpmath import *
mp.dps=200; mp.pretty=True
da=0.000000001
a=pi/2
def rectangles(f,a,b,n) :
h=(b-a)/float(n)
z=0
for i in range(n) :
z=z+f(a+i*h)
return h*z
def sx(x):
return (x-sin(x)) *sin(x)
def cx(x):
return (1-cos(x)) *cos(x)
def sxp(x):
return (x+pi - sin(x+pi)) *sin(x+pi)
def cxp(x):
return (1 - cos(x+pi)) *cos(x+pi)
m1=rectangles(sx,a,a+da,10000)
m2=rectangles(cx,a,a+da,10000)
m3=rectangles(sxp,a,a+da,10000)
m4=rectangles(cxp,a,a+da,10000)
print m1
print m2
print m3
print m4
print m1-m2+m3-m4
The result is:
Sum of works = 0.00000000514
But I'm not sure about my integration
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxMATH INTEGRATIONxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Here all math integrations:
x=pi/2.0
p1=1/4*(-2*x+4*sin(x)+sin(2*x)-4*x*cos(x))
p2=-x/2+sin(x)-1/4*sin(2*x)
x=pi/2.0+da
p3=1/4*(-2*x+4*sin(x)+sin(2*x)-4*x*cos(x))
p4=-x/2+sin(x)-1/4*sin(2*x)
x=pi/2.0
p5=-x/2-sin(x)+1/4*sin(2*x)+(x+pi)*cos(x)
p6=1/2*(-x-2*sin(x)-sin(x)*cos(x))
x=pi/2.0+da
p7=-x/2-sin(x)+1/4*sin(2*x)+(x+pi)*cos(x)
p8=1/2*(-x-2*sin(x)-sin(x)*cos(x))
The result is 0.00000000514
The difference is small because the angle is small, if I take $$\alpha$$ at 0.1 rd the result is 0.5
I change the math integrals and I obtain this :
x=pi/2.0
p1=1/4.0*(-2.0*x+4.0*sin(x)+sin(2.0*x)-4.0*x*cos(x))
p2=-x/2.0+sin(x)-1/4.0*sin(2.0*x)
x=pi/2.0+da
p3=1/4.0*(-2.0*x+4.0*sin(x)+sin(2.0*x)-4.0*x*cos(x))
p4=-x/2.0+sin(x)-1/4.0*sin(2.0*x)
x=3.0*pi/2.0
p5=1/4.0*(-2.0*x+4.0*sin(x)+sin(2.0*x)-4.0*x*cos(x))
p6=-x/2.0+sin(x)-1/4.0*sin(2.0*x)
x=3.0*pi/2.0+da
p7=1/4.0*(-2.0*x+4.0*sin(x)+sin(2.0*x)-4.0*x*cos(x))
p8=-x/2.0+sin(x)-1/4.0*sin(2.0*x)
print p1
print p2
print p3
print p4
print p5
print p6
print p7
print p8
z1=-(p3-p1)+p4-p2
z2=p6-p5+p7-p8
print abs(z1)-abs(z2)
It is the same result. And if I take the angle à pi the sum is 0.
#### EOW
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• Posts: 383
##### Re: Sum of torque
« Reply #203 on: August 26, 2015, 07:58:16 AM »
The idea is to use small balls and springs to have a pressure inside a container like a fluid do with the gravity. The black container is full of small balls and springs, the pressure is from 0 to pmax in the top view (not the side view). One spring for one ball, I didn't draw all springs, only three. Balls are compressible or there are in random position (not like I drawn), I need the lateral force from the pressure. I drawn big balls but there are smaller in reality. In the top view I drawn the disk like an ellipse because there is an angle on the side view.
I attrack the disk with the springs like the balls. IT IS THE DIFFERENCE with older ideas. The red disk has 2 axis of rotation, one around the black axis and the other around itself. It is very important to have this axis of rotation around itself too, because the energy come from this axis.
The container is an object. The disk is an object.
Look at the forces on the red disk. I attrack the red disk like the blue balls. This gives a torque on the red disk, it's logical. Like it's a disk there is no other torque on it.
Look at the forces on the red arm. Like the disk is attracked like the balls it's like an object with the same density in water with gravity, the force on the red arm is F2. F2 is only located on the arm, it is the sum of forces on the disk. F2 can't give a torque on the red arm.
Look at the forces on the container: only F1. There are lot of F1 because they are forces from the pressure but each F1 is parallel to the axis of rotation, so no torque.
Like I attrack the balls AND the red disk the force on bottom (top view) from pressure is cancelled by the force from springs, so there is no force on the side where there are green points.
The sum of torques is not 0, but the device is unstable. If I let the device like that, the springs will move and lost their potential energy. So it is necessary to accelerate more and more to increase the kinetic energy of the device. The red disk has a mass so like there is a torque on it, it will accelerate but I accelerate the black arm (the container) at the same acceleration EXACTLY the same acceleration. Like that the red disk keeps its position inside the black container. I need an energy to accelerate the container but this energy can be recover later. I win the energy from the acceleration of the red disk.
I can put several disks inside the container and no balls between disks like that I don't have torque on the red arm.
« Last Edit: August 26, 2015, 07:51:17 PM by EOW »
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##### Re: Sum of torque
« Reply #203 on: August 26, 2015, 07:58:16 AM »
#### EOW
• Sr. Member
• Posts: 383
##### Re: Sum of torque
« Reply #204 on: August 26, 2015, 09:54:11 AM »
To look of 2 axis of rotation of the red disk. Like before each spring attrack.
I noted all surfaces Sx:
The torque on the container:
S1 and S2 cancel themselves the torques on the recipient
S3 and S4 give the forces F1: F1 are parallel to the axis of rotation: no torque
S5: no pressure (top)
S6: the force from pressure is cancelled by the force from springs BECAUSE I attrack the disk too
The torque on the disk around itself S7 to S9:
S7: it's a sum of segments, at final the torque from springs is always there
S8 and S9 give forces parallel to the axis of rotation (red), there is a difference of surface between S8 and S9
The sum of forces on the disk:
There are some forces like F2, but these forces can't give a torque on the red arm because they are parallel on it.
Torque on the red arm:
With severals disks inside the container I can suppress balls between disks. And if I reduce the size of the container I can suppress the difference from S8/S9
« Last Edit: August 26, 2015, 07:58:00 PM by EOW »
#### EOW
• Sr. Member
• Posts: 383
##### Re: Sum of torque
« Reply #205 on: August 26, 2015, 10:29:57 PM »
The problem on the red arm come from the difference of surface S8/S9, so if I put balls inside the part of the red disk (only the part that is inside the container) I can cancel the pressure from S8/S9 (I attrack the balls inside the red disk from springs from the container). The force on the curve part (circle) don't change the torque on the disk. The red disk has always its torque around itself.
« Last Edit: August 27, 2015, 07:51:31 AM by EOW »
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##### Re: Sum of torque
« Reply #205 on: August 26, 2015, 10:29:57 PM »
#### EOW
• Sr. Member
• Posts: 383
##### Re: Sum of torque
« Reply #206 on: August 27, 2015, 04:01:38 PM »
First image: if I take this simple device the force F will lost an altitude so a potential energy will be lost. But the disk will turn more and more. The sum of energy is constant or not... ?
Second image: I put balls inside the part of disk that is inside the container. I attrack balls with springs. The disk will have the force F so a torque. But the container don't have any torque. The container has balls +springs.
« Last Edit: August 27, 2015, 06:18:59 PM by EOW »
#### EOW
• Sr. Member
• Posts: 383
##### Re: Sum of torque
« Reply #207 on: August 30, 2015, 10:03:58 AM »
If I take this device:
(http://s5.postimg.org/kzk1nvd8z/a70.jpg)
The support turns at w0 and the purple disk can turn around itself, at start the angular velocity of the purple disk is 0.
I can adjust the angle alpha like I want before start but the angle is constant after start.
If alpha=0° a point A fixed on the purple disk will be at differents position with time:
(http://s5.postimg.org/91n2npyvn/a73.jpg)
If alpha=2 or 3° it's the same but the point moves up/down like that:
(http://s5.postimg.org/rej2kjjyr/a71.jpg)
And like that:
(http://s5.postimg.org/67t41rbgz/a72.jpg)
The angle can be increased more at 20° for example.
So if the point A moves down or up, it's possible to attach a mass, this mass will move up (under gravity) and the purple will turn a little. I win the energy of the rotation of the purple disk and the mass that moves up. THERE IS NO TORQUE ON THE SUPPORT.
« Last Edit: August 30, 2015, 01:27:13 PM by EOW »
#### EOW
• Sr. Member
• Posts: 383
##### Re: Sum of torque
« Reply #208 on: September 03, 2015, 03:17:18 PM »
The video of the device, I simulated with Ansys, 10 seconds, it's possible to look at the rotation of the disk, at start the disk has no rotation around itself, a small black circle is fixed on the disk to look at the rotation :
https://youtu.be/Pjc4dIf1aWI
If with this device the sum of energy is constant. I can use the last device with the balls+springs but WITHOUT MASS, or a mass as lower as possible. Like that I don't have these problems of inertia. I can recover the torque on the red axis from the pressure and accelerate more and more the device.
« Last Edit: September 03, 2015, 10:46:28 PM by EOW »
#### EOW
• Sr. Member
• Posts: 383
##### Re: Sum of torque
« Reply #209 on: September 04, 2015, 08:04:40 PM »
I can place the disks like that:
http://s5.postimg.org/i21birqon/image.png
to increase the angular velocity of each disk | 2021-04-23 17:24:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.6822475790977478, "perplexity": 2563.139203795407}, "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-2021-17/segments/1618039596883.98/warc/CC-MAIN-20210423161713-20210423191713-00513.warc.gz"} |
https://wiki.seg.org/wiki/Loop_layout_for_a_3D_survey | Loop layout for a 3D survey
Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 12 469 - 484 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store
Problem 12.8
In one 3D technique, source points (${\displaystyle \times }$) and geophones (${\displaystyle \circ }$) are laid out around a loop such as the square shown in Figure 12.8a and all of the geophones are recorded for each source point. This example employs 48 geophone stations spaced 50 m apart and 24 source points spaced 100 m apart. Locate the midpoints and determine their multiplicity.
Solution
The minimum bin size has sides half the geophone spacing; the lines in Figure 12.8b indicate the bin centers.
Ignoring the sources outside the square for the moment, the sources and geophones along the top of the square will produce inline multiplicity along this line. The source and geophone in the upper left-hand corner will produce a zero-offset trace at that midpoint. This source and geophone #2 will give a midpoint trace between geophones 1 and 2. This source and geophone #3 will give a midpoint trace at geophone #2 as will the source at geophone 3 into geophone #1, giving 2-fold data here. However, both of these involve the same travelpath and so do not produce independent information and so only one counts as increasing the multiplicity. The midpoint at geophone #3 will have multiplicity of 2, once for source-geophone locations at 1 and 5 and once for coincident source and geophone at location 3. The sources along the west side of the area and geophones along the north side will give single-fold coverage over much of the interior of the square, but will leave many bins empty because there are only half as many source locations as geophone locations. Sources along the west side and geophone locations along the east side will provide multiplicity along the north-south bisector of the square. Most of these will not duplicate the raypaths from sources on the east and geophones on the west, and hence, the multiplicity down this bisector will be larger than that along the edges of the square. The multiplicity achieved is shown in Figure 12.8b.
Figure 12.8a. Loop of geophones (${\displaystyle \circ }$), sources (${\displaystyle \times }$).
Figure 12.2b. Multiplicity achieved in one quadrant ignoring sources outside the square.
The sources outside the square will expand the coverage area and also increase the multiplicity along the edges of the square, as indicated in Figure 12.8c.
If adjacent squares are shot by repeating locations along an edge of the square, the midpoints outside one square will fall inside another square. But, in general, this will not increase the multiplicity because reciprocal raypaths will be involved.
Figure 12.8c. Multiplicity achieved in one quadrant from all sources.
If the loop had been irregular rather than square, the irregularity would have produced irregularities in the distribution of the midpoints and changed the multiplicity somewhat.
Working out the multiplicity pattern longhand is not only tedious but also very subject to errors. Ordinarily a computer is used to make maps showing the multiplicity. The mixture of offsets involved is often at least as important as the multiplicity, and so, maps are also made showing the offset distribution. Likewise maps are often made showing the distribution of azimuths. | 2020-02-21 02:58:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "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": 0.5162631869316101, "perplexity": 1594.8449981278284}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145438.12/warc/CC-MAIN-20200221014826-20200221044826-00221.warc.gz"} |
https://villavu.com/forum/showthread.php?t=116894&s=f33dc8c81466a0e68840aaa5992966c7&p=1393860 | Okay, so this is a sloppy script I wrote up that can assist you on making overloads.
This script requires that you have some basic geometric knowledge, such as knowing what an X and Y axis is. An X axis is left to right, and a Y axis is up and down. As you move further to the right, the number on the X axis increases. As you move further to the left, the number on the X axis deceases.
The Y axis works the same way. As you move up the Y axis deceases. As you move down the number increases.
Quick example: We have a point A (also called a ordered pair) (5, 5) and another point B at (5, 7). Which one is further to the right? Remember, order pairs are formated X, Y.
Point B is further to the right
Fake out, Point A is further to the right
fake out, Point A and B are on the same point on the X axis
At the top of the script there are two variables named and . Right here you input the place of where your first ingredient is.
For example mine is the Extreme attack potion(3) located at (188, 147)
The rest of your ingredients should follow to the right of the first.
Next, there is a variable called . This is the key ID that you press that will trigger the script to withdraw the ingredients. It is set to 116, which is F7. You can find a list of other possible options with the graphic I have provided here:
Copy and paste this script in simba.
Simba Code:
const START_X = 188; START_Y = 147; USE_KEY = 118; BANK_X = 491; BANK_Y = 73; CHECK_EVERY = 25;function string.contains(s: string): Boolean;begin if ((self <> '') and (s <> '')) then result := (pos(s, self) > 0) else result := False;end;function activateRsClient():boolean;var processes: TSysProcArr; i: integer;begin result := false; processes := GetProcesses(); writeln('Scanning for ' , 'Runique' , 'Client..'); for i := 0 to high(processes) do if (processes[i].title.contains('Runique')) and (processes[i].width = 765) and (processes[i].height = 525) then begin writeln('Found client'); writeln('Target set to: ' , toStr(processes[i])); setTarget(processes[i]); ActivateClient(); exit(true); end;end;procedure mouseFast(tx, ty:integer);begin moveMouse(tx, ty); wait(50); clickMouse(tx, ty, 1);end;procedure withDrawIngr;var slotX, i:integer;begin for slotX := 0 to 5 do begin for i := 0 to 3 do begin mouseFast(START_X + (slotX * 44), START_Y); wait(45); end; wait(100); end;end;procedure closeBank;begin mouseFast(BANK_X, BANK_Y);end;begin if activateRsClient() then messageBox('Found Runique Client Press key ID ' + toStr(USE_KEY), 'Client found', 0) else messageBox('Did not find Runique Client be sure to double check Coords!', 'Client not found', 7) repeat wait(CHECK_EVERY); if (isKeyDown(USE_KEY)) then begin withDrawIngr; wait(150); closeBank; wait(3000); end; until false;end.
2. Nice little script, hopefully this motivates someone to automate the process completely. I don't get how point b is farther to the right though, because they have the same x coordinate of 5
3. Originally Posted by jstemper
Nice little script, hopefully this motivates someone to automate the process completely. I don't get how point b is farther to the right though, because they have the same x coordinate of 5
I guess that explains why I can't set up the script properly
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Originally Posted by jstemper
Nice little script, hopefully this motivates someone to automate the process completely. I don't get how point b is farther to the right though, because they have the same x coordinate of 5
i guess it's in how you read it =p point a --> point b
cough cough* @jstemper
5. Originally Posted by goodgamescript
i guess it's in how you read it =p point a --> point b
cough cough* @jstemper
god damn it
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oh okay this makes sense | 2019-01-23 13:17:39 | {"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": 0.4975500702857971, "perplexity": 8954.395792084555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547584332824.92/warc/CC-MAIN-20190123130602-20190123152602-00290.warc.gz"} |
http://libros.duhnnae.com/2017/jul6/150048531588-On-universality-in-aging-ferromagnets-1-.php | # On universality in aging ferromagnets
On universality in aging ferromagnets - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.
1 LPM - Laboratoire de physique des matériaux
Abstract : This work is a contribution to the study of universality in out-of-equilibrium lattice models undergoing a second-order phase transition at equilibrium. The experimental protocol that we have chosen is the following: the system is prepared in its high-temperature phase and then quenched at the critical temperature $T c$. We investigated by mean of Monte Carlo simulations two quantities that are believed to take universal values: the exponent $\lambda-z$ obtained from the decay of autocorrelation functions and the asymptotic value $X \infty$ of the fluctuation-dissipation ratio $Xt,s$. This protocol was applied to the Ising model, the 3-state clock model and the 4-state Potts model on square, triangular and honeycomb lattices and to the Ashkin-Teller model at the point belonging at equilibrium to the 3-state Potts model universality class and to a multispin Ising model and the Baxter-Wu model both belonging to the 4-state Potts model universality class at equilibrium.
Keywords : aging universality out-of-equilibrium processes lattice models
Autor: Christophe Chatelain -
Fuente: https://hal.archives-ouvertes.fr/
DESCARGAR PDF | 2018-05-22 23:58:27 | {"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": 0.7864596247673035, "perplexity": 1880.1535712461555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794864999.62/warc/CC-MAIN-20180522225116-20180523005116-00292.warc.gz"} |
http://www.karanahuja.in/2014-05-28-my-motivation/ | The fight is with the enemy within .
The enemy on my shoulders.
It will tell me 10,000 things daily and show me multiple paths of easy workarounds .
The question at every step is - is this all i have got ???
I shall keep working until the answer is Yes .
I am a lion in a field of Lions .
Victory will kiss my feet .
The game begins . | 2019-01-21 11:58:51 | {"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": 0.9317762851715088, "perplexity": 2569.8154561357055}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583792338.50/warc/CC-MAIN-20190121111139-20190121133139-00163.warc.gz"} |
https://echelon.bastelhalde.de/post/how-to-set-up-a-btm-222-and-connect-it-to-android | Ben,
# How to set up a BTM 222 and connect it to Android
Because I had some problems to get it working I will show you how to connect a BTM222 bluetooth module (or any similar device) to an Android mobile phone.
### 1. What you need
• obviously a Bluetooth Module
• something to power the module and communicate with it (I use Arduino)
• a level-shifter (I use the 74HC4050N but it can be done with 2 resistors)
• an Android mobile phone
### 2. Setting up the Bluetooth module
At first have a look at the Datasheet. To use the BTM222 you need to hook up 11 pins. 6 of them should have the GND signal. It is very important to tie every single one of the GND-pads to the GND-signal because in the mysterious world of high frequency applications devices don't work when they are not properly grounded. Also remember that the BTM222 needs a supply voltage of 3.3 V when hooking up the module. Of the three remaining pads of the module there are two used for UART communication and one for the antenna. If you check the antenna against GND with a circuit indicator don't wonder if there is a connection - but it works. I found it very hard to solder wires to the BTM222. The pads of the devices are very small and you need a calm hand to get it done. I don't have a calm hand. Thats why my device looks like this:
In my case I have an Arduino which I want to connect to the bluetooth module. So this is the schematic for my setup. Please remember that you can use a voltage divider instead of the 74HC4050N to get the conversion between 5 V and 3.3V for the UART done.
Please check if you really have connected "UART Data Output" (Pin 27) to the TX input of the Arduino board (so its TX -> TX).
This is how my wiring looks like:
### 3. Use the UART to connect to the module
Now its time to communicate with the BTM222. Make sure you Arduino does not use the UART interface (you can burn the "Blink" demo on the device or just remove the chip of the board). Now connect your PC to the Arduino Board. Use 19200 Baud, 8 data bits, no flow control, no parity and 1 stop bit for the connection. Since I am running Ubuntu it is very simple to initiate a connection.
screen /dev/ttyUSB0 19200
You can also use your Arduino IDE serial console window. When you do that make sure to send a carriage return after each command. If the connection was established successfully the BTM222 will echo all chars you send to it. Send "ATI1" to get some information of the device.
### 4. Connecting the module to the Android device
At first you need to pair the mobile phone with the BTM222. Just activate your bluetooth and search for devices. The default name of the module ist "Serial Adapter". If you don't find it check your wiring. Don't forget the antenna or to set all grounds. Use the pin "1234" for pairing. Now it is time to write the Android App which communicates with the devices. Google created a very nice example application called "Bluetooth Chat". You will find a download of the Eclipse project at the bottom of the article if you do not want to search for the above mentioned code and create a project by yourself. So I started an eclipse android project with the "Bluetooth Chat" demo app code by Google. The application can list bluetooth devices, connect to them, run a server socket and send and recieve strings. The problem ist that it will only work for other mobiles devices but not for the BTM222 which only supports the bluetooth SPP profile. There is only one line of code to change to get a communication between your mobile and the BTM 222 working. Just replace the UUID definition in the file "BluetoothChatService.java" in line 49 with the UUID "00001101-0000-1000-8000-00805F9B34FB". So it looks like this in the end:
// Unique UUID for this application
private static final UUID MY_UUID = UUID.fromString("00001101-0000-1000-8000-00805F9B34F");
Thats it. Now you should be able to connect your BTM222 via the menu and send messages. To do it, execute the App, press the Menu button and then click "Connect a device". The rest is self explaining.
This is how it looks in the serial console of arduino and on the Android device:
To come to the end here you find the download of the eclipse project: BluetoothChat (100 kb .zip) An apk is included if you do not want to compile the project again. Have fun.
Jonas wrote on
Thank you really, that's exactly, what I need!
ozoderm wrote on
**ozoderm...** Thanks for taking the time to debate this, I feel strongly about it and love studying extra on this topic. If possible, as you acquire expertise, would you mind updating your blog with further information? It is extremely useful for me....
couzty wrote on
What antenna should i use ?
Ben wrote on
Hi! Thanks for your question. There are many factors which affect the performance of an antenna but in general you can remember this: The optimal antenna is a piece of wire with the length of 1/2 * λ (wavelength). So for Bluetooth you have a radio frequency from 2402 to 2480 MHz. You calculate the wavelength like this: λ = c/f (c is speed of light, f is your frequency) I assumed a frequency of 2450 MHz for calculation. So you have a wavelength of approximately 12,4 cm. Take the half of it (6,2cm) and you will probably get a good antenna. But remember that many other effects influence the antenna (like material of wire) especially in these high frequency application.
Mihai wrote on
Hello, I have a question about the antenna and ground pins of the btm222 module (pins number 37 and 38 respectively). I soldered the module onto a PCB and when I checked for unintended connections I found that those two pins are connected. I don't know whether they are connected internally through an inductor or something, or maybe I've made some error in my PCB which I can't seem to find. So my question is this: can you check if those two pins are connected internally with a multimeter on your board (which is working properly), because I don't want to risk damaging my device if I have made some mistake. Thank you in advance
Ben wrote on
Hi. They are also connected on my board. Very strange but it works. Greetings
Mihai wrote on
Thank you very much for your help. I managed to get my btm222 working. I've sent characters through a serial to usb converter to the microcontroller, which sent those characters back to the PC through the btm222 paired with a bluetooth dongle attached to the PC. Tip: If you ever need a high speed level converter look for the sn74cbtd series form texas instruments, it can convert between many voltage levels, it's pretty cheap, but it's not designed to drive loads.
Tilman wrote on
Hey tolles Projekt! habe sowas ähnliches vor ;) hat mir schon ein bisschen weitergeholfen ;) habe vor meine zimmerbeleuchtung über ein Android app, einen Microcontroller, Relais und den BTM22 zu steuern. Dass XML Layout des Apps ist schon fertig mit Checkboxen zum an und ausmachen der lichter. Doch ich kenne mich mit Java kein bisschen aus und hoffe du könntest mir ein paar Tipps geben wie ich dass ganze programmieren könnte. Außerdem weiß ich nicht welchen Microcontoller ich benützen soll, und hoffe daher dass du vielleicht schon Erfahrungen mit welchen gemacht hast und mir einen nennen kannst. Mfg Tilman
Ben wrote on
Hallo, danke für deinen Kommentar. Witzigerweise haben wir genau gestern auf einer Messe in München genau das präsentiert, was du da vorhast. Wir haben einen ATmega 8 mit einem selbstgebauten Relaiscontroller verwendet. Ich werde in den nächsten Tagen über die Konstruktion einen Artikel schreiben, geht denn dein BTM222 und funktioniert die Kommunikation zu deinem Android-Gerät schon? Grüße
Zdzisław wrote on
Bardzo dziękuję autorowi za umieszczenie cennych informacji, zwłaszcza że dopiero zaczynam zabawę z Androidem i w końcu mogę wykorzystać swoje moduły Bluetooth. [Thank you in Polish :) ]
Andreas wrote on
hi suppppper seite ! ist die version BluetoothChat von deiner Seite schon für BTM-222 gefixt ? Danke
Ben wrote on
Ja sollte sie sein.
Martin wrote on
Nope, mir hat's nichts gebracht. Ich finde den BTM222 auch ziemlich schlecht gemacht bzw. ein eindeutiges Datasheet vermisse ich (sogar im Internet) ebenfalls. Ich habe nur welche gefunden, bei denen die Grounds gaaaanz anders belegt sind, als bei dir. Ich check nicht mal ob ich das Teil von unten oder oben löten muss. Trotzdem danke. Martin | 2019-12-08 05:21:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2786990702152252, "perplexity": 5283.6803303768875}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540506459.47/warc/CC-MAIN-20191208044407-20191208072407-00011.warc.gz"} |
http://mathhelpforum.com/calculus/166293-maximum-minimum.html | # Math Help - maximum and minimum
1. ## maximum and minimum
given $g(x,y)=ax + by^2$
$a>0,b>0,x+y=1, 0<=x>=1, 0<=y>=1$
$a=1$ and $b=5/6$
What is the smallest value y can have? can anyone show how to solve?
2. Originally Posted by EricTu
given $g(x,y)=ax + by^2$
$a>0,b>0,x+y=1, 0<=x>=1, 0<=y>=1$
$a=1$ and $b=5/6$
What is the smallest value y can have? can anyone show how to solve?
CB
3. I assume you are asking for the minimum value $ax+ by^2$ on a regions bounded by some lines buy $0<= x>= 1$ and $0<=y>=1$ make no sense. If $x>= 0$ then x must be " $>= 0$. Did you mean " $0<= x< = 1$ and $0<= y<= 1$?
If so then the minimum value must occur:
1) in the interior of the set where the gradient is 0 (or undefined) or
2) on the boundary of the set.
$\nabla ax+ by^2= a\vec{i}+ 2by\vec{j}$ which is never 0 because of the " $a\vec{i}$" term. On the line x= 0, the y axis, the function is $by^2$ which has its minimum at y= 0. On the line y= 0, the x-axis, the function is $ax$ which has its minimum at x= 0. On the line x+ y= 1, x= 1- y so the function is $a- ay+ by^2$. The derivative of that is $-a+ 2by$ and is 0 at $y=\frac{a}{2b}$. Then $x= 1- \frac{a}{2b}= \frac{2b- a}{2b}$.
Of course, we also need to consider the vertices (1, 0) and (0, 1). That is, the minimum value must occur at one of (0, 0), (1, 0), (0, 1), or $\left(\frac{2b-a}{2b}, \frac{a}{2b}\right)$. Evaluate $ax+ by^2$ at each of those points to decide where it is smallest.
4. Originally Posted by HallsofIvy
I assume you are asking for the minimum value $ax+ by^2$ on a regions bounded by some lines buy $0<= x>= 1$ and $0<=y>=1$ make no sense. If $x>= 0$ then x must be " $>= 0$. Did you mean " $0<= x< = 1$ and $0<= y<= 1$?
If so then the minimum value must occur:
1) in the interior of the set where the gradient is 0 (or undefined) or
2) on the boundary of the set.
$\nabla ax+ by^2= a\vec{i}+ 2by\vec{j}$ which is never 0 because of the " $a\vec{i}$" term. On the line x= 0, the y axis, the function is $by^2$ which has its minimum at y= 0. On the line y= 0, the x-axis, the function is $ax$ which has its minimum at x= 0. On the line x+ y= 1, x= 1- y so the function is $a- ay+ by^2$. The derivative of that is $-a+ 2by$ and is 0 at $y=\frac{a}{2b}$. Then $x= 1- \frac{a}{2b}= \frac{2b- a}{2b}$.
Of course, we also need to consider the vertices (1, 0) and (0, 1). That is, the minimum value must occur at one of (0, 0), (1, 0), (0, 1), or $\left(\frac{2b-a}{2b}, \frac{a}{2b}\right)$. Evaluate $ax+ by^2$ at each of those points to decide where it is smallest.
Yes, sorry - it was a typo. The way you write it is correct.
5. The question is: Assume that a=1 and b=5/6. What is the minimum value y can have in given boundaries (0<=y<=1)? | 2014-07-25 11:35:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 42, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8312921524047852, "perplexity": 184.6167982183162}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997894250.40/warc/CC-MAIN-20140722025814-00198-ip-10-33-131-23.ec2.internal.warc.gz"} |
http://www.ck12.org/geometry/Vertical-Angles/lesson/Vertical-Angles-Grade-7/r9/ | You are viewing an older version of this Concept. Go to the latest version.
# Vertical Angles
## Two congruent, non-adjacent angles formed by intersecting lines.
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### Notes/Highlights Having trouble? Report an issue.
Color Highlighted Text Notes
### Vocabulary Language: English
TermDefinition
Adjacent Angles Two angles are adjacent if they share a side and vertex. The word 'adjacent' means 'beside' or 'next-to'.
Corresponding Angles Corresponding angles are two angles that are in the same position with respect to the transversal, but on different lines.
Intersecting lines Intersecting lines are lines that cross or meet at some point.
Parallel Two or more lines are parallel when they lie in the same plane and never intersect. These lines will always have the same slope.
Perpendicular lines Perpendicular lines are lines that intersect at a $90^{\circ}$ angle.
Supplementary angles Supplementary angles are two angles whose sum is 180 degrees.
Vertical Angles Vertical angles are a pair of opposite angles created by intersecting lines.
Vertical Angles Theorem The Vertical Angles Theorem states that if two angles are vertical, then they are congruent. | 2017-04-27 18:50:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 1, "texerror": 0, "math_score": 0.5197577476501465, "perplexity": 1454.3168253783147}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917122619.71/warc/CC-MAIN-20170423031202-00273-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://or.stackexchange.com/questions/416/sensible-and-realistic-way-to-model-truck-based-transport-costs-depending-on-amo/417#417 | # Sensible and realistic way to model truck based transport costs depending on amount
Different kinds of problems involve transporting an amount $$x$$ from A to B which results in a cost $$c(x)$$ in the objective function.
Traditionally, often linearized costs are used to get an easy, linear model. This might lead to unrealistic solutions if transport volumes are small.
What are more realistic ways to model transport costs (considering especially road transport, to be more specific) which nevertheless lead to manageable and practically solvable mixed integer programs?
• Might be interesting to include an example problem in which you would like to come with a better approximation of transport costs? Jun 12, 2019 at 14:34
• So, your question is not about how to use a more realistic cost c(x) function (because this is just a replacement of a simple function with a more complicated one in your code) and rather, what are some examples of more realistic cost functions, correct?
– EhsanK
Jun 12, 2019 at 14:38
• Examples of cost functions that include the main features of real world costs without being intractable by MIP solvers. I came to this by looking a hub location problems, but any problems that include amount dependant transportation costs (that cannot be preprocessed) is also interesting. (BTW: I managed to write question number 100. Yeah!). Jun 12, 2019 at 14:46
• Congrats, #100! :) Jun 12, 2019 at 15:06
• Can you be more specific about what features of real world costs you have in mind? Nonconvexities? Step functions? Usually things like that are modeled using piecewise linear functions; is that the kind of thing you’re asking for? Jun 12, 2019 at 15:09
In my experience, there are two types of logistics modeling questions that require truck transportation costs.
In the first, a shipper will operate a fleet of its own vehicles and would like to estimate costs. In such cases, it is usually important to model both fixed costs of using additional vehicles and the variable costs of operating vehicles (which vary with distance travelled and/or time operated). Any cost approximation should be careful to appropriately model these fixed costs.
In the second type of problem, a shipper will outsource truck transportation to meet its shipping needs. Here is a bit more detail about that scenario:
Purchased truck transportation:
In this case, a shipper needs to purchase freight transportation from third parties (like trucking companies (carriers) or from third-party logistics (3PL) firms). Different types of truck service are available for purchase, most notably truckload, less-than-truckload, or small package services. Each type of service is priced somewhat differently. It is quite common that truckload services on a lane $$(i,j)$$ are priced per trailerload, and thus a step function for cost is often needed. LTL and package services on a lane are typically priced per weight (or per some weight-and-cubic-volume measure like dimensional weight), and there are often rate breaks where the price per weight decreases at larger weights making piecewise-linear cost functions necessary. Often these functions do not have strictly-decreasing slopes since they may include shoulders where cost is fixed for a certain weight range.
One common $$c(x)$$ function is a "cost per distance unit (mile/km)" like $2/mile, which is just distance-dependent. My 2 cents for more realistic cost functions: For distance-dependent costs: • Use a different distance function. If you are using linear distance (Euclidean or Manhattan), replace it with a better approximation like Haversine distance or the real distance obtained from, for example, Google map API. • Of course, better depending on your use case. In Manhattan, Manhattan distance is good! • Generally, there should be a minimum amount on the route cost. Assuming the simple \$2/mile charge, you won't charge just \\$20 if you drive for 10 miles. So, $$\text{Cost}=\max(\min{\rm charge}, c(x))$$
For amount-dependent costs:
Use some piecewise approximation so that you can pre-process the cost. For example, if you have cost of $$c(q)$$ (where $$q$$ is a decision variable for the quantity on arc $$(i,j)$$, your cost function in your code, should be conditional on the value of $$q$$.
Note, although I used the word "approximation", this is more or less how it works in real-world instances. Your cost does not depend on every increment of $$q$$, rather it goes up and down depending on the range or bin where $$q$$ falls in. | 2022-06-28 10:13:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 11, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.27924343943595886, "perplexity": 1283.750424488359}, "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/1656103360935.27/warc/CC-MAIN-20220628081102-20220628111102-00700.warc.gz"} |
http://mathhelpforum.com/statistics/184007-longest-run-50-tosses.html | # Thread: Longest run in 50 tosses
1. ## Longest run in 50 tosses
Hello....
How do I find the longest run in fifty tosses of a coin?
To add to this question, how do I compare two sequences of coin tosses to differentiate between real toss and the fake toss?
2. ## Re: Longest run in 50 tosses
Longest run of what? Is the coin fair? What exactly is a "fake toss"?
3. ## Re: Longest run in 50 tosses
the coin is fair. A fake toss is what I predict the sample space can be like.
4. ## Re: Longest run in 50 tosses
Longest run of Heads, I mean...
5. ## Re: Longest run in 50 tosses
Originally Posted by hsetima
Longest run of Heads, I mean...
The longest possible run of heads in 50 tosses of a fair coin is ..... 50 heads. (Unlikely, yes. But impossible, no.)
6. ## Re: Longest run in 50 tosses
It may be possible. But is there any way of calculating it - maybe by relative frequencies based on an experiment?
But I managed to find a formula - log of n in base 2. In my case it gave me the longest run to be 5.6 - it can be either be of heads or tails, but I cant explain how.....
7. ## Re: Longest run in 50 tosses
Originally Posted by hsetima
It may be possible. But is there any way of calculating it - maybe by relative frequencies based on an experiment?
$\text{(longest possible run of heads)}=\text{(total of times that you flip the coin)}$
The possibility that you will flip a heads every single time always exists. If you flip a coin one million times in a row, then the longest possible run of heads is one million (extreeemely unlikely, but possible nevertheless).
8. ## Re: Longest run in 50 tosses
Originally Posted by hsetima
It may be possible. But is there any way of calculating it - maybe by relative frequencies based on an experiment?
But I managed to find a formula - log of n in base 2. In my case it gave me the longest run to be 5.6 - it can be either be of heads or tails, but I cant explain how.....
The longest possible run is 50 heads. The probability of it occuring is (1/2)^50.
But I think what you want and what you asked are two different things. Perhaps you want the longest expected run ....
9. ## Re: Longest run in 50 tosses
Yes Mr. Fantastic,
I want the longest expected run
10. ## Re: Longest run in 50 tosses
Originally Posted by hsetima
Hello....
How do I find the longest run in fifty tosses of a coin?
To add to this question, how do I compare two sequences of coin tosses to differentiate between real toss and the fake toss? | 2017-06-27 18:17:39 | {"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": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6014796495437622, "perplexity": 1015.4940576736706}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128321497.77/warc/CC-MAIN-20170627170831-20170627190831-00351.warc.gz"} |
http://ncatlab.org/nlab/show/simplicial+homotopy | model category
for ∞-groupoids
# Contents
## Idea
A simplicial homotopy is a homotopy in the classical model structure on simplicial sets. It can also be defined combinatorially; in that form one can define a homotopy 2-cell between morphisms of simplicial objects in any category $C$.
## Definition via cylinders
SSet has a cylinder functor given by cartesian product with the standard 1-simplex, $I := \Delta[1]$. (In fact, one can define simplicial cylinders, $\Delta[1]\odot X$, more generally, for example for $X$ being a simplicial object in an cocomplete category $C$,(see below).)
Therefore for $f,g : X \to Y$ two morphisms of simplicial sets, a homotopy $\eta : f \Rightarrow g$ is a morphism $\eta : X \times \Delta[1] \to Y$ such that the diagram
$\array{ X \simeq X\times \Delta[0] &\stackrel{Id \times \delta^1}{\to}& X \times \Delta[1] & \stackrel{Id \times \delta^0}{\leftarrow}& X \times \Delta[0] \simeq X \\ & {}_f\searrow &\downarrow^\eta& \swarrow_{g} \\ && Y }$
commutes.
Remark: Since in the standard model structure on simplicial sets every simplicial set is cofibrant, this indeed defines left homotopies.
## Combinatorial definition
Given morphisms $f,g,:X\to Y$ of simplicial objects in any category $C$, a simplicial homotopy is a family of morphisms, $h_i:X_n\to Y_{n+1}$, $i= 0,\ldots,n$ of $C$, such that $d_0 h_0 = f_n$, $d_{n+1} h_n = g_n$ and
$d_i h_j = \left\lbrace\array{ h_{j-1}d_i, & i\lt j \\ d_i h_{i-1}, &i=j\neq 0\\ h_j d_{i-1}, & i\gt j+1. }\right.$
$s_i h_j = \left\lbrace\array{ h_{j+1} s_i, & i\le j\\ h_j s_{i-1}, & i\gt j. }\right.$
## Remark / warning on conventions
The above formulae give one of the, at least, two forms of the combinatorial specfication of a homotopy between $f$ and $g$. (When trying to construct a specific homotopy using a combinatorial form, check which convention is being used!) The two forms correspond to different conventions such as saying that this is a homotopy from $g$ to $f$, or reversing the labelling of the $h_i$.
## Commentary
It is fairly easy to prove that the combinatorial definition of homotopy agrees with the one via the cylinder both for simplicial sets and for simplicial objects in any finitely cocomplete category, $C$. This uses the fact that the category of simplicial objects in a cocomplete category, $C$, has copowers with finite simplicial sets and hence in particular with $\Delta[1]$. (As there are explicit formulae for the construction of copowers …)
Tim: With only my own resources available, I was unable to find them so was hoping someone kind would come up with them. They derive from the coend formulae/Kan extension formulae using some combinatorics to discuss the indexing sets. I think Quillen gave some form of them, but have not got a copy of HA. I needed them recently and could not find them in any of the usual sources, and did not manage to work them out using the Kan extension idea either (Help please anyone). We could do with those formulae or with a reference to them at least.
In the case of the category of (not necessarily abelian) groups, the combinatorial definition equals the one via cylinder only if the role of “cylinder” for a group $G$ is played by a simplicial object in the category of groups which in degree $n$ equals the free product of $(n+2)$ copies of $G$, indexed by the set $\Delta[1]_n$ (noted by Swan and quoted in exercise 8.3.5 of Weibel: Homological algebra).
Tim Porter: Perhaps we need an explicit description of copowers in simplicial objects also. I pointed out in an edit above that the combinatorial description is much more general than just for simplicial objects in an abelian category.
Can specific references to Swan be given, anyone?
Zoran Škoda: I agree that one should talk about copowers etc.
## Properties
###### Lemma
Precisely when $Y$ is a Kan complex, the relation
$(f \sim g) \Leftrightarrow (\exists simplicial homotopy f \Rightarrow g : X \to Y )$
is an equivalence relation.
###### Proof
Since Kan complexes are precisely the fibrant objects with respect to the standard model structure on simplicial sets this follows from general statements about homotopy in model categories.
The following is a direct proof.
We first show that the homotopy between points $x,y : \Delta[0] \to Y$ is an equivalence relation when $Y$ is a Kan complex.
We identify in the following $x$ and $y$ with vertices in the image of these maps.
• -reflexivity- For every vertex $x \in Y_0$, the degenerate 1-simplex $s_0 x \in S_1$ has, by the simplicial identities, 0-faces $d_0 s_0 x = x$ and $d_1 s_0 x = x$.
$(d_1 s_0 x) \stackrel{s_0 x}{\to} (d_0 s_0 x)$
Therefore the morphism $\eta : \Delta[0] \times \Delta[1] \to Y$ that takes the unique non-degenerate 1-simplex in $\Delta[1]$ to $s_0 x$ constitutes a homotopy from $x$ to itself.
• -transitivity- let $v_2 : x \to y$ and $v_0 : y \to z$ in $Y_1$ be 1-cells. Together they determine a map from the horn $\Lambda^2_1$ to $Y$,
$(v_2, v_2) : \Lambda^2_1 \to Y \,.$
By the Kan complex property there is an extension $\theta$ of this morphism through the 2-simplex $Delta^2$
$\array{ \Lambda^2_1 &\stackrel{(v_0,v_2)}{\to}& Y \\ \downarrow & \nearrow_{\theta} \\ \Delta[2] } \,.$
If we again identify $\theta$ with its image (the image of its unique non-degenerate 2-cell) in $Y_2$, then using the simplicial identities we find
$\array{ && (d_0 d_2 \theta) = (d_1 d_0 \theta) \\ & {}^{d_2 \theta }\nearrow & \Downarrow \theta & \searrow^{d_0 \theta} \\ (d_1 d_2 \theta) = (d_1 d_1 \theta) && \stackrel{d_1 \theta}{\to} && (d_0 d_1 \theta) = (d_0 d_1 \theta) }$
that the 1-cell boundary bit $d_1 \theta$ in turn has 0-cell boundaries
$d_0 d_1 \theta = d_0 d_0 \theta = z$
and
$d_1 d_1 \theta = d_1 d_2 \theta = x \,.$
This means that $d_1 \theta$ is a homotopy $x \to z$.
• -symmetry- In a similar manner, suppose that $v_2 : x \to y$ is a 1-cell in $Y_1$ that constitutes a homotopy from $x$ to $y$. Let $v_1 := s_0 x$ be the degenerate 1-cell on $x$. Since $d_1 v_1 = d_1 v_2$ together they define a map $\Lambda^2_0 \stackrel{v_1, v_2}{\to} Y$ which by the Kan property of $Y$ we may extend to a map $\theta'$
$\array{ \Lambda^2_0 &\stackrel{v_1, v_2}{\to}& Y \\ \downarrow & \nearrow_{\theta'} \\ \Delta[2] }$
on the full 2-simplex.
Now the 1-cell boundary $d_0 \theta'$ has, using the simplicial identities, 0-cell boundaries
$d_0 d_0 \theta' = d_0 d_1 \theta' = x$
and
$d_1 d_0 \theta' = d_0 d_2 \theta' = y$
and hence yields a homotopy $y \to x$. So being homotopic is a symmetric relation on vertices in a Kan complex.
Finally we use the fact that SSet is a cartesian closed category to deduce from this statements about vertices the corresponding statement for all map:
a morphism $f : X \to Y$ is the Hom-adjunct of a morphism $\bar f : \Delta[0] \to [X,Y]$, and a homotopy $\eta : X \times \Delta[1] \to Y$ is the adjunct of a morphism $\bar \eta : \Delta[1] \to [X,Y]$. Therefore homotopies $\eta : f \Rightarrow g$ are in bijection with homotopies $\bar \eta : \bar f \to \bar g$.
###### Proposition
Let $A$ be an abelian category and $f,g : X\to Y$ homotopic morphisms of simplicial objects in $A$. Then the induced maps $f_*, g_* : N(X)\to N(Y)$ of their normalized chain complexes are chain homotopic.
###### Proposition
Let $A$ be an abelian category. The morphisms of simplicial objects (variant: of unbounded chain complexes) in $A$, which are homotopic to zero, form an ideal. More precisely
• being homotopic is an equivalence relation on the class of morphism,
• $f_1\sim 0$ and $f_2\sim 0$ implies $f_1+f_2 \sim 0$,
• if $f\circ h$ (resp. $h\circ f$) exists and if $f\sim 0$ then $f\circ h\sim 0$ (resp. $h\circ f\sim 0$).
## References
• Goerss, Jardine, Simplicial homotopy theory (ps)
Cylinder based homotopy is also discussed extensively in
• K. H. Kamps and T. Porter, Abstract Homotopy and Simple Homotopy Theory, World Scientific Publishing Co. Inc., River Edge, NJ.
Revised on January 26, 2014 12:48:14 by Anonymous Coward (134.76.62.130) | 2015-07-29 00:43:40 | {"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": 0.9576473236083984, "perplexity": 362.23363378587595}, "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-2015-32/segments/1438042985140.15/warc/CC-MAIN-20150728002305-00087-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://docs.ergoplatform.com/dev/protocol/scaling/ | # Scaling
There are three generations of blockchain, each capable of more complex behaviours than the last. The first generation refers to the original use BTC was intended for, that is, to provide a reliable and accurate method of maintaining a public transaction ledger. Second-generation blockchains were created with the implementation of smart contracts as a priority, with the most popular one being ETH (Ethereum). One of the big differences in ETH is the language used to code smart-contracts. ETH utilizes a turing complete language (known as Solidity), meaning it is computationally universal.
The third generation of blockchain technology now focuses on solving issues with congestion and scalability. As this technology becomes more decentralized, there will inevitably be an exponential increase in the number of users interacting with the blockchain. DOT, ADA and ERG are third-generation blockchains, meaning they have smart-contract capabilities while proposing solutions to scalability for a global audience. Third-generation blockchains generally can process off-chain transactions, helping speed up transactions significantly. Ergo also has various innovations to address scalability. NIPoPows on Ergo allow for connection to side chains and light chains, opening up a whole world of possibilities.
Ergo Platform has a research-based approach for long-term success and has a lot in its toolbox to tackle scaling as we grow, which scaling options we implement will depend on the needs of applications building on top of Ergo, as well as the success of the solutions in other protocols.
## Transaction Speed
Transaction speed is generally measured in Transactions per Block/Second (TPS). Below are approximated TPS values for other blockchains:
• BTC - ~7 TPS (Gobbel, 2017).
• ETH - ~15 TPS (Clincy et al (table1), 2019)
• XRP - ~ 1500 TPS (Clincy et al (table1), 2019)
• ADA - ~ ~7 TPS (~250 in controlled tests) (Stamoulis, 2021)
• DOT - ~1500 TPS (Hiemstra et al., 2021)
Transaction speed, specifically TPS, measures the transactions per block/second a blockchain can perform, quantifying how quickly a blockchain can complete transactions.
It's important to remember that TPS is mostly a vanity metric. It's not about how many transactions you can do but rather the weight of those transactions and the computational cost limit per block. This cost limit depends on the hardware miners have, the size of the network, and other dynamic factors, but there is no concrete formula to calculate scalability. The best path forward is to log transaction profiles once DeFi on Ergo is established and perform load testing in the testnet. With the release of v5, the raw TPS numbers should bring us to around 47.5tx/s - improvements on top of this are still possible. The focus is on raising TPS without compromising classic blockchain assumptions and guarantees.
## Settlement Layer
Below you will find a developer harnessing the power of eUTXO to airdrop native tokens to 10,000 addresses at once
Thanks to the high flexibility of the ErgoScript programming model, large chunks of transactions can happen on layer two and be settled in Ergo using a single transaction.
ErgoScript adds several improvements such as time-weighted data, Turing completeness, read-only data inputs, multi-stage contracts, sigma protocols and NIPoPoWs that make many different protocols will possible on Layer 2, each one solving scalability problems in a specific domain (like simple payment transactions, sped up with sub-block confirmation protocols).
Ergo can be considered a common settlement layer for many Level-2 protocols and applications.
## Layer 0 (Network Layer)
The network or peer to peer layer. The Ergo Node Client has greatly improved since v4.0.8 and still has room to grow. Quick bootstrapping using NiPoPoWs proofs and UTXO set snapshots in development
Stateless Clients: Then light clients: You can have full-node guarantees in Ergo without storing the full UTXO set. Bringing improved bootstrapping and block validation times.
State Bloat: One of Ergo's major strengths when scaling is to avoid bloat without compromising functionality. E.g. persistent updateable storage is possible, with updates to be checked by a blockchain contract. However, only the digest of the authenticated data structure (and some additional bytes, less than 40) are stored in the UTXO set regardless of data set size. Ergo utilizes a Storage Rent Fee to prevent spam and recirculate unused data bytes, known as dust. Storage Rent Fee helps clean the network pollution and encourages users to be more active.
Block size: Parameters like block size, etc., are not set in stone; miners can adjust them. So if a miner is experiencing low full block validation time (as hardware is getting better with time and software), he may propose or vote to increase the block size. Currently set to 8MB.
Transaction size: As of node 4.0.23, there is a transaction size limit of 96kb for the mempool. Larger transactions can only be included manually by miners.
Logarithmic space mining: allows for light miners. Similar to light clients, light miners can bootstrap with block headers without downloading the entire blockchain. Integrating logarithmic space mining in Ergo is possible via a velvet (soft) fork; see this video from Dionysis Zindros from The University of Athens for a introduction and their progress so far.
## Layer 1 (Blockchain)
Ergo supports multiple on-chain scalability solutions such as Sharding.
Sub-block confirmation protocols: such as (Bitcoin-NG or Flux are an active topic for research in 2022. Ergo blocks have extension sections with mandatory and arbitrary key-value data; by putting certain anchors there, it is possible to do BitcoinNG-style micro blocks, Aspen-like service-chains or generic sidechains with just velvet or soft forks. Also see Flux: Revisiting Near Blocks for Proof-of-Work Blockchains
## Layer 2 (Off-Chain)
Ergo can utilize multiple off-chain solutions, such as Hydra and sidechains to compress blockchain bloat and provide similar benefits as zk-rollups. Ergo can also be compatible with other UTXO Layer 2 solutions, such as Bitcoin's Lightning Network. The implementation here will depend on the applications being built on Ergo.
### Plasma
Plasma The ledger is stored as an AVL tree, a balanced and sorted binary tree. Users perform off-chain transactions with the bank, and the ledger keeps changing. Occasionally, the bank publishes a compact snapshot of the ledger on the blockchain.
You can use Plasma to create Plasma chains and make a full L2 solution. Right now it’s mostly just used for data compression and simplifying contracts, though Plasma chains will likely come in the future.
Plasma tutorials for Ergo have now been released. Please see Bank & AVLTrees
### NIPoPoWs
NIPoPoWs: Non-interactive proofs of proof of work are essential for two reasons: Light Clients and Side Chains. Light clients, which consist of light nodes and light wallets, are efficient clients that do not need to hold the whole blockchain to verify transactions and enable efficient mobile wallets and faster miner bootstrapping. Clients can interact using only the block headers, thus reducing computational resources. Ergo has enabled NIPoPoW support since the genesis block. They can be applied to Ergo's blockchain with an easy-to-implement velvet fork. NIPoPoWs can also be deployed to support PoW and PoS cross-chain communication. NIPoPoW implementations via Velvet soft forks enable infinite scalability via sidechains on top of Ergo.
State Channels (Hydra): is a peer-to-peer signing model, and the design can work well for payment channels for simple purposes. The problem, however, is the state channels are pre-set contracts for which the participants are defined at the launch. New contract creation is needed each time a new participant wants to use the channel. In return, there is higher privacy and security but little flexibility for an open system. IOHK has published a new model called Hydra: Isomorphic State Channels that introduces multi-party state channels by utilizing both on-chain and off-chain computations powered by the eUTXO design. Other novel state channel constructions should be possible as well. It would be good to apply off-chain techniques to applications like ErgoMixer. Ergo is mentioned in the Hydra whitepaper. Research and discussions are underway.
### Other Possibilities
#### Lightning Network:
possible due to the shared UTXO architecture utilizing Bitcoins, the Lightning network is also possible. Basically, in a lightning channel, two participants send their funds to a specific type of joint multi-sig wallet that allows them to create and enforce off-chain agreements. The network itself is just a bunch of these channels connected. You can then structure an off-chain payment across many channels, where none of the funds leaves any individual channel but shuffles around like an abacus.
#### Rainbow Network:
as described in this paper
#### Rollups:
Rollups are also possible via AVL trees. This involves rolling up collections of transactions. The only concern is posting the data on-chain, not verification.
There are two types of Rollups.
• Optimistic Rollups: compute the transactions on a parallel compatible chain that communicates with the main chain. The model is optimistic because it relies on the Fraud-Proof principle, where the aggregators are not actively verifying layer two. Still, they interfere in the event of a fraud dispute. Disputes in optimistic rollups when computations are done only on data whose validity is disputed
• ZK-Rollups utilize zkSNARKs (zero-knowledge succinct non-interactive arguments of knowledge), they can decrease network load by taking hundreds of transfers off-chain and combining or "rolling" them up into a single transaction. The security of the transactions relies directly on the main chain secured by adding mathematical proofs to validate transactions. However, it is relatively harder than hybrid approaches to implement all the functionalities of the mainnet with full security. Various projects are attempting to implement zkSNARKs.
Zk rollups have a lot of issues in practice, and pairing compatible curves support in the core protocol would likely be required.
#### Zero-Knowledge Contingent Payments:
It's possible to make payments that are released if and only if the payee discloses some knowledge (in a trustless manner where neither the payer nor payee can cheat). Achieved using a combination of a hash-locked transaction and an external protocol to ensure the correct data is revealed in the hash lock release.
#### FairSwap/FastSwap protocols:
As described in this paper
#### Coinpools:
Another L2 solution for the UTXO model to consider as described in this paper
## Roadmap (Dec, 2021)
Ergo protocol research and client development roadmap (Dec 2021
We are researching different scalability proposals for Bitcoin, Cardano, and Ethereum, such as sidechains (which are also nice for testing new features), commit-chains, rollups, isomorphic state channels, FairSwap etc. For some solutions for Bitcoin, new opcodes are needed (so a little chance to see things in the real world), while Ergo allows for such constructions with no forks, it seems. Ergo will not just be a chain but a king of chains (which will improve the cryptoeconomic security of the protocol as miners will get additional rewards from sidechains).
for improving the performance of the network, the reference protocol client (Ergo node) is getting different performance improvements in the p2p layer and not only right now
bootstrapping via UTXO set snapshot and NiPoPoWs are in progress now. Should allow for a client to have much faster bootstrapping with no compromising security
5.0 soft-fork is going to be proposed to miners soon; the main change is about switching to just-in-time-costing in ErgoTree evaluation which is giving a 5-6x boost in scripts processing (on real blockchain data)
time to consider long-term cryptoeconomic security of the protocol; discussions have already started: https://www.ergoforum.org/t/ergo-emission-details-retargeting-via-a-soft-fork/2778/7
for application development, more frameworks and ready apps are needed; there are some results to be announced already
Plans for supporting the different application is out of the scope of this text and would be a topic of another piece. | 2022-09-30 19:29:14 | {"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": 0.3038347065448761, "perplexity": 4741.00681261755}, "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-2022-40/segments/1664030335504.22/warc/CC-MAIN-20220930181143-20220930211143-00212.warc.gz"} |
https://www.taylorfrancis.com/chapters/mono/10.1201/b13868-26/bioactive-compounds-marine-sources-gastrointestinal-cancer-treatment-se-kwon-kim | ## ABSTRACT
Gastrointestinal track cancer is the malignant condition of the gastrointestinal tract and is a major health problem worldwide. The prognosis for patients with gastrointestinal tract cancer is grim. Most of the gastrointestinal cancer cases are reported as metastatic cancer. Surgery is an option for gastrointestinal cancer, but results vary depending on the affected organ. For example, it is a curative option in 50% of colorectal cancers, whereas it is less effective in gastric cancers where the overall 5-year survival rate is less than 10% (Hasegawa et al. 2003). | 2022-10-01 16:55:08 | {"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": 0.8658769726753235, "perplexity": 4105.233153138812}, "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/1664030336880.89/warc/CC-MAIN-20221001163826-20221001193826-00479.warc.gz"} |
http://en.wikipedia.org/wiki/Dielectric_spectroscopy | # Dielectric spectroscopy
A dielectric permittivity spectrum over a wide range of frequencies. The real and imaginary parts of permittivity are shown, and various processes are depicted: ionic and dipolar relaxation, and atomic and electronic resonances at higher energies. From the Dielectric spectroscopy page of the research group of Dr. Kenneth A. Mauritz.
Dielectric spectroscopy (sometimes called impedance spectroscopy), and also known as electrochemical impedance spectroscopy (EIS), measures the dielectric properties of a medium as a function of frequency.[1][2][3][4] It is based on the interaction of an external field with the electric dipole moment of the sample, often expressed by permittivity.
It is also an experimental method of characterizing electrochemical systems. This technique measures the impedance of a system over a range of frequencies, and therefore the frequency response of the system, including the energy storage and dissipation properties, is revealed. Often, data obtained by EIS is expressed graphically in a Bode plot or a Nyquist plot.
Impedance is the opposition to the flow of alternating current (AC) in a complex system. A passive complex electrical system comprises both energy dissipater (resistor) and energy storage (capacitor) elements. If the system is purely resistive, then the opposition to AC or direct current (DC) is simply resistance.
Almost any physico-chemical system, such as electrochemical cells, mass-beam oscillators, and even biological tissue possesses energy storage and dissipation properties. EIS examines them.
This technique has grown tremendously in stature over the past few years and is now being widely employed in a wide variety of scientific fields such as fuel cell testing, biomolecular interaction, and microstructural characterization. Often, EIS reveals information about the reaction mechanism of an electrochemical process: different reaction steps will dominate at certain frequencies, and the frequency response shown by EIS can help identify the rate limiting step.
## Dielectric mechanisms
There are a number of different dielectric mechanisms, connected to the way a studied medium reacts to the applied field (see the figure illustration). Each dielectric mechanism is centered around its characteristic frequency, which is the reciprocal of the characteristic time of the process. In general, dielectric mechanisms can be divided into relaxation and resonance processes. The most common, starting from high frequencies, are:
### Electronic polarization
This resonant process occurs in a neutral atom when the electric field displaces the electron density relative to the nucleus it surrounds.
This displacement occurs due to the equilibrium between restoration and electric forces. Electronic polarization may be understood by assuming an atom as a point nucleus surrounded by spherical electron cloud of uniform charge density.
### Atomic polarization
Atomic polarization is observed when the nucleus of the atom reorients in response to the electric field. This is a resonant process. Atomic polarization is intrinsic to the nature of the atom and is a consequence of an applied field. Electronic polarization refers to the electron density and is a consequence of an applied field. Atomic polarization is usually small compared to electronic polarization.
### Dipole relaxation
This originates from permanent and induced dipoles aligning to an electric field. Their orientation polarisation is disturbed by thermal noise (which mis-aligns the dipole vectors from the direction of the field), and the time needed for dipoles to relax is determined by the local viscosity. These two facts make dipole relaxation heavily dependent on temperature, pressure[5] and chemical surrounding.
### Ionic relaxation
Ionic relaxation comprises ionic conductivity and interfacial and space charge relaxation. Ionic conductivity predominates at low frequencies and introduces only losses to the system. Interfacial relaxation occurs when charge carriers are trapped at interfaces of heterogeneous systems. A related effect is Maxwell-Wagner-Sillars polarization, where charge carriers blocked at inner dielectric boundary layers (on the mesoscopic scale) or external electrodes (on a macroscopic scale) lead to a separation of charges. The charges may be separated by a considerable distance and therefore make contributions to the dielectric loss that are orders of magnitude larger than the response due to molecular fluctuations.[1]
### Dielectric relaxation
Dielectric relaxation as a whole is the result of the movement of dipoles (dipole relaxation) and electric charges (ionic relaxation) due to an applied alternating field, and is usually observed in the frequency range 102-1010 Hz. Relaxation mechanisms are relatively slow compared to resonant electronic transitions or molecular vibrations, which usually have frequencies above 1012 Hz.
## Principles
For a redox reaction R $\leftrightarrow$ O + e, without mass-transfer limitation, the relationship between the current density and the electrode overpotential is given by the Butler-Volmer equation:
$j_{\text{t}}=j_0\left(\exp(\alpha_{\text{o}}\,f\, \eta)-\exp(-\alpha_{\text{r}}\,f\,\eta)\right)$
with
$\eta=E-E_{\text{eq}} ,\;f=F/(R\,T),\;\alpha_{\text{o}}+\alpha_{\text{r}}=1$. $j_0$ is the exchange current density and $\alpha_{\text{o}}$ and $\alpha_{\text{r}}$ are the symmetry factors.
Fig. 1 : Steady-state current density vs. overpotential for a redox reaction.
The curve $j_{\text{t}}\; vs.\; E$ is not a straight line (Fig. 1), therefore a redox reaction is not a linear system.[6]
### Dynamic behavior
Let us suppose that the Butler-Volmer relationship correctly describes the dynamic behavior of the redox reaction :
$j_{\text{t}}(t)=j_{\text{t}}(\eta(t))=j_0\,\left(\exp(\alpha_{\text{o}}\,f\, \eta(t))-\exp(-\alpha_{\text{r}}\,f\,\eta(t))\right)$
Dynamic behavior of the redox reaction is characterized by the so-called charge transfer resistance defined by :
$R_{\text{ct}}=\frac{1}{\partial j_{\text{t}}/\partial \eta }= \frac{1}{f\,j_0\,\left(\alpha_{\text{o}}\,\exp(\alpha_{\text{o}}\,f\, \eta)+\alpha_{\text{r}}\,\exp(-\alpha_{\text{r}}\,f\, \eta) \right)}$
The value of the charge transfer resistance changes with the overpotential. For this simplest example the Faradaic impedance is reduced to a resistance. It is worthwhile to notice that:
$R_{\text{ct}} = \frac{1}{f\,j_0}$
for $\eta = 0$ .
#### Double layer capacitance
An electrode $|$ electrolyte interface behaves like a capacitance called electrochemical double-layer capacitance $C_{\text{dl}}$. The equivalent electrical circuit for the redox reaction taking account of the double-layer capacitance is shown in Fig. 2. Another analog circuit commonly used to model the electrochemical double-layer is called a constant phase element.
Fig. 2 : Equivalent circuit for a redox reaction without mass-transfer limitation.
The electrical impedance of this circuit is easily obtained remembering the impedance of a capacitance which is given by :
$Z_{\text{dl}}(\omega) =\frac{1}{\text{i}\,\omega\, C_{\text{dl}}}$
where $\omega$ is the angular frequency of a sinusoidal signal (rd/s), and $\scriptstyle{ \text{i}=\sqrt{-1}}$. It is obtained:
$Z(\omega)=\frac{R_{\text{t}}}{1+R_{\text{t}}\,C_{\text{dl}}\,\text{i} \,\omega}$
Nyquist diagram of the impedance of the circuit shown in Fig. 3 is a semicircle with a diameter $\scriptstyle{R_{\text{t}}}$ and an angular frequency at the apex equal to $\scriptstyle{1/(R_{\text{t}}\,C_{\text{dc}})}$ (Fig. 3). Others representations, Bode or Black plans can be used.[7]
Fig. 3 : Electrochemists Nyquist diagram of a RC parallel circuit. The arrow indicates increasing angular frequencies.
#### Ohmic resistance
The ohmic resistance $R_\Omega$ appears in series with the electrode impedance of the reaction and the Nyquist diagram is translated to the right.
## Measurement of the impedance parameters
Plotting the Nyquist diagram with a potentiostat[8] and an impedance analyzer, most often included in modern potentiostats, allows the user to determine charge transfer resistance, double layer capacitance and ohmic resistance. The exchange current density $j_0$ can be easily determined measuring the impedance of a redox reaction for $\eta=0$.
Nyquist diagrams are made of several arcs for reactions more complex than redox reactions and with mass-transfer limitations. | 2014-08-21 18:24:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 22, "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": 0.6242117285728455, "perplexity": 1131.5402844684847}, "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/1408500820886.32/warc/CC-MAIN-20140820021340-00332-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://sam.pikesley.org/projects/wen/ | # Wen, the eternally surprised
Building a clock out of Maple, Pi and Ruby
This story begins with a man named Frank Howarth. I’ve spent a lot of time over the last few years, obsessively watching woodturning videos by Frank and others, and then earlier this year I went on a woodturning course (with my friend Chris), where I used a lathe for the first time. I came back sufficiently enthused that my girlfriend bought me a mini-lathe for my birthday, and I’ve spent most weekends since then turning nightlight holders and other things in the garden.
## Everything is a circle
When all you have is a lathe, everything you make is round. It occurred to me that I might be able to fashion a clock, using some Neopixel rings for the dials, and driven by a Raspberry Pi. So I bought some Neopixels and a Pi Zero, turned a simple clock body out of a maple disc, and this is what I came up with…
## The hardware
I was given to understand, from reading the specs, that a Pi would not be able to handle the timing requirements of the Neopixels, and I would need an Arduino in between Pi and pixels. However, after a conversation with Chris at a very weird bar in South London I discovered that there is at least one PWM pin on a Pi with enough grunt to drive the Neopixels, and that even better, there’s a Ruby Gem. So, no Arduino required.
### Wiring it up
It turns out you can chain the Neopixels together and then address them as one long string of lights, so I ran a wire from Physical Pin 12 (which appears as Pin 18 to the client for reasons I don’t fully understand but have ranted about elsewhere) to Data In on the big ring, and then from its Data Out to Data In on the little one. The box marked 5v there is just an old USB cable into which I spliced some wires to power the rings and the Pi, and it all seems to work just fine.
## Software
### Installation
The code is all on Github. To get it up and running from a clean install of NOOBS 1.9 Raspbian Jessie Lite on a Pi Zero, you should look at my Chef cookbook
To play with it locally, you’ll need redis-server running, then 3 different processes in a terminal each:
• bundle exec rake run:queue
• bundle exec rake run:sass
• bundle exec rake run:app
and it should be live at http://localhost:9292
### Internals
Getting a computer to tell you the time is very easy: in Ruby, we just do DateTime.now and there it is. Translating that into lighting the correct lights on a string of LEDs is slightly more challenging, but not much, and I had a crude script working in about an hour. But I quickly fell into the over-engineering rabbit hole, so here’s what I have now:
#### Two rings, with two hands
The inner, hour ring has 12 pixels, which is the correct number of pixels for an hour ring to have, and conceptually, Wen thinks about it as having a hand (the single pixel which indicates the hour) and a face, the other 11 pixels. The outer minutes ring, however, has 24 pixels, which is a little trickier to deal with: a single pixel accounts for 2.5 minutes of actual time, and lighting just that one light to indicate the minutes didn’t really work very well. My strategies for dealing with this are explained below.
It should be noted that 60-pin Neopixels do exist but they’re too big to mount on anything I’d be able to turn on my mini-lathe. I might make one out of concrete, though…
#### Everything gets a RESTful API eventually
Why the hell does a clock need a RESTful API? Well, here’s how I justified it to myself: my first script ran a while true loop, updating the LEDs every 10 seconds, which worked OK, but then I started thinking about how I might be able to get the clock to show patterns and so on, and because of the way my mind works now, I reached for Sinatra (via my Skellington gem) and started wrapping some HTTP around everything.
So it now has four main endpoints:
##### /colours
If you hit this with a GET and an Accept: text/html header (i.e. with a browser), it returns a colour picker
(which I lashed together with Spectrum and some poorly-written d3)
It also Accepts a POST with some JSON like
{
hours: {
hand:
[0, 255, 0]
}
}
}
to change the colour of the specified clock element (this is what the jQuery does behind the picker). There’s also a sub-endpoint
###### /colours/:wheel/:layer
which you can GET to return the current colour of the specified element:
{
colour: [127, 0, 255]
}
(this is used for populating the picker)
##### /modes
This with a GET and Accept: text/html returns the clock-mode picker:
The available modes are:
• range: the minute lights accumulate around the clock throughout the hour, then reset to zero
• vague: the lights on either side of the principal pin are illuminated
• strict: only the principal pin is illuminated
A GET with Accept: application/json will return the current mode:
{
mode: "range"
}
It also accepts a POST with some JSON:
{
mode: "vague"
}
to set the mode (which is what the jQuery does, behind the buttons)
##### /tricks
If you hit this with a browser, it will return a list of available trick modes:
It also Accepts a POST with some JSON like
{
mode: 'shuffle'
}
##### /time
A (empty) POST to this causes the clock to show the current time, and this is how the clock actually works: this systemd config calls this cURL script which hits this URL every 10 seconds.
All of these POST requests then get pushed onto the Sidekiq queue for asynchro…
#### Wait, there’s a queue in here too?
How else would you do this? The ClockWorker pulls the jobs off the queue and throws them at the Clock class, which passes them to the Neopixels singleton, which talks to PixelPi, which does some PWM magic to make the lights come on. I’m actually genuinely amazed at how much bullshit a 4 quid, 15-square-cm computer can handle.
## What else?
This is definitely a prototype: I can certainly turn a better clock body, and there’s always room to improve code…
## There’s a movie, too
It’s a bit out of date now, but:
Tags: | 2018-03-21 02:47:08 | {"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": 0.23397482931613922, "perplexity": 2433.2418858723354}, "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/1521257647567.36/warc/CC-MAIN-20180321023951-20180321043951-00523.warc.gz"} |
https://www.reference.com/web?q=derivative+of+cos+x+2&qo=contentPageRelatedSearch&o=600605&l=dir | Web Results
www.symbolab.com/solver/first-derivative-calculator/\frac{d}{dx}\cos^{2}(x)?or=ex
first-derivative-calculator \frac{d}{dx}\cos^{2}(x) en. image/svg+xml. Related Symbolab blog posts. High School Math Solutions – Derivative Calculator, Logarithms & Exponents . In the previous post we covered trigonometric functions derivatives (click here). We can continue to logarithms...
socratic.org/questions/how-do-find-the-derivative-of-y-cos-2-x
Calculus Differentiating Trigonometric Functions Derivative Rules for y=cos(x) and y=tan(x)
How to find the derivative of cos^2x? I seem to be confused. I was told to use the chain rule, but I don't see how it can apply. Can't I just substitute (cos x)(cos x) in its place and then use the product rule? Isn't cos^2 x = (cos x)(cos x)? Please let me know if I am right. If not, then please explain how to use the chain rule in...
www.easycalculation.com/differentiation/derivative-calculator.php
An online derivative calculator that differentiates a given function with respect to a given variable by using analytical differentiation. A useful mathematical differentiation calculator to simplify the functions.
www.derivative-calculator.net
In doing this, the Derivative Calculator has to respect the order of operations. A specialty in mathematical expressions is that the multiplication sign can be left out sometimes, for example we write "5x" instead of "5*x". The Derivative Calculator has to detect these cases and insert the multiplication sign.
cosx^2 differentiates too 2(cosx)^1 x the differential of cos which is -sin so u get -2sinxcosx use the chain rule! Asked in Math and Arithmetic , Algebra , Calculus Is cos squared x the same as ...
www.intmath.com/differentiation-transcendental/1-derivative-sine-cosine...
The derivative of cos x is −sin x (note the negative sign!) and The derivative of tan x is sec 2 x. Now, if u = f(x) is a function of x, then by using the chain rule, we have: ... The first one, y = cos x 2 + 3, or y = (cos x 2) + 3, means take the curve y = cos x 2 and move it up by 3 units. | 2019-12-16 00:14:14 | {"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": 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": 0.9164657592773438, "perplexity": 678.6930165764365}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541310970.85/warc/CC-MAIN-20191215225643-20191216013643-00415.warc.gz"} |
http://www.iam.fmph.uniba.sk/institute/forum/viewtopic.php?f=2&t=89 | ## Seminar 26.8.2015: Andrej Zlatos
Seminar on Qualitative Theory of Differential Equations
organized by P.Quittner, M.Fila and R.Kollar
Moderator: sevcovic
### Seminar 26.8.2015: Andrej Zlatos
Seminár z kvalitatívnej teórie diferenciálnych rovníc
Seminar on Qualitative Theory of Differential Equations
Wednesday 26.8.2015 at 10:30 Lecture room M-223
Andrej Zlatoš (University of Wisconsin):
Finite time blow-up for the $\alpha$-patch model
Abstract:
The global regularity vs finite time blow-up question remains open for many
fundamental equations of fluid dynamics. In two dimensions, the solutions
to the incompressible Euler equation have been known to be globally regular
since the 1930s. On the other hand, this question has not yet been resolved
for the less regular (by one derivative) surface quasi-geostrophic (SQG)
equation. The latter state of affairs is also true for a natural family
of PDE which interpolate between these two equations. They involve
a parameter $\alpha$, which appears as a power in the kernel of their
Biot-Savart laws and describes the degree of regularity of the equation,
with the values $\alpha=0$ and $\alpha=\frac 12$ corresponding
to the Euler and SQG cases, respectively.
In this talk I will present two results about the patch dynamics version
of these equations in the half-plane. The first is global-in-time regularity
for the Euler patch model, even if the patches initially touch the boundary
of the half-plane. The second is local regularity and existence of solutions
which blow up in finite time for the $\alpha$-patch model with any small
enough $\alpha>0$. The latter appears to be the first rigorous proof
of finite time blow-up in this type of fluid dynamics models.
quittner
Posts: 79
Joined: Fri Oct 12, 2012 11:21 am | 2018-06-18 11:53:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6756044030189514, "perplexity": 1896.9886296592215}, "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-2018-26/segments/1529267859766.6/warc/CC-MAIN-20180618105733-20180618125733-00274.warc.gz"} |
https://math.stackexchange.com/questions/2017662/how-to-get-nabla-p-nabla-delta-1-nabla-cdot-u-cdot-nabla-u-from | # How to get $\nabla p=\nabla (-\Delta)^{-1} \nabla \cdot (u\cdot \nabla u)$ from the Navier-Stokes equations?
The incompressible Navier-Stokes equations are given by:
$$u_t + u \cdot \nabla u -\Delta u + \nabla p=0,\qquad \text{div } u=0,$$ where $u$ is the velocity field and $p$ is the pressure.
Here is a fact that I don't know why it's true from the above equations: $$\nabla p=\nabla (-\Delta)^{-1} \nabla \cdot (u\cdot \nabla u).$$
Thanks!
• Do you know what is $(-\Delta)^{-1}$? – Jack Nov 17 '16 at 0:22
• @Jack I just know it's the inverse of Laplacian operator. Maybe it's not enough, though. – Feipi Wen Nov 17 '16 at 0:25
Formally, you have the so called pressure poisson equation: $$-\Delta p=\nabla\cdot((u\cdot\nabla)u)\tag{1}$$
Now applying the inverse of $-\Delta$ on both sides, you get $$p=(-\Delta)^{-1}\nabla\cdot((u\cdot\nabla)u)$$ and thus $$\nabla p=\nabla(-\Delta)^{-1}\nabla\cdot((u\cdot\nabla)u)$$
To see how to get (1), consider $$\nabla\cdot(u_t+(u\cdot\nabla)u)=\nabla\cdot(\Delta u-\nabla p)$$ and use the divergence-free condition for $u$. On the LHS, you have $$\partial_t(\nabla\cdot u)+\nabla\cdot((u\cdot\nabla)u)=0+\nabla\cdot((u\cdot\nabla)u)$$ On the RHS, you have $$\Delta(\nabla\cdot u)-\Delta p=-\Delta p.$$ | 2019-06-24 18:01:10 | {"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": 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": 0.9155758023262024, "perplexity": 161.87863026006474}, "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/1560627999620.99/warc/CC-MAIN-20190624171058-20190624193058-00542.warc.gz"} |
https://www.groundai.com/project/blind-multi-band-signal-reconstruction-compressed-sensing-for-analog-signals/ | Blind Multi-Band Signal Reconstruction: Compressed Sensing for Analog Signals The authors are with the Technion—Israel Institute of Technology, Haifa Israel. Email: moshiko@tx.technion.ac.il, yonina@ee.technion.ac.il.
# Blind Multi-Band Signal Reconstruction: Compressed Sensing for Analog Signals
## Abstract
We address the problem of reconstructing a multi-band signal from its sub-Nyquist point-wise samples. To date, all reconstruction methods proposed for this class of signals assumed knowledge of the band locations. In this paper, we develop a non-linear blind perfect reconstruction scheme for multi-band signals which does not require the band locations. Our approach assumes an existing blind multi-coset sampling method. The sparse structure of multi-band signals in the continuous frequency domain is used to replace the continuous reconstruction with a single finite dimensional problem without the need for discretization. The resulting problem can be formulated within the framework of compressed sensing, and thus can be solved efficiently using known tractable algorithms from this emerging area. We also develop a theoretical lower bound on the average sampling rate required for blind signal reconstruction, which is twice the minimal rate of known-spectrum recovery. Our method ensures perfect reconstruction for a wide class of signals sampled at the minimal rate. Numerical experiments are presented demonstrating blind sampling and reconstruction with minimal sampling rate.
\setstretch
1.44
{keywords}
Kruskal-rank, Landau-Nyquist rate, multiband, multiple measurement vectors (MMV), nonuniform periodic sampling, orthogonal matching pursuit (OMP), signal representation, sparsity.
## I Introduction
\PARstart
The well known Whittaker, Kotelńikov, and Shannon (WKS) theorem links analog signals with a discrete representation, allowing the transfer of the signal processing to a digital framework. The theorem states that a real-valued signal bandlimited to Hertz can be perfectly reconstructed from its uniform samples if the sampling rate is at least samples per second. This minimal rate is called the Nyquist rate of the signal.
Multi-band signals are bandlimited signals that posses an additional structure in the frequency domain. The spectral support of a multi-band signal is restricted to several continuous intervals. Each of these intervals is called a band and it is assumed that no information resides outside the bands. The design of sampling and reconstruction systems for these signals involves three major considerations. One is the sampling rate. The other is the set of multi-band signals that the system can perfectly reconstruct. The last one is blindness, namely a design that does not assume knowledge of the band locations. Blindness is a desirable property as signals with different band locations are processed in the same way. Landau [1] developed a minimal sampling rate for an arbitrary sampling method that allows perfect reconstruction. For multi-band signals, the Landau rate is the sum of the band widths, which is below the corresponding Nyquist rate.
Uniform sampling of a real bandpass signal with a total width of Hertz on both sides of the spectrum was studied in [2]. It was shown that only special cases of bandpass signals can be perfectly reconstructed from their uniform samples at the minimal rate of samples/sec. Kohlenberg [3] suggested periodic non-uniform sampling with an average sampling rate of . He also provided a reconstruction scheme that recovers any bandpass signal exactly. Lin and Vaidyanathan [4] extended his work to multi-band signals. Their method ensures perfect reconstruction from periodic non uniform sampling with an average sampling rate equal to the Landau rate. Both of these works lack the blindness property as the information about the band locations is used in the design of both the sampling and the reconstruction stages.
Herley and Wong [5] and Venkataramani and Bresler [8] suggested a blind multi-coset sampling strategy that is called universal in [8]. The authors of [8] also developed a detailed reconstruction scheme for this sampling strategy, which is not blind as its design requires information about the spectral support of the signal. Blind multi-coset sampling renders the reconstruction applicable to a wide set of multi-band signals but not to all of them.
Although spectrum-blind reconstruction was mentioned in two conference papers in 1996 [6],[7], a full spectrum-blind reconstruction scheme was not developed in these papers. It appears that spectrum-blind reconstruction has not been handled since then.
We begin by developing a lower bound on the minimal sampling rate required for blind perfect reconstruction with arbitrary sampling and reconstruction. As we show the lower bound is twice the Landau rate and no more than the Nyquist rate. This result is based on recent work of Lue and Do [20] on sampling signals from a union of subspaces.
The heart of this paper is the development of a spectrum-blind reconstruction (SBR) scheme for multi-band signals. We assume a blind multi-coset sampling satisfying the minimal rate requirement. Theoretical tools are developed in order to transform the continuous nature of the reconstruction problem into a finite dimensional problem without any discretization. We then prove that the solution can be obtained by finding the unique sparsest solution matrix from Multiple-Measurement-Vectors (MMV). This set of operations is grouped under a block we name Continuous to Finite (CTF). This block is the cornerstone of two SBR algorithms we develop to reconstruct the signal. One is entitled SBR4 and enables perfect reconstruction using only one instance of the CTF block but requires twice the minimal sampling rate. The other is referred to as SBR2 and allows for sampling at the minimal rate, but involves a bi-section process and several uses of the CTF block. Other differences between the algorithms are also discussed. Both SBR4 and SBR2 can easily be implemented in DSP processors or in software environments.
Our proposed reconstruction approach is applicable to a broad class of multi-band signals. This class is the blind version of the set of signals considered in [8]. In particular, we characterize a subset of this class by the maximal number of bands and the width of the widest band. We then show how to choose the parameters of the multi-coset stage so that perfect reconstruction is possible for every signal in . This parameter selection is also valid for known-spectrum reconstruction with half the sampling rate. The set represents a natural characterization of multi-band signals based on their intrinsic parameters which are usually known in advance. We prove that the SBR4 algorithm ensures perfect reconstruction for all signals in . The SBR2 approach works for almost all signals in but may fail in some very special cases (which typically will not occur). As our strategy is applicable also for signals that do not lie in , we present a nice feature of a success recovery indication. Thus, if a signal cannot be recovered this indication prevents further processing of invalid data.
The CTF block requires finding a sparsest solution matrix which is an NP-hard problem [12]. Several sub-optimal efficient methods have been developed for this problem in the compressed sensing (CS) literature [15],[16]. In our algorithms, any of these techniques can be used. Numerical experiments on random constructions of multi-band signals show that both SBR4 and SBR2 maintain a satisfactory exact recovery rate when the average sampling rate approaches their theoretical minimum rate requirement and sub-optimal implementations of the CTF block are used. Moreover, the average runtime is shown to be fast enough for practical usage.
Our work differs from other main stream CS papers in two aspects. The first is that we aim to recover a continuous signal, while the classical problem addressed in the CS literature is the recovery of discrete and finite vectors. An adaptation of CS results to continuous signals was also considered in a set of conferences papers (see [21],[22] and the references therein). However, these papers did not address the case of multi-band signals. In [22] an underlying discrete model was assumed so that the signal is a linear combination of a finite number of known functions. Here, there is no discrete model as the signals are treated in a continuous framework without any discretization. The second aspect is that we assume a deterministic sampling stage and our theorems and results do not involve any probability model. In contrast, the common approach in compressive sensing assumes random sampling operators and typical results are valid with some probability less than 1 [13],[19],[21],[22].
The paper is organized as follows. In Section II we formulate our reconstruction problem. The minimal density theorem for blind reconstruction is stated and proved in Section III. A brief overview of multi-coset sampling is presented in Section IV. We develop our main theoretical results on spectrum-blind reconstruction and present the CTF block in Section V. Based on these results, in Section VI, we design and compare the SBR4 and the SBR2 algorithms. Numerical experiments are described in Section VII.
## Ii Preliminaries and Problem formulation
### Ii-a Notation
Common notation, as summarized in Table I, is used throughout the paper. Exceptions to this notation are indicated in the text.
In addition, the following abbreviations are used. The norm of a vector is defined as
∥v∥pp=∑i|vi|p,p≥0.
The default value for is 2, so that denotes the norm of . The standard norm is used for continuous signals. The th column of is written as , the th row is written as a column vector.
Indicator sets for vectors and matrices are defined respectively as
I(v)={k|v(k)≠0},I(A)={k|(AT)k≠0}.
The set contains the indices of non-zero values in the vector . The set contains the indices of the non-identical zero rows of .
Finally, is the matrix that contains the columns of with indices belonging to the set . The matrix is referred to as the (columns) restriction of to . Formally,
(AS)i=(A)Si,1≤i≤|S|.
Similarly, is referred to as the rows restriction of to .
### Ii-B Multi-band signals
In this work our prime focus is on the set of all complex-valued multi-band signals bandlimited to with no more than bands where each of the band widths is upper bounded by . Fig. 1 depicts a typical spectral support for .
The Nyquist rate corresponding to any is . The Fourier transform of a multi-band signal has support on a finite union of disjoint intervals in . Each interval is called a band and is uniquely represented by its edges . Without loss of generality it is assumed that the bands are not overlapping.
Although our interest is mainly in signals , our results are applicable to a broader class of signals, as explained in the relevant sections. In addition, the results of the paper are easily adopted to real-valued signals supported on . The required modifications are explained in Appendix A and are based on the equations derived in Section IV-A.
### Ii-C Problem formulation
We wish to perfectly reconstruct from its point-wise samples under two constraints. One is blindness, so that the information about the band locations is not used while acquiring the samples and neither can it be used in the reconstruction process. The other is that the sampling rate required to guarantee perfect reconstruction should be minimal.
This problem is solved if either of its constraints is removed. Without the rate constraint, the WKS theorem allows perfect blind-reconstruction for every signal bandlimited to from its uniform samples at the Nyquist rate . Alternatively, if the exact number of bands and their locations are known, then the method of [4] allows perfect reconstruction for every multi-band signal at the minimal sampling rate provided by Landau’s theorem [1].
In this paper, we first develop the minimal sampling rate required for blind reconstruction. We then use a multi-coset sampling strategy to acquire the samples at an average sampling rate satisfying the minimal requirement. The design of this sampling method does not require knowledge of the band locations. We provide a spectrum-blind reconstruction scheme for this sampling strategy in the form of two different algorithms, named SBR4 and SBR2. It is shown that if the sampling rate is twice the minimal rate then algorithm SBR4 guarantees perfect reconstruction for every . The SBR2 algorithm requires the minimal sampling rate and guarantees perfect reconstruction for most signals in . However, some special signals from , discussed in Section VI-B, cannot be perfectly reconstructed by this approach. Excluding these special cases, our proposed method satisfies both constraints of the problem formulation.
## Iii Minimal sampling rate
We begin by quoting Landau’s theorem for the minimal sampling rate of an arbitrary sampling method that allows known-spectrum perfect reconstruction. It is then proved that blind perfect-reconstruction requires a minimal sampling rate that is twice the Landau rate.
### Iii-a Known spectrum support
Consider the space of bandlimited functions restricted to a known support :
BT={x(t)∈L2(R)|suppX(f)⊆T}. (1)
A classical sampling scheme takes the values of on a known countable set of locations . The set is called a sampling set for if can be perfectly reconstructed in a stable way from the sequence of samples . The stability constraint requires the existence of constants and such that:
α∥x−y∥2≤∥xR−yR∥2≤β∥x−y∥2,∀x,y∈BT. (2)
Landau [1] proved that if is a sampling set for then it must have a density , where
D−(R)=limr→∞infy∈R|R∩[y,y+r]|r (3)
is the lower Beurling density, and is the Lebesgue measure of . The numerator in (3) counts the number of points from in every interval of width of the real axis2. This result is usually interpreted as a minimal average sampling rate requirement for , and is called the Landau rate.
### Iii-B Unknown spectrum support
Consider the set of signals bandlimited to with bandwidth occupation no more than , so that
λ(suppX(f))≤ΩT,∀x(t)∈NΩ.
The Nyquist rate for is . Note that is not a subspace so that the Landau theorem is not valid here. Nevertheless, it is intuitive to argue that the minimal sampling rate for cannot be below as this value is the Landau rate had the spectrum support been known.
A blind sampling set for is a sampling set whose design does not assume knowledge of . Similarly to (2) the stability of requires the existence of and such that:
α∥x−y∥2≤∥xR−yR∥2≤β∥x−y∥2,∀x,y∈NΩ. (4)
###### Theorem 1 (Minimal sampling rate)
Let be a blind sampling set for . Then,
D−(R)≥min{2ΩT,1T}. (5)
{proof}
The set is of the form
NΩ=⋃T∈ΓBT, (6)
where
Γ={T|T⊆F,λ(T)≤Ω/T}. (7)
Clearly, is a non-countable union of subspaces. Sampling signals that lie in a union of subspaces has been recently treated in [20]. For every define the subspaces
Bγ,θ=Bγ+Bθ={x+y|x∈Bγ,y∈Bθ}. (8)
Since is a sampling set for , (4) holds for some constants . It was proved in [20, Proposition 2] that (4) is valid if and only if
α∥x−y∥2≤∥xR−yR∥2≤β∥x−y∥2,∀x,y∈Bγ,θ (9)
holds for every . In particular, is a sampling set for every with .
Observe that the space is of the form (1) with . Applying Landau’s density theorem for each results in
D−(R)≥λ(γ∪θ),∀γ,θ∈Γ. (10)
Choosing
γ=[0,ΩT],θ=[1−ΩT,1T],
we have that for ,
D−(R)≥λ(γ∪θ)=λ(γ)+λ(θ)=2ΩT. (11)
If then and
D−(R)≥λ(γ∪θ)=1T. (12)
Combining (11) and (12) completes the proof.
In [20], the authors consider minimal sampling requirements for a union of shift-invariant subspaces, with a particular structure of sampling functions. Specifically, they view the samples as inner products with sampling functions of the form , which includes multi-coset sampling. Theorem 1 extends this result to an arbitrary point-wise sampling operator. In particular, it is valid for non periodic sampling sets that are not covered by [20].
An immediate corollary of Theorem 1 is that if then uniform sampling at the Nyquist rate with an ideal low pass filter satisfies the requirements of our problem formulation. Namely, both the sampling and the reconstruction do not use the information about the band locations, and the sampling rate is minimal according to Theorem 1. As is contained in the space of bandlimited signals, this choice also provides perfect reconstruction for every . Therefore, in the sequel we assume that so that the minimal sampling rate of Theorem 1 is exactly twice the Landau rate.
It is easy to see that for . Therefore, for known spectral support, the Landau rate is . Despite the fact that is a true subset of , the proof of Theorem 1 can be adopted to show that a minimal density of is required so that stable perfect reconstruction is possible for signals from .
We point out that both Landau’s and Theorem 1 state a lower bound but do not provide a method to achieve the bounds. The rest of the paper is devoted to developing a reconstruction method that approaches the minimal sampling rate of Theorem 1.
## Iv Universal Sampling
This section reviews multi-coset sampling which is used in our development. We also briefly explain the fundamentals of known-spectrum reconstruction as derived in [8].
### Iv-a Multi-coset sampling
Uniform sampling of at the Nyquist rate results in samples that contain all the information about . Multi-coset sampling is a selection of certain samples from this grid. The uniform grid is divided into blocks of consecutive samples. A constant set of length describes the indices of samples that are kept in each block while the rest are zeroed out. The set is referred to as the sampling pattern where
0≤c1
Define the th sampling sequence for as
xci[n]= x(t=nT) n=mL+ci, for some m∈Z xci[n]= 0 otherwise. (14)
The sampling stage is implemented by uniform sampling sequences with period , where the th sampling sequence is shifted by from the origin. Therefore, a multi-coset system is uniquely characterized by the parameters and the sampling pattern .
Direct calculations show that [8]
Xci(ej2πfT)=1LTL−1∑r=0exp(j2πLcir)X(f+rLT), (15) ∀f∈F0=[0,1LT),1≤i≤p,
where is the discrete-time Fourier transform (DTFT) of . Thus, the goal is to choose parameters such that can be recovered from (15).
For our purposes it is convenient to express (15) in a matrix form as
y(f)=Ax(f),∀f∈F0, (16)
where is a vector of length whose th element is , and the vector contains unknowns for each
xi(f)=X(f+iLT),0≤i≤L−1,f∈F0. (17)
The matrix depends on the parameters and the set but not on and is defined by
Aik=1LTexp(j2πLcik). (18)
Dealing with real-valued multi-band signals requires simple modifications to (16). These adjustments are detailed in Appendix A.
The Beurling lower density (i.e. the average sampling rate) of a multi-coset sampling set is
1TAVG=pLT, (19)
which is lower than the Nyquist rate for . However, an average sampling rate above the Landau rate is not sufficient for known-spectrum reconstruction. Additional conditions are needed as explained in the next section.
### Iv-B Known-spectrum reconstruction and universality
The presentation of the reconstruction is simplified using CS sparsity notation. A vector is called -sparse if the number of non-zero values in is no greater than . Using the pseudo-norm the sparsity of is expressed as . We use the following definition of the Kruskal-rank of a matrix [14]:
###### Definition 1
The Kruskal-rank of , denoted as , is the maximal number such that every set of columns of is linearly independent.
Observe that for every the system of (16) has less equations than unknowns. Therefore, a prior on must be used to allow for recovery. In [8] it is assumed that the information about the band locations is available in the reconstruction stage. This information supplies the set for every . Without any additional prior the following condition is necessary for known-spectrum perfect reconstruction
x(f) is p-sparse ,∀f∈F0. (20)
Using the Kruskal-rank of a sufficient condition is formulated as
x(f) is σ(A)-sparse ,∀f∈F0. (21)
The known-spectrum reconstruction of [8] basically restricts the columns of to and inverts the resulting matrix in order to recover .
A sampling pattern that yields a fully Kruskal-rank is called universal and corresponds to . Therefore, the set of signals that are consistent with (21) is the broadest possible if a universal sampling pattern is used. As we show later, choosing , and a universal pattern makes (21) valid for every signal .
Finding a universal pattern , namely one that results in a fully Kruskal-rank , is a combinatorial process. Several specific constructions of sampling patterns that are proved to be universal are given in [8],[10]. In particular, choosing to be prime renders every pattern universal [10].
To summarize, choosing a universal pattern allows recovery of any satisfying (20) when the band locations are known in the reconstruction. We next consider blind signal recovery using universal sampling patterns.
## V Spectrum-Blind Reconstruction
In this section we develop the theory needed for SBR. These results are then used in the next section to construct two efficient algorithms for blind signal reconstruction.
The theoretical results are devoted in the following steps: We first note that when considering blind-reconstruction, we cannot use the prior of [8]. In Section V-A we present a different prior that does not assume the information about the band locations. Using this prior we develop a sufficient condition for blind perfect reconstruction which is very similar to (21). Furthermore, we prove that under certain conditions on , perfect reconstruction is possible for every signal in . We then present the basic SBR paradigm in Section V-B. The main result of the paper is transforming the continuous system of (16) into a finite dimensional problem without using discretization. In Section V-C we develop two propositions for this purpose, and present the CTF block.
### V-a Conditions for blind perfect reconstruction
Recall that for every the system of (16) is undetermined since there are fewer equations than unknowns. The prior assumed in this paper is that for every the vector is sparse but in contrast to [8] the location of the non-zero values is unknown. Clearly, in this case (20) is still necessary for blind perfect reconstruction. The following theorem from the CS literature is used to provide a sufficient condition.
###### Theorem 2
Suppose is a solution of . If then is the unique sparsest solution of the system.
Theorem 2 and its proof are given in [11], [15] with a slightly different notation of instead of the Kruskal-rank of . Note that the condition of the theorem is not necessary as there are examples that the sparsest solution of is unique while .
Using Theorem 2, it is evident that perfect reconstruction is possible for every signal satisfying
x(f) is σ(A)2 -sparse ,∀f∈F0. (22)
As before, choosing a universal pattern makes the set of signals that conform with (22) the widest possible. Note that a factor of two distinguishes between the sufficient conditions of (21) and of (22), and results from the fact that here we do not know the locations of the non-zero values in .
Note that (22) provides a condition under which perfect reconstruction is possible, however, it is still unclear how to find the original signal. Although the problem is similar to that described in the CS literature, here finding the unique sparse vector must be solved for each value in the continuous interval , which clearly cannot be implemented.
In practice, conditions (21) and (22) are hard to verify since they require knowledge of and depend on the parameters of the multi-coset sampling. We therefore prefer to develop conditions on the class which characterizes multi-band signals based on their intrinsic properties: the number of bands and their widths. It is more likely to know the values of and in advance than to know if the signals to be sampled satisfy (21) or (22). The following theorem describes how to choose the parameters and so that the sufficient conditions for perfect reconstruction hold true for every , namely it is a unique solution of (16). The theorem is valid for both known and blind reconstruction with a slight difference resulting from the factor of two in the sufficient conditions.
###### Theorem 3 (Uniqueness)
Let be a multi-band signal. If:
1. The value of is limited by
L≤1BT, (23)
2. for known reconstruction or for blind,
3. is a universal pattern,
then, for every , the vector is the unique solution of (16).
{proof}
If is limited by (23) then for the th band we have
λ(Ti)≤B≤1LT,1≤i≤N.
Therefore, implies
f+kLT∉Ti,∀k≠0.
According to (17) for every the vector takes the values of on a set of points spaced by . Consequently, the number of non-zero values in is no greater than the number of the bands, namely is -sparse.
Since is a universal pattern, . This implies that conditions (21) and (22) are satisfied.
Note that the condition on the value of implies the minimal sampling rate requirement. To see this, substitute (23) into (19):
1TAVG=pLT≥pB. (24)
As pointed out in the end of Section III-B, if the signals are known to lie in then the Landau rate is , which is implied by . Theorem 1 requires an average sampling rate of , which can be guaranteed if .
### V-B Reconstruction paradigm
The goal of our reconstruction scheme is to recover the signal from the set of sequences . Equivalently, the aim is to reconstruct of (16) for every from the input data .
A straight forward approach is to find the sparsest solution on a dense grid of . However, this discretization strategy cannot guarantee perfect reconstruction. In contrast, our approach is exact and does not require discretization.
Our reconstruction paradigm is targeted at finding the diversity set which depends on and is defined as
S=⋃f∈F0I(x(f)). (25)
The SBR algorithms we develop in Section VI are aimed at recovering the set . With the knowledge of perfect reconstruction of is possible for every by noting that (16) can be written as
y(f)=ASxS(f). (26)
If the diversity set of satisfies
|S|≤σ(A), (27)
then
(AS)†AS=I, (28)
where is of size . Multiplying both sides of (26) by results in:
xS(f)=(AS)†y(f),∀f∈F0. (29)
From (25),
xi(f)=0,∀f∈F0,i∉S. (30)
Thus, once is known, and as long as (27) holds, perfect reconstruction can be obtained by (29)-(30).
As we shall see later on (27) is implied by the condition required to transform the problem into a finite dimensional one. Furthermore, the following proposition shows that for , (27) is implied by the parameter selection of Theorem 3.
###### Proposition 1
If is limited by (23) then . If in addition and is universal then for every , the set satisfies (27).
{proof}
The bands are continuous intervals upper bounded by . From (17) it follows that is constructed by dividing into equal intervals of length . Therefore if is limited by (23) then each band can either be fully contained in one of these intervals or it can be split between two consecutive intervals. Since the number of bands is no more than it follows that . With the additional conditions we have that .
As we described, our general strategy is to determine the diversity set and then recover via (29)-(30). In the non-blind setting, is known, and therefore if it satisfies (27) then the same equations can be used to recover . However, note that when the band locations are known, we may use a value of that is smaller than since the sampling rate can be reduced. Therefore, (27) may not hold. Nonetheless, it is shown in [8], that the frequency axis can be divided into intervals such that this approach can be used over each frequency interval. Therefore, once the set is recovered there is no essential difference between known and blind reconstruction.
### V-C Formulation of a finite dimensional problem
The set of equations of (16) consists of an infinite number of linear systems because of the continuous variable . Furthermore, the expression for the diversity set given in (25) involves a union over the same continuous variable. The main result of this paper is that can be recovered exactly using only one finite dimensional problem. In this section we develop the underlying theoretical results that are used for this purpose.
Consider a given . Multiplying each side of (16) by its conjugate transpose we have
(31)
Integrating both sides over the continuous variable gives
Q=AZ0AH, (32)
with the matrix
Q=∫f∈Ty(f)yH(f)df⪰0, (33)
and the matrix
Z0=∫f∈Tx(f)xH(f)df⪰0. (34)
Define the diversity set of the interval as
ST=⋃f∈TI(x(f)). (35)
Now,
(Z0)ii=∫f∈T|xi(f)|2df.
This means that if and only if , which implies that .
The next proposition is used to determine whether can be found by a finite dimensional problem. The proposition is stated for general matrices .
###### Proposition 2
Suppose of size and are given matrices. Let be any matrix satisfying
Q=AZAH, (36a) Z⪰0, (36b) |I(Z)|≤σ(A). (36c) Then, rank(Z)=rank(Q). If, in addition, |I(Z)|≤σ(A)2, (36d)
then, is the unique solution of (36a)-(36d).
{proof}
Let satisfy (36a)-(36c). Define . Since it can be decomposed as with of size having orthogonal columns. From (36a),
Q=(AP)(AP)H. (37)
It can be easily be concluded that , and thus . The following lemma whose proof is given in Appendix B ensures that the matrix of size also has full column rank.
###### Lemma 1
For every two matrices , if then .
Since for every matrix it is true that , (37) implies .
For the second part of Proposition 2 suppose that both satisfy (36a),(36b),(36d). From the first part,
rank(Z)=rank(~Z)=rQ.
Following the earlier decompositions we write
(38) ~Z=~P~PH,I(~Z)=I(~P).
|I(P)|≤σ(A)2,|I(~P)|≤σ(A)2. (39)
From (36a),
Q=(AP)(AP)H=(A~P)(A~P)H, (40)
which implies that
A(P−~PR)=0, (41)
for some unitary matrix . It is easy to see that (39) results in . Therefore the matrix has at most non-identical zero rows. Applying Lemma 1 to (41) results in . Substituting this into (38) we have that .
The following proposition shows how to construct the matrix by finding the sparsest solution of a linear system.
###### Proposition 3
Consider the setting of Proposition 2 and assume satisfies (36d). Let and define a matrix of size using the decomposition , such that has orthogonal columns. Then the linear system
V=AU (42)
has a unique sparsest solution matrix . Namely, and is minimal. Moreover, .
{proof}
Substitute the decomposition into (36a) and let . The result is for some unitary . Therefore, the linear system of (42) has a solution . It is easy to see that , thus (36d) results in . Applying Theorem 2 to each of the columns of provides the uniqueness of . It is trivial that .
Using the same arguments as in the proof it is easy to conclude that , so that can be found directly from the solution matrix . In particular, we develop the Continuous to Finite (CTF) block which determines the diversity set of a given frequency interval . Fig. 2 presents the CTF block that contains the flow of transforming the continuous linear system of (16) on the interval into the finite dimensional problem of (42) and then to the recovery of . The role of Propositions 2 and 3 is also illustrated. The CTF block is the heart of the SBR scheme which we discuss next.
In the CS literature, the linear system of (42) is referred to as an MMV system. Theoretical results regarding the sparsest solution matrix of an MMV system are given in [15]. Finding the solution matrix is known to be NP-hard [12]. Several sub-optimal efficient algorithms for finding are given in [16]. Some of them can indicate a success recovery of . We explain which class of algorithms has this property in Section VI-A.
## Vi SBR algorithms
The theoretical results developed in the previous section are now used in order to construct the diversity set which enables the recovery of via (29)-(30).
We begin by defining a class of signals. The SBR4 algorithm is then presented and is proved to guarantee perfect reconstruction for signals in . We then show that in order to ensure that the sampling rate must be at least , which is twice the minimal rate stated in Theorem 1. To improve on this result, we define a class of signals, and introduce a conceptual method to perfectly reconstruct this class. The SBR2 algorithm is developed so that it ensures exact recovery for a subset of . We then prove that is contained in this subset even for sampling at the minimal rate. However, the computational complexity of SBR2 is higher than that of SBR4. Since universal patterns lead to the largest sets and , we assume throughout this section that universal patterns are used, which results in .
### Vi-a The SBR4 algorithm
Define the class of signals
AK={suppX(f)⊆F and |S|≤K}, (43)
with given by (25). Let , and observe that a multi-coset system with ensures that all the conditions of Proposition 2 are valid for every . Thus, applying the CTF block on results in a unique sparsest solution , with . The reconstruction of the signal is then carried out by (29)-(30). We note that (27) is valid as it represents the class that contains for .
Algorithm 1, named SBR4, follows the steps of the CTF block in Fig. 2 to recover the diversity set from , for any . The algorithm also outputs an indication flag which we discuss later on.
The SBR4 algorithm guarantees perfect reconstruction of signals in from samples at twice the Landau rate, which is also the lower bound stated in Theorem 1. To see this, observe that (25) implies that every must satisfy
λ(suppX(f))≤KLT. (44)
Although is not a subspace, we use (44) to say that the Landau rate for is as it contains subspaces whose widest support is . As we proved, ensures perfect reconstruction for . Substituting the smallest possible value into (19) results in an average sampling rate of .
It is easy to see that flag is equal to 1 for every signal in . However, when a sub-optimal algorithm is used to solve the MMV in step 1 we cannot guarantee a correct solution . Thus, flag=0 indicates that the particular MMV method we used failed, and we may try a different MMV approach.
Existing algorithms for MMV systems can be classified into two groups. The first group contains algorithms that seek the sparsest solution matrix , e.g. Basis Pursuit [17] or Matching Pursuit [18] with a termination criterion based on the residual. The other contains methods that approximate a sparse solution according to user specification, e.g. Matching Pursuit with a predetermined number of iterations. Using a technique from the latter group neutralizes the indication flag as the approximation is always sparse. Therefore, this set of algorithms should be avoided if an indication is desired.
An important advantage of algorithm SBR4 is that the matrix can be computed in the time domain from the known sequences . The computation involves a set of digital filters that do not depend on the signal and thus can be designed in advance. The exact details are given in Appendix C.
The drawback of the set , is that typically we do not know the value of . Moreover, even if is known, then usually we do not know in advance whether as does not characterize the signals according to the number of bands and their widths. Therefore, we would like to determine conditions that ensure . Proposition 1 shows that for the set satisfies if . Thus, under this condition on we have , which in turn implies as a minimal value for . Consequently, SBR4 guarantees perfect reconstruction for under the restrictions and . However, the Landau rate for is , while implies a minimal sampling rate of . Indeed, substituting and into (19) we have
pLT≥4NT1BT=4NB. (45)
In contrast, it follows from Theorem 3 that is sufficient for uniqueness of the solution. The reason for the factor of two in the sampling rate is that is -sparse for each specific ; however, when combining the frequencies, the maximal size of is . The SBR2 algorithm, developed in the next section, capitalizes on this difference to regain the factor of two in the sampling rate, and thus achieves the minimal rate, at the expense of a more complicated reconstruction method.
### Vi-B The SBR2 algorithm
We now would like to reduce the sampling rate required for signals of to its minimum, i.e. twice the Landau rate. To this end, we introduce a set for which SBR2 guarantees perfect reconstruction, and then prove that if .
Consider a partition of into consecutive intervals defined by
0=¯d1<¯d2<⋯<¯dM+1=1LT.
For a given partition set we define the set of signals
BK,¯D={suppX(f)⊆F % and |S[¯di,¯di+1]|≤K,1≤i≤M}.
Clearly, if then we can perfectly reconstruct every by applying the CTF block to each of the intervals . We now define the set as
BK=⋃¯DBK,¯D, (46)
which is the union of over all choices of partition sets and integers . Note that neither nor is a subspace. If we are able to find a partition such that , then can be perfectly reconstructed using . Since the Landau rate for is , this approach requires the minimal sampling rate3.
The following proposition shows that if the parameters are chosen properly, then . Thus, and a method to find of is sufficient for perfect reconstruction of .
###### Proposition 4
If are selected according to Theorem 3 then .
{proof}
In the proof of Theorem 3 we showed that under the conditions of the theorem, is -sparse for every . The proof of the proposition then follows from the following lemma [8]:
###### Lemma 2
If is a multi-band signal with bands sampled by a multi-coset system then there exists a partition set with intervals such that is a constant set over the interval for .
Lemma 2 implies that for every which means that .
So far we showed that , however to recover we need a method to find in practice; Lemma 2 only ensures its existence. Given the data , our strategy is aimed at finding any partition set such that
^S=|D|−1⋃i=0S[di,di+1] (47)
is equal to , and such that for every . As long as (27) holds, once we find the solution is exactly recovered via (29)-(30). To find , we apply the CTF block on each interval . If , then the conditions of Proposition 2 are valid, a unique solution is guaranteed for each interval. Since for (27) is valid for , our method guarantees perfect reconstruction of signals in . As always, using a universal pattern makes the set of signals the largest. Since the Landau rate for is this approach allows for the minimal sampling rate when .
In order to find we suggest a bi-section process on . We initialize and seek . If does not satisfy some condition explained below, then we halve into and and determine and . The bi-section process is repeated several times until the conditions are met, or until it reaches an interval width of no more than . The set is then determined according to (47).
We now describe the conditions for which a given is halved. The matrix of (34) satisfies the constraints (36a)-(36b). Since and (36c) is also valid. However, the last constraint (36d) of Proposition 2 is not guaranteed as it requires a stronger condition . Note that this condition is satisfied immediately if since . We suggest to approximate the value by , and solve the MMV system for the sparsest solution only if . This approximation is motivated by the fact that for any it is true that . From Proposition 2 we have that which results in
rank(Q)≤|I(Z)|. (48)
However, only special multi-band signals result in strict inequality in (48). Therefore, an interval that produces is halved. Otherwise, we apply the CTF block for this assuming that (48) holds with equality. As in SBR4 the flag indicates a correct solution for . Therefore, if the flag is 0 we halve . These reconstruction steps are detailed in Algorithm 2, named SBR2.
It is important to note that SVR2 is sub-optimal, since the final output of the algorithm may not be equal to even for . One reason this can happen is if strict inequality holds in (48) for some interval . In this scenario step 2 is executed even though does not satisfy (36d). For example, a signal with two equal width bands and such that
(49)
and . If also satisfies
X(f−a1)=X(f−a2),∀f∈[0,W], (50)
then it can be verified that while on the interval . This is of course a rare special case. Another reason is a signal for which the algorithm reached the termination step 2 for some small enough interval. This scenario can happen if two or more points of reside in an interval width of . As an empty set is returned for this interval, the final output may be missing some of the elements of . Clearly, the value of influences the amount of cases of this type. We note that since we do not rely on the missing values are typically recovered from other intervals. Thus, both of these sources of error are very uncommon.
The most common case in which SBR2 can fail is due to the use of sub-optimal algorithms to find ; this issue also occurs in SBR4. As explained before, we assume that flag=0 means an incorrect solution and halves the interval . An interesting behavior of MMV methods is that even if cannot be found for , the algorithm may still find a sparse solution for each of its subsections. Thus, the indication flag is also a way to partially overcome the practical limitations of MMV techniques. Note that the indication property is crucial for SBR2 as it helps to refine the partition and reduce the sub-optimality resulting from the MMV algorithm.
We point out that Proposition 4 shows that . We also have that from Proposition 1, which motivates our approach. The SBR2 algorithm itself does not impose any additional limitations on other than those of Theorem 3 required to ensure the uniqueness of the solution. Therefore, theoretically, perfect reconstruction for is guaranteed if the samples are acquired at the minimal rate, with the exception of the special cases discussed before.
The complexity of SBR2 is dictated by the number of iterations of the bi-section process, which is also affected by the behavior of the MMV algorithm that is used. Numerical experiments in Section VII show that empirically SBR2 converges sufficiently fast for practical usage.
Finally, we emphasize that SBR2 does not provide an indication on the success recovery of even for since there is no way to know in advance if is a signal of the special type that SBR2 cannot recover.
### Vi-C Comparison between SBR4 and SBR2
Table II compares the properties of SBR4 and SBR2. We added the WKS theorem as it also offers spectrum-blind reconstruction. Both SBR4 and SBR2 algorithms recover the set according to the paradigm stated in Section V-B. Observe that an indication property is available only for SBR4 and only if the signals are known to lie in . Although both SBR4 and SBR2 can operate at the minimal sampling rate, SBR2 guarantees perfect reconstruction for a wider set of signals as is a true subset of .
Considering signals from we have to restrict the parameter selection. The specific behavior of SBR4 and SBR2 for this scenario is compared in Table III. In particular, SBR4 requires twice the minimal rate.
In the tables, perfect reconstruction refers to reconstruction with a brute-force MMV method that finds the correct solution. In practice, sub-optimal MMV algorithms may result in failure of recovery even when the other requirements are met. The indication flag is intended to discover these cases.
The entire reconstruction scheme is presented in Fig. 3. The scheme together with the tables allow for a wise decision on the particular implementation of the system. Clearly, for it should be preferred to sample at the Nyquist rate and to reconstruct with an ideal low pass filter. For we have to choose between SBR4 and SBR2 according to our prior on the signal. Typically, it is natural to assume | 2021-03-09 03:45:23 | {"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": 0.8365954756736755, "perplexity": 530.6081208535936}, "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/1614178385984.79/warc/CC-MAIN-20210309030723-20210309060723-00291.warc.gz"} |
https://mechcontent.com/ntu-method-heat-exchanger/ | # NTU method in Heat exchanger: Formula, Example, Effectiveness [with Pdf]
Contents
## What is NTU method?
NTU (Number of transfer units) gives the heat transfer capacity of the heat exchanger. In the NTU method, the effectiveness of the heat exchanger is expressed in the form of three non-dimensional terms.
The NTU method is useful when minimum data is provided for calculating heat transfer.
## NTU method formula:
The method has different effectiveness formulae for parallel flow and counterflow heat exchangers.
1) For parallel-flow heat exchanger:
The effectiveness of The parallel-flow heat exchanger is given by,
\varepsilon _{PARALLEL}=\frac{1-exp[-NTU(1+R)]}{1+R}
Where,
R (specific heat ratio) = \frac{C_{min}}{C_{max}}
Where Cmin is the minimum between Cc and Ch while the Cmax is the maximum between the same.
Where,
C_{c}=\dot{m}C_{pc}
C_{h}=\dot{m}C_{ph}
And, NTU = \frac{UA}{C_{min}}
Where U = Overall heat transfer coefficient
A = Area of the heat exchanger.
2) For counter-flow heat exchanger:
The effectiveness of the counterflow heat exchanger is given by,
\varepsilon _{counter}=\frac{1-exp[-NTU(1+R)]}{1-R[-NTU(1-R)]}
Where, R = \frac{C_{min}}{C_{max}}
and, NTU = \frac{C_{UA}}{C_{min}}
## NTU special cases:
Case 1] For condenser and evaporators:
For evaporating and condensing liquid the value of specific heat is considered as infinity.
Therefore, Cmax= ∞
∴ R = \frac{C_{min}}{C_{max}}=\frac{C_{min}}{\infty} =0
For R=0, the effectiveness is given by,
\varepsilon = 1-exp(-NTU)
Case 2] For regenators:
For regenerators the specific heat of both fluid is same
\therefore R=\frac{C_{min}}{C_{max}}=1
∴ a] For parallel flow heat exchanger:-
\varepsilon _{PARALLEL}=\frac{1-exp(-2NTU)}{2}
b] For counter flow heat exchanger:-
\varepsilon _{counter}=\frac{NTU}{1+NTU}
## Effectiveness:
The effectiveness of the heat exchanger is in the ratio of actual heat transfer to the maximum heat transfer.
\therefore \varepsilon =\frac{Q_{actual}}{Q_{max}}
## Numerical on NTU method:
Hot process fluid with cp = 2.6 Kj/Kg.°C enters in a parallel-flow heat exchanger at 100 °C at a mass flow rate of 25000 Kg/hr while the cooling water with cp = 4.2 Kj/Kg.°C enters into the heat exchanger at 10°C at a mass flow rate of 45000 Kg/hr. The heat exchanger has a heat transfer area of 11 m² with an overall heat transfer coefficient equal to 1000 W/m².°C.
Find,
1] Hot fluid outlet temperature
2]Cold fluid outlet temperature
Solution:-
Given:-
Cph = 2.6 KJ/Kg °C
Thi = 100 °C
\dot{m}_{h} = 25000 Kg/hr
Cpc = 4.2 KJ/Kg °C
Tci = 10 °C
\dot{m}_{c} = 45000 Kg/hr
A = 11 m²
U = 1000 W/m² °C
Step-1) Find Cmin, Cmax:-
Ch = \dot{m}_{h} Cph = 25000 x 2.5 = 62500 KJ/hr °C
Cc = \dot{m}_{c} Cpc = 45000 x 4.2 = 189000 KJ/hr °C
∴ Cmax = Cc = 189000 KJ/hr °C and
Cmin = Ch = 62500 KJ/hr °C
Step-2) Find R & NTU:-
R = \frac{C_{min}}{C_{max}} =\frac{62500}{189000}= 0.3306
NTU = \frac{UA}{C_{min}}=\frac{1000\times 11}{62500}=0.176
NTU = 0.176
Step-3) Find effectiveness (ε):-
For parallelflow heat exchanger,
\epsilon_{PARALLEL}=\frac{1-exp[-NTU(1+R)]}{1+R}
\epsilon_{PARALLEL}=\frac{1-exp[-0.176(1+0.3306)]}{1+0.3306}
\epsilon_{PARALLEL}= 0.1569
Step- 5) Exit temperature of hot fluid:-
By formula of effectiveness,
\epsilon=\frac{C_{h}(T_{hi}-T_{ho})}{C_{min}(T_{hi}-T_{ci})}
0.1569=\frac{62500(100-T_{ho})}{62500(100-10)}
T_{ho}= 85.879 °C —Answer 1]
This is the exit temperature of the hot fluid.
Step-6) Exit temperature of cold fluid:-
By using the formula of effectiveness,
\epsilon=\frac{C_{c}(T_{co}-T_{ci})}{C_{min}(T_{hi}-T_{ci})}
0.1569=\frac{189000(T_{co}-10)}{62500(100-10)}
T_{co}= 14.66 °C —Answer 2]
This is the exit temperature of the cold fluid.
Pratik is a Graduated Mechanical engineer. He enjoys sharing the engineering knowledge learned by him with people. | 2023-02-05 00:39:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.4458090364933014, "perplexity": 9570.39066329524}, "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-2023-06/segments/1674764500158.5/warc/CC-MAIN-20230205000727-20230205030727-00671.warc.gz"} |
https://electronics.stackexchange.com/questions/447417/converting-signed-hex-integer-to-display-negative-decimal-integer-on-terminal-in | Converting signed hex integer to display negative decimal integer on terminal in C code
I am trying to take a signed hex number and display it on the terminal as a negative decimal number so that it can be easily read. I am having a hard time working through the logic on how to make this happen. I am using a PIC16F15345 and Real Term as my terminal. My UART is up and running correctly.
Problem Example:
Hex: B6
Display dec value: -74
I have thought about first converting the hex number to binary by creating a large if statement for each byte and then concatenate them together, but I am not quite sure on where to go next for converting to decimal. I need this code to be efficient and easily readable as it is going into some documentation I have and all of my ideas seem very complex and long. I feel like I am overthinking this.
Thank you for the help
• When you refer to the number you want to 'convert' and display as a "hex integer", do you mean that you have 2 ASCII characters, for example a B and a 6, or do you really just have a number in an 8-bit variable in your code? Jul 8 '19 at 22:50
• I am trying to display the decimal value, and what I have is a received 8-bit value from my radio module that I am storing in a variable in my code. I then want to convert that 8-bit number to its correct negative decimal number for easy readability on the terminal. Jul 8 '19 at 22:55
• Ok - here's the thing - it's not a "hex number". It's just a number in a variable. If the variable type you've put the number into is signed, then the number is signed. What you're really wanting to do is convert or print it as decimal. Typically, printf will suffice - or to be safe and correct, probably snprintf would be better. If you really want to be a sadist (or you think you can/need to optimize things) then you could do the conversion 'manually' with an if for the sign bit and a couple of / division and % modulus operations for the digits. Jul 8 '19 at 23:01
• ... and once you've found the 100's, 10's and 1's using / and %, don't forget to add the ASCII 0 character to make them printable. Jul 8 '19 at 23:14 | 2021-11-27 20:40:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3367341458797455, "perplexity": 684.4741760241036}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358233.7/warc/CC-MAIN-20211127193525-20211127223525-00564.warc.gz"} |
https://ask.pinoybix.org/1500/consisting-constant-pressure-constant-isentropic-processes | 181 views
A cycle consisting of one constant pressure, one constant volume and two isentropic processes is known as
| 181 views
+1 vote | 2020-05-29 14:47:30 | {"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": 0.8361234068870544, "perplexity": 9868.242361936274}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347404857.23/warc/CC-MAIN-20200529121120-20200529151120-00331.warc.gz"} |
http://koreascience.or.kr/article/JAKO201711553398744.page?lang=en | # Adaptive group of ink drop spread: a computer code to unfold neutron noise sources in reactor cores
• Accepted : 2017.05.30
• Published : 2017.10.25
#### Abstract
The present paper reports the development of a computational code based on the Adaptive Group of Ink Drop Spread (AGIDS) for reconstruction of the neutron noise sources in reactor cores. AGIDS algorithm was developed as a fuzzy inference system based on the active learning method. The main idea of the active learning method is to break a multiple input-single output system into a single input-single output system. This leads to the ability to simulate a large system with high accuracy. In the present study, vibrating absorber-type neutron noise source in an International Atomic Energy Agency-two dimensional reactor core is considered in neutron noise calculation. The neutron noise distribution in the detectors was calculated using the Galerkin finite element method. Linear approximation of the shape function in each triangle element was used in the Galerkin finite element method. Both the real and imaginary parts of the calculated neutron distribution of the detectors were considered input data in the developed computational code based on AGIDS. The output of the computational code is the strength, frequency, and position (X and Y coordinates) of the neutron noise sources. The calculated fraction of variance unexplained error for output parameters including strength, frequency, and X and Y coordinates of the considered neutron noise sources were $0.002682{\sharp}/cm^3s$, 0.002682 Hz, and 0.004254 cm and 0.006140 cm, respectively.
#### Acknowledgement
Supported by : Iran National Science Foundation
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7. S.A. Hosseini, N. Vosoughi, Neutron noise simulation by GFEM and unstructured triangle elements, Nucl. Eng. Des. 253 (2012) 238-258. https://doi.org/10.1016/j.nucengdes.2012.08.023
8. S. Itoh, A fundamental study of neutron spectra unfolding based on the maximum likelihood method, Nucl, Instrum. Methods Phys. Res. Sect. A 251 (1986) 144-155. https://doi.org/10.1016/0168-9002(86)91161-7
9. L. Zedeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Syst. Man Cybern. 3 (1973) 28-44.
10. E.H. Mamdani, Application of fuzzy logic to approximate reasoning using linguistic synthesis, in: Proceedings of the Sixth International Symposium on Multiple-valued Logic, IEEE Computer Society Press, 1976.
11. T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst. Man Cybern. SCM-15 (1985) 116-132.
12. I.E.P. Afrakoti, S.B. Shouraki, F.M. Bayat, M. Gholami, Using a memristor crossbar structure to implement a novel adaptive real-time fuzzy modeling algorithm, Fuzzy Sets Syst. 307 (2016) 115-128.
13. S.B. Shouraki, N. Honda, Recursive fuzzy modeling based on fuzzy interpolation, J. Adv. Comput. Intell. 3 (1999) 114-125.
14. M. Murakami, N. Honda, A study on the modeling ability of the IDS method: a soft computing technique using pattern-based information processing, Int. J. Approx. Reason 45 (2007) 470-487. https://doi.org/10.1016/j.ijar.2006.06.022
15. M. Firouzi, S.B. Shouraki, I.E.P. Afrakoti, Pattern analysis by active learning method classifier, J. Intell. Fuzzy Syst. 26 (2014) 49-62.
16. Y. Sakurai, A Study of the Learning Control Method Using PBALM-a Nonlinear Modeling Method, PhD, University of Electro-Communications, Tokyo, 2005.
17. I.E.P. Afrakoti, S.B. Shouraki, B. Haghighat, An optimal hardware implementation for active learning method based on memristor crossbar structures, IEEE Syst. J. 8 (2014) 1190-1199. https://doi.org/10.1109/JSYST.2013.2295963
18. I.E.P. Afrakoti, A. Ghaffari, S.B. Shouraki, Effective partitioning of input domains for ALM algorithm in pattern recognition and image analysis (PRIA), in: 2013 First Iranian Conference on, IEEE, 2013.
19. H. Sagha, S.B. Shouraki, H. Beigy, H. Khasteh, E. Enayati, Genetic ink drop spread, in: Intelligent Information Technology Application, IITA'08, Second International Symposium on, IEEE, 2008.
20. S.A. Hosseini, N. Vosoughi, On a various noise source reconstruction algorithms in VVER-1000 reactor core, Nucl. Eng. Des. 261 (2013) 132-143. https://doi.org/10.1016/j.nucengdes.2013.03.034
21. S.A. Hosseini, N. Vosoughi, Development of 3D neutron noise simulator based on GFEM with unstructured tetrahedron elements, Ann. Nucl. Energy 97 (2016) 132-141. https://doi.org/10.1016/j.anucene.2016.07.006 | 2020-09-23 06:23:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.6021941304206848, "perplexity": 9655.12802581969}, "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/1600400209999.57/warc/CC-MAIN-20200923050545-20200923080545-00694.warc.gz"} |
https://firmfunda.com/maths/differential-calculus | maths
Differential Calculus
The differentiation or derivative calculus is explained in astonishingly simple and clear thought process. The differentiation is covered in the following topics.
• application scenario of differentiation
• first principles of differentiation
• graphical meaning of derivatives
• differentiability of a function
• algebra of derivatives
• standard results in derivatives
The details explained are ingenious and found nowhere else.
maths > differential-calculus > differentiation-application-scenario
Differential Calculus : Understanding Application Scenarios
In this page, the application scenario of derivatives is explained with examples. We consider pair of related quantities. One quantity is the cause and the other is the effect. These are related by a function involving instentaneous rate of change.
maths > differential-calculus > differentiation-first-principles
Differentiation: First Principles
In this page, differentiation is defined in first principles : instantaneous rate of change is the change in a quantity for a small change $\delta \to 0$$\delta \to 0$ in the variable.
Derivative or Differentiation of a function
For a small change in variable $x$$x$, the rate of change in the function $f\left(x\right)$$f \left(x\right)$ is the derivative of the function.
$\frac{d}{dx}f\left(x\right)$$\frac{d}{\mathrm{dx}} f \left(x\right)$
$=f\prime \left(x\right)$
$=\underset{\delta \to 0}{lim}\frac{f\left(x+\delta \right)-f\left(x\right)}{\delta }$$= {\lim}_{\delta \to 0} \frac{f \left(x + \delta\right) - f \left(x\right)}{\delta}$
maths > differential-calculus > differentiation-graphical-meaning
Differentiation: Graphical Meaning
In this page, graphical meaning of differentiation is discussed with examples: derivative of a function at a point is the slope of tangent at that point.
maths > differential-calculus > differentiation-differentiability
Differentiability of Functions
In this page, the conditions under which derivative is defined for functions is discussed.
A function is differentiable at a point, if the instantaneous rate of change at that point is continuous.
A function is continuously differentiable, if the instantaneous rate of change is continuous for any and all points from $-\infty \to +\infty$$- \infty \to + \infty$
A function is continuously differentiable in a range `a
maths > differential-calculus > differentiation-algebra-of-derivatives
Understanding Algebra of Derivatives
In this page, what is algebra of derivatives and conditions under which it is applicable are discussed. Then the derivatives of functions that are given as arithmetic operation of multiple functions is discussed. The arithmetic operations are addition, subtraction, multiplication, and division. The derivatives of functions in composite form and parametric forms are discussed.
Differentiation under Basic Arithmetic Operations
→ $\left(au\right)\prime =au\prime$
→ $\left(u+v\right)\prime =u\prime +v\prime$
→ $\left(u-v\right)\prime =u\prime -v\prime$
→ $\left(uv\right)\prime =u\prime v+uv\prime$
→ $\left(\frac{u}{v}\right)\prime =\frac{u\prime v-uv\prime }{{v}^{2}}$
» Differentiation under Function Operations
→ Composite form and Chain rule: given $v\left(u\right)$$v \left(u\right)$ and $u\left(x\right)$$u \left(x\right)$ (i.e. $v\left(u\left(x\right)\right)$$v \left(u \left(x\right)\right)$) then
$\frac{dv}{dx}=\frac{dv}{du}\frac{du}{dx}$$\frac{\mathrm{dv}}{\mathrm{dx}} = \frac{\mathrm{dv}}{\mathrm{du}} \frac{\mathrm{du}}{\mathrm{dx}}$
→ Parametric form : given $g\left(x\right)$$g \left(x\right)$ and $h\left(x\right)$$h \left(x\right)$ then
$\frac{dg}{dh}=\frac{dg/dx}{dh/dx}$$\frac{\mathrm{dg}}{\mathrm{dh}} = \frac{\mathrm{dg} / \mathrm{dx}}{\mathrm{dh} / \mathrm{dx}}$
maths > differential-calculus > differentiation-standard-results
Derivatives of Algebraic Expressions
In this page, Derivatives of standard functions in algebraic expressions are explained. Then the derivatives of trigonometric functions sin, cos, tan etc. are discussed. Then the derivatives of inverse trigonometric functions such as $\mathrm{arcsin}$$\arcsin$, $\mathrm{arccos}$$\arccos$, etc. are discussed. Then the derivatives of exponents and logarithmic functions such as ${e}^{x}$${e}^{x}$, ${a}^{x}$${a}^{x}$, and $\mathrm{ln}x$$\ln x$ | 2020-08-08 05:29:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 18, "math_annotations": 0, "math_alttext": 0, "mathml": 24, "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": 0.9501492381095886, "perplexity": 3698.454809929705}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439737289.75/warc/CC-MAIN-20200808051116-20200808081116-00546.warc.gz"} |
https://www.askiitians.com/forums/Differential-Calculus/i-ve-attached-an-image-which-contains-the-question_260403.htm | # I’ve attached an image which contains the question. Hoope to get a reply and answer from you as soon as possible!
Arun
25758 Points
2 years ago
Dear student
(a) for limit at 0- (means limit in the LHS of 0) = -2
(b) for limit at 0+ (means limit in the RHS of 0) = -1
(c) for limit at 2- (means limit in the LHS of 2) = 7
Vikas TU
14149 Points
2 years ago
Dear student | 2022-08-16 15:50:04 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9762876629829407, "perplexity": 3120.9505777759423}, "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/1659882572408.31/warc/CC-MAIN-20220816151008-20220816181008-00589.warc.gz"} |
https://www.gamedev.net/forums/topic/651806-300-000-fps/ | # OpenGL 300 000 fps
This topic is 1593 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
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i never run newer version of opengl, back then i was used only OGl 1 and i never got fps higher than about 900 for a some cube test with that, but yesterday i did it with my new OGL/freeglut framevork and it seems that i got 300 000 fps - at least the succesive timer cals in the called in display method raports that delta time 3 microsecond on each display - is this really flushing 300 thousands of screens per second or I mestaken something (something asynchronous is called or something) and should measure it in a differend way?
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and should measure it in a differend way?
This in the first place! FPS is a reciprocal measure and as such not useful if the range becomes bigger than some ten FPS, perhaps up to e.g. 100 FPS or so. The 900 is already a less meaningful number. Its better to use a linear measure: Compute the mean time per frame over a couple of frames as an absolute measure, and perhaps a percentage between such values for a comparative one.
Also: A cube is most probably not a meaningful test at all. In a real world you may encounter several limits: DMA transfer, texture sampling, pixel fill rate, shader complexity, …; none of them is in peril with a cube example (assuming you don't mean a Koch cube ;)).
Regarding the question of performance boost itself: OpenGL 1 is very, VERY old. None of the (more or less) modern techniques was supported. If you use a modern OpenGL it is much more adapted to existing graphics cards. So yes, it is principally possible, of course.
Edited by haegarr
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Try measuring the FPS across multiple frames, it might be that the timer resolution is not enough for the tiny duration of a single frame and is for some reason giving results smaller than reality.
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Try measuring the FPS across multiple frames, it might be that the timer resolution is not enough for the tiny duration of a single frame and is for some reason giving results smaller than reality.
no the timer is good, i think that it gives good results it is when i have
IdleLoop()
{
double timeDelta = TakeTime();
displayScene();
}
the time delta (3 microseconds) is given properly but I am maybe not sure if the DisplayScene here is doing all the work of making whole
new pixelbuffer and show it or maybe not?
the display code itself is
void draw()
{
glEnableClientState(GL_NORMAL_ARRAY);
glEnableClientState(GL_COLOR_ARRAY);
glEnableClientState(GL_VERTEX_ARRAY);
glNormalPointer(GL_FLOAT, 0, normals2);
glColorPointer(3, GL_FLOAT, 0, colors2);
glVertexPointer(3, GL_FLOAT, 0, vertices2);
glPushMatrix();
glDrawElements(GL_TRIANGLES, 36, GL_UNSIGNED_BYTE, indices);
glPopMatrix();
glDisableClientState(GL_VERTEX_ARRAY);
glDisableClientState(GL_COLOR_ARRAY);
glDisableClientState(GL_NORMAL_ARRAY);
}
////////////////////////////////////////
void display()
{
frame++;
glClear(GL_COLOR_BUFFER_BIT);
glPushMatrix();
glRotatef(frame/1000 , 1, 0, 0);
draw();
glPopMatrix();
glFlush();
}
can it be so fast? does it really whole frame generation (as a result
i see rotating rectangle (some tears on the sufface too) but ofc i cannot be sure if this is 300 tys frames per second or maybe just 300 or so
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You are measuring the speed at which you can submit render commands, not the speed at which your scene is drawn and displayed. Basically what you measure is the memcpy that OpenGL does on your vertex array (to a vertex buffer that you don't know about) when you call glDrawElements, plus the overhead of a dozen library calls. It's not very surprising that this is fast.
You are not swapping buffers, so there is really no notion of a "frame" at all. You do call glFlush, but that isn't the same thing (for the most part, glFlush is pretty useless).
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You are measuring the speed at which you can submit render commands, not the speed at which your scene is drawn and displayed. Basically what you measure is the memcpy that OpenGL does on your vertex array (to a vertex buffer that you don't know about) when you call glDrawElements, plus the overhead of a dozen library calls. It's not very surprising that this is fast.
You are not swapping buffers, so there is really no notion of a "frame" at all. You do call glFlush, but that isn't the same thing (for the most part, glFlush is pretty useless).
atlight tnx, (i suspected thah things releted to asynchronicity) so how to measure real frame making speed - i am using ogl + freeglut, but not got much experience with this yet
I used glFinish and got only 9000fps :C
with glutSwapBuffers() i got 75 fps-es (oscillating 74.7 - 75.3)
this is my screen refresh rate setting
I use lcd right now - has this monitor the refresh rate like crt ones-
it is better to set 60 or 75?
Edited by fir
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glFinish is much closer to what one would want to use since it blocks until all command execution has finished (glFlush doesn't wait for anything). It comes with a serious performance impact, however, since it causes a pipeline stall.
glutSwapBuffers, on the other hand, is the real, true thing. It actually swaps buffers, so there is really a notion of "frame". It also blocks, but synchronized to the actual hardware update frequency, and in a somewhat less rigid way (usually drivers will let you pre-render 2 or 3 frames or will only block at the next draw command after swap, or something else).
The reason why you only see 75 fps is that you have vertical sync enabled (in your driver settings). If you can "comfortably" get those 75 fps at all times (i.e. your frame time (worst, not average) is below 13.3 ms), it doesn't really matter how much faster you can render since that's all the monitor will display anyway. Rendering more frames than those displayed is only a waste of energy (and wearing down components due to heat development).
Now of course, if you only ever get at most 75 (or 60 on other monitors) frames per second displayed, it seems a bit hard to measure the actual frame time accurately. You might have a frame time of 13.3 ms or 10ms or 8ms and it would be no difference since it all comes out as 75fps because the driver syncs to that after finishing your drawing commands
glQueryCounter can be of help here. It lets you get accurate timing without having to stall as when using glFinish. So you can measure the actual time it takes to draw, regardless of how long the driver blocks thereafter to sync.
(Another less elegant but nevertheless effective solution would be to disable vertical sync during development.)
Edited by samoth
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glFinish is much closer to what one would want to use since it blocks until all command execution has finished (glFlush doesn't wait for anything). It comes with a serious performance impact, however, since it causes a pipeline stall.
glutSwapBuffers, on the other hand, is the real, true thing. It actually swaps buffers, so there is really a notion of "frame". It also blocks, but synchronized to the actual hardware update frequency, and in a somewhat less rigid way (usually drivers will let you pre-render 2 or 3 frames or will only block at the next draw command after swap, or something else).
The reason why you only see 75 fps is that you have vertical sync enabled (in your driver settings). If you can "comfortably" get those 75 fps at all times (i.e. your frame time (worst, not average) is below 13.3 ms), it doesn't really matter how much faster you can render since that's all the monitor will display anyway. Rendering more frames than those displayed is only a waste of energy (and wearing down components due to heat development).
Now of course, if you only ever get at most 75 (or 60 on other monitors) frames per second displayed, it seems a bit hard to measure the actual frame time accurately. You might have a frame time of 13.3 ms or 10ms or 8ms and it would be no difference since it all comes out as 75fps because the driver syncs to that after finishing your drawing commands
glQueryCounter can be of help here. It lets you get accurate timing without having to stall as when using glFinish. So you can measure the actual time it takes to draw, regardless of how long the driver blocks thereafter to sync.
(Another less elegant but nevertheless effective solution would be to disable vertical sync during development.)
alright tnx for explanation, i disabled vsync in nvconsole (do not know why perprogram not working but global disable working) and got about 9000 fps with swap buffer close to the same as with glFinish
yet some doubt if glFlush is not drawing all the calls what it is doing with such calls? skips or queues?
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glFlush is supposed to mean "start processing all pending GL commands now and return immediately". It doesn't wait for the commands to finish processing, it just signals to the driver that it can start processing them. There are actually a lot of implicit glFlush cases in normal code, with the most obvious one being when the command buffer is full - the driver must start emptying the buffer before new commands can go in.
I see that Carmack has noted on his Twitter that with some drivers glFlush is a nop. If this is the case, then calling glFlush at the end of a frame (or wherever in the frame) will have no effect and the actual flush won't occur until the command buffer fills. Depending on how much work you do in a frame, and on how big the command buffer is (that's driver-dependent so don't ask) it means that you may get 10, 20, or even hundreds of frames worth of commands in there before anything actually happens.
It's easy to see how this kind of behaviour can seriously mislead you into thinking that you're running crazy-fast. A large part of the blame here must seriously go to old GLUT tutorials that always create a single-buffered context. That's just so unrepresentative of how things work in real programs.
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glutSwapBuffers, on the other hand, is the real, true thing. It actually swaps buffers, so there is really a notion of "frame". It also blocks, but synchronized to the actual hardware update frequency, and in a somewhat less rigid way (usually drivers will let you pre-render 2 or 3 frames or will only block at the next draw command after swap, or something else).
When timing with just SwapBuffers though, be careful. The problem ends up being the driver typically queues up the request quickly on the CPU and returns immediately (i.e. CPU does not block), after which it lets you start queuing up render commands for future frames. At some random point in the middle of queuing one of those frames when the FIFO fills, "then" the CPU blocks, waiting on some VSYNC event in the middle of a frame. This causes really odd timing spikes leaving you puzzled as to what's going on.
If you want reasonable full-frame timings, after SwapBuffers(), put a glFinish(), and then stop your timer.
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Sprite Class
(You mostly need to see the Constructor, the Render Method and the Move Method)
#include "Brain.h" #include <glm/gtc/matrix_transform.hpp> #include <vector> struct Sprite::Implementation { //Position. struct pos pos; //Tag. std::string tag; //Texture. Texture *texture; //Model matrix. glm::mat4 model; //Vertex Array Object. VertexArray *vao; //Vertex Buffer Object. VertexBuffer *vbo; //Layout. VertexBufferLayout *layout; //Index Buffer Object. IndexBuffer *ibo; //Shader. Shader *program; //Brains. std::vector<Brain *> brains; //Deconstructor. ~Implementation(); }; Sprite::Sprite(std::string image_path, std::string tag, float x, float y) { //Create Pointer To Implementaion. m_Impl = new Implementation(); //Set the Position of the Sprite object. m_Impl->pos.x = x; m_Impl->pos.y = y; //Set the tag. m_Impl->tag = tag; //Create The Texture. m_Impl->texture = new Texture(image_path); //Initialize the model Matrix. m_Impl->model = glm::mat4(1.0f); //Get the Width and the Height of the Texture. int width = m_Impl->texture->GetWidth(); int height = m_Impl->texture->GetHeight(); //Create the Verticies. float verticies[] = { //Positions //Texture Coordinates. x, y, 0.0f, 0.0f, x + width, y, 1.0f, 0.0f, x + width, y + height, 1.0f, 1.0f, x, y + height, 0.0f, 1.0f }; //Create the Indicies. unsigned int indicies[] = { 0, 1, 2, 2, 3, 0 }; //Create Vertex Array. m_Impl->vao = new VertexArray(); //Create the Vertex Buffer. m_Impl->vbo = new VertexBuffer((void *)verticies, sizeof(verticies)); //Create The Layout. m_Impl->layout = new VertexBufferLayout(); m_Impl->layout->PushFloat(2); m_Impl->layout->PushFloat(2); m_Impl->vao->AddBuffer(m_Impl->vbo, m_Impl->layout); //Create the Index Buffer. m_Impl->ibo = new IndexBuffer(indicies, 6); //Create the new shader. m_Impl->program = new Shader("Shaders/SpriteShader.shader"); } //Render. void Sprite::Render(Window * window) { //Create the projection Matrix based on the current window width and height. glm::mat4 proj = glm::ortho(0.0f, (float)window->GetWidth(), 0.0f, (float)window->GetHeight(), -1.0f, 1.0f); //Set the MVP Uniform. m_Impl->program->setUniformMat4f("u_MVP", proj * m_Impl->model); //Run All The Brains (Scripts) of this game object (sprite). for (unsigned int i = 0; i < m_Impl->brains.size(); i++) { //Get Current Brain. Brain *brain = m_Impl->brains[i]; //Call the start function only once! if (brain->GetStart()) { brain->SetStart(false); brain->Start(); } //Call the update function every frame. brain->Update(); } //Render. window->GetRenderer()->Draw(m_Impl->vao, m_Impl->ibo, m_Impl->texture, m_Impl->program); } void Sprite::Move(float speed, bool left, bool right, bool up, bool down) { if (left) { m_Impl->pos.x -= speed; m_Impl->model = glm::translate(m_Impl->model, glm::vec3(-speed, 0, 0)); } if (right) { m_Impl->pos.x += speed; m_Impl->model = glm::translate(m_Impl->model, glm::vec3(speed, 0, 0)); } if (up) { m_Impl->pos.y += speed; m_Impl->model = glm::translate(m_Impl->model, glm::vec3(0, speed, 0)); } if (down) { m_Impl->pos.y -= speed; m_Impl->model = glm::translate(m_Impl->model, glm::vec3(0, -speed, 0)); } } void Sprite::AddBrain(Brain * brain) { //Push back the brain object. m_Impl->brains.push_back(brain); } pos *Sprite::GetPos() { return &m_Impl->pos; } std::string Sprite::GetTag() { return m_Impl->tag; } int Sprite::GetWidth() { return m_Impl->texture->GetWidth(); } int Sprite::GetHeight() { return m_Impl->texture->GetHeight(); } Sprite::~Sprite() { delete m_Impl; } //Implementation Deconstructor. Sprite::Implementation::~Implementation() { delete texture; delete vao; delete vbo; delete layout; delete ibo; delete program; }
Renderer Class
#include "Renderer.h" #include "Error.h" Renderer::Renderer() { } Renderer::~Renderer() { } void Renderer::Draw(VertexArray * vao, IndexBuffer * ibo, Texture *texture, Shader * program) { vao->Bind(); ibo->Bind(); program->Bind(); if (texture != NULL) texture->Bind(); GLCall(glDrawElements(GL_TRIANGLES, ibo->GetCount(), GL_UNSIGNED_INT, NULL)); } void Renderer::Clear(float r, float g, float b) { GLCall(glClearColor(r, g, b, 1.0)); GLCall(glClear(GL_COLOR_BUFFER_BIT)); } void Renderer::Update(GLFWwindow *window) { /* Swap front and back buffers */ glfwSwapBuffers(window); /* Poll for and process events */ glfwPollEvents(); }
#shader vertex #version 330 core layout(location = 0) in vec4 aPos; layout(location = 1) in vec2 aTexCoord; out vec2 t_TexCoord; uniform mat4 u_MVP; void main() { gl_Position = u_MVP * aPos; t_TexCoord = aTexCoord; } #shader fragment #version 330 core out vec4 aColor; in vec2 t_TexCoord; uniform sampler2D u_Texture; void main() { aColor = texture(u_Texture, t_TexCoord); } Also i'm pretty sure that every time i'm hitting the up, down, left and right arrows on the keyboard, i'm changing the model Matrix of the Player and not the others.
Window Class:
#include "Window.h" #include <GL/glew.h> #include <GLFW/glfw3.h> #include "Error.h" #include "Renderer.h" #include "Scene.h" #include "Input.h" //Global Variables. int screen_width, screen_height; //On Window Resize. void OnWindowResize(GLFWwindow *window, int width, int height); //Implementation Structure. struct Window::Implementation { //GLFW Window. GLFWwindow *GLFW_window; //Renderer. Renderer *renderer; //Delta Time. double delta_time; //Frames Per Second. int fps; //Scene. Scene *scnene; //Input. Input *input; //Deconstructor. ~Implementation(); }; //Window Constructor. Window::Window(std::string title, int width, int height) { //Initializing width and height. screen_width = width; screen_height = height; //Create Pointer To Implementation. m_Impl = new Implementation(); //Try initializing GLFW. if (!glfwInit()) { std::cout << "GLFW could not be initialized!" << std::endl; std::cout << "Press ENTER to exit..." << std::endl; std::cin.get(); exit(-1); } //Setting up OpenGL Version 3.3 Core Profile. glfwWindowHint(GLFW_CONTEXT_VERSION_MAJOR, 3); glfwWindowHint(GLFW_CONTEXT_VERSION_MINOR, 3); glfwWindowHint(GLFW_OPENGL_PROFILE, GLFW_OPENGL_CORE_PROFILE); /* Create a windowed mode window and its OpenGL context */ m_Impl->GLFW_window = glfwCreateWindow(width, height, title.c_str(), NULL, NULL); if (!m_Impl->GLFW_window) { std::cout << "GLFW could not create a window!" << std::endl; std::cout << "Press ENTER to exit..." << std::endl; std::cin.get(); glfwTerminate(); exit(-1); } /* Make the window's context current */ glfwMakeContextCurrent(m_Impl->GLFW_window); //Initialize GLEW. if(glewInit() != GLEW_OK) { std::cout << "GLEW could not be initialized!" << std::endl; std::cout << "Press ENTER to exit..." << std::endl; std::cin.get(); glfwTerminate(); exit(-1); } //Enabling Blending. GLCall(glEnable(GL_BLEND)); GLCall(glBlendFunc(GL_SRC_ALPHA, GL_ONE_MINUS_SRC_ALPHA)); //Setting the ViewPort. GLCall(glViewport(0, 0, width, height)); //**********Initializing Implementation**********// m_Impl->renderer = new Renderer(); m_Impl->delta_time = 0.0; m_Impl->fps = 0; m_Impl->input = new Input(this); //**********Initializing Implementation**********// //Set Frame Buffer Size Callback. glfwSetFramebufferSizeCallback(m_Impl->GLFW_window, OnWindowResize); } //Window Deconstructor. Window::~Window() { delete m_Impl; } //Window Main Loop. void Window::MainLoop() { //Time Variables. double start_time = 0, end_time = 0, old_time = 0, total_time = 0; //Frames Counter. int frames = 0; /* Loop until the user closes the window */ while (!glfwWindowShouldClose(m_Impl->GLFW_window)) { old_time = start_time; //Total time of previous frame. start_time = glfwGetTime(); //Current frame start time. //Calculate the Delta Time. m_Impl->delta_time = start_time - old_time; //Get Frames Per Second. if (total_time >= 1) { m_Impl->fps = frames; total_time = 0; frames = 0; } //Clearing The Screen. m_Impl->renderer->Clear(0, 0, 0); //Render The Scene. if (m_Impl->scnene != NULL) m_Impl->scnene->Render(this); //Updating the Screen. m_Impl->renderer->Update(m_Impl->GLFW_window); //Increasing frames counter. frames++; //End Time. end_time = glfwGetTime(); //Total time after the frame completed. total_time += end_time - start_time; } //Terminate GLFW. glfwTerminate(); } //Load Scene. void Window::LoadScene(Scene * scene) { //Set the scene. m_Impl->scnene = scene; } //Get Delta Time. double Window::GetDeltaTime() { return m_Impl->delta_time; } //Get FPS. int Window::GetFPS() { return m_Impl->fps; } //Get Width. int Window::GetWidth() { return screen_width; } //Get Height. int Window::GetHeight() { return screen_height; } //Get Input. Input * Window::GetInput() { return m_Impl->input; } Renderer * Window::GetRenderer() { return m_Impl->renderer; } GLFWwindow * Window::GetGLFWindow() { return m_Impl->GLFW_window; } //Implementation Deconstructor. Window::Implementation::~Implementation() { delete renderer; delete input; } //OnWindowResize void OnWindowResize(GLFWwindow *window, int width, int height) { screen_width = width; screen_height = height; //Updating the ViewPort. GLCall(glViewport(0, 0, width, height)); }
Brain Class
#include "Brain.h" #include "Sprite.h" #include "Window.h" struct Brain::Implementation { //Just A Flag. bool started; //Window Pointer. Window *window; //Sprite Pointer. Sprite *sprite; }; Brain::Brain(Window *window, Sprite *sprite) { //Create Pointer To Implementation. m_Impl = new Implementation(); //Initialize Implementation. m_Impl->started = true; m_Impl->window = window; m_Impl->sprite = sprite; } Brain::~Brain() { //Delete Pointer To Implementation. delete m_Impl; } void Brain::Start() { } void Brain::Update() { } Window * Brain::GetWindow() { return m_Impl->window; } Sprite * Brain::GetSprite() { return m_Impl->sprite; } bool Brain::GetStart() { return m_Impl->started; } void Brain::SetStart(bool value) { m_Impl->started = value; } Script Class (Its a Brain Subclass!!!)
#include "Script.h" Script::Script(Window *window, Sprite *sprite) : Brain(window, sprite) { } Script::~Script() { } void Script::Start() { std::cout << "Game Started!" << std::endl; } void Script::Update() { Input *input = this->GetWindow()->GetInput(); Sprite *sp = this->GetSprite(); //Move this sprite. this->GetSprite()->Move(200 * this->GetWindow()->GetDeltaTime(), input->GetKeyDown("left"), input->GetKeyDown("right"), input->GetKeyDown("up"), input->GetKeyDown("down")); std::cout << sp->GetTag().c_str() << ".x = " << sp->GetPos()->x << ", " << sp->GetTag().c_str() << ".y = " << sp->GetPos()->y << std::endl; }
Main:
#include "SpaceShooterEngine.h" #include "Script.h" int main() { Window w("title", 600,600); Scene *scene = new Scene(); Sprite *player = new Sprite("Resources/Images/player.png", "Player", 100,100); Sprite *other = new Sprite("Resources/Images/cherno.png", "Other", 400, 100); Sprite *other2 = new Sprite("Resources/Images/cherno.png", "Other", 300, 400); Brain *brain = new Script(&w, player); player->AddBrain(brain); scene->AddSprite(player); scene->AddSprite(other); scene->AddSprite(other2); w.LoadScene(scene); w.MainLoop(); return 0; }
I literally can't find what is wrong. If you need more code, ask me to post it. I will also attach all the source files.
Brain.cpp
Error.cpp
IndexBuffer.cpp
Input.cpp
Renderer.cpp
Scene.cpp
Sprite.cpp
Texture.cpp
VertexArray.cpp
VertexBuffer.cpp
VertexBufferLayout.cpp
Window.cpp
Brain.h
Error.h
IndexBuffer.h
Input.h
Renderer.h
Scene.h
SpaceShooterEngine.h
Sprite.h
Texture.h
VertexArray.h
VertexBuffer.h
VertexBufferLayout.h
Window.h
• Hello fellow programmers,
For a couple of days now i've decided to build my own planet renderer just to see how floating point precision issues
can be tackled. As you probably imagine, i've quickly faced FPP issues when trying to render absurdly large planets.
I have used the classical quadtree LOD approach;
I've generated my grids with 33 vertices, (x: -1 to 1, y: -1 to 1, z = 0).
Each grid is managed by a TerrainNode class that, depending on the side it represents (top, bottom, left right, front, back),
creates a special rotation-translation matrix that moves and rotates the grid away from the origin so that when i finally
normalize all the vertices on my vertex shader i can get a perfect sphere.
T = glm::translate(glm::dmat4(1.0), glm::dvec3(0.0, 0.0, 1.0)); R = glm::rotate(glm::dmat4(1.0), glm::radians(180.0), glm::dvec3(1.0, 0.0, 0.0)); sides[0] = new TerrainNode(1.0, radius, T * R, glm::dvec2(0.0, 0.0), new TerrainTile(1.0, SIDE_FRONT)); T = glm::translate(glm::dmat4(1.0), glm::dvec3(0.0, 0.0, -1.0)); R = glm::rotate(glm::dmat4(1.0), glm::radians(0.0), glm::dvec3(1.0, 0.0, 0.0)); sides[1] = new TerrainNode(1.0, radius, R * T, glm::dvec2(0.0, 0.0), new TerrainTile(1.0, SIDE_BACK)); // So on and so forth for the rest of the sides As you can see, for the front side grid, i rotate it 180 degrees to make it face the camera and push it towards the eye;
the back side is handled almost the same way only that i don't need to rotate it but simply push it away from the eye.
The same technique is applied for the rest of the faces (obviously, with the proper rotations / translations).
The matrix that result from the multiplication of R and T (in that particular order) is send to my vertex shader as r_Grid'.
// spherify vec3 V = normalize((r_Grid * vec4(r_Vertex, 1.0)).xyz); gl_Position = r_ModelViewProjection * vec4(V, 1.0); The r_ModelViewProjection' matrix is generated on the CPU in this manner.
// No the most efficient way, but it works. glm::dmat4 Camera::getMatrix() { // Create the view matrix // Roll, Yaw and Pitch are all quaternions. glm::dmat4 View = glm::toMat4(Roll) * glm::toMat4(Pitch) * glm::toMat4(Yaw); // The model matrix is generated by translating in the oposite direction of the camera. glm::dmat4 Model = glm::translate(glm::dmat4(1.0), -Position); // Projection = glm::perspective(fovY, aspect, zNear, zFar); // zNear = 0.1, zFar = 1.0995116e12 return Projection * View * Model; } I managed to get rid of z-fighting by using a technique called Logarithmic Depth Buffer described in this article; it works amazingly well, no z-fighting at all, at least not visible.
Each frame i'm rendering each node by sending the generated matrices this way.
// set the r_ModelViewProjection uniform // Sneak in the mRadiusMatrix which is a matrix that contains the radius of my planet. Shader::setUniform(0, Camera::getInstance()->getMatrix() * mRadiusMatrix); // set the r_Grid matrix uniform i created earlier. Shader::setUniform(1, r_Grid); grid->render(); My planet's radius is around 6400000.0 units, absurdly large, but that's what i really want to achieve;
Everything works well, the node's split and merge as you'd expect, however whenever i get close to the surface
of the planet the rounding errors start to kick in giving me that lovely stairs effect.
I've read that if i could render each grid relative to the camera i could get better precision on the surface, effectively
getting rid of those rounding errors.
My question is how can i achieve this relative to camera rendering in my scenario here?
I know that i have to do most of the work on the CPU with double, and that's exactly what i'm doing.
I only use double on the CPU side where i also do most of the matrix multiplications.
As you can see from my vertex shader i only do the usual r_ModelViewProjection * (some vertex coords). | 2018-05-23 09:26:55 | {"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": 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": 0.2114366590976715, "perplexity": 7819.3462286397225}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794865468.19/warc/CC-MAIN-20180523082914-20180523102914-00101.warc.gz"} |
http://www.particle.cz/conferences/c2cr2005/prog.html | HOME
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! NEW !
C2CR 2005, 7-13 September 2005, Prague
Tentative Programme
9:00 12:00 14:00 - 17:00
7.9.
Wednesday
a r r i v a l
(registration 16:00 - 20:00)
8.9.
Thursday
chairman: J. Horejsi
Carlos Lourenco
Recent results on heavy-ion physics from the SPS
Hongyan Yang
Hadron production at forward rapidity in nuclear collisions at RHIC
Aneta Iordanova
PHOBOS
Jaap Panman
HARP
##### lunch
chairman: J. W. Cronin
Leonid Frankfurt
Martin Block
The Elusive p-air Cross Section - tying together accelerator sigma_pp and cosmic ray sigma_p-air^inel measurements
##### coffee
Mark Strikman
Small x physics and Forward dynamics in pp/pA ultrahigh energy collisions
Alexander Kiselev
Gravi-Reggeons and Cosmic Neutrino-Nucleon Interactions
Welcome drink
(18:30)
9.9.
Friday
chairman: J. Chyla
Albert de Roeck
Diffraction and forward physics at the LHC
The LHCf experiment at LHC
Katevi Assamagan
Discovery Potential of the LHC
##### coffee
chairman: C. Lourenco
Terry Sloan
The Structure of the Nucleon and Simulation of Cosmic Ray Showers
Rainer Mankel
The hadronic final state at HERA
Alexandr Kupco
D0/CDF
##### lunch
chairman: E. Zas
Sergey Ostapchenko
Models for cosmic ray interactions
Tanguy Pierog
Impact of uncertainties in hadron production on air shower predictions
##### coffee
chairman: C. Jui
H. P. Wellish
Geant4
Mani Tripathi
A Search for Dark Matter with CACTUS
10.9.
Saturday
chairman: R. Engel
Petr Travnicek
Detection of cosmic rays by LEP experiments
Lawrence Jones
Cosmic Ray Studies with the L3-Cosmics Program at CERN
Christine Meurer
Muon production in extensive air showers and its relation to accelerator measurements
##### coffee
chairman: A. de Roeck
Markus Risse
Geometric structures in hadronic cores of extensive air showers observed by KASCADE
Search for nuclearites with the SLIM detector
##### lunch
E X C U R S I O N
(castle Konopiste)
11.9.
Sunday
chairman: A. Haungs
Dmitri Semikoz
Ultra-High Energy Cosmic Rays: status of the problem
Charles Jui
Latest Results from the High Resolution Fly's Eye (HiRes) Experiment
Enrique Zas
The Auger Observatory: Status and first results
##### coffee
chairman: G. Varner
Andreas Haungs
Holger Ulrich
Reconstruction of energy spectra of elemental groups with KASCADE: sensitivity to hadronic interaction models
##### lunch
chairman: M. Tripathi
Kimihiro Okumura
SuperKamiokande
Christopher Wiebusch
AMANDA
##### coffee
chairman: T. Sloan
Morihiro Honda
Atmospheric muon and neutrino fluxes and hadronic interaction
Miriam Giorgini
Limits on Lorentz invariance violation in atmospheric neutrino oscillations using MACRO data
Guey-Lin Lin
The Shower Event Rates of Ultrahigh Energy Tau Neutrinos in the Rock Salt
12.9.
Monday
chairman: M. Strikman
Gary Varner
Radio Detection of Ultra High Energy Neutrinos: ANITA and SalSA for both Astrophysics and Particle Physics
Opher Ganel
CREAM
CREAM web
K. Belov
HiRes
##### coffee
chairman: M. Honda
Markus Risse
Photon air showers at ultra-high energy and the photonuclear cross-section
Dan Hooper
Exotic Physics With Cosmic Neutrinos
##### lunch
chairman: M. Risse
Ralph Wischnewski
BAIKAL
Mauro Taiuti
NEMO - toward the km3 Neutrino Telescope in the Mediterranean Sea
##### coffee
chairman: L. Jones
d i s c u s s i o n
Conference dinner
(19:00)
13.9.
Tuesday
d e p a r t u r e
Abstracts
Search for nuclearites with the SLIM detector
The strange quark matter (SQM) may be the ground state of GCD; if so, nuggets of SQM could be present in cosmic rays (CR), as nuclearites. SLIM is a large area experiment, using CR39 and Makrofol track etch detectors, presently deployed at the high altitude CR Laboratory of Chacaltaya, Bolivia. We discuss the expected properties of nuclearites, from the point of view of their search with SLIM. We present calculations concerning their propagation in the Earth atmosphere and the possibility to detect them with the CR39 and Makrofol nuclear track detectors, as well as some preliminary results from SLIM.
back to programme
A Search for Dark Matter with CACTUS
Mani Tripathi
CACTUS is a ground-based ACT at the Solar 2 facility located near Barstow, California and operated by UC, Davis. It uses an array of 160 large solar tracking mirrors (heliostats) and a camera with 80 photomultiplier tubes to collect atmospheric Cherenkov radiation produced by intermediate energy air showers. CACTUS has incorporated novel techniques of time projection imaging and triggering to improve upon the first generation sampling arrays of its kind. Draco is a dwarf spheroidal galaxy within the Milky Way local group and is known to be rich in Dark Matter content. Various theoretical models in high energy physics, mainly Supersymmetry, include particles that could be the basic constituent of Dark Matter. Furthermore, these particles could annihilate with each other and produce gamma rays. An excess of gamma ray emissions from Draco, for example, would lend support to such theories or else, constraint the parameters embedded in these models. We have mounted a campaign to survey the region around Draco using CACTUS. Gamma-rays from Dark Matter annihilations are expected to be in the low energy (50-200 GeV) regime. In earlier measurements of the gamma-ray emissions from the Crab nebula, we have established that CACTUS is sensitive in this energy range, thus making it well-suited for this survey. We will describe our calibrations and present an energy spectrum for gamma-rays from the Crab and for background cosmic-rays. We will also present the first results from our Draco campaign.
back to programme
Limits on Lorentz invariance violation in atmospheric neutrino oscillations using MACRO data
Miriam Giorgini
The energy spectrum of neutrino-induced upwardgoing muons in MACRO has been analysed in terms of relativity principle violating effects, keeping standard mass-induced atmospheric neutrino oscillations as the dominant source of numu --> nutau transitions. The data disfavor these exotic possibilities even at subdominant level and stringent 90% C.L. limits are placed on the Lorentz invariance violation parameter. These limits can also be reinterpreted as upper bounds on the parameters describing the violation of the equivalence principle.
back to programme
Indirect Dark Matter searches with AMS-02
Rosier Lees Sylvie on behalf of AMS02
The Alpha Magnetic Spectrometer (AMS), to be placed on ISS, will provide data on cosmic radiation s in a large range of energy from 0.5 GeV to 3 TeV. The main physics goals in the astroparticle domain are the anti-matter and the dark matter searches. Observations and cosmology indicate that the Universe may include a large amount of unknown Dark Matter. It should be composed of non baryonic Weakly Interactive Massive Particles (WIMP).A good WIMP candidate being the Lightest Susy Particle in R-Parity conserving models. AMS offers a unique opportunity to study simultaneously SUSY dark matter in three decay channels from the neutralino annihilation: e+, antiproton and gamma. The supersymmetric theory frame is considered together with alternative scenarios (extra dimensions). The expected flux sensitivities in 3 year exposure for the e+/e- ratio, antiproton and gamma yields as a function of energy are presented and compared to other direct and indirect searches.
back to programme
Gravi-Reggeons and Cosmic Neutrino-Nucleon Interactions
Alexandre Kisselev
The gravity effects in high-energy scattering of the SM particles confined to a brane in a space-time with extra dimensions are calculated. We consider interactions induced by gravi-Reggeon exchanges in both flat and warped metrics. The cross section for cosmic neutrino scattering off the atmospheric nucleons is estimated.
back to programme
Muon production in extensive air showers and its relation to accelerator measurements
Christine Meurer
Using CORSIKA for simulating extensive air showers, we study the relation between the shower characteristics and features of hadronic multiparticle production at low energies. We report about investigations of typical energies and phase space regions of secondary particles which are important for muon production in extensive air showers. Possibilities to measure relevant quantities of hadron production in existing and planned accelerator experiments are discussed.
back to programme
Impact of uncertainties in hadron production on air shower predictions
Tanguy Pierog
Main observables of high energy air showers mainly depend on the nature and energy of the primary particles. As a result, simulation of showers initiated by hadronic particles will have a strong dependence on the hadronic interaction model used. Although the main hadronic air shower development can be understood within a simplified model of hadron production, detailed simulations and a realistic description of hadronic interactions are needed to calculate observables relevant for air shower experiments. We will study the effect of using different hadronic models available for cosmic ray calculation on extensive air shower predictions within CORSIKA and CONEX.
back to programme
Models for cosmic ray interactions
S. Ostapchenko
Contemporary models of hadronic interactions are reviewed. Basic phenomenological approaches are compared, with an emphasises on the predicted air shower characteristics. A special attention is payed to the remaining discrepances between present hadronic MC generators and cosmic ray data. Finally, future prospects concerning model improvements are discussed, in particular, regarding the possibilities to discriminate between different models on the basis of accelerator and cosmic ray measurements.
back to programme
Photon air showers at ultra-high energy and the photonuclear cross-section
Markus Risse
The origin of cosmic rays with energies of 10^20 eV and more is one of the large puzzles in contemporary astroparticle physics. In some non-acceleration source models, a large fraction of these particles are predicted to be photons. One can check whether cosmic-ray events observed by large air shower experiments are compatible with expectations for primary photons by comparing to simulations of photon-initiated air showers. Among other processes, such simulations require the photonuclear cross-section to be extrapolated over several decades in energy. On one hand, the corresponding theoretical uncertainty affects the photon shower simulations and translates into a systematic uncertainty when deriving experimental limits on the primary photon flux. On the other hand, an upper limit on the photonuclear cross-section could be given if photon-induced showers were observed. The status of primary photon simulations, recent results from comparisons to data, and the influence when changing the extrapolation of the photonuclear cross-section will be presented.
back to programme
Cosmic Ray Studies with the L3-Cosmics Program at CERN
Lawrence W. Jones
The L3 detector at the CERN electron-positron collider had a muon spectrometer which was employed during the last years of LEP operation for cosmic ray studies. The results of this program include a precise muon spectrum and charge ratio between 20 GeV and 3 TeV, a limit on TeV primary antiprotons from a study of the Moon's shadow, the (possible) observation of a flare signal from a fixed position in the northern hemisphere through muons, a solar anisotropy of 200 GeV protons, and other results. Negative limits on muons associated with known H.E. gamma sources or gamma bursts, the search for anomalous H.E. interaction or decay events such as reported from the Kolar Gold Fields, and a search for muons correlated with a solar flare are also noted.
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Geometric structures in hadronic cores of extensive air showers observed by KASCADE
Markus Risse
The geometric distribution of high-energy hadrons >100 GeV in shower cores measured with the KASCADE calorimeter is analyzed. The data are checked for sensitivity to hadronic interaction features and indications of new physics as discussed in the literature. The angular correlation of the most energetic hadrons and in particular the fraction of events with hadrons being aligned are quantified by means of the commonly used parameter lambda_4. The analysis shows that the observed lambda_4 distribution is compatible with that predicted by simulations and is not linked to an angular correlation from hadronic jet production at high energy. Another parameter, d_4^max, describing distances between hadrons measured in the detector, is found to be sensitive both to the transverse momenta in secondary hadron production and the primary particle type. Transverse momenta in high-energy hadron interactions differing by a factor two or more from what is assumed in the standard simulations are disfavoured by the measured d_4^max distribution.
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Atmospheric muon and neutrino fluxes and hadronic interaction
Morihiro Honda
The muon fluxes at several altitudes are examined with the simulation code used by HKKM04. This study makes it possible to estimate the uncertainty in the atmospheric neutrino flux calculated by HKKM04, which is less than 10~\% in 1--10 GeV. It also provides us a method to calibrate the interaction model, and we find that DPMJET-III, the interaction model used by HKKM04, needs to be modified at high energies. We propose a modification of DPMJET-III, and present the results with the modified interaction model.
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The Double-Bang Shower Event Rates of Ultrahigh Energy Tau Neutrinos in the Rock Salt: Results from Standard Model and Beyond Standard Model Physics
Guey-Lin Lin
We perform a simulation on the double-bang shower event rates of GZK tau neutrinos in the rock salt, taking into account propagations and energy losses of tau neutrinos and tau leptons in various density components of the earth. As an illustration, the rock salt region is taken as spherical in shape with a radius R=5 km. Such a region is treated as about 2 km below the earth surface. We present the simulation results based upon Standard Model Particle Physics and the TeV Scale Gravity Theory. The angular distributions of the events are studied and the criteria for distinguishing the above two models are proposed.
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A measurement of shower lateral and longitudinal profiles using the 1.5 GeV electron beams on aluminum targets at National Synchrotron Radiation Research Center (NSRRC) in Taiwan: A Status Report
Guey-Lin Lin
We perform the shower profile measurements by shooting the 1.5 GeV electron beams in NSRRC on targets made of 15 aluminum blocks, each with a thickness of 2.9 cm. A scintillator (Al2O3: Cr)) is placed behind the targets, converting the secondary shower particles into light. The light from the central region of the shower is recorded by a CCD camera while the light from the outer region is monitored by PMTs. The CCD system has been successfully implemented in the run 2 and run 3 of the experiment FLASH at SLAC. A test run of the current experiment has been conducted in NSRRC in the past June to study the scintillator spectra. Two more runs are planned from now to the year 2006 to measure the shower lateral and longitudinal profiles.
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Hadron production at forward rapidity in nuclear collisions at RHIC
Hongyan Yang
We will present hadron production in ultra-relativistic heavy ion collisions, which has been measured by BRAHMS experiment at RHIC. The rapidity dependence of the nuclear modification factor will be compared to models in order to differentiate between initial state gluon saturation effect and final state effect like parton recombination. The net-proton distribution in p+p, d+Au and Au+Au collisions will be shown to study the high energy collision scenario - transparency and stopping. Very preliminary results on radial and elliptic flow measured by BRAHMS will be included.
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Reconstruction of energy spectra of elemental groups with KASCADE: sensitivity to hadronic interaction models
Holger Ulrich
The large field array of the KASCADE experiment (located at Forschungszentrum Karlsruhe, Karlsruhe, Germany) measures with high accuracy the electromagnetic and muonic component of extensive air showers. The two-dimensional frequency distribution of these observables is used for reconstructing the energy spectra of five elemental groups by means of unfolding methods. These elemental groups represent the chemical composition of primary cosmic rays around the knee. The results of the analysis are based on a vast number of Monte Carlo simulations using different high-energy hadronic interaction models. As it turns out, the description of the measured data by the simulations shows to be sensitive to the characteristics of the used interaction models. Moreover, none of the used models (QGSJet and SIBYLL) is capable to describe the data consistently over the whole measurement range. Results of the analysis and their implications will be discussed.
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NEMO - a km3 Neutrino Telescope in the Mediterranean Sea
Mauro Taiuti
The NEMO (NEutrino Mediterranean Observatory) Project aims at the construction of a km3-scale neutrino telescope in the Mediterranean Sea. After extensive deep-sea surveys performed in several sites in the Mediterranean, an optimal installation site for the apparatus has been identified at a depth of 3500 m, about 80 km off Capo Passero, at the SE corner of Sicily, Italy. In this talk, we will illustrate the apparatus design and discuss its simulated capabilities. We will then illustrate the results of our long-lasting R&D efforts toward the construction of the apparatus. The NEMO Project has recently entered its Phase 1, aimed at the construction, deployment and operation of a subsystem which includes all the key elements of the apparatus at a test site located at a depth of 2000 m, about 25 km off the coast of Catania. A prototype hydrophone station for acoustic observations of high-energy neutrinos (ONDE) is currently in operation at this test site. The cable for connecting the Capo Passero deep-sea site to the shore is under design, and the work to set up the shore station has started. The plan is to install a full-size tower at this location in two years.
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The LHCf experiment at LHC
The LHCf experiment will be installed in 2007 on the LHC collider, 140 m away from one of the interaction points, to precisely measure the pion production cross section at a very small angle through the measurement of the photons produced in the neutral pion decay. This measurement is crucial for the simulation of the showers induced in the atmosphere by very high energy cosmic rays; the 14 TeV energy available in the center of mass frame corresponds in fact to an equivalent energy of 10^17 eV in the laboratory system. The presentation will mainly be focused on the description of the proposed experiment and on the physics results that we expect from it.
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Discovery Potential of the LHC
Ketevi A. Assamagan
The physics program at the LHC includes precision tests of the Standard Model (SM), the search for the SM Higgs boson up to 1 TeV, the search for the MSSM Higgs bosons in the entire parameter space, the search for Super Symmetry, sensitivity to alternative scenarios such as compositeness, large extra dimensions, etc. This requires general purpose detector with excellent performance. ATLAS is one such detectors under construction for the LHC. Data taking is expected to start in April 2007. The detector performance and the prospects for discoveries are studied in various physics working groups. In this talk, we will review discovery potential of the LHC.
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EMMA - Experiment with MultiMuon Array
Sarkamo Juho
We are developing a new underground cosmic ray experiment EMMA (Experiment with MultiMuon Array). The EMMA-experiment will measure the multiplicity and lateral distribution of high energetic muons originated near the primary collision point of a cosmic ray air shower. The experiment will be set up at shallow depths underground in Pyhäsalmi Mine in Central Finland and it will be optimized to study air showers with primary energies around the knee region. At the first phase EMMA will consist of two or three separate detector units with an area of about 50 m2 each with a mutual distance of 20 m. We will use old Delphi Muon Barrel Chambers for the muon multiplicity and position measurements. Our simulations show that the measurements of lateral distribution of high energy muons can be used to deduce the primary composition. Hence EMMA can shed light on the composition and air shower development from a different point of view.
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Latest Results from the High Resolution Fly's Eye (HiRes) Experiment
Charles Jui
The High-Resolution Fly's Eye (HiRes) is an air-fluorescence observatory located in Utah, USA with the mission of measuring the energy spectrum, chemical composition, and arrival directions of the highest energy cosmic rays. HiRes has been collecting monocular data since 1997 and stereo data since 1999, and has accumulated the largest integrated aperture at the highest energies of any UHECR observatory to date. We present the latest monocular and stereo spectra, results of searches for arrival direction anisotropy, and composition measurements.
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Detection of cosmic rays by LEP experiments
Petr Travnicek
Accelerator experiments located at LEP collider have been used also to detect cosmic ray events. Various results obtained by Cosmo-ALEPH, DELPHI and L3+C collaborations will be described. Emphasis will be put on detection of multi-muon bundles and comparisons of the measurements with Monte-Carlo simulations based on current models of primary hadron-hadron interactions. The analysis of inclusive muon energy spectra and muon charge ratio by Cosmo-Aleph and L3+C collaborations will be also explained. Other topics such as source searches, limits on anti-proton/proton flux calculated from moon shadow, solar flare studies etc. will be briefly mentioned.
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Radio Detection of Ultra High Energy Neutrinos: ANITA and SalSA for both Astrophysics and Particle Physics
Gary S. Varner
The primary objective of the ANtarctic Impulsive Transient Antenna (ANITA) mission is to investigate and constrain the nature of the sources of high energy cosmic ray particles above 100 EeV, by measurements of neutrinos that are strongly believed to be both spectrally and spatially correlated to them. Observation over the last 40 years of several dozen cosmic ray events with energies exceeding the Greisen-Zatsepin-Kuzmin (GZK) cutoff poses among the most intriguing and intractable problems in high energy astrophysics. Operating from a long-duration balloon at an altitude of 37km, ANITA will synoptically observe the Antarctic ice sheet out to a horizon of more than 600km, giving a detection volume of order 1 million cubic kilometers. ANITA will search for radio pulses that arise from electromagnetic cascade interactions of high energy neurinos within the ice. Such radio pulses, recently confirmed in accelerator experiments, easily propagate through the ice due to its remarkable radio transparency. A preliminary design and technical challenges of the low-power radio frequency instrumentation will be presented. Time permitting, the application of this Askaryan-effect detection technique to naturally occuring salt domes (Salt dome Shower Array [SalSA] will be mentioned. Such a large detector would enable the use of GZK neutrinos to search for new physics at center-of-mass energies well beyond the reach of the LHC. If there is interest, results from a detector prototype, will be provided.
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Leonid Frankfurt
We will explain that blackening of interactions in the high energy collisions at central impact parameters is direct consequence of compositeness of projectile in QCD and increase with energy of cross sections. Relationship between this property of high energy processes and phenomenon of spontaneous violation of continious symmetries will be visualised. Account of these phenomena leads to increase with energy of radius of interactions and to observable effects at achievable energies and at superhigh energies to universal limit for all cross sections: $\sigma(pp):\sigma(pA):\sigma(AA)> \to 1$ The same universality is expected for structure functions of nucleons and nuclei,for cross section of photodesintegration of nucleons and nuclei etc.
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Exotic Physics With Cosmic Neutrinos
Dan Hooper
Although collider experiments are often thought of as the primary tool in testing physics beyond the Standard Model, astroparticle physics experiments have characteristics which make them highly complementary probes. In this talk, I will review the ability of neutrino telescopes and cosmic ray experiments to measure the interaction cross sections of neutrinos at energies well beyond those accessible to colliders. I will also discuss the use of high and ultra-high energy cosmic neutrinos as an ultra-long baseline measurement. Such experiments can be used to constrain -- or potentially discover -- signatures of exotic physics, including low scale quantum gravity, Standard Model electroweak instanton induced interactions, neutrino decay and quantum decoherence.
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The Elusive p-air Cross Section - tying together accelerator sigma_pp and cosmic ray sigma_p-air^inel measurements
Martin Block
We fit {\em all} of the extensive accelerator data of the Particle Data Group ($\sigma_{pp},\sigma_{\bar pp},\rho_{pp},\rho_{\bar p p}$) with c.m.s. energy $6 \ {\rm GeV}\le\sqrt s\le 1800$ GeV, using real analytic amplitudes that saturate the Froissart bound, {\em i.e.}, the cross section rises as $\ln^2 s$. However, we then subject these data to a screening process, the Sieve'' algorithm[M.M. Block, physics/0506010], in order to eliminate outliers'' that can skew a $\chi^2$ fit. With the Sieve'' algorithm, a robust fit using a Lorentzian distribution is first made to all of the data to sieve out abnormally high $\delchi$, the individual i$^{\rm th}$ point's contribution to the total $\chi^2$. The $\chi^2$ fits are then made to the sieved data. We show that both the nucleon-nucleon and $\pi p$ systems fit exceedingly well to a high energy $\ln^2s$ fit of the form: $\sigma^{\pm}=c_0 +c_1{\ln }\left({\nu\over m}\right)+c_2{\ln }^2\left ({\nu\over m}\right)+\beta_{\cal P'}\left({\nu\over m}\right)^{\mu -1}\pm \delta\left({\nu\over m}\right)^{\alpha -1}$, using the equivalent of finite energy sum rules to anchor the fits to the low energy data just above the resonance regions[M. M. Block and F. Halzen, Phys. Rev. D {\bf72}, 036006 (2005)]. The upper sign is for $pp$ ($\pi^+p$) and the lower sign is for $\bar pp$ ($\pi^-p$) scattering, where $\nu$ is the laboratory proton (pion ) energy, and $m$ is the proton (pion) mass. The fitting technique allows us to make very accurate predictions at cosmic ray energies, {\em e.g.}, $\sigma_{pp}=131.1\pm1.7$ mb at $\sqrt s=50$ Tev. The $\ln^2s$ fit that we find, along with a QCD-inspired eikonal fit[M. M. Block, F. Halzen and T. Stanev, Phys. Rev. D {\bf 62}, 077501 (2000)] for B, the nuclear slope in the forward direction, allows us to make a Glauber calculation[M. Block and R. Engel, unpublished] using a two-channel inelastic screening model, to predict $\sigma_{\rm p-air}^{\rm inel}$ at cosmic ray energies. These results are compared to the new HiRes experimental value, $\sigma_{\rm p-air}^{\rm inel}=456\pm17_{\rm stat}+39_{\rm syst}-11_{\rm syst}$, along with AGASA and Fly's eye data, giving us a benchmark for tying together accelerator and cosmic ray data.
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The Auger Observatory: Status and first results
Enrique Zas
More than half of the Auger Observatory has been constructed and is opertating in Malargue, Mendoza, Argentina. We present a review of the observatory, its goals and perspectives and a summary of the first scientific results.
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CREAM
Opher Ganel
Pushing the High Energy Frontier of Directly Measured Cosmic Rays After nearly a century since their discovery, much is still uncertain about cosmic rays, including their source, acceleration mechanism, and propagation. Direct measurement of cosmic ray elemental spectra in the critical energy range of 1012 - 1015 eV offers a promising avenue to answering these questions. The Cosmic Ray Energetics And Mass (CREAM) experiment was launched from McMurdo Station, Antarctica, under a NASA research balloon on December 16, 2004. Floating for nearly 42 days at altitudes between 36 and 39 km, CREAM collected over 4x107 events. Preliminary results show great promise for this dataset. CREAM-II is set to launch in December 2005, with a refurbished CREAM planned to fly again in 2006. This presentation will examine the experimental context of CREAM, describe the detector, and show preliminary results from accelerator beam tests and the first flight.
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The Structure of the Nucleon and Simulation of Cosmic Ray Showers"
Terry Sloan
In this talk the structure of the nucleon will be reviewed using the picture that has emerged from studies of deep inelastic lepton-nucleon scattering. The successes of the quark-parton model of the nucleon in interpreting the data and what they reveal about the strong force, QCD, will be described. Some of the remaining problems will be described. Finally the difficulties implied for the simulation of high energy cosmic showers will be discussed.
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Diffraction and Forward physics at the LHC
Albert de Roeck
The central detectors ATLAS and CMS can measure particles up to pseudo-rapidity values of 5. The TOTEM detector, which uses the same interaction point as CMS, can measure particles in the range 3< eta <5 and has roman pots to measure leading protons. CMS is furthermore considering additional calorimeters in the forward region, with an acceptance of up to 7 in pseudorapidity. Also ATLAS is developing plans for additional forward detectors. The combination of e.g. CMS and TOTEM and their common data taking will give a detector with unprecedented acceptance in the combined central and forward region. This talk will present these possible upgrades and the physics potential of the LHC detectors in the forward region, with for example QCD measurements in the area of diffraction and low-x physics. Moreover recently central exclusive production of the Higgs bosons has gained a lot of interest and will be discussed.
back to programme | 2017-09-25 18:54:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.5628002285957336, "perplexity": 2563.045039654298}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818693240.90/warc/CC-MAIN-20170925182814-20170925202814-00609.warc.gz"} |
https://www.jobilize.com/online/course/6-2-line-integrals-vector-calculus-by-openstax?qcr=www.quizover.com&page=8 | # 6.2 Line integrals (Page 9/20)
Page 9 / 20
We now give a formula for calculating the flux across a curve. This formula is analogous to the formula used to calculate a vector line integral (see [link] ).
## Calculating flux across a curve
Let F be a vector field and let C be a smooth curve with parameterization $\text{r}\left(t\right)=⟨x\left(t\right),y\left(t\right)⟩,a\le t\le b.$ Let $\text{n}\left(t\right)=⟨{y}^{\prime }\left(t\right),\text{−}{x}^{\prime }\left(t\right)⟩.$ The flux of F across C is
${\int }_{C}\text{F}·\text{N}ds={\int }_{a}^{b}\text{F}\left(\text{r}\left(t\right)\right)·\text{n}\left(t\right)dt$
## Proof
The proof of [link] is similar to the proof of [link] . Before deriving the formula, note that $‖\text{n}\left(t\right)‖=‖⟨y\prime \left(t\right),\text{−}x\prime \left(t\right)⟩‖=\sqrt{{\left(y\prime \left(t\right)\right)}^{2}+{\left(x\prime \left(t\right)\right)}^{2}}=‖{r}^{\prime }\left(t\right)‖.$ Therefore,
$\begin{array}{cc}\hfill {\int }_{C}\text{F}·\text{N}ds& ={\int }_{C}\text{F}·\frac{\text{n}\left(t\right)}{‖\text{n}\left(t\right)‖}ds\hfill \\ & ={\int }_{a}^{b}\text{F}·\frac{\text{n}\left(t\right)}{‖\text{n}\left(t\right)‖}‖{r}^{\prime }\left(t\right)‖dt\hfill \\ & ={\int }_{a}^{b}\text{F}\left(\text{r}\left(t\right)\right)·\text{n}\left(t\right)dt.\hfill \end{array}$
## Flux across a curve
Calculate the flux of $\text{F}=⟨2x,2y⟩$ across a unit circle oriented counterclockwise ( [link] ).
To compute the flux, we first need a parameterization of the unit circle. We can use the standard parameterization $\text{r}\left(t\right)=⟨\text{cos}\phantom{\rule{0.2em}{0ex}}t,\text{sin}\phantom{\rule{0.2em}{0ex}}t⟩,$ $0\le t\le 2\pi .$ The normal vector to a unit circle is $⟨\text{cos}\phantom{\rule{0.2em}{0ex}}t,\text{sin}\phantom{\rule{0.2em}{0ex}}t⟩.$ Therefore, the flux is
$\begin{array}{cc}\hfill {\int }_{C}\text{F}·\text{N}ds& ={\int }_{0}^{2\pi }⟨2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t,2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t⟩·⟨\text{cos}\phantom{\rule{0.2em}{0ex}}t,\text{sin}\phantom{\rule{0.2em}{0ex}}t⟩\phantom{\rule{0.2em}{0ex}}dt\hfill \\ & ={\int }_{0}^{2\pi }\left(2\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{2}t+2\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{2}t\right)\phantom{\rule{0.2em}{0ex}}dt=2{\int }_{0}^{2\pi }\left({\text{cos}}^{2}t+{\text{sin}}^{2}t\right)\phantom{\rule{0.2em}{0ex}}dt\hfill \\ & =2{\int }_{0}^{2\pi }dt=4\pi .\hfill \end{array}$
Calculate the flux of $\text{F}=⟨x+y,2y⟩$ across the line segment from $\left(0,0\right)$ to $\left(2,3\right),$ where the curve is oriented from left to right.
3/2
Let $\text{F}\left(x,y\right)=⟨P\left(x,y\right),Q\left(x,y\right)⟩$ be a two-dimensional vector field. Recall that integral ${\int }_{C}\text{F}·\text{T}ds$ is sometimes written as ${\int }_{C}Pdx+Qdy.$ Analogously, flux ${\int }_{C}\text{F}·\text{N}ds$ is sometimes written in the notation ${\int }_{C}\text{−}Qdx+Pdy,$ because the unit normal vector N is perpendicular to the unit tangent T . Rotating the vector $d\text{r}=⟨dx,dy⟩$ by 90° results in vector $⟨dy,\text{−}dx⟩.$ Therefore, the line integral in [link] can be written as ${\int }_{C}-2ydx+2xdy.$
Now that we have defined flux, we can turn our attention to circulation. The line integral of vector field F along an oriented closed curve is called the circulation of F along C . Circulation line integrals have their own notation: ${\oint }_{C}\text{F}·\text{T}ds.$ The circle on the integral symbol denotes that C is “circular” in that it has no endpoints. [link] shows a calculation of circulation.
To see where the term circulation comes from and what it measures, let v represent the velocity field of a fluid and let C be an oriented closed curve. At a particular point P , the closer the direction of v ( P ) is to the direction of T ( P ), the larger the value of the dot product $\text{v}\left(P\right)·\text{T}\left(P\right).$ The maximum value of $\text{v}\left(P\right)·\text{T}\left(P\right)$ occurs when the two vectors are pointing in the exact same direction; the minimum value of $\text{v}\left(P\right)·\text{T}\left(P\right)$ occurs when the two vectors are pointing in opposite directions. Thus, the value of the circulation ${\oint }_{C}\text{v}·\text{T}ds$ measures the tendency of the fluid to move in the direction of C .
## Calculating circulation
Let $\text{F}=⟨-y,x⟩$ be the vector field from [link] and let C represent the unit circle oriented counterclockwise. Calculate the circulation of F along C .
We use the standard parameterization of the unit circle: $\text{r}\left(t\right)=⟨\text{cos}\phantom{\rule{0.2em}{0ex}}t,\text{sin}\phantom{\rule{0.2em}{0ex}}t⟩,0\le t\le 2\pi .$ Then, $\text{F}\left(\text{r}\left(t\right)\right)=⟨\text{−}\text{sin}\phantom{\rule{0.2em}{0ex}}t,\text{cos}\phantom{\rule{0.2em}{0ex}}t⟩$ and ${r}^{\prime }\left(t\right)=⟨\text{−}\text{sin}\phantom{\rule{0.2em}{0ex}}t,\text{cos}\phantom{\rule{0.2em}{0ex}}t⟩.$ Therefore, the circulation of F along C is
$\begin{array}{cc}\hfill {\oint }_{C}\text{F}·\text{T}ds& ={\int }_{0}^{2\pi }⟨\text{−}\text{sin}\phantom{\rule{0.2em}{0ex}}t,\text{cos}\phantom{\rule{0.2em}{0ex}}t⟩·⟨\text{−}\text{sin}\phantom{\rule{0.2em}{0ex}}t,\text{cos}\phantom{\rule{0.2em}{0ex}}t⟩dt\hfill \\ & ={\int }_{0}^{2\pi }\left({\text{sin}}^{2}t+{\text{cos}}^{2}t\right)\phantom{\rule{0.2em}{0ex}}dt\hfill \\ & ={\int }_{0}^{2\pi }dt=2\pi .\hfill \end{array}$
Notice that the circulation is positive. The reason for this is that the orientation of C “flows” with the direction of F . At any point along the circle, the tangent vector and the vector from F form an angle of less than 90°, and therefore the corresponding dot product is positive.
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations? | 2020-07-03 10:57:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 32, "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": 0.845796525478363, "perplexity": 875.9346207860694}, "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/1593655881984.34/warc/CC-MAIN-20200703091148-20200703121148-00380.warc.gz"} |
https://ncertmcq.com/selina-concise-mathematics-class-10-icse-solutions-chapter-5-quadratic-equations-ex-5b/ | ## Selina Concise Mathematics Class 10 ICSE Solutions Chapter 5 Quadratic Equations Ex 5B
These Solutions are part of Selina Concise Mathematics Class 10 ICSE Solutions. Here we have given Selina Concise Mathematics Class 10 ICSE Solutions Chapter 5 Quadratic Equations Ex 5B.
Other Exercises
Solve equations, number 1 to number 20, given below, using factorisation method :
Question 1.
x² – 10x – 24 = 0
Solution:
x² – 12x + 2x – 24 = 0
⇒ x (x – 12) + 2 (x – 12) = 0
⇒ (x – 12) (x + 2) = 0
Either x – 12 = 0, then x = 12
or x + 2 = 0, then x = – 2
x = 12, – 2
Question 2.
x² – 16 = 0
Solution:
⇒ x² – (4)² = 0
⇒ (x + 4) (x – 4) = 0
Either x + 4 = 0, then x = – 4
or x – 4 = 0, then x = 4
x = 4, – 4
Question 3.
2x² – $$\frac { 1 }{ 2 }$$ x = 0
Solution:
⇒ 4x² – x = 0
⇒ x (4x – 1) = 0
Either x = 0,
or 4x – 1 = 0, then 4x = 1 ⇒ x = $$\frac { 1 }{ 4 }$$
x = 0, $$\frac { 1 }{ 4 }$$
Question 4.
x (x – 5) = 24
Solution:
⇒ x² – 5x – 24 = 0
⇒ x² – 8x + 3x – 24 = 0
⇒ x (x – 8) + 3 (x – 8) = 0
⇒ (x – 8) (x + 3) = 0
Either x – 8 = 0, then x = 8
or x + 3 = 0, then x = – 3
x = 8, – 3
Question 5.
$$\frac { 9 }{ 2 }$$ x = 5 + x²
Solution:
⇒ 9x = 10 + 2x²
⇒ 2x² – 9x + 10 = 0
⇒ 2x² – 4x – 5x + 10 = 0
⇒ 2x (x – 2) – 5 (x – 2) = 0
⇒ (x – 2) (2x – 5) = 0
Either x – 2 = 0, then x = 2
or 2x – 5 = 0, then 2x = 5 ⇒ x = $$\frac { 5 }{ 2 }$$
x = 2, $$\frac { 5 }{ 2 }$$
Question 6.
$$\frac { 6 }{ x }$$ = 1 + x
Solution:
⇒ 6 = x + x²
⇒ x² + x – 6 = 0
⇒ x² + 3x – 2x – 6 = 0
⇒ x (x + 3) – 2 (x + 3) = 0
⇒ (x + 3) (x – 2) = 0
Either x + 3 = 0, then x = – 3
or x – 2 = 0, then x = 2
x = 2, – 3
Question 7.
x = $$\frac { 3x + 1 }{ 4x }$$
Solution:
⇒ 4x² = 3x + 1
⇒ 4x² – 3x – 1 = 0
⇒ 4x² – 4x + x – 1 = 0
⇒ 4x (x – 1) + 1 (x – 1) = 0
⇒ (x – 1) (4x + 1) = 0
Either x – 1 = 0, then x = 1
or 4x + 1 = 0, then 4x = -1 ⇒ x = $$\frac { -1 }{ 4 }$$
x = 1, $$\frac { -1 }{ 4 }$$
Question 8.
x + $$\frac { 1 }{ x }$$ = 2.5
Solution:
Question 9.
(2x – 3)² = 49
Solution:
⇒ 4x² – 12x + 9 = 49
⇒ 4x² – 12x + 9 – 49 = 0
⇒ 4x² – 12x – 40 = 0
⇒ x² – 3x – 10 = 0 (Dividing by 4)
⇒ x² – 5x + 2x – 10 = 0
⇒ x (x – 5) + 2 (x – 5) = 0
⇒ (x – 5) (x + 2) = 0
Either x – 5 = 0, then x = 5
or x + 2 = 0, then x = – 2
x = 5, – 2
Question 10.
2 (x² – 6) = 3 (x – 4)
Solution:
⇒ 2x² – 12 = 3x- 12
⇒ 2x² – 3x – 12 + 12 = 0
⇒ 2x² – 3x = 0
⇒ x (2x – 3) = 0
Either A = 0,
or 2x – 3 = 0, then 2x = 3 ⇒ x = $$\frac { 3 }{ 2 }$$
x = 0, $$\frac { 3 }{ 2 }$$
Question 11.
(x + 1) (2x + 8) = (x + 7) (x + 3)
Solution:
Question 12.
x² – (a + b) x + ab = 0
Solution:
⇒ x² – ax – bx + ab = 0
⇒ x (x – a) – b (x – a) = 0
⇒ (x – a) (x – b) = 0
Either x – a = 0, then x = a
or x – b = 0, then x = b
x = a, b
Question 13.
(x + 3)² – 4 (x + 3) – 5 = 0
Solution:
Let x + 3 = y, then
⇒ y² – 4y – 5 = 0
⇒y² – 5y + y – 5 = 0
⇒ y (y – 5) + 1 (y – 5) = 0
⇒ (y – 5) (y + 1) = 0
Substituting the value of y,
⇒ (x + 3 – 5) (x + 3 + 1) = 0
⇒ (x – 2) (x + 4) = 0
Either x – 2 = 0, then x = 2
or x + 4 = 0, then x = – 4
x = 2, -4
Question 14.
4 (2x – 3)² – (2x – 3) – 14 = 0
Solution:
Let 2x – 3 = y, then
⇒ 4y² – y – 14 = 0
⇒ 4y² – 8y + 7y – 14 = 0
⇒ 4y (y – 2) + 7 (y – 2) = 0
⇒ (y – 2) (4y + 7) = 0
Substituting the value of y,
⇒ (2x – 3 – 2) (8x – 12 + 7) = 0
⇒ (2x – 5) (8x – 5) = 0
Either 2x – 5 = 0, then 2x = 5 ⇒ x = $$\frac { 5 }{ 2 }$$
or 8x – 5 = 0, then 8x = 5 ⇒ x = $$\frac { 5 }{ 8 }$$
x = $$\frac { 5 }{ 2 }$$ , $$\frac { 5 }{ 8 }$$
Question 15.
Solution:
⇒ (3x – 2) (x + 4) = (3x – 8) (2x – 3)
⇒ 3x² + 12x – 2x – 8 = 6x² – 9x – 16x + 24
⇒ 3x² + 12x – 2x – 8 – 6x² + 9x + 16x – 24 = 0
⇒ – 3x² + 35x – 32 = 0
⇒ 3x² – 35x + 32 = 0
⇒ 3x² – 3x – 32x + 32 = 0
⇒ 3x (x – 1) – 32 (x – 1) = 0
⇒ (x – 1) (3x – 32) = 0
⇒ 3x (x – 1) – 32 (x – 1) = 0
⇒ (x – 1) (3x – 32) – 0
Either x – 1 = 0, then x = 1
or 3x – 32 = 0, then 3x = 32 ⇒ x = $$\frac { 32 }{ 3 }$$
x = 1, $$\frac { 32 }{ 3 }$$ or 1, 10$$\frac { 2 }{ 3 }$$
Question 16.
2x² – 9x + 10 = 0, when :
(i) x ∈ N
(ii) x ∈ Q.
Solution:
2x² – 9x + 10 = 0
⇒ 2x² – 4x – 5x + 10 = 0
⇒ 2x (x – 2) – 5 (x – 2) = 0
⇒ (x – 2) (2 – 5) = 0
Either x – 2 = 0, then x = 2
or 2x – 5 = 0, then 2x = 5 ⇒ x = $$\frac { 5 }{ 2 }$$
(i) When x ∈ N, then x = 2
(ii) When x ∈ Q, then x = 2 , $$\frac { 5 }{ 2 }$$
Question 17.
Solution:
Question 18.
Solution:
Question 19.
Solution:
Question 20.
Solution:
Question 21.
Find the quadratic equation, whose solution set is :
(i) {3, 5}
(ii) {-2, 3}
Solution:
(i) Solution set is {3, 5} or x = 3 and x = 5
Equation will be
(x – 3) (x – 5) = 0
⇒ x² – 5x – 3x + 15 = 0
⇒ x² – 8x + 15 = 0
(ii) Solution set is {-2, 3} or x = -2, x = 3
Equation will be
(x + 2) (x – 3) = 0
⇒ x² – 3x + 2x – 6 = 0
⇒ x² – x – 6 = 0
Question 22.
Solution:
Roots are not real.
Hence there is no possible real value of x.
Question 23.
Find the value of x, if a + 1 = 0 and x² + ax – 6 = 0.
Solution:
a + 1 = 0 ⇒ a = -1
Now the equation x² + ax – 6 = 0 will be x² + (-1) x – 6 = 0
⇒ x² – x – 6 = 0
⇒ x² – 3x + 2x – 6 = 0
⇒ x (x – 3) + 2 (x – 3) = 0
Either x – 3 = 0, then x = 3
or x + 2 = 0, then x = – 2
x = 3, – 2
Question 24.
Find the value of x, if a + 7 = 0; b + 10 = 0 and 12x² = ax – b.
Solution:
a + 7 = 0, then a = -7
and b + 10 = 0, then b = -10
Now, substituting the value of a and b in
12x² = ax – b
⇒ 12x² = – 7x – (-10)
⇒ 12x² = – 7x + 10
⇒ 12x² + 7x – 10 = 0
⇒ 12x² + 15x – 8x – 10 = 0
⇒ 3x (4x + 5) – 2 (4x + 5) = 0
⇒ (4x + 5) (3x – 2) = 0
Either 4x + 5 = 0, then 4x = -5 ⇒ x = $$\frac { 5 }{ 4 }$$
or 3x – 2 = 0, then 3x = 2 ⇒ x = $$\frac { 2 }{ 3 }$$
x = $$\frac { 5 }{ 4 }$$, $$\frac { 2 }{ 3 }$$
Question 25.
Use the substitution y = 2x + 3 to solve for x, if 4 (2x + 3)² – (2x + 3) – 14 = 0.
Solution:
y = 2x + 3, then equation
4 (2x + 3)² – (2x + 3) – 14 = 0 will be 4y² – y – 14 = 0
⇒ 4y² – 8y + 7y – 14 = 0
⇒ 4y (y – 2) + 7 (y – 2) = 0
⇒ (y – 2) (4y + 7) = 0
Either y – 2 = 0, then y = 2
Question 26.
Without solving the quadratic equation 6x² – x – 2 = 0, find whether x = $$\frac { 2 }{ 3 }$$ is a solution of this equation or not.
Solution:
Question 27.
Determine whether x = -1 is a root of the equation x² – 3x + 2 = 0 or not.
Solution:
x² – 3x + 2 = 0
x = -1
Substituting the value of x = -1, in the quadratic equation
L.H.S. = x² – 3x + 2 = (-1)² – 3(-1) + 2 = 1 + 3 + 2 = 6 ≠ 0
Remainder is not equal to zero
x = -1 is not its root.
Question 28.
If x = $$\frac { 2 }{ 3 }$$ is a solution of the quadratic equation 7x² + mx – 3 = 0; find the value of m.
Solution:
Question 29.
If x = -3 and x = $$\frac { 2 }{ 3 }$$ are solutions of quadratic equation mx² + 7x + n = 0, find the values of m and n.
Solution:
x = -3, x = $$\frac { 2 }{ 3 }$$ are the solution of the quadratic equation, mx² + 7x + n = 0
Then these values of x will satisfy it
(i) If x = -3, then mx² + 7x + n = 0
⇒ m(-3)² + 7(-3) + n = 0
⇒ 9m – 21 + n = 0
⇒ n = 21 – 9m ……(i)
Question 30.
If quadratic equation x² – (m + 1) x + 6 = 0 has one root as x = 3; find the value of m and the other root of the equation.
Solution:
In equation, x² – (m + 1) x + 6 = 0
x = 3 is its root, then it will satisfy it
⇒ (3)² – (m + 1) x 3 + 6 = 0
⇒ 9 – 3m – 3 + 6 = 0
⇒ -3m + 12 = 0
Question 31.
Give that 2 is a root of the equation 3x² – p (x + 1) = 0 and that the equation px² – qx + 9 = 0 has equal roots, find the values of p and q.
Solution:
3x² – p (x + 1) = 0
⇒ 3x² – px – p = 0
2 is a root of the equal It will satisfy it
3(2)² – p(2) – p = 0
⇒ 3 x 4 – 2p – p = 0
⇒ 12 – 3p = 0
⇒ 3p = 12
⇒ p = 4
px² – qx + 9 = 0
Here, a = p, b = -q, c = 9
D = b² – 4ac = (-q)² – 4 x p x 9 = q² – 36p
Roots are equal.
D = 0
⇒ q² – 36p = 0
⇒ q² – 36 x 4 = 0
⇒ q² = 144
⇒ q² = (±12)²
⇒ q = ± 12
Hence, p = 4 and q = ±12
Question 32.
Solution:
Question 33.
Solve: ( $$\frac { 1200 }{ x }$$ + 2 ) (x – 10) – 1200 = 60
Solution:
By cross multiplication,
⇒ 1200x – 12000 + 2x² – 20x – 1260x = 0
⇒ 2x² + 1200x – 20x – 1260x – 12000 = 0
⇒ 2x² – 80x – 12000 = 0
⇒ x² – 40x – 6000 = 0
⇒ x² – 100x + 60x – 6000 = 0
⇒ x (x – 100) + 60 (x – 100) = 0
⇒ (x – 100) (x + 60) = 0
Either x – 100 = 0, then x = 100
or x + 60 = 0, then x = -60
x = 100, -60
Question 34.
If -1 and 3 are the roots of x² + px + q = 0, find the values of p and q.
Solution:
-1 and 3 are the roots of the equation
x² + px + q = 0
Substituting the value of x = -1 and also x = 3, then
(-1 )² + p(-1) + q = 0
⇒ 1 – p + q = 0
⇒ p – q = 1
⇒ p = 1 + q …(i)
and (3)² + p x 3 + q = 0
⇒ 9 + 3p + q = 0
⇒ 9 + 3 (1 + q) + q = 0 [From(i)]
⇒ 9 + 3 + 3q + q = 0
⇒ 12 + 4q = 0
⇒ 4q = -12
⇒ q = -3
p = 1 + q = 1 – 3 = -2
Hence, p = -2, q = -3
Hope given Selina Concise Mathematics Class 10 ICSE Solutions Chapter 5 Quadratic Equations Ex 5B are helpful to complete your math homework.
If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you. | 2022-12-07 07:57:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.6225011348724365, "perplexity": 534.4215509146998}, "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/1669446711150.61/warc/CC-MAIN-20221207053157-20221207083157-00872.warc.gz"} |
https://planetmath.org/OrderedRing | # ordered ring
An ordered ring is a commutative ring $R$ with a total ordering $\leq$ such that, for every $a,b,c\in R$:
1. 1.
If $a\leq b$, then $a+c\leq b+c$
2. 2.
If $a\leq b$ and $0\leq c$, then $c\cdot a\leq c\cdot b$
An ordered field is an ordered ring $(R,\leq)$ where $R$ is also a field.
Examples of ordered rings include:
• The integers $\mathbb{Z}$, under the standard ordering $\leq$.
• The real numbers $\mathbb{R}$ under the standard ordering.
• The polynomial ring $\mathbb{R}[x]$ in one variable over $\mathbb{R}$, under the relation $f\leq g$ if and only if $g-f$ has nonnegative leading coefficient.
Examples of rings which do not admit any ordering relation making them into an ordered ring include:
• The complex numbers $\mathbb{C}$.
• The finite field $\mathbb{Z}/p\mathbb{Z}$, where $p$ is any prime.
Title ordered ring Canonical name OrderedRing Date of creation 2013-03-22 11:52:06 Last modified on 2013-03-22 11:52:06 Owner djao (24) Last modified by djao (24) Numerical id 13 Author djao (24) Entry type Definition Classification msc 06F25 Classification msc 12J15 Classification msc 13J25 Classification msc 11D41 Related topic TotalOrder Related topic OrderingRelation Defines ordered field | 2019-09-17 00:11:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 20, "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": 0.8207217454910278, "perplexity": 1609.0080893586971}, "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-2019-39/segments/1568514572980.56/warc/CC-MAIN-20190917000820-20190917022820-00260.warc.gz"} |
https://docs.duckietown.org/daffy/opmanual_developer/draft/dt_way_nodes.html | build details
# Structuring ROS Nodes
Modified 2020-07-08 by Aleksandar Petrov
Aleksandar Petrov
Maintainer: Aleksandar Petrov
This section deals with how you should write the code in a ROS node. In particular, how to structure it. Writing the code of a node goes hand-in-hand with documenting it but this will be discussed in more detail in Unit C-4 - Documenting your code.
## General structure
Modified today by Andrea F. Daniele
All ROS nodes should be in the `src` directory of the respective package. If the node is called `some_name`, then the file that has its implementation should be called `some_name_node.py`. This file should always be executable. Furthermore, all the logic of the node should be implemented in a Python class called `SomeNameNode`.
The structure of the `some_name_node.py` should generally look like the following example (without the comments):
``````#!/usr/bin/env python
# import external libraries
import rospy
# import libraries which are part of the package (i.e. in the include dir)
import library
# import DTROS-related classes
from duckietown.dtros import \
DTROS, \
NodeType, \
TopicType, \
DTReminder,\
DTParam, \
ParamType
# import messages and services
from std_msgs.msg import Float32
from duckietown_msgs.msg import \
SegmentList, \
Segment, \
BoolStamped
class SomeNameNode(DTROS):
def __init__(self, node_name):
# class implementation
if __name__ == '__main__':
some_name_node = SomeNameNode(node_name='same_name_node')
rospy.spin()
``````
Observe that all nodes in Duckietown should inherit from the super class `DTROS`. This is a hard requirement. `DTROS` provides a lot of functionalities on top of the standard ROS nodes which make writing and debugging your node easier, and also sometimes comes with performance improvements.
In Python code, never ever do universal imports like `from somepackage import *`. This is an extremely bad practice. Instead, specify exactly what you are importing, i.e. `from somepackage import somefunction`. It is fine if you do it in `__init__.py` files but even there try to avoid it if possible.
When using a package that has a common practice alias, use it, e.g. `import numpy as np`, `import matplotlib.pyplot as plt`, etc. However, refrain from defining your own aliases.
The code in this node definition should be restricted as much as possible to ROS-related functionalities. If your node is performing some complex computation or has any logic that can be separated from the node itself, implement it as a separate library and put it in the `include` directory of the package.
## Node initialization
Modified today by Andrea F. Daniele
There are a lot of the details regarding the initalization of the node so let’s take a look at an example structure of the `__init__` method of our sample node.
``````class SomeNameNode(DTROS):
def __init__(self, node_name):
super(SomeNameNode, self).__init__(
node_name=node_name,
node_type=NodeType.PERCEPTION
)
# Setting up parameters
self.detection_freq = DTParam(
'~detection_freq',
param_type=ParamType.INT,
min_value=-1,
max_value=30
)
# ...
# Generic attributes
self.something_happened = None
self.arbitrary_counter = 0
# ...
# Subscribers
self.sub_img = rospy.Subscriber(
'image_rect',
Image,
self.cb_img
)
self.sub_cinfo = rospy.Subscriber(
'camera_info',
CameraInfo,
self.cb_cinfo
)
# ...
# Publishers
self.pub_img = rospy.Publisher(
'tag_detections_image/compressed',
CompressedImage,
queue_size=1,
dt_topic_type=TopicType.VISUALIZATION
)
self.pub_tag = rospy.Publisher(
'tag_detections',
AprilTagDetectionArray,
queue_size=1,
dt_topic_type=TopicType.PERCEPTION
)
# ...
``````
Now, let’s take a look at it section by section.
### Node Creation
Modified today by Andrea F. Daniele
In classic ROS nodes, you would initialize a ROS node with the function `rospy.init_node(...)`. DTROS does that for you, you simply need to pass the node name that you want to the super constructor as shown above.
DTROS supports node categorization, this is useful when you want to visualize the ROS network as a graph, where graph nodes represent ROS nodes and graph edges represent ROS topics. In such a graph, you mught want to group all the nodes working on the `PERCEPTION` problem together, say, to clear the clutter and make the graph easier to read. Use the parameter `node_type` in the super constructor of your node to do so. Use the values from the `NodeType` enumeration. Possible node types are the following,
``````GENERIC
DRIVER
PERCEPTION
CONTROL
PLANNING
LOCALIZATION
MAPPING
SWARM
BEHAVIOR
VISUALIZATION
INFRASTRUCTURE
COMMUNICATION
DIAGNOSTICS
DEBUG
``````
### Node Parameters
Modified today by Andrea F. Daniele
All parameters should have names relative to the namespace of the node, i.e. they should start with `~`. Also, all parameters should be in the scope of the instance, not the method, so they should always be declared inside the constructor and start with `self.`.
The parameters should never have default values set in the code. All default values should be in the configuration file! This makes sure that we don’t end up in a situation where there are two different default values in two different files related to the node.
In classic ROS, you get the value of a parameter with `rospy.get_param(...)`. One of the issues of the ROS implementation of parameters is that a node cannot request to be notified when a parameter’s value changes at runtime. Common solutions to this problem employ a polling strategy (which consists of querying the parameter server for changes in value at regular intervals). This is highly inefficient and does not scale. The `dtros` library provides a solution to this. Alternatively to using `rospy.get_param(...)` which simply returns you the current value of a paramter, you can create a `DTParam` object that automatically updates when a new value is set. Use `self.my_param = DTParam("~my_param")` to create a `DTParam` object and `self.my_param.value` to read its value.
### Generic attributes
Modified today by Andrea F. Daniele
Then we initialize all the non-ROS attributes that we will need for this class. Note that this is done before initializing the Publishers and Subscribers. The reason is that if a subscriber’s callback depends on one of these attributes, we need to define it before we use it. Here’s an example that might fail:
``````class CoolNode(DTROS):
def __init__(...):
self.sub_a = rospy.Subscriber(..., callback=cb_sth, ...)
self.important_variable = 3.1415
def cb_sth(self):
self.important_variable *= 1.0
``````
And something that is better:
``````class CoolNode(DTROS):
def __init__(...):
self.important_variable = 3.1415
sub_a = rospy.Subscriber(..., callback=cb_sth, ...)
def cb_sth(self):
self.important_variable *= 1.0
``````
### Publishers and Subscribers
Modified today by Andrea F. Daniele
Finally, we initialize all the Subscribers and Publishers as shown above. The `dtros` library automatically decorates the methods `rospy.Publisher` and `rospy.Subscriber`. By doing so, new parameters are added. All the parameters added by `dtros` have the prefix `dt_` (e.g., `dt_topic_type`). Use the values from the `TopicType` enumeration. Possible types list is identical to the node types list above.
Only declare a topic type in a `rospy.Publisher` call.
## Naming of variables and functions
Modified 2020-07-08 by Aleksandar Petrov
All functions, methods, and variables in Duckietown code should be named using `snake_case`. In other words, only lowercase letters with spaces replaced by underscored. Do not use `CamelCase`. This is to be used only for class names.
The names of all subscribers should start with `sub_` as in the example above. Similarly, names of publishers should start with `pub_` and names of callback functions should start with `cb_`.
Initalizing publishers and subscribers should again always be in the scope of the instance, hence starting with `self.`.
## Switching nodes on and off
Modified 2020-07-08 by Aleksandar Petrov
## Custom behavior on shutdown
Modified 2020-07-08 by Aleksandar Petrov
If you need to take care of something before when ROS tries to shutdown the node but before it actually shuts it down, you can implement the `on_shutdown` method. This is useful if you are running threads in the background, there are some files that you need to close, resources to release, or to put the robot into a safe state (e.g. to stop the wheels).
## Handling debug topics
Modified today by Andrea F. Daniele
Often we want to publish some information which helps us analyze the behavior and performance of the node but which does not contribute to the behavior itself. For example, in order to check how well the lane filter works, you might want to plot all the detected segments on a map of the road. However, this can be quite computationally expensive and is needed only on the rare occasion that someone wants to take a look at it.
A frequent (but bad design) way of handling that is to have a topic, to which one can publish a message, which when received will induce the node to start building a publishing the debug message. A much better way, and the one that should be used in Duckietown is to create and publish the debug message only if someone has subscribed to the debug topic. This is very easy to achieve with the help of `dtros`. Publishers created within a DTROS node exports the utility function `anybody_listening()`. Here’s an example:
``````if self.pub_debug_img.anybody_listening():
debug_img = self.very_expensive_function()
debug_image_msg = self.bridge.cv2_to_compressed_imgmsg(debug_img)
self.pub_debug_img.publish(debug_image_msg)
``````
Note also that all debug topics should be in the `debug` namespace of the node, i.e. `~debug/debug_topic_name`.
Similarly, a Subscribers created within a DTROS node exports the utility function `anybody_publishing()` that checks whether there are nodes that are currently publishing messages.
## Timed sections
Modified today by Andrea F. Daniele
If you have operations that might take non-trivial amount of computational time, you can profile them in order to be able to analyze the performance of your node. `DTROS` has a special context for that which uses the same mechanism as the debug topics. Hence, if you do not subscribe to the topic with the timing information, there would be no overhead to your performance. Therefore, be generous with the use of timed sections.
The syntax looks like that:
``````with self.time_phase("Step 1"):
run_step_1()
...
with self.time_phase("Step 2"):
run_step_2()
``````
Then, if you subscribe to `~debug/phase_times` you will be able to see for each separate section detailed information about the frequency of executing it, the average time it takes, and also the exact lines of code and the file in which this section appears.
## Config files
Modified today by Andrea F. Daniele
If your node has at least one parameter, then it should have a configuration file. If there is a single configuration (as is the case with most nodes) this file should be called `default.yaml`. Assuming that our node is called `some_node`, the configuration files for the node should be in the `config/some_node/` directory.
Every parameter used in the implementation of the node should have a default value in the configuration file. Furthermore, there should be no default values in the code. The only place where they should be defined is the configuration file.
## Launch files
Modified today by Andrea F. Daniele
Assuming that our node is called `some_node` then in the `launch` directory of the package there should be an atomic launch file with the name `some_node.launch` which launches the node in the correct namespace and loads its configuration parameters.
The launch file content of most node will be identical to the following, with only the node name and package name being changed.
``````<launch>
<arg name="veh"/>
<arg name="pkg_name" value="some_package"/>
<arg name="node_name" default="some_node"/>
<arg name="param_file_name" default="default" doc="Specify a param file"/>
<group ns="$(arg veh)"> <node name="$(arg node_name)" pkg="$(arg pkg_name)" type="$(arg node_name).py" output="screen">
<rosparam command="load" file="$(find some_package)/config/$(arg node_name)/\$(arg param_file_name).yaml"/>
</node>
</group>
</launch>
``````
## Ask the community
Modified 2020-07-08 by Aleksandar Petrov
If you have any questions about good practices in software development, join the Slack channel #info-developers. | 2021-04-22 16:42:07 | {"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": 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": 0.3681216537952423, "perplexity": 1876.3837004711172}, "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-2021-17/segments/1618039594341.91/warc/CC-MAIN-20210422160833-20210422190833-00446.warc.gz"} |
https://www.physicsforums.com/threads/how-do-i-solve-the-following-circuit.516977/ | # How do i solve the following circuit?
1. Jul 25, 2011
### cshum00
I have the following circuit and i want to look for the current and voltage across R1:
Let's ignore the numeric value for v1, v2 and R1 and generalize them.
If i apply Mesh Analysis i get the following formulas:
$$v_1 = (i_1 + i_2) R_1$$
$$v_2 = (i_1 + i_2) R_1$$
And if i apply Crammer's rule i get:
$$\Delta = R_1^2 - R_1^2 = 0$$
$$\Delta_1 = v_1 R_1 - v_2 R_1$$
$$\Delta_2 = v_1 R_1 - v_2 R_1$$
So, if i want to get i1, i2:
$$i_1 = \frac{\Delta_1}{\Delta} = \frac{v_1 R_1 - v_2 R}{0}$$
$$i_2 = \frac{\Delta_2}{\Delta} = \frac{v_2 R_1 - v_1 R_1}{0}$$
And if v1, v2 are equal it comes even worse!:
$$i_1 = i_2 = \frac{(0)(0) - (0)(0)}{0}$$
As for the voltage across R1:
$$v_R1 = (i_1 + i_2) R_1 = (\frac{v_1 R_1 - v_2 R}{0} + \frac{v_2 R_1 - v_1 R_1}{0}) R_1 = \frac{0}{0} R_1$$
2. Jul 26, 2011
### KingNothing
I am not sure what you are trying to do here. V1 and V2 are in parallel - they will always be the same. Putting voltage sources in parallel is a bad idea! The current through the resistor is simply V/R.
Plugging in zeros for everything just makes the current undefined.
3. Jul 26, 2011
### EWH
If V1 does not equal V2, then the non-grounded node of the circuit will have two different voltages at the same time, which seems impossible for ideal voltage sources and current analysis. In real life there would be a big current through the wire, limited only by the internal battery source resistances and the resistance of the wire. If V1=V2, then just remove one or the other. (Perhaps finding the equivalent parallel source resistance if the problem calls for it.)
4. Jul 26, 2011
### cshum00
I have actually done this experiment myself with two or more batteries. I know that the answer happens to be the average of all the input voltages. However, i can't get the formulas to come out as the averages of the sources (regardless of the methods i use for solving the equations). This is the reason why i created this topic.
I believe that result happens to be the average of the inputs because each source is trying to make the voltage difference to be the amount each battery of them supply. In doing so, they play a tug of war until they settle onto a value that represents all of them which is also the definition of average.
It is generalized that components parallel to a voltage source will have the same voltage. However, it doesn't mean that two or more voltage sources can't be in parallel. Although, it is true that it doesn't make sense to have multiple voltage sources in parallel; i want to be able to calculate and analyze the effects of it.
If you actually look at it closely, i didn't use zero or any specific numeric value for the components. Ignore the drawing saying 0V because i could not get rid of those labels on the schematics. I used general variables and the equations without plugging any values gave me 0/0.
Last edited: Jul 26, 2011
5. Jul 26, 2011
### Staff: Mentor
You cannot put two ideal voltage sources in parallel, because they have zero internal output resistance, so the "tug of war" current would be infinite.
You can put two real voltage sources in parallel, but you need to be careful, because the lower the output resistance of the voltage sources, the higher the fighting current will be. This can easily lead to a fire or the bursting of batteries. Please do not do this kind of experiment without understanding the basics of circuit operation.
6. Jul 26, 2011
### cshum00
Thanks for the explanations and allow me to be a little sarcastic. It is not that we "cannot" put two ideal voltage sources in parallel. But that the result in putting two ideal voltage sources in parallel could be hazardous (like having infinte output current which you pointed out). On the other hand, because real voltage souces have internal resistance and they want the output voltage to be the same as their own; they end up messing around with the internal resistance of the parallel voltage source(s).
And since I wanted to see how it worked out mathematically I created a modified circuit. It includes R1, R2 as the internal resistance of the voltage sources and R3 as the reistance of the load:
$$v_1 = i_1 R_1 + (i_1 + i_2) R3 = i_1 (R_1 + R_3) + i_2 R$$
$$v_2 = i_2 R_2 + (i_1 + i_2) R3 = i_1 R3 + i_2 (R_2 + R_3)$$
$$\Delta = (R_1 + R_3)(R_2 + R_3) - R_3^2 = R_1 R_2 + R_2 R_3 + R_1 R_3$$
$$\Delta_1 = v_1 R_2 + v_1 R_3 - v_2 R_3$$
$$\Delta_2 = v_2 R_1 + v_2 R_3 - v_1 R_3$$
$$i_1 = \frac{v_1 R_2 + v_1 R_3 - v_2 R_3}{R_1 R_2 + R_2 R_3 + R_1 R_3}$$
$$i_2 = \frac{v_2 R_1 + v_2 R_3 - v_1 R_3}{R_1 R_2 + R_2 R_3 + R_1 R_3}$$
From here we can see:
$$i_3 = i_1 + i_2 = \frac{v_1 R_2 + v_2 R_1}{R_1 R_2 + R_2 R_3 + R_1 R_3}$$
On the top of the fraction we can see that v1 messing with v2's resistance and vice-versa.
While on the bottom of the fraction we can see that it is R1||R2||R3, which indicates a simple voltage to current source transformation:
So assuming that both voltage sources have the same internal resistance R1 = R2 = R and let the limit go to zero:
$$\lim {R \to 0} = i_3 = \frac{R (v_1 + v_2)}{R^2 + 2(R)R_3}$$
Without simplifying any further, we can see that the bottom part dominates which leads to infinite current.
However, if we do a proper simplification:
$$\lim {R \to 0} = i_3 = \frac{v_1 + v_2}{R + 2R_3} = \frac{v_1 + v_2}{2R_3}$$
Voala! The average of both voltage sources and the voltage across R3 is simply:
$$v_3 = \frac{v_1 + v_2}{2R_3}R_3 = \frac{v_1 + v_2}{2}$$
Last edited: Jul 26, 2011
7. Jul 26, 2011
### Staff: Mentor
Real power supplies typically can only pull *up* on their voltage outputs. If you look at the equivalent circuit of a LM7805 linear voltage regulator, for example, you will see that the pass transistor can source current into the load, but there is no mechanism to sink current from the load if another power supply pulls the output up higher than 5V.
So in that situation, the highest power supply would determine the combined output voltage.
8. Jul 26, 2011
### cshum00
I am not quite sure how transistors work. But do the voltage supplies on the transistors end up parallel and not in series for the output voltage to be a "pull up"?
9. Jul 26, 2011
### EWH
With LTSpice, it throws an error of "overdetermined circuit" if there are no source resistances. With e.g. V1=10V V2=5V, R=1K, and V_s for both V1 and V2 = 0.1 Ohm, then:
V(n001): 7.49963 voltage
I(R1): 0.00749962 device_current
I(V2): 24.9963 device_current
I(V1): -25.0037 device_current
With real batteries, one will be trying to charge the other and hydrogen could be produced, perhaps bursting the battery and even causing a fire or explosion. Also 25 A could easily melt most wires, again potentially leading to a fire.
With a more reasonable situation V1=9.2V V2=9.1V, R=1K, and V_s for both V1 and V2 = 0.25 Ohm:
V(n001): 9.14886 voltage
I(R1): 0.00914886 device_current
I(V2): 0.195426 device_current
I(V1): -0.204574 device_current
Which still means that ~95% of the current is not going through the resistor.
10. Jul 26, 2011
### Staff: Mentor
I don't quite understand your question. A transistor like the pass elelment NPN transistor in a LM7805 linear regulator is a "pullup" element controlled by a feedback circuit. If the output voltage is below the 5V that is desired, the pullup transistor is turned on harder to try to pull up the output voltage.
If the output voltage is above the desired 5V, all that the pullup transistor can do is shut off, and hope that the load pulls down the 5V until it is back in regulation.
BTW, never anthropomorphise transistors. They hate it when you do that. :tongue2:
11. Jul 26, 2011
### cshum00
Well you see, we are talking about the effects of voltage regulation when two voltage sources are in parallel. Then you come up with the analogy of transistors as an explanation on how the behavior on the parallel voltage sources work. What i understood from your analogy is that there is no such thing as a "pull downs" but only "pull ups". In other words, what i understood from you is that in transistors a lower voltage can be amplified and not de-amplified.
But that is when i say, is the configuration on the transistor exactly like having two power sources in parallel? Like i mentioned, i don't know much about transistors. All i know is that there are three terminals. One terminal has a fixed voltage value the collector. The other terminal is the base which only if the current flowing in there is high enough then current flows on the emitter terminal.
But then i don't know exactly the internals of a transistor is configured. But in my opinion, only if some part of the transistor can be substituted by two parallel voltage sources then we can use the transistor as a explanation on how parallel voltage sources work.
The reason i say that is because it is hard to say whether there really is just a "pull up" or "pull down" dominated by only one of the parallel voltage sources. Let's take for example V1>V2 and V1||V2. In V1's point of view, there is a pull down but in V2's point of view there is a pull up. However, because both V1 and V2 have the exact same terminals, when we try to measure one source's pull we end up measuring the other one too.
12. Jul 27, 2011
### Staff: Mentor
All I was pointing out is that there are different kinds of "voltage sources" in real life. Batteries (depending on what type) will be able to both source and sink current. Power supplies will generally only be able to source current (think of a battery with a diode in series with the output). Power amplifiers are able to both source and sink current, because their outputs have both pull-up and pull-down transistors. | 2018-06-19 18:31:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.44630369544029236, "perplexity": 743.5347051759167}, "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-2018-26/segments/1529267863109.60/warc/CC-MAIN-20180619173519-20180619193519-00364.warc.gz"} |
https://math.stackexchange.com/questions/3510233/solve-inverse-trigonometric-equation-sin-left-operatornamecot-1x-1-r | # Solve inverse trigonometric equation $\sin\left(\operatorname{cot^{-1}}(x + 1)\right) = \cos\left(\tan^{-1}x\right)$
If $$\sin\left(\operatorname{cot^{-1}}(x + 1)\right) = \cos\left(\tan^{-1}x\right)$$, then find the value of $$x$$.
Please solve this question by using $$\cos\left(\dfrac\pi2 - \theta\right) = \sin\theta$$ by changing $$\cos\left(\tan^{-1}x\right) = \sin\left(\dfrac\pi2 - \tan^{-1}x\right)$$ and then equate both LHS and RHS. If not then why? How does the contradiction below occur?
• $\sin x = \sin y$ does not imply $x = y$ – ab123 Jan 15 at 16:39
• $\sin$ isn't an injective function... – Don Thousand Jan 15 at 16:44
Replace $$x$$ with $$\dfrac1h$$ to find $$h=0$$
Alternatively
$$\cos(\arctan x)=\sin(\text{arccot}(x+1))=\cdots=\cos(\arctan(x+1))$$
$$\arctan(x)=2m\pi\pm\arctan(x+1)$$ where $$m$$ is an integer
$$m=0$$
Now replace $$x$$ with $$\dfrac1h$$
and consider +/- sign one by one
$$+$$ sign will give $$h=0$$
By using $$-$$ sign, $$h=-2$$
Note that there are multiple possibilities to the equation you derived below,
$$\sin(\cot^{-1}(1+x))=\sin(\frac\pi2-\tan^{-1} x)$$
You only considered
$$\cot^{-1}(1+x) = \frac\pi2-\tan^{-1} x$$ which leads to contradiction, or, no solutions. In addition, you also need to examine
$$\cot^{-1}(1+x) = \pi - (\frac\pi2-\tan^{-1} x)$$
which leads to $$x+1=-x$$, hence the valid solution $$x=-\frac12$$. | 2020-09-26 16:19:09 | {"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": 22, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8180912137031555, "perplexity": 605.7482937598663}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400244231.61/warc/CC-MAIN-20200926134026-20200926164026-00175.warc.gz"} |
https://en.wikipedia.org/wiki/Derivative | # Derivative
The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.
The derivative of a function of a real variable measures the sensitivity to change of the function value (output value)—with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change": the ratio of the instantaneous change in the dependent variable to that of the independent variable.
Derivatives may be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation, whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation—with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.
The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.[Note 1]
## Differentiation
Differentiation is the action of computing a derivative. The derivative of a function y = f(x) of a variable x is a measure of the rate at which the value y of the function changes—with respect to the change of the variable x. It is called the derivative of f with respect to x. If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point.
Slope of a linear function: ${\displaystyle m={\frac {\Delta y}{\Delta x}}}$
The simplest case, apart from the trivial case of a constant function, is when y is a linear function of x, meaning that the graph of y is a line. In this case, y = f(x) = mx + b for real numbers m and b, and the slope m is given by
${\displaystyle m={\frac {{\text{change in }}y}{{\text{change in }}x}}={\frac {\Delta y}{\Delta x}},}$
where the symbol Δ (Delta) is an abbreviation for "change in", and the combinations ${\displaystyle \Delta x}$ and ${\displaystyle \Delta y}$ refer to corresponding changes[1] (i.e., ${\displaystyle \Delta y=f(x+\Delta x)-f(x)}$). The above formula holds because
{\displaystyle {\begin{aligned}y+\Delta y&=f\left(x+\Delta x\right)\\&=m\left(x+\Delta x\right)+b=mx+m\Delta x+b\\&=y+m\Delta x.\end{aligned}}}
Thus
${\displaystyle \Delta y=m\Delta x.}$
This gives the value for the slope of a line.
If the function f is not linear (i.e., its graph is not a straight line), then the change in y divided by the change in x varies over the considered range: differentiation is a method to find a unique value for this rate of change, not across a certain range ${\displaystyle (\Delta x),}$ but at any given value of x.
Rate of change as a limit value
Figure 1. The tangent line at (x, f(x))
Figure 2. The secant to curve y= f(x) determined by points (x, f(x)) and (x + h, f(x + h))
Figure 3. The tangent line as limit of secants
Figure 4. Animated illustration: the tangent line (derivative) as the limit of secants
The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the limit value of the ratio of the differences Δy / Δx, as Δx tends towards 0.
### Notation
Two distinct notations are commonly used for the derivative, one deriving from Gottfried Wilhelm Leibniz, and the other from Joseph Louis Lagrange. A third notation, first used by Isaac Newton, is sometimes seen in physics.
In Leibniz's notation, an infinitesimal change in x is denoted by dx, and the derivative of y with respect to x is written[2][3]
${\displaystyle {\frac {dy}{dx}}}$
suggesting the ratio of two infinitesimal quantities. (The above expression is read as "the derivative of y with respect to x", "dy by dx", or "dy over dx". The oral form "dy dx" is often used conversationally, although it may lead to confusion.)
In Lagrange's notation, the derivative with respect to x of a function f(x) is denoted f'(x)[2][3] (read as "f prime of x") or fx′(x) (read as "f prime x of x"), in case of ambiguity of the variable implied by the differentiation. Lagrange's notation is sometimes incorrectly attributed to Newton.
Newton's notation for differentiation (also called the dot notation for differentiation) places a dot over the dependent variable. That is, if y is a function of t, then the derivative of y with respect to t is[2][3]
${\displaystyle {\dot {y}}}$
Higher derivatives are represented using multiple dots, as in[2]
${\displaystyle {\ddot {y}},{\overset {...}{y}}}$
Newton's notation is generally used when the independent variable denotes time. For example. if location y is a function of t, then ${\displaystyle {\dot {y}}}$ denotes velocity[4] and ${\displaystyle {\ddot {y}}}$ denotes acceleration.[5]
### Rigorous definition
A secant approaches a tangent when ${\displaystyle \Delta x\to 0}$.
The most common approach to turn this intuitive idea into a precise definition is to define the derivative as a limit of difference quotients of real numbers.[6] This is the approach described below.
Let f be a real valued function defined in an open neighborhood of a real number a. In classical geometry, the tangent line to the graph of the function f at a was the unique line through the point (a, f(a)) that did not meet the graph of f transversally, meaning that the line did not pass straight through the graph. The derivative of y with respect to x at a is, geometrically, the slope of the tangent line to the graph of f at (a, f(a)).[7] The slope of the tangent line is very close to the slope of the line through (a, f(a)) and a nearby point on the graph, for example (a + h, f(a + h)). These lines are called secant lines. A value of h close to zero gives a good approximation to the slope of the tangent line, and smaller values (in absolute value) of h will, in general, give better approximations. The slope m of the secant line is the difference between the y values of these points—divided by the difference between the x values. That is,[7]
${\displaystyle m={\frac {\Delta f(a)}{\Delta a}}={\frac {f(a+h)-f(a)}{(a+h)-(a)}}={\frac {f(a+h)-f(a)}{h}}.}$
This expression is Newton's difference quotient.[7] Passing from an approximation to an exact answer is done using a limit. Geometrically, the limit of the secant lines is the tangent line. Therefore, the limit of the difference quotient as h approaches zero, if it exists, should represent the slope of the tangent line to (a, f(a)). This limit is defined to be the derivative of the function f at a:[3][7]
${\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}.}$
When the limit exists, f is said to be differentiable at a. Here, f(a) is one of several common notations for the derivative (see § Notations for differentiation below). From this definition, it is obvious that a differentiable function f is increasing if and only if its derivative is positive, and is decreasing if and only if its derivative is negative. This fact is used extensively when analyzing function behavior (e.g., when finding local extrema).
Equivalently, the derivative satisfies the property that
${\displaystyle \lim _{h\to 0}{\frac {f(a+h)-(f(a)+f'(a)\cdot h)}{h}}=0,}$
which has the intuitive interpretation (see Figure 1) that the tangent line to f at a gives the best linear approximation
${\displaystyle f(a+h)\approx f(a)+f'(a)h}$
to f near a (i.e., for small h). This interpretation is the easiest to generalize to other settings (see § Total derivative, total differential and Jacobian matrix below).
Substituting 0 for h in the difference quotient causes division by zero, so the slope of the tangent line cannot be found directly using this method. Instead, one defines Q(h) to be the difference quotient as a function of h:
${\displaystyle Q(h)={\frac {f(a+h)-f(a)}{h}}.}$
Geometrically, Q(h) is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). If f is a continuous function, meaning that its graph is an unbroken curve with no gaps, then Q is a continuous function away from h = 0. If the limit limh→0Q(h) exists, meaning that there is a way of choosing a value for Q(0) that makes Q a continuous function, then the function f is differentiable at a, and its derivative at a equals Q(0).
In practice, the existence of a continuous extension of the difference quotient Q(h) to h = 0 is shown by modifying the numerator to cancel h in the denominator. Such manipulations can make the limit value of Q for small h clear—even though Q is still not defined at h = 0. This process can be long and tedious for complicated functions, and many shortcuts are commonly used to simplify the process.
### Definition over the hyperreals
Relative to a hyperreal extension RR of the real numbers, the derivative of a real function y = f(x) at a real point x can be defined as the shadow of the quotient y/x for infinitesimal x, where y = f(x + ∆x) − f(x). Here, the natural extension of f to the hyperreals is still denoted f, and the derivative is said to exist if the shadow is independent of the infinitesimal chosen.
### Example
The square function
The square function given by f(x) = x2 is differentiable at x = 3, and its derivative there is 6. This result is established by calculating the limit of the difference quotient of f(3)—as h approaches zero :
{\displaystyle {\begin{aligned}f'(3)&=\lim _{h\to 0}{\frac {f(3+h)-f(3)}{h}}=\lim _{h\to 0}{\frac {(3+h)^{2}-3^{2}}{h}}\\[10pt]&=\lim _{h\to 0}{\frac {9+6h+h^{2}-9}{h}}=\lim _{h\to 0}{\frac {6h+h^{2}}{h}}=\lim _{h\to 0}{(6+h)}.\end{aligned}}}
The last expression shows that the difference quotient equals 6 + h when h ≠ 0, and is undefined when h = 0 (due to the definition of the difference quotient). However, the definition of the limit says the difference quotient does not need to be defined when h = 0. The limit is the result of letting h go to zero, meaning that it is the value 6 + h tends to as h becomes very small:
${\displaystyle \lim _{h\to 0}{(6+h)}=6+0=6.}$
Hence the slope of the graph of the square function at the point (3, 9) is 6, and so its derivative at x = 3 is f(3) = 6.
More generally, a similar computation shows that the derivative of the square function at x = a is f(a) = 2a:[7]
{\displaystyle {\begin{aligned}f'(a)&=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}=\lim _{h\to 0}{\frac {(a+h)^{2}-a^{2}}{h}}\\[0.3em]&=\lim _{h\to 0}{\frac {a^{2}+2ah+h^{2}-a^{2}}{h}}=\lim _{h\to 0}{\frac {2ah+h^{2}}{h}}\\[0.3em]&=\lim _{h\to 0}{(2a+h)}=2a\end{aligned}}}
### Continuity and differentiability
This function does not have a derivative at the marked point, as the function is not continuous there (specifically, it has a jump discontinuity).
If f is differentiable at a, then f must also be continuous at a. As an example, choose a point a and let f be the step function that returns the value 1 for all x less than a, and returns a different value 10 for all x greater than or equal to a. Defined this way, f cannot have a derivative at a: if h is negative, then a + h is on the low part of the step, so the secant line from a to a + h is very steep, and as h tends to zero the slope tends to infinity. If h is positive, then a + h is on the high part of the step, so the secant line from a to a + h has slope zero. Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist.
The absolute value function is continuous, but fails to be differentiable at x = 0 since the tangent slopes do not approach the same value from the left as they do from the right.
However, even if a function is continuous at a point, it may not be differentiable there. For example, the absolute value function given by f(x) = |x| is continuous at x = 0, but it is not differentiable there:[3] if h is positive, then the slope of the secant line from 0 to h is one, whereas if h is negative, then the slope of the secant line from 0 to h is negative one. This can be seen graphically as a "kink" or a "cusp" in the graph at x = 0. Even a function with a smooth graph is not differentiable at a point where its tangent is vertical: For instance, the function given by f(x) = x1/3 is not differentiable at x = 0.
In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative.
Most functions that occur in practice have derivatives at all points or at almost every point. Early in the history of calculus, many mathematicians assumed that a continuous function was differentiable at most points. Under mild conditions, if the function is for example a monotone function or a Lipschitz function, then this is true. In 1872, however, Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere. This example is now known as the Weierstrass function. In 1931, Stefan Banach proved that the set of functions that have a derivative at some point is a meager set in the space of all continuous functions.[8] Informally, this means that hardly do any random continuous functions have a derivative at even one point.
### The derivative as a function
The derivative at different points of a differentiable function. In this case, the derivative is equal to:${\displaystyle \sin \left(x^{2}\right)+2x^{2}\cos \left(x^{2}\right)}$
Let f be a function that has a derivative at every point in its domain. We can then define a function that maps every point ${\displaystyle x}$ to the value of the derivative of ${\displaystyle f}$ at ${\displaystyle x}$. This function is written f and is called the derivative function or the derivative of f.
Sometimes, f has a derivative at most but not all points of its domain. The function whose value at a equals f(a) whenever f(a) is defined and elsewhere is undefined is also called the derivative of f. It is still a function, but its domain is strictly smaller than the domain of f.
Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions that have derivatives at every point of their domain, and whose range is a set of functions. If we denote this operator by D, then D(f) is the function f. Since D(f) is a function, it can be evaluated at a point a. By the definition of the derivative function, D(f)(a) = f(a).
For comparison, consider the doubling function given by f(x) = 2x. Here, we assume that f is a real-valued function of a real number, meaning that it takes numbers as inputs and has numbers as outputs:
{\displaystyle {\begin{aligned}1&{}\mapsto 2,\\2&{}\mapsto 4,\\3&{}\mapsto 6.\end{aligned}}}
The operator D, however, is not defined on individual numbers. It is only defined on functions:
{\displaystyle {\begin{aligned}D(x\mapsto 1)&=(x\mapsto 0),\\D(x\mapsto x)&=(x\mapsto 1),\\D\left(x\mapsto x^{2}\right)&=(x\mapsto 2\cdot x).\end{aligned}}}
Because the output of D is a function, the output of D can be evaluated at a point. For instance, when D is applied to the square function, xx2, D outputs the doubling function x ↦ 2x, which we named f(x). This output function can then be evaluated to get f(1) = 2, f(2) = 4, and so on.
### Higher derivatives
Let f be a differentiable function, and let f be its derivative. The derivative of f (if it has one) is written f ′′ and is called the second derivative of f. Similarly, the derivative of the second derivative, if it exists, is written f ′′′ and is called the third derivative of f. Continuing this process, one can define, if it exists, the nth derivative as the derivative of the (n-1)th derivative. These repeated derivatives are called higher-order derivatives. The nth derivative is also called the derivative of order n, denoted ${\displaystyle f^{(n)}}$.[2]
If x(t) represents the position of an object at time t, then the higher-order derivatives of x have specific interpretations in physics. The first derivative of x is the object's velocity. The second derivative of x is the acceleration. The third derivative of x is the jerk. And finally, the fourth through sixth derivatives of x are snap, crackle, and pop; most applicable to astrophysics.
A function f need not have a derivative (for example, if it is not continuous). Similarly, even if f does have a derivative, it may not have a second derivative. For example, let
${\displaystyle f(x)={\begin{cases}+x^{2},&{\text{if }}x\geq 0\\-x^{2},&{\text{if }}x\leq 0.\end{cases}}}$
Calculation shows that f is a differentiable function whose derivative at ${\displaystyle x}$ is given by
${\displaystyle f'(x)={\begin{cases}+2x,&{\text{if }}x\geq 0\\-2x,&{\text{if }}x\leq 0.\end{cases}}}$
f'(x) is twice the absolute value function at ${\displaystyle x}$, and it does not have a derivative at zero. Similar examples show that a function can have a kth derivative for each non-negative integer k, but not a (k + 1)th derivative. A function that has k successive derivatives is called k times differentiable. If the kth derivative is in addition continuous, then the function is said to be of differentiability class Ck. (This is a stronger condition than having k derivatives, as shown by the second example of Smoothness § Examples.) A function that has infinitely many derivatives is called infinitely differentiable or smooth.
On the real line, every polynomial function is infinitely differentiable. By standard differentiation rules, if a polynomial of degree n is differentiated n times, then it becomes a constant function. All of its subsequent derivatives are identically zero. In particular, they exist, so polynomials are smooth functions.
The derivatives of a function f at a point x provide polynomial approximations to that function near x. For example, if f is twice differentiable, then
${\displaystyle f(x+h)\approx f(x)+f'(x)h+{\tfrac {1}{2}}f''(x)h^{2}}$
in the sense that
${\displaystyle \lim _{h\to 0}{\frac {f(x+h)-f(x)-f'(x)h-{\frac {1}{2}}f''(x)h^{2}}{h^{2}}}=0.}$
If f is infinitely differentiable, then this is the beginning of the Taylor series for f evaluated at x + h around x.
### Inflection point
A point where the second derivative of a function changes sign is called an inflection point.[9] At an inflection point, the second derivative may be zero, as in the case of the inflection point x = 0 of the function given by ${\displaystyle f(x)=x^{3}}$, or it may fail to exist, as in the case of the inflection point x = 0 of the function given by ${\displaystyle f(x)=x^{\frac {1}{3}}}$. At an inflection point, a function switches from being a convex function to being a concave function, or vice versa.
## Notation (details)
### Leibniz's notation
The symbols ${\displaystyle dx}$, ${\displaystyle dy}$, and ${\displaystyle {\frac {dy}{dx}}}$ were introduced by Gottfried Wilhelm Leibniz in 1675.[10] It is still commonly used when the equation y = f(x) is viewed as a functional relationship between dependent and independent variables. Then the first derivative is denoted by[2]
${\displaystyle {\frac {dy}{dx}},\quad {\frac {df}{dx}},{\text{ or }}{\frac {d}{dx}}f,}$
and was once thought of as an infinitesimal quotient. Higher derivatives are expressed using the notation[2]
${\displaystyle {\frac {d^{n}y}{dx^{n}}},\quad {\frac {d^{n}f}{dx^{n}}},{\text{ or }}{\frac {d^{n}}{dx^{n}}}f}$
for the nth derivative of ${\displaystyle y=f(x)}$. These are abbreviations for multiple applications of the derivative operator. For example,
${\displaystyle {\frac {d^{2}y}{dx^{2}}}={\frac {d}{dx}}\left({\frac {dy}{dx}}\right).}$
With Leibniz's notation, we can write the derivative of ${\displaystyle y}$ at the point ${\displaystyle x=a}$ in two different ways:
${\displaystyle \left.{\frac {dy}{dx}}\right|_{x=a}={\frac {dy}{dx}}(a).}$
Leibniz's notation allows one to specify the variable for differentiation (in the denominator), which is relevant in partial differentiation. It also can be used to write the chain rule as[Note 2]
${\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}.}$
### Lagrange's notation
One of the most common modern notation for differentiation, sometimes referred to as prime notation,[11] is due to Joseph-Louis Lagrange and uses the prime mark, so that the derivative of a function ${\displaystyle f}$ is denoted ${\displaystyle f'}$. Similarly, the second and third derivatives are denoted
${\displaystyle (f')'=f''}$ and ${\displaystyle (f'')'=f'''.}$
To denote the number of derivatives beyond this point, some authors use Roman numerals in superscript, whereas others place the number in parentheses:
${\displaystyle f^{\mathrm {iv} }}$ or ${\displaystyle f^{(4)}.}$
The latter notation generalizes to yield the notation ${\displaystyle f^{(n)}}$ for the nth derivative of ${\displaystyle f}$[2] – this notation is most useful when we wish to talk about the derivative as being a function itself, as in this case the Leibniz notation can become cumbersome.
### Newton's notation
Newton's notation for differentiation, also called the dot notation, places a dot over the function name to represent a time derivative. If ${\displaystyle y=f(t)}$, then
${\displaystyle {\dot {y}}}$ and ${\displaystyle {\ddot {y}}}$
denote, respectively, the first and second derivatives of ${\displaystyle y}$.[2][3] This notation is used exclusively for derivatives with respect to time or arc length. It is typically used in differential equations in physics and differential geometry.[12][13] The dot notation, however, becomes unmanageable for high-order derivatives (order 4 or more), and cannot deal with multiple independent variables.
### Euler's notation
Euler's notation uses a differential operator ${\displaystyle D}$, which is applied to a function ${\displaystyle f}$ to give the first derivative ${\displaystyle Df}$. The nth derivative is denoted ${\displaystyle D^{n}f}$.
If y = f(x) is a dependent variable, then often the subscript x is attached to the D to clarify the independent variable x. Euler's notation is then written
${\displaystyle D_{x}y}$ or ${\displaystyle D_{x}f(x)}$,
although this subscript is often omitted when the variable x is understood, for instance when this is the only independent variable present in the expression.
Euler's notation is useful for stating and solving linear differential equations.
## Rules of computation
The derivative of a function can, in principle, be computed from the definition by considering the difference quotient, and computing its limit. In practice, once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using rules for obtaining derivatives of more complicated functions from simpler ones.
### Rules for basic functions
Here are the rules for the derivatives of the most common basic functions, where a is a real number.[3]
${\displaystyle {\frac {d}{dx}}x^{a}=ax^{a-1}.}$
${\displaystyle {\frac {d}{dx}}e^{x}=e^{x}.}$
${\displaystyle {\frac {d}{dx}}a^{x}=a^{x}\ln(a),\qquad a>0}$
${\displaystyle {\frac {d}{dx}}\ln(x)={\frac {1}{x}},\qquad x>0.}$
${\displaystyle {\frac {d}{dx}}\log _{a}(x)={\frac {1}{x\ln(a)}},\qquad x,a>0}$
${\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x).}$
${\displaystyle {\frac {d}{dx}}\cos(x)=-\sin(x).}$
${\displaystyle {\frac {d}{dx}}\tan(x)=\sec ^{2}(x)={\frac {1}{\cos ^{2}(x)}}=1+\tan ^{2}(x).}$
${\displaystyle {\frac {d}{dx}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}},\qquad -1
${\displaystyle {\frac {d}{dx}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}},\qquad -1
${\displaystyle {\frac {d}{dx}}\arctan(x)={\frac {1}{1+x^{2}}}}$
### Rules for combined functions
Here are some of the most basic rules for deducing the derivative of a compound function from derivatives of basic functions.[3]
• Constant rule: if f(x) is constant, then
${\displaystyle f'(x)=0.}$
${\displaystyle (\alpha f+\beta g)'=\alpha f'+\beta g'}$ for all functions f and g and all real numbers ${\displaystyle \alpha }$ and ${\displaystyle \beta }$.
${\displaystyle (fg)'=f'g+fg'}$ for all functions f and g. As a special case, this rule includes the fact ${\displaystyle (\alpha f)'=\alpha f'}$ whenever ${\displaystyle \alpha }$ is a constant, because ${\displaystyle \alpha 'f=0\cdot f=0}$ by the constant rule.
${\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-fg'}{g^{2}}}}$ for all functions f and g at all inputs where g ≠ 0.
• Chain rule for composite functions: If ${\displaystyle f(x)=h(g(x))}$, then
${\displaystyle f'(x)=h'(g(x))\cdot g'(x).}$
### Computation example
The derivative of the function given by
${\displaystyle f(x)=x^{4}+\sin \left(x^{2}\right)-\ln(x)e^{x}+7}$
is
{\displaystyle {\begin{aligned}f'(x)&=4x^{(4-1)}+{\frac {d\left(x^{2}\right)}{dx}}\cos \left(x^{2}\right)-{\frac {d\left(\ln {x}\right)}{dx}}e^{x}-\ln(x){\frac {d\left(e^{x}\right)}{dx}}+0\\&=4x^{3}+2x\cos \left(x^{2}\right)-{\frac {1}{x}}e^{x}-\ln(x)e^{x}.\end{aligned}}}
Here, the second term was computed using the chain rule and the third using the product rule. The known derivatives of the elementary functions x2, x4, sin(x), ln(x) and exp(x) = ex, as well as the constant 7, were also used.
## In higher dimensions
### Vector-valued functions
A vector-valued function y of a real variable sends real numbers to vectors in some vector space Rn. A vector-valued function can be split up into its coordinate functions y1(t), y2(t), ..., yn(t), meaning that y(t) = (y1(t), ..., yn(t)). This includes, for example, parametric curves in R2 or R3. The coordinate functions are real valued functions, so the above definition of derivative applies to them. The derivative of y(t) is defined to be the vector, called the tangent vector, whose coordinates are the derivatives of the coordinate functions. That is,
${\displaystyle \mathbf {y} '(t)=(y'_{1}(t),\ldots ,y'_{n}(t)).}$
Equivalently,
${\displaystyle \mathbf {y} '(t)=\lim _{h\to 0}{\frac {\mathbf {y} (t+h)-\mathbf {y} (t)}{h}},}$
if the limit exists. The subtraction in the numerator is the subtraction of vectors, not scalars. If the derivative of y exists for every value of t, then y′ is another vector-valued function.
If e1, ..., en is the standard basis for Rn, then y(t) can also be written as y1(t)e1 + … + yn(t)en. If we assume that the derivative of a vector-valued function retains the linearity property, then the derivative of y(t) must be
${\displaystyle y'_{1}(t)\mathbf {e} _{1}+\cdots +y'_{n}(t)\mathbf {e} _{n}}$
because each of the basis vectors is a constant.
This generalization is useful, for example, if y(t) is the position vector of a particle at time t; then the derivative y′(t) is the velocity vector of the particle at time t.
### Partial derivatives
Suppose that f is a function that depends on more than one variable—for instance,
${\displaystyle f(x,y)=x^{2}+xy+y^{2}.}$
then f can be reinterpreted as a family of functions of one variable indexed by the other variables:
${\displaystyle f(x,y)=f_{x}(y)=x^{2}+xy+y^{2}.}$
In other words, every value of x chooses a function, denoted fx, which is a function of one real number.[Note 3] That is,
${\displaystyle x\mapsto f_{x},}$
${\displaystyle f_{x}(y)=x^{2}+xy+y^{2}.}$
Once a value of x is chosen, say a, then f(x, y) determines a function fa that sends y to a2 + ay + y2:
${\displaystyle f_{a}(y)=a^{2}+ay+y^{2}.}$
In this expression, a is a constant, not a variable, so fa is a function of only one real variable. Consequently, the definition of the derivative for a function of one variable applies:
${\displaystyle f_{a}'(y)=a+2y.}$
The above procedure can be performed for any choice of a. Assembling the derivatives together into a function gives a function that describes the variation of f in the y direction:
${\displaystyle {\frac {\partial f}{\partial y}}(x,y)=x+2y.}$
This is the partial derivative of f with respect to y. Here is a rounded d called the partial derivative symbol. To distinguish it from the letter d, ∂ is sometimes pronounced "der", "del", or "partial" instead of "dee".
In general, the partial derivative of a function f(x1, …, xn) in the direction xi at the point (a1, ..., an) is defined to be:
${\displaystyle {\frac {\partial f}{\partial x_{i}}}(a_{1},\ldots ,a_{n})=\lim _{h\to 0}{\frac {f(a_{1},\ldots ,a_{i}+h,\ldots ,a_{n})-f(a_{1},\ldots ,a_{i},\ldots ,a_{n})}{h}}.}$
In the above difference quotient, all the variables except xi are held fixed. That choice of fixed values determines a function of one variable
${\displaystyle f_{a_{1},\ldots ,a_{i-1},a_{i+1},\ldots ,a_{n}}(x_{i})=f(a_{1},\ldots ,a_{i-1},x_{i},a_{i+1},\ldots ,a_{n}),}$
and, by definition,
${\displaystyle {\frac {df_{a_{1},\ldots ,a_{i-1},a_{i+1},\ldots ,a_{n}}}{dx_{i}}}(a_{i})={\frac {\partial f}{\partial x_{i}}}(a_{1},\ldots ,a_{n}).}$
In other words, the different choices of a index a family of one-variable functions just as in the example above. This expression also shows that the computation of partial derivatives reduces to the computation of one-variable derivatives.
An important example of a function of several variables is the case of a scalar-valued function f(x1, ..., xn) on a domain in Euclidean space Rn (e.g., on R2 or R3). In this case f has a partial derivative ∂f/∂xj with respect to each variable xj. At the point (a1, ..., an), these partial derivatives define the vector
${\displaystyle \nabla f(a_{1},\ldots ,a_{n})=\left({\frac {\partial f}{\partial x_{1}}}(a_{1},\ldots ,a_{n}),\ldots ,{\frac {\partial f}{\partial x_{n}}}(a_{1},\ldots ,a_{n})\right).}$
This vector is called the gradient of f at a. If f is differentiable at every point in some domain, then the gradient is a vector-valued function ∇f that takes the point (a1, ..., an) to the vector ∇f(a1, ..., an). Consequently, the gradient determines a vector field.
### Directional derivatives
If f is a real-valued function on Rn, then the partial derivatives of f measure its variation in the direction of the coordinate axes. For example, if f is a function of x and y, then its partial derivatives measure the variation in f in the x direction and the y direction. They do not, however, directly measure the variation of f in any other direction, such as along the diagonal line y = x. These are measured using directional derivatives. Choose a vector
${\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n}).}$
The directional derivative of f in the direction of v at the point x is the limit[2]
${\displaystyle D_{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\rightarrow 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h}}.}$
In some cases, it may be easier to compute or estimate the directional derivative after changing the length of the vector. Often, this is done to turn the problem into the computation of a directional derivative in the direction of a unit vector. To see how this works, suppose that v = λu. By substituting h = k/λ into the difference quotient, the difference quotient becomes:
${\displaystyle {\frac {f(\mathbf {x} +(k/\lambda )(\lambda \mathbf {u} ))-f(\mathbf {x} )}{k/\lambda }}=\lambda \cdot {\frac {f(\mathbf {x} +k\mathbf {u} )-f(\mathbf {x} )}{k}}.}$
This is λ times the difference quotient for the directional derivative of f with respect to u. Furthermore, taking the limit as h tends to zero is the same as taking the limit as k tends to zero, because h and k are multiples of each other. Therefore, Dv(f) = λDu(f). Because of this rescaling property, directional derivatives are frequently considered only for unit vectors.
If all the partial derivatives of f exist and are continuous at x, then they determine the directional derivative of f in the direction v by the formula:
${\displaystyle D_{\mathbf {v} }{f}({\boldsymbol {x}})=\sum _{j=1}^{n}v_{j}{\frac {\partial f}{\partial x_{j}}}.}$
This is a consequence of the definition of the total derivative. It follows that the directional derivative is linear in v, meaning that Dv + w(f) = Dv(f) + Dw(f).
The same definition also works when f is a function with values in Rm. The above definition is applied to each component of the vectors. In this case, the directional derivative is a vector in Rm.
### Total derivative, total differential and Jacobian matrix
When f is a function from an open subset of Rn to Rm, then the directional derivative of f in a chosen direction is the best linear approximation to f at that point and in that direction. But when n > 1, no single directional derivative can give a complete picture of the behavior of f. The total derivative gives a complete picture by considering all directions at once. That is, for any vector v starting at a, the linear approximation formula holds:
${\displaystyle f(\mathbf {a} +\mathbf {v} )\approx f(\mathbf {a} )+f'(\mathbf {a} )\mathbf {v} .}$
Just like the single-variable derivative, f ′(a) is chosen so that the error in this approximation is as small as possible.
If n and m are both one, then the derivative f ′(a) is a number and the expression f ′(a)v is the product of two numbers. But in higher dimensions, it is impossible for f ′(a) to be a number. If it were a number, then f ′(a)v would be a vector in Rn while the other terms would be vectors in Rm, and therefore the formula would not make sense. For the linear approximation formula to make sense, f ′(a) must be a function that sends vectors in Rn to vectors in Rm, and f ′(a)v must denote this function evaluated at v.
To determine what kind of function it is, notice that the linear approximation formula can be rewritten as
${\displaystyle f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} )\approx f'(\mathbf {a} )\mathbf {v} .}$
Notice that if we choose another vector w, then this approximate equation determines another approximate equation by substituting w for v. It determines a third approximate equation by substituting both w for v and a + v for a. By subtracting these two new equations, we get
${\displaystyle f(\mathbf {a} +\mathbf {v} +\mathbf {w} )-f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} +\mathbf {w} )+f(\mathbf {a} )\approx f'(\mathbf {a} +\mathbf {v} )\mathbf {w} -f'(\mathbf {a} )\mathbf {w} .}$
If we assume that v is small and that the derivative varies continuously in a, then f ′(a + v) is approximately equal to f ′(a), and therefore the right-hand side is approximately zero. The left-hand side can be rewritten in a different way using the linear approximation formula with v + w substituted for v. The linear approximation formula implies:
{\displaystyle {\begin{aligned}0&\approx f(\mathbf {a} +\mathbf {v} +\mathbf {w} )-f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} +\mathbf {w} )+f(\mathbf {a} )\\&=(f(\mathbf {a} +\mathbf {v} +\mathbf {w} )-f(\mathbf {a} ))-(f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} ))-(f(\mathbf {a} +\mathbf {w} )-f(\mathbf {a} ))\\&\approx f'(\mathbf {a} )(\mathbf {v} +\mathbf {w} )-f'(\mathbf {a} )\mathbf {v} -f'(\mathbf {a} )\mathbf {w} .\end{aligned}}}
This suggests that f ′(a) is a linear transformation from the vector space Rn to the vector space Rm. In fact, it is possible to make this a precise derivation by measuring the error in the approximations. Assume that the error in these linear approximation formula is bounded by a constant times ||v||, where the constant is independent of v but depends continuously on a. Then, after adding an appropriate error term, all of the above approximate equalities can be rephrased as inequalities. In particular, f ′(a) is a linear transformation up to a small error term. In the limit as v and w tend to zero, it must therefore be a linear transformation. Since we define the total derivative by taking a limit as v goes to zero, f ′(a) must be a linear transformation.
In one variable, the fact that the derivative is the best linear approximation is expressed by the fact that it is the limit of difference quotients. However, the usual difference quotient does not make sense in higher dimensions because it is not usually possible to divide vectors. In particular, the numerator and denominator of the difference quotient are not even in the same vector space: The numerator lies in the codomain Rm while the denominator lies in the domain Rn. Furthermore, the derivative is a linear transformation, a different type of object from both the numerator and denominator. To make precise the idea that f ′(a) is the best linear approximation, it is necessary to adapt a different formula for the one-variable derivative in which these problems disappear. If f : RR, then the usual definition of the derivative may be manipulated to show that the derivative of f at a is the unique number f ′(a) such that
${\displaystyle \lim _{h\to 0}{\frac {f(a+h)-(f(a)+f'(a)h)}{h}}=0.}$
This is equivalent to
${\displaystyle \lim _{h\to 0}{\frac {|f(a+h)-(f(a)+f'(a)h)|}{|h|}}=0}$
because the limit of a function tends to zero if and only if the limit of the absolute value of the function tends to zero. This last formula can be adapted to the many-variable situation by replacing the absolute values with norms.
The definition of the total derivative of f at a, therefore, is that it is the unique linear transformation f ′(a) : RnRm such that
${\displaystyle \lim _{\mathbf {h} \to 0}{\frac {\lVert f(\mathbf {a} +\mathbf {h} )-(f(\mathbf {a} )+f'(\mathbf {a} )\mathbf {h} )\rVert }{\lVert \mathbf {h} \rVert }}=0.}$
Here, h is a vector in Rn, so the norm in the denominator is the standard length on Rn. However, f′(a)h is a vector in Rm, and the norm in the numerator is the standard length on Rm. If v is a vector starting at a, then f ′(a)v is called the pushforward of v by f, and is sometimes written fv.
If the total derivative exists at a, then all the partial derivatives and directional derivatives of f exist at a, and for all v, f ′(a)v is the directional derivative of f in the direction v. If we write f using coordinate functions, so that f = (f1, f2, ..., fm), then the total derivative can be expressed using the partial derivatives as a matrix. This matrix is called the Jacobian matrix of f at a:
${\displaystyle f'(\mathbf {a} )=\operatorname {Jac} _{\mathbf {a} }=\left({\frac {\partial f_{i}}{\partial x_{j}}}\right)_{ij}.}$
The existence of the total derivative f′(a) is strictly stronger than the existence of all the partial derivatives, but if the partial derivatives exist and are continuous, then the total derivative exists, is given by the Jacobian, and depends continuously on a.
The definition of the total derivative subsumes the definition of the derivative in one variable. That is, if f is a real-valued function of a real variable, then the total derivative exists if and only if the usual derivative exists. The Jacobian matrix reduces to a 1×1 matrix whose only entry is the derivative f′(x). This 1×1 matrix satisfies the property that f(a + h) − (f(a) + f ′(a)h) is approximately zero. In other words,
${\displaystyle f(a+h)\approx f(a)+f'(a)h.}$
Up to changing variables, this is the statement that the function ${\displaystyle x\mapsto f(a)+f'(a)(x-a)}$ is the best linear approximation to f at a.
The total derivative of a function does not give another function in the same way as the one-variable case. This is because the total derivative of a multivariable function has to encode much more information than the derivative of a single-variable function. Instead, the total derivative gives a function from the tangent bundle of the source to the tangent bundle of the target.
The natural analog of second, third, and higher-order total derivatives is not a linear transformation, is not a function on the tangent bundle, and is not built by repeatedly taking the total derivative. The analog of a higher-order derivative, called a jet, cannot be a linear transformation because higher-order derivatives reflect subtle geometric information, such as concavity, which cannot be described in terms of linear data such as vectors. It cannot be a function on the tangent bundle because the tangent bundle only has room for the base space and the directional derivatives. Because jets capture higher-order information, they take as arguments additional coordinates representing higher-order changes in direction. The space determined by these additional coordinates is called the jet bundle. The relation between the total derivative and the partial derivatives of a function is paralleled in the relation between the kth order jet of a function and its partial derivatives of order less than or equal to k.
By repeatedly taking the total derivative, one obtains higher versions of the Fréchet derivative, specialized to Rp. The kth order total derivative may be interpreted as a map
${\displaystyle D^{k}f:\mathbb {R} ^{n}\to L^{k}(\mathbb {R} ^{n}\times \cdots \times \mathbb {R} ^{n},\mathbb {R} ^{m})}$
which takes a point x in Rn and assigns to it an element of the space of k-linear maps from Rn to Rm – the "best" (in a certain precise sense) k-linear approximation to f at that point. By precomposing it with the diagonal map Δ, x → (x, x), a generalized Taylor series may be begun as
{\displaystyle {\begin{aligned}f(\mathbf {x} )&\approx f(\mathbf {a} )+(Df)(\mathbf {x-a} )+\left(D^{2}f\right)(\Delta (\mathbf {x-a} ))+\cdots \\&=f(\mathbf {a} )+(Df)(\mathbf {x-a} )+\left(D^{2}f\right)(\mathbf {x-a} ,\mathbf {x-a} )+\cdots \\&=f(\mathbf {a} )+\sum _{i}(Df)_{i}(x_{i}-a_{i})+\sum _{j,k}\left(D^{2}f\right)_{jk}(x_{j}-a_{j})(x_{k}-a_{k})+\cdots \end{aligned}}}
where f(a) is identified with a constant function, xiai are the components of the vector xa, and (Df)i and (D2f)jk are the components of Df and D2f as linear transformations.
## Generalizations
The concept of a derivative can be extended to many other settings. The common thread is that the derivative of a function at a point serves as a linear approximation of the function at that point.
• An important generalization of the derivative concerns complex functions of complex variables, such as functions from (a domain in) the complex numbers C to C. The notion of the derivative of such a function is obtained by replacing real variables with complex variables in the definition. If C is identified with R2 by writing a complex number z as x + iy, then a differentiable function from C to C is certainly differentiable as a function from R2 to R2 (in the sense that its partial derivatives all exist), but the converse is not true in general: the complex derivative only exists if the real derivative is complex linear and this imposes relations between the partial derivatives called the Cauchy–Riemann equations – see holomorphic functions.
• Another generalization concerns functions between differentiable or smooth manifolds. Intuitively speaking such a manifold M is a space that can be approximated near each point x by a vector space called its tangent space: the prototypical example is a smooth surface in R3. The derivative (or differential) of a (differentiable) map f: MN between manifolds, at a point x in M, is then a linear map from the tangent space of M at x to the tangent space of N at f(x). The derivative function becomes a map between the tangent bundles of M and N. This definition is fundamental in differential geometry and has many uses – see pushforward (differential) and pullback (differential geometry).
• Differentiation can also be defined for maps between infinite dimensional vector spaces such as Banach spaces and Fréchet spaces. There is a generalization both of the directional derivative, called the Gateaux derivative, and of the differential, called the Fréchet derivative.
• One deficiency of the classical derivative is that very many functions are not differentiable. Nevertheless, there is a way of extending the notion of the derivative so that all continuous functions and many other functions can be differentiated using a concept known as the weak derivative. The idea is to embed the continuous functions in a larger space called the space of distributions and only require that a function is differentiable "on average".
• The properties of the derivative have inspired the introduction and study of many similar objects in algebra and topology — see, for example, differential algebra.
• The discrete equivalent of differentiation is finite differences. The study of differential calculus is unified with the calculus of finite differences in time scale calculus.
• Also see arithmetic derivative.
## History
Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Isaac Newton and Gottfried Leibniz independently discovered calculus in the mid-17th century. However, each inventor claimed that the other stole his work—in a bitter dispute that continued until the end of their lives.
## Notes
1. ^ Differential calculus, as discussed in this article, is a very well established mathematical discipline for which there are many sources. See Apostol 1967, Apostol 1969, and Spivak 1994.
2. ^ In the formulation of calculus in terms of limits, the du symbol has been assigned various meanings by various authors. Some authors do not assign a meaning to du by itself, but only as part of the symbol du/dx. Others define dx as an independent variable, and define du by du = dxf(x). In non-standard analysis du is defined as an infinitesimal. It is also interpreted as the exterior derivative of a function u. See differential (infinitesimal) for further information.
3. ^ This can also be expressed as the operation known as currying.
## References
1. ^ "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-09-15.
2. "List of Calculus and Analysis Symbols". Math Vault. 2020-05-11. Retrieved 2020-09-15.
3. Weisstein, Eric W. "Derivative". mathworld.wolfram.com. Retrieved 2020-09-15.
4. ^ Weisstein, Eric W. "Overdot." From MathWorld--A Wolfram Web Resource. "Archived copy". Archived from the original on 2015-09-05. Retrieved 2016-02-05.CS1 maint: archived copy as title (link)
5. ^ Weisstein, Eric W. "Double Dot." From MathWorld--A Wolfram Web Resource. "Archived copy". Archived from the original on 2016-03-03. Retrieved 2016-02-05.CS1 maint: archived copy as title (link)
6. ^ Spivak 1994, chapter 10.
7. "The meaning of the derivative - An approach to calculus". themathpage.com. Retrieved 2020-09-15.
8. ^ Banach, S. (1931), "Uber die Baire'sche Kategorie gewisser Funktionenmengen", Studia Math., 3 (3): 174–179, doi:10.4064/sm-3-1-174-179.. Cited by Hewitt, E; Stromberg, K (1963), Real and abstract analysis, Springer-Verlag, Theorem 17.8
9. ^ Apostol 1967, §4.18
10. ^ Manuscript of November 11, 1675 (Cajori vol. 2, page 204)
11. ^ "The Notation of Differentiation". MIT. 1998. Retrieved 24 October 2012.
12. ^ Evans, Lawrence (1999). Partial Differential Equations. American Mathematical Society. p. 63. ISBN 0-8218-0772-2.
13. ^ Kreyszig, Erwin (1991). Differential Geometry. New York: Dover. p. 1. ISBN 0-486-66721-9. | 2020-09-19 22:45:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 120, "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": 0.9349117875099182, "perplexity": 300.2005155144897}, "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/1600400192887.19/warc/CC-MAIN-20200919204805-20200919234805-00431.warc.gz"} |
https://space.stackexchange.com/tags/saturn/hot | Podcast #128: We chat with Kent C Dodds about why he loves React and discuss what life was like in the dark days before Git. Listen now.
# Tag Info
21
tl;dr: No chance, not even close! The escape velocity from the surface of a round (spherically symmetric) body is given by $$v_{esc} = \sqrt{\left(\frac{2 GM}{r_0} \right)},$$ showing that it is the $\frac{mass}{radius}$ ratio that's key here, not just the surface gravity given by $$a_{g} = -\frac{GM}{r_0^2}.$$ So since v_{esc} = \sqrt{a_g r_0}, $... 19 Could they be harvested? Sure. Could they be harvested in an economically profitable way? Probably not, for all the reasons you listed. There's a lot of unknowns to this question — while the chemistry they mention certainly makes sense, there's lots of stuff about the atmosphere we don't know. A harvesting ship would need to find exactly what ... 16 Cassini's INMS, the Ion Neutral Mass Spectrometer, is an in situ instrument that measures the neutral and plasma gas composition of what it ingests. It was intended for the measurement of Titan's atmosphere, Saturn's magnetosphere plasma, ring composition, and in fact the composition of icy satellite effluents. Here is a good presentation on the basics of ... 15 If we were going to send a probe into Saturn's atmosphere and were concerned about contamination of a potential ecosphere there, we would sterilize the probe first, e.g. with dry heat microbial reduction, to make sure that nothing viable was on the probe. Cassini has no protection from the entry heat like a probe would, and will be entering at an incredible ... 13 For Jovian and Saturnine moons, the simplest answer is no, at least not much more habitable than our own Moon is, because none of these moons have their own magnetic field or sufficient mass and their atmosphere would eventually thin out via the ionospheric hydrogen loss due to the solar wind. A bit more difficult answer is how many of these moons would be ... 12 No. Fermi estimate: Mimas' gravity is 0.064 m/s2, you need gravity to be about 1/20 of that to escape using a bike and a ramp (going by Deimos' surface gravity of 1/20 that of Mimas), lower still to escape by jumping : 11 For a Hohmann transfer to Saturn, I get 15.7 km/s for both burns. The transfer time is also a simple formula. I obtain roughly 6 years. Compare to the lunar ice. It is roughly 2.8 km/s to get to, and the trip time would be a few days, even from Low Earth Orbit. As suggested by the other answer, you could compare to Earth's surface. If we're using some ... 11 Can Dragonfly make it to one of Titan's Lakes? tl;dr: Yes! It could be doable in 2-3 years. According to your linked document Dragonfly will use a Multi-mission radioisotope thermoelectric generator or MMRTG: The MMRTG design incorporates PbTe/TAGS thermoelectric couples (from Teledyne Energy Systems), where the TAGS material is a material incorporating ... 10 Define "destroyed". It will no longer be able to maintain attitude control due to torque from the atmosphere at a radius of about 61,700 km. From there we will get no more data from the spacecraft since it can't point the antenna. From our point of view, at that moment, Cassini is buh bye. 10 It is not being overlooked. There are studies which are being carried out about the feasibility of life on Titan based on methane. It is true that Titan has all the necessary conditions for life: It is not in thermodynamic equilibrium It has more than enough of carbon based molecules It has a fluid environment (ethane/methane) for chemical reactions to ... 9 To answer the question, 'is Cassini running out of fuel': As you'd expect, NASA has monitored Cassini's fuel levels. In 2014 JPL published a study, 'Ensuring Cassini’s End-of-Mission Propellant Margins': With three years left and only 2.5% of its loaded bipropellant and 37% of its loaded monopropellant remaining, the Cassini project actively manages the ... 9 The solar flux at 1 AU (Earth's distance from the sun) is ~1361 W/m2. Flux falls off as the square of distance. Saturn, on average -- and thus Titan, on average -- is about 9.58 AU from the sun, so the falloff factor is 1 / 9.582 or 0.0109. Titan should average about 1361 x 0.0109 = 14.8 W/m2 when not shadowed by the gas giant. When Saturn is at its ... 8 3-5% of the speed of light would only be achievable for a really long duration mission. It just isn't efficient enough to do that kind of acceleration, and it takes a really long time to achieve. Okay, so how long would it take? The math is pretty easy, it's just speed/ acceleration. c=300,000,000 m/s, g= 9.8 m/s^2, so just find the speed, divide by 9.8, and ... 8 After reading this blog post and skimming through this paper1, it seems like it's just the stabilization of a standing waves caused by the difference in angular velocity. Basically, as you go outwards from the poles, there is an increase in the velocity as the system moves with a constant angular velocity. This creates some shear on the gases present ... 8 This is purely a speculation, as I have not found any official sources to confirm it, but I think they want to be able to view Saturn's pole during the fly by. The trigonometry works out at least: If you want to get a close-up of the polar regions with a lower inclination, like 40, 50 or 60 degrees, the altitude required is larger than the gap between ... 8 Let's turn the question on its head and see what exhaust velocity we need to if Titan's entire (mostly nitrogen) atmosphere were used as a propellant.$\Delta v = v_e log(m_i / m_f)$Wikipedia tells us that the atmosphere of Titan is about 1.19 times as massive as that of Earth so we get about 6.13e18 kg of atmosphere (propellant) in a total mass of about ... 7 Let's do a Fermi estimate: Rockets bring about 2-5% of their start mass to orbital velocity. To cancel out Titan's orbital velocity, you're looking at two orders of magnitude more fuel and oxidizer than Titan's mass. Earth's atmosphere weighs$10^{18}\$ kg, or 1/200,000 of Earth's total mass. Titan's is 1.5 times as dense, so if Titan's atmosphere were ...
6
No, you can't fly through the rings you can see without hitting lots and lots of dust-sized ice particles at high velocity. Your vehicle will not fare well. The ring material is not sparse in that sense. There is a wide distribution of particle sizes from boulders to dust. There's a lot more dust-sized, so that's what you need to worry about. (I wouldn'...
6
I'm going to interpret "surface" to mean Saturn's upper atmosphere. It was recently discovered that ring particles are raining down on Saturn. It’s raining on Saturn. Each second, the planet’s rings shed perhaps thousands of pounds of water ice, organic molecules, and other tiny particles into the gas giant’s clouds. This is caused by particles ...
5
Concise version from the pre-flyby media teleconference announcement: Cassini scientists are hopeful the flyby will provide insights into how much hydrothermal activity is occurring within Enceladus, and how this hot-water chemistry might impact the ocean’s potential habitability for simple forms of life. If the spacecraft’s ion and neutral mass ...
5
Well, it's the sugarcoating blowing off the Saturn's doughnut, of course! Not far from the truth, but jokes aside still, the color dichotomy of Iapetus is due to the darker half, the Cassini Regio, being a result of the moon's accumulation of the dust in the Saturn's largest, yet extremely tenuous, diffuse dust ring called the Phoebe ring depositing onto ...
5
I don't know about multiple mission proposals, but I was able to find a fairly well spelled-out plan so I'll answer based on that. The proposal is based on the IceMole melting probe but I don't know if there's a name for the proposal itself. The best source of data I've found is this PDF from 2014. It indicates that at least for this proposed mission they ...
5
It is modeled that the dunes must form from chips of Titan's bedrock. Whether those chips are primarily hydrocarbons or primarily water ice, depends on what Titan's crust is presumed to be composed of. There are varying models, depending on what is known of Titan (such as the moment of inertia), and one model with a reasonable level of support involves a ...
5
Any picture of Cassini in space is an artist's conception, probably a computer rendering, although the Saturn backdrop may come from real Cassini photographs. I had thought it possible that the Huygens probe carried by Cassini to Titan might have taken one or more pictures of Cassini as it departed, but it's not possible, as the camera was enclosed under ...
5
The raw images of Cassini can be viewed online in reverse order https://saturn.jpl.nasa.gov/galleries/raw-images?order=earth_date+desc&per_page=50&page=0
4
Although I and many other authors of the 2007 Enceladus study would love to see such a mission happen, that seems very unlikely in the near future. The first issue is that Enceladus orbits only four Saturn radii out, so is in a very deep gravitational well. Second, assuming the probe orbits with its pariapsis at Enceladus and its apoapsis at Titan, the ...
4
The rings became flat over time as the trillions of particles in them collided over and over, slowly causing their vectors (direction of motion) to average out until they were all aligned in the same direction. See this short video from Minute Physics. A flat disk rotating in one direction is the only arrangement of these particles that is stable. The ...
4
No. The surface of Saturn is not solid. Saturn is too hot to support solid ice, and not just because of solar radiation: Also like Jupiter, Saturn gives off almost twice as much energy as it receives from the Sun, because it has its own internal heat source, powered by the slow gravitational collapse that started when the planet first formed. Given ...
4
(This image shows that the rear side of Saturn ring is dark ) (this image shows the variation of rings brightness as the distance from earth the more closer the more brighter) The Saturn ring consist of 99% of water and the remaining impurities** The water is in the form of water ice since ice is a crystalline structure it reflection and refracts light ...
4
The effect you are seeing is simply wind, not a twist. The reason for the diagonal direction is that the smoke is moving horizontally while the rocket is moving upwards, like this: How can I be so sure about this? Well, viewing the other side of the rocket is not necessary: If the smoke was indeed ejected sideways, it would continue in the same direction ...
Only top voted, non community-wiki answers of a minimum length are eligible | 2019-11-22 17:26:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4888497591018677, "perplexity": 1519.5828642291594}, "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/1573496671411.14/warc/CC-MAIN-20191122171140-20191122200140-00534.warc.gz"} |
https://www.physicsforums.com/threads/integrate-2exp-2-jot.708184/ | # Integrate 2exp(2+jωt)
1. Sep 1, 2013
### hogrampage
I do not understand the following integral:
$\int^{\infty}_{0}2e^{2+jωt}dt$ = $\frac{j2e^{2}}{\omega}$
Why is it not ∞? Here are my steps:
Let u = 2+jωt, du = jωdt, dt = $\frac{1}{jω}$du = -$\frac{j}{ω}$du
$\int^{\infty}_{0}2e^{2+jωt}dt$
= -$\frac{2j}{ω}$$\int^{\infty}_{2}2e^{u}du$
= -$\frac{2j}{ω}$$\stackrel{lim}{h\rightarrow∞}$$\int^{h}_{2}2e^{u}du$
= -$\frac{2j}{ω}$$\stackrel{lim}{h\rightarrow∞}$($e^{h}-e^{2}$)
To me, this limit does not exist, so why is the answer $\frac{j2e^{2}}{\omega}$?
2. Sep 2, 2013
### CompuChip
You also need to change your integration limits, if $t \to \infty$ then $u \to 2 + j \infty$.
First thing I would do is separate $e^{2 + j \omega t} = e^2 e^{j \omega t}$ and pull the e² in front of the integration sign. The remaining integral can be solved using for example contour integration.
3. Sep 2, 2013
### jackmell
That's cus you don't know what $\omega$ is. Let's just look at:
$$\int_0^{\infty} e^{iwt}dt=\frac{1}{iw}e^{iwt}\biggr|_0^{\infty}$$
Let $\omega=a+bi$ then we have
$$\frac{1}{i(a+bi)}e^{i(a+bi)t}\biggr|_0^{\infty}$$
Now, for what values of a and b will that expression converge?
Last edited: Sep 2, 2013
4. Sep 2, 2013
### hogrampage
a, b < 0 which means ω < 0. So:
$\frac{1}{iw}e^{iwt}\biggr|_0^{\infty} = -\frac{1}{iω}$ for ω < 0.
From that, the complete answer to the original integral is:
$-\frac{2e^{2}}{jω} = \frac{j2e^{2}}{ω}$ for ω < 0.
Thanks for the help! If I made any mistakes above, let me know.
Last edited: Sep 2, 2013
5. Sep 2, 2013
### jackmell
Your analysis of a and b is not correct. for w=a+bi, in order for the integral to converge, b has to be less than zero. a can be any real number. Go through that to make sure you understand it. | 2018-02-20 10:22:09 | {"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": 2, "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": 0.9526685476303101, "perplexity": 1103.2608019069316}, "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/1518891812932.26/warc/CC-MAIN-20180220090216-20180220110216-00049.warc.gz"} |
https://discuss.codechef.com/questions/85985/cl16ba-editorial | ×
# CL16BA - Editorial
Author: Subham Das
Tester: Soumik Sarkar
Editorialist: Soumik Sarkar
CAKEWALK
# PREREQUISITES:
Basic knowledge of arrays or lists.
# PROBLEM:
Determine the largest value in an array and compare it to a given value.
# EXPLANATION:
An array of N values is provided along with a value M. The maximum value in the array needs to be determined, let's call it maxval. The simple algorithm given below can be used to obtain maxval.
maxval = 0
for each val in N
if maxval < val
maxval = val
Now we have obtained maxval. The next step we perform is to check whether maxval is less than M. If it is less than M, we output "RESIGN" else we output the value of maxval.
if maxval < M
print "RESIGN"
else
print maxval
Complexity of this approach is $\mathcal{O}(N)$.
# AUTHOR'S AND TESTER'S SOLUTIONS:
Author's solution can be found here.
Tester's solution can be found here.
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• to add a line break simply add two spaces to where you would like the new line to be.
• basic HTML tags are also supported
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question asked: 14 Oct '16, 22:48
question was seen: 462 times
last updated: 07 Jun '17, 17:28 | 2019-03-25 07:47:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.4237661063671112, "perplexity": 9862.019767072077}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203842.71/warc/CC-MAIN-20190325072024-20190325094024-00410.warc.gz"} |
https://www.physicsforums.com/threads/race-car-velocity-problem.63020/ | # Race car velocity problem
A race car moves such that its position fits the relationship
x(t) = (4.4 m/s) t + (0.91 m/s^3) t^3 (yes cubed)
where x is measured in meters and t in seconds.Approximate the instantaneous velocity at t = 2.6 s, using a time interval of 0.2 s. Answer
in units of m/s.
i am just stuck and dont even know where to start
Related Introductory Physics Homework Help News on Phys.org
ok, the velocity is
$$\frac{\Delta x} {\Delta t}$$
now the question come down to what is delta x and delta t
hope you can come out the answer
dextercioby
Homework Helper
Take the time origin $t_{0}$ to be 0.
Daniel.
but what is Xo and X?
dextercioby
Homework Helper
Kay.
$$x_{0}=x(t_{0})=x(0)$$
$$x=x(2.6)$$
Daniel.
ok i got all that, but what is still confusing me is the "time interval" what do i do with that
dextercioby said:
Kay.
$$x_{0}=x(t_{0})=x(0)$$
$$x=x(2.6)$$
Daniel.
what is the time interval for??? | 2020-10-27 09:45:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4749355912208557, "perplexity": 1906.8714961363294}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107893845.76/warc/CC-MAIN-20201027082056-20201027112056-00280.warc.gz"} |
https://math.stackexchange.com/questions/3913333/symmetries-of-a-circle-and-the-o-2-group-why-there-are-both-rotations-and-ref | # symmetries of a circle and the $O_2$ group: why there are both rotations AND reflections, and not either rotations OR reflections?(geometric view)
I'm reading "Camila Jordan, David Jordan Groups - Modular Mathematics series", section 1.2 Symmetries of a circle.
In this section we consider the unit circle. A typical point on the unit circle has coordinates $$(\cos\theta, \sin\theta)$$ for some $$\theta$$. In contrast to the square, this circle has infinitely many symmetries. They are all the rotations $$rot_{\theta}$$ together with all the reflections $$ref_{\theta}$$. As for the square, these symmetries form a group, the group of symmetries of the circle, but this time the group is infinite. The neutral element is $$rot_0$$. Each rotation $$rot_{\theta}$$ has the rotation $$rot_{-\theta}$$ as inverse while each reflection $$ref_{\theta}$$ is its own inverse. This group is known as orthogonal group $$O_2$$. The group $$O_2$$ acts on the circle by rotations and reflections. Now let us consider the stabilizer of an arbitrary point $$P=(\cos\theta, \sin\theta)$$ on the circle. Which elements of the group $$O_2$$ send $$P$$ to itself? There are just two, namely $$rot_0$$ and $$ref_{2\theta}$$
My question is: why does this $$O_2$$ consist of both rotations and reflections? As if we take any arc on a circle, we can cover all the circle by rotating this arc. Why we would need reflections then?
• It is not a question of needing reflections. The authors are just observing that reflections exist and are symmetries of the circle. And the fact that reflections fix two points of the circle and rotations fix none (except the identity, which fixes everything) shows that reflections are distinct from rotations. Nov 18 '20 at 22:33
• Well, as I said the set of reflections generates the whole group $\mathcal{O}(2)$. However all nontrivial rotations (which are elements of $\mathcal{O}(2)$!) are not reflections. It is an astonishing fact that given any isometry of the circle it is either a reflection or a rotation. Nov 25 '20 at 12:45
• ok, so if the set of reflections generates the whole group $O_2$, even though rotations are not reflections(but any rotation can be acquired as a combination of 2 reflections), why we need rotations in the group itself? Aren't the rotations just aliases for combinations of reflections? Nov 25 '20 at 12:59 | 2021-12-07 22:51:04 | {"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": 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": 16, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8243464827537537, "perplexity": 166.33971848275138}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363418.83/warc/CC-MAIN-20211207201422-20211207231422-00282.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-10-radical-expressions-and-equations-10-2-simplifying-radicals-practice-and-problem-solving-exercises-page-611/70 | ## Algebra 1
$x= -3 ± 3\sqrt 2$
The trinomial cannot be factored so we use the quadratic formula to calculate the x. $x= \frac{-b ± \sqrt (b^{2} - 4ac)}{2a}$ $x^{2} + 6x -9$ In this trinomial a = 1, b= 6 and c= -9 $x= \frac{-(6) ± \sqrt (6^{2} - 4(1)(-9))}{2(1)}$ $x= \frac{-6 ± \sqrt (36 + 36)}{2}$ $x= \frac{-6 ± \sqrt (72)}{2}$ Square root of 72 is $6\sqrt 2$ because the factors of 72 are 36 and 2 and 36 is a perfect square of 6. $x= \frac{-6 ± 6\sqrt 2}{2}$ We simplify the numbers to get the final x value. $x= -3 ± 3\sqrt 2$ | 2018-10-23 03:42:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.597569465637207, "perplexity": 173.99287832147925}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583516003.73/warc/CC-MAIN-20181023023542-20181023045042-00149.warc.gz"} |
https://www.eclipse.org/n4js/spec/classifiers.html | ## 5. Classifiers
### 5.1. N4JS Specific Classifiers
N4JS provides three new metatypes: class, interface, and enums. In this section we describe classes and interfaces. These metatypes, called classifiers, share some common properties which are described before type specific properties are outlined in the following sections.
All of these metatypes can be marked with type access modifiers:
enum N4JSTypeAccessModifier: project | public;
#### 5.1.1. Properties
Properties defined by syntactic elements:
annotations
Arbitrary annotations, see Annotations for details.
accessModifier
N4JS type access modifier: public, or project; public can be combined with @Internal; if export is true the default is else the default is private.
name
The simple name of a classifier. If the classifier is defined by an anonymous class expression, an artificial but unique name is created. The name needs to be a valid identifier, see Valid Names.
typePars
Collection of type parameters of a generic classifier; empty by default.
ownedMembers
Collection of owned members, i.e. methods and fields defined directly in the classifier and, if present, the explicitly defined constructor. Depending on the concrete classifier, additional constraints are defined.
typingStrategy
The definition-site typing strategy. By default nominal typing is used. See Structural Typing for details.
The following pseudo properties are defined via annotations:
export
Boolean property set to true if the export modifier is set. If value is true, the classifier may be accessible outside the project.
final
Boolean property which is set to final if annotation @Final is set. Also see Final Modifier
deprecated
Boolean property set to true if annotation @Deprecated is set.
Version 0.4, not implemented in Version 0.3
We additionally define the following pseudo properties:
acc
Type access modifier as described in Accessibility of Types, Top-Level Variables and Function Declarations, it is the aggregated value of the accessModifier and the export property.
owned{Fields|Methods|Getters|Setters|Accessors}
Filters ownedMembers by metatype, short for
$x\in ownedMembers,\mu \left(x\right)=\text{Field}$ etc.
members
Reflexive transitive closure of all members of a classifier and its super classifiers, see Common Semantics of Classifiers on how this is calculated.
fields|methods|getters|setters|accessors
Filters members by metatype, short for
$x\in members,\mu \left(x\right)=\text{Field}$ etc.
superClassifiers
Classes and interface may extend or implement classes or interfaces. Any class or interface extended or interface implemented is called super classifier. We distinguish the directly subtyped classifiers and from the transitive closure of supertypes $superClassifier{s}^{*}$
#### 5.1.2. Common Semantics of Classifiers
Req. IDE-42: Subtyping of Classifiers (ver. 1)
For a given type C, and supertypes $superClassifiers=\left\{{S}_{1},.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}},{S}_{n}\right\}$ directly subtyped C, the following constraints must be true:
1. The supertypes must be accessible to the subtype:
${S}_{1},.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}},{S}_{n}$ must be accessible to C.
2. All type parameters of the direct supertypes have to be bound by type arguments in the subtype and the type arguments have to be substitutable types of the type parameters.
$\forall 0
$\exists A\in {C}_{.}typeArgs:bind\left(A,P\right)\wedge A.upperBound<:P.upperBound$
3. Wildcards may not be used as type argument when binding a supertype’s type parameters.
4. A classifier cannot be directly subtyped directly multiple times:
$\forall {S}_{i},{S}_{j}\left(i,j\in \left\{1..n\right\}\right):{S}_{i}={S}_{j}\to i=j$
In order to simplify the following constraints, we use the pseudo property $members$ to refer to all members of a classifier. This includes all members directly declared by the classifier itself, i.e. the $ownedMember$, and all members inherited from its super classifiers. The concrete mechanisms for inheriting a member are different and further constraint (cf. Redefinition of Members). A classifier only inherits its members from its direct supertypes, although the supertypes may contains members inherited from their supertypes.
#### 5.1.3. Classes
##### 5.1.3.1. Definition of Classes
Classes are either declared with a class declaration on top level, or they can be used as anonymous classes in expressions. The latter may have a name, which may be used for error messages and reflection.
At the current stage, class expressions are effectively disabled at least until the semantics of them are finalized in ECMAScript 6.
In N4JS (as in many other languages) multi-inheritance of classes is not supported. Although the diamond problem (of functions being defined in both superclasses) could be solved via union and intersection types, this would lead to problems when calling these super implementations. This is particularly an issue due to JavaScript not supporting multiple prototypes.[27] Interfaces, however, allow for multi-inheritance. Since the former can also define functions with bodies, this is not a hard restriction.
###### 5.1.3.1.1. Syntax
Syntax N4 Class Declaration and Expression
N4ClassDeclaration <Yield>:
=>(
{N4ClassDeclaration}
annotations+=Annotation*
(declaredModifiers+=N4Modifier)*
'class' typingStrategy=TypingStrategyDefSiteOperator? name=BindingIdentifier<Yield>?
)
TypeVariables?
ClassExtendsClause<Yield>?
Members<Yield>
;
N4ClassExpression <Yield>:
{N4ClassExpression}
'class' name=BindingIdentifier<Yield>?
ClassExtendsClause<Yield>?
Members<Yield>;
fragment ClassExtendsClause <Yield>*:
'extends' (
=>superClassRef=ParameterizedTypeRefNominal ('implements' ClassImplementsList)?
| superClassExpression=LeftHandSideExpression<Yield>
)
| 'implements' ClassImplementsList
;
fragment ClassImplementsList*:
implementedInterfaceRefs+=ParameterizedTypeRefNominal
(',' implementedInterfaceRefs+=ParameterizedTypeRefNominal)*
;
fragment Members <Yield>*:
'{'
ownedMembers+=N4MemberDeclaration<Yield>*
'}'
;
###### 5.1.3.1.2. Properties
These are the properties of class, which can be specified by the user: Syntax N4 Class Declaration and Expression
abstract
Boolean flag indicating whether class may be instantiable; default is false, see Abstract Classes.
external
Boolean flag indicating whether class is a declaration without implementation or with an external (non-N4JS) implementation; default is false, see Definition Site Structural Typing.
defStructural
Boolean flag indicating whether subtype relation uses nominal or structural typing, see Definition Site Structural Typing for details.
superType/sup
The type referenced by $superType$ is called direct superclass of a class, and vice versa the class is a direct subclass of $superType$. Instead of $superType$, we sometimes simply write $sup$. The derived set $su{p}^{+}$ is defined as the transitive closures of all direct and indirect superclasses of a class. If no supertype is explicitly stated, classes are derived from N4Object.
implementedInterfaces/interfaces
Collection of interfaces directly implemented by the class; empty by default. Instead of implementedInterfaces, we simply write interfaces.
ownedCtor
Explicit constructor of a class (if any), see Constructor and Classifier Type.
And we additionally define the following pseudo properties:
ctor
Explicit or implicit constructor of a class, see Constructor and Classifier Type.
fields
Further derived properties for retrieving all methods (property $methods$), fields (property $fields$), static members (property $staticOwnedMembers$), etc. can easily be added by filtering properties $members$ or $ownedMembers$.
###### 5.1.3.1.3. Type Inference
The type of a class declaration or class expression C (i.e., a class definition in general) is of type constructor{C} if it is not abstract, that is if it can be instantiated. If it is abstract, the type of the definition simply is type{C}:
$\begin{array}{c}\frac{¬C.abstract}{\Gamma ⊢C:\text{constructor}\left\{C\right\}}\\ \frac{C.abstract}{\Gamma ⊢C:\text{type}\left\{C\right\}}\end{array}$
Req. IDE-43: Structural and Nominal Supertypes (ver. 1)
The type of supertypes and implemented interfaces is always the nominal type, even if the supertype is declared structurally.
##### 5.1.3.2. Semantics
This section deals with the (more or less) type-independent constraints on classes.
Class expressions are not fully supported at the moment.
The reflexive transitive closure of members of a class is indirectly defined by the override and implementation constraints defined in Redefinition of Members.
Note that since overloading is forbidden, the following constraint is true [28]:
$\forall {m}_{1},{m}_{2}\in members:{m}_{1}.name={m}_{2}.name⇔{m}_{1}={m}_{2}\vee accessorPair\left({m}_{1},{m}_{2}\right)$
Remarks: Class and method definition is quite similar to the proposed ECMAScript version 6 draft [ECMA15a(p.S13.5)], except that an N4 class and members may contain
• annotations, abstract and access modifiers
• fields
• types
• implemented interfaces
Note that even static is used in ECMAScript 6.
Mixing in members (i.e. interface’s methods with default implementation or fields) is similar to mixing in members from roles as defined in [Dart13a(p.S9.1)]. It is also similar to default implementations in Java 8 [Gosling15a]. In Java, however, more constraints exist, (for example, methods of interfaces must be public).
Example 30. Simple Class
This first example shows a very simple class with a field, a constructor and a method.
class C {
data: any;
constructor(data: any) {
this.data = data;
}
foo(): void { }
}
Example 31. Extend and implement
The following example demonstrate how a class can extend a superclass and implement an interface.
interface I {
foo(): void
}
class C{}
class X extends C implements I {
@Override
foo(): void {}
}
A class C is a subtype of another classifier S (which can be a class or interface) if the other classifier S is (transitively) contained in the supertypes (superclasses or implemented interfaces) of the class:
Req. IDE-44: Implicit Supertype of Classes (ver. 1)
1. The implicit supertype of all classes is N4Object. All classes with no explicit supertype are inherited from N4Object.
2. If the supertype is explicitly set to Object, then the class is not derived from N4Object. Meta-information is created similar to an N4Object-derived class. Usually, there is no reason to explicitly derive a class from Object.
3. External classes are implicitly derived from , unless they are annotated with @N4JS(cf.External Declarations).
##### 5.1.3.3. Final Modifier
Extensibility refers to whether a given classifier can be subtyped. Accessibility is a prerequisite for extensibility. If a type cannot be seen, it cannot be subclassed. The only modifier influencing the extensibility directly is the annotation @Final, which prevents all subtyping. The following table shows how to prevent other projects or vendors from subtyping by also restricting the accessibility of the constructor:
Table 6. Extensibility of Types
Type C Settings Subclassed in
Project
Vendor
World
C.final
no
no
no
C.ctor.accessModifier=\lenum{project}
yes
no
no
C.ctor.accessModifier=\lenum{public@Internal}
yes
yes
no
Since interfaces are always to be implemented, they must not be declared final.
##### 5.1.3.4. Abstract Classes
A class with modifier abstract is called an abstract class and has its $abstract$ property set to true. Other classes are called concrete classes.
Req. IDE-45: Abstract Class (ver. 1)
1. A class C must be declared abstract if it owns or inherits one or more abstract members and neither C nor any interfaces implemented by C implements these members. A concrete class has to, therefore, implement all abstract members of its superclasses’ implemented interfaces. Note that a class may implement fields with field accessors and vice versa.
2. An abstract class may not be instantiated.
3. An abstract class cannot be set to final (with annotation @Final).
Req. IDE-46: Abstract Member (ver. 1)
1. A member declared as abstract must not have a method body (in contrary a method not declared as abstract have to have a method body).
2. Only methods, getters and setters can be declared as abstract (fields cannot be abstract).
3. It is not possible to inherit from an abstract class which contains abstract members which are not visible in the subclass.
4. An abstract member must not be set to final (with annotation @Final).
5. Static members must not be declared abstract.
Remarks:
• We decided to disallow abstract static members, since we cannot guarantee that a static members is not accessed in all cases
• Only static members can override static members and only instance members can override other instance members of course.
• An abstract member must not be declared in a final class (i.e. a class annotated with @Final). This is not explicitly defined as constraint in [Req-IDE-46] since abstract classes must not defined final anyway. We also do not produce error message for abstract members in final classes since these errors would be consequential errors.
Abstract members might be declared private, as they can be accessed from within the module. This is to be changed in order to be aligned with TypeScript, cf. #1221. However we also want to add class expressions — and then the abstract members may be accessed (and overridden) in nested classes created by means of class expressions.
##### 5.1.3.5. Non-Instantiable Classes
To make a class non-instantiable outside a defining compilation unit, i.e. disallow creation of instances for this class, simply declare the constructor as private. This can be used for singletons.
##### 5.1.3.6. Superclass
Req. IDE-47: Superclass (ver. 1)
For a class C with a supertype $S=C.sup$, the following constraints must hold;
• $C.sup$ must reference a class declaration S
• S must be be extendable in the project of C
• $C\notin C.su{p}^{+}$
• All abstract members in S must be accessible from C:
$\forall M\in S.members:M.abstract⇒$
M is accessible from C.
Note that M need not be an owned member of S and that this constraint applies even if C is abstract).
All members of superclasses become members of a class. This is true even if the owning classes are not directly accessible to a class. The member-specific access control is not changed.
#### 5.1.4. Interfaces
##### 5.1.4.1. Definition of Interfaces
###### 5.1.4.1.1. Syntax
Syntax N4 Interface Declaration
N4InterfaceDeclaration <Yield>:
=> (
{N4InterfaceDeclaration}
annotations+=Annotation*
(declaredModifiers+=N4Modifier)*
'interface' typingStrategy=TypingStrategyDefSiteOperator? name=BindingIdentifier<Yield>?
)
TypeVariables?
InterfaceImplementsList?
Members<Yield>
;
fragment InterfaceImplementsList*:
'implements' superInterfaceRefs+=ParameterizedTypeRefNominal
(',' superInterfaceRefs+=ParameterizedTypeRefNominal)*
;
###### 5.1.4.1.2. Properties
These are the additional properties of interfaces, which can be specified by the user:
superInterfaces
Collection of interfaces extended by this interface; empty by default. Instead of superInterfaces, we simply write interfaces.
###### 5.1.4.1.3. Type Inference
The type of an interface declaration I is of type type{I}:
$\frac{}{\Gamma ⊢I:\text{type}\left\{I\right\}}$
###### 5.1.4.1.4. Semantics
Interfaces are used to describe the public API of a classifier. The main requirement is that the instance of an interface, which must be an instance of a class since interfaces cannot have instances, provides all members declared in the interface. Thus, a (concrete) class implementing an interface must provide implementations for all the fields, methods, getters and setters of the interface (otherwise it the class must be declared abstract). The implementations have to be provided either directly in the class itself, through a superclass, or by the interface if the member has a default implementation.
A field declaration in an interface denotes that all implementing classes can either provide a field of the same name and the same(!) type or corresponding field accessors. If no such members are defined in the class or a (transitive) superclass, the field is mixed in from the interface automatically. This is also true for the initializer of the field.
All instance methods, getters and setters declared in an interface are implicitly abstract if they do not provide a default implementation. The modifier abstract is not required, therefore, in the source code. The following constraints apply:
Req. IDE-48: Interfaces (ver. 1)
For any interface I, the following must hold:
1. Interfaces may not be instantiated.
2. Interfaces cannot be set to final (with annotation @Final): $¬I.final$.
3. Members of an interface must not be declared private. The default access modifier in interfaces is the the type’s visibility or project, if the type’s visibility is private.
4. Members of an interface, except methods, must not be declared @Final:
$\forall m\in I.member:m.final\to m\in I.methods$
not allowing field accessors to be declared final was a deliberate decision, because it would complicate the internal handling of member redefinition; might be reconsidered at a later time
5. The literal may not be used in the initializer expression of a field of an interface.
This restriction is required, because the order of implementation of these fields in an implementing class cannot be guaranteed. This applies to both instance and static fields in interfaces, but in case of static fields, this is also disallowed due to Static Members of Interfaces.
It is possible to declare members in interfaces with a smaller visibility as the interface itself. In that case, clients of the interface may be able to use the interface but not to implement it.
In order to simplify modeling of runtime types, such as elements, interfaces do not only support the notation of static methods but constant data fields as well. Since IDL [OMG14a] is used to describe these elements in specifications (and mapped to JavaScript via rules described in [W3C12a]) constant data fields are an often-used technique there and they can be modeled in N4JS 1:1.
As specified in [Req-IDE-56], interfaces cannot contain a constructor i.e.
$\forall m\in I.ownedMethods:m.name{\ne }^{\text{'}}constructo{r}^{\text{'}}$.
Example 32. Simple Interfaces
The following example shows the syntax for defining interfaces. The second interface extends the first one. Note that methods are implicitly defined abstract in interfaces.
interface I {
foo(): void
}
interface I2 extends I {
someText: string;
bar(): void
}
If a classifier C implements an interface I, we say I is implemented by C. If C redefines members declared in I, we say that these members are implemented by C. Members not redefined by C but with a default implementations are mixed in or consumed by C. We all cases we call C the implementor.
Besides the general constraints described in Common Semantics of Classifiers, the following constraints must hold for extending or implementing interfaces:
Req. IDE-49: Extending Interfaces (ver. 1)
For a given type I, and $\left\{{I}_{1},.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}},{I}_{n}\right\}$ directly extended by I, the following constraints must be true:
1. Only interfaces can extend interfaces: $I,{I}_{1},.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}},{I}_{n}$ must be interfaces.
2. An interface may not directly extend the same interface more than once:
${I}_{i}={I}_{j}\to i=j$ for any $i,j\in \left\{1.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}n\right\}$.
3. An interface may (indirectly) extend the same interface $J$ more than once only if
1. $J$ is not parameterized, or
2. in all cases $J$ is extended with the same type arguments for all invariant type parameters.
Note that for type parameters of $J$ that are declared covariant or contravariant on definition site, different type arguments may be used.
4. All abstract members in ${I}_{i}$, $i\in \left\{1,.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}},n\right\}$, must be accessible from I:
$\forall i\in \left\{1,.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}},n\right\}:M\in {I}_{i}.members\wedge M.abstract\to$ M is accessible from I.
Note that M need not be an owned member of ${I}_{i}$.
Req. IDE-50: Implementing Interfaces (ver. 1)
For a given type C, and $\left\{{I}_{1},.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}},{I}_{n}\right\}$ directly implemented by C, the following constraints must be true:
1. Only classes can implement interfaces: C must be a Class.
2. A class can only implement interfaces: ${I}_{1},.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}},{I}_{n}$ must be interfaces.
3. A class may not directly implement the same interface more than once:
${I}_{i}={I}_{j}⇒i=j$ for any $i,j\in \left\{1,.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}},n\right\}$.
4. A class may (indirectly) implement the same interface $J$ more than once only if
1. $J$ is not parameterized, or
2. in all cases $J$ is implemented with the same type arguments for all invariant type parameters.
Note that for type parameters of $J$ that are declared covariant or contravariant on definition site, different type arguments may be used.
5. All abstract members in ${I}_{i}$, $i\in \left\{1,.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}},n\right\}$, must be accessible from C:
$\forall i\in \left\{1,.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}},n\right\}:M\in {I}_{i}.members\wedge M.abstract\to$ M is accessible from C.
Note that M need not be an owned member of ${I}_{i}$.
For default methods in interfaces, see Default Methods in Interfaces.
#### 5.1.5. Generic Classifiers
Classifiers can be declared generic by defining a type parameter via type-param.
Definition: Generic Classifiers
A generic classifier is a classifier with at least one type parameter. That is, a given classifier C is generic if and only if $|C.typePars|\ge 1$.
If a classifier does not define any type parameters, it is not generic, even if its superclass or any implemented interface is generic.
The format of the type parameter expression is described in Parameterized Types. The type variable defined by the type parameter’s type expression can be used just like a normal type inside the class definition.
If using a generic classifier as type of a variable, it may be parameterized. This is usually done via a type expression (cf. Parameterized Types) or via typearg in case of supertypes. If a generic classifier defines multiple type variables, these variables are bound in the order of their definition. In any case, all type variables have to be bound. That means in particular that raw types are not allowed. (cf Parameterized Types for details).
If a generic classifier is used as super classifier, the type arguments can be type variables. Note that the type variable of the super classifier is not lifted, that is to say that all type variables are to be explicitly bound in the type references used in the extend, with, or implements section using typearg. If a type variable is used in typearg to bound a type variable of a type parameter, it has to fulfil possible type constraints (upper/lower bound) specified in the type parameter.
Example 33. Generic Type Definition and Usage as Type of Variable
This example demonstrates how to define a generic type and how to refer to it in a variable definition.
export class Container<T> {
private item: T;
getItem(): T {
return this.item;
}
setItem(item: T): void {
this.item = item;
}
}
This type can now be used as a type of a variable as follows
import Container from "p/Container"
var stringContainer: Container<string> = new Container<string>();
stringContainer.setItem("Hello");
var s: string = stringContainer.getItem();
In line 3, the type variable T of the generic class Container is bound to string.
Example 34. Binding of type variables with multiple types
For a given generic class G
class A{}
class B{}
class C extends A{}
class G<S, T extends A, U extends B> {
}
the variable definition
var x: G<Number,C,B>;
would bind the type variables as follows:
S Number Bound by first type argument, no bound constraints defined for S. T C Bound by second type argument, C must be a subtype of in order to fulfill the type constraint. U B Bound by third type argument, extends is reflexive, that is B fulfills the type constraint.
For a given generic superclass SuperClass
class SuperClass<S, T extends A, U extends B> {};
and a generic subclass SubClass
class SubClass<X extends A> extends SuperClass<Number, X, B> {..};
the variable definition
var s: SubClass<C>;
would bind the type variables as follows:
TypeVariable Bound to Explanation
SuperClass.S
Number
Type variable s of supertype SuperClass is bound to Number.
SuperClass.T
SubClass.X=C
Type variable T of supertype SuperClass is bound to type variable X of SubClass. It gets then indirectly bound to C as specified by the type argument of the variable definition.
SuperClass.U
B
Type variable U of supertype SuperClass is auto-bound to C as no explicit binding for the third type variable is specified.
SubClass.X
C
Bound by first type argument specified in variable definition.
#### 5.1.6. Definition-Site Variance
In addition to use-site declaration of variance in the form of Java-like wildcards, N4JS provides support for definition-site declaration of variance as known from languages such as C# and Scala.
The variance of a parameterized type states how its subtyping relates to its type arguments’ subtyping. For example, given a parameterized type G<T> and plain types A and B, we know
• if G is covariant w.r.t. its parameter T, then
$\text{B}\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}\text{A}\to \text{G}<\text{B}>\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}\text{G}<\text{A}>$
• if G is contravariant w.r.t. its parameter T, then
$\text{B}\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}\text{A}\to \text{G}<\text{A}>\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}\text{G}<\text{B}>$
• if G is invariant w.r.t. its parameter T, then
$\text{B}\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}\text{A}\to \text{G}<\text{A}>\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}\text{G}<\text{B}>$
$\text{B}\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}\text{A}\to \text{G}<\text{A}>\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}\text{G}<\text{B}>$
Note that variance is declared per type parameter, so a single parameterized type with more than one type parameter may be, for example, covariant w.r.t. one type parameter and contravariant w.r.t. another.
Strictly speaking, a type parameter/variable itself is not co- or contravariant;
however, for the sake of simplicity we say T is covariant as a short form for G is covariant with respect to its type parameter T (for contravariant and invariant accordingly).
To declare the variance of a parameterized classifier on definition site, simply add keyword in or out before the corresponding type parameter:
class ReadOnlyList<out T> { // covariance
// ...
}
interface Consumer<in T> { // contravariance
// ...
}
In such cases, the following constraints apply.
Req. IDE-174: Definition-Site Variance (ver. 1)
Given a parameterized type with a type parameter , the following must hold:
1. T may only appear in variance-compatible positions:
1. if T is declared on definition site to be covariant, then it may only appear in covariant positions within the type’s non-private member declarations.
2. if T is declared on definition site to be contravariant, then it may only appear in contravariant positions within the type’s non-private member declarations.
3. if T is invariant, i.e. neither declared covariant nor declared contravariant on definition site, then it may appear in any position (where type variables are allowed).
Thus, no restrictions apply within the declaration of private members and within the body of field accessors and methods.
2. definition-site variance may not be combined with incompatible use-site variance:
1. if T is declared on definition site to be covariant, then no wildcard with a lower bound may be provided as type argument for T.
2. if T is declared on definition site to be contravariant, then no wildcard with an upper bound may be provided as type argument for T.
3. if T is invariant, i.e. neither declared covariant nor declared contravariant on definition site, then any kind of wildcard may be provided as type argument.
Unbounded wildcards are allowed in all cases.
Example 35. Use-site declaration of variance
For illustration purposes, let’s compare use-site and definition-site declaration of variance. Since use-site variance is more familiar to the Java developer, we start with this flavor.
class Person {
name: string;
}
class Employee extends Person {}
interface List<T> {
}
function getNameOfFirstPerson(list: List<? extends Person>): string {
}
Function getNameOfFirstPerson below takes a list and returns the name of the first person in the list. Since it never adds new elements to the given list, it could accept Lists of any subtype of Person, for example a List<Employee>. To allow this, its formal parameter has a type of List<? extends Person> instead of List<Person>. Such use-site variance is useful whenever an invariant type, like List above, is being used in a way such that it can be treated as if it were co- or contravariant.
Sometimes, however, we are dealing with types that are inherently covariant or contravariant, for example an ImmutableList from which we can only read elements would be covariant. In such a case, use-site declaration of variance is tedious and error-prone: we would have to declare the variance wherever the type is being used and would have to make sure not to forget the declaration or otherwise limit the flexibility and reusability of the code (for example, in the above code we could not call getNameOfFirstPerson with a List<Employee>).
The solution is to declare the variance on declaration site, as in the following code sample:
interface ImmutableList<out T> {
// add(elem: T) // error: such a method would now be disallowed
}
function getNameOfFirstPerson2(list: ImmutableList<Person>): string {
}
Now we can invoke getNameOfFirstPerson2 with a List<Employee> even though the implementor of getNameOfFirstPerson2 did not add a use-site declaration of covariance, because the type ImmutableList is declared to be covariant with respect to its parameter T, and this applies globally throughout the program.
### 5.2. Members
A member is either a method (which may be a special constructor function), a data field, or a getter or a setter. The latter two implicitly define an accessor field. Similar to object literals, there must be no data field with the same name as a getter or setter.
(overriding, implementation and consumption) is described in Redefinition of Members.
#### 5.2.1. Syntax
Syntax N4JS member access modifier
enum N4JSMemberAccessModifier: private | project | protected | public;
N4MemberDeclaration: N4MethodDeclaration | N4FieldDeclaration | N4GetterDeclaration | N4SetterDeclaration;
##### 5.2.1.1. Properties
Members share the following properties:
annotations
Arbitrary annotations, see Annotations for details.
accessModifier
N4JS member access modifier: private, project, potected, or public; the latter two can be combined with @Internal; default is project for classes and private interfaces. For a non-private interface defaults to the interface’s visibility.
name
The simple name of the member, that is an identifier name (cf. Valid Names).
static
Boolean property to distinguish instance from classifier members, see Static Members.
The following pseudo properties are defined via annotations:
deprecated
Boolean property set to true if annotation @Deprecated is set. [29]
And we additionally define the following pseudo properties:
acc
Member access modifier as described in Accessibility of Members, it is the aggregated value of the $accessModifier$ and the $export$ property.
owner
Owner classifier of the member.
typeRef
Type of the member—this is the type of a field or the type of the method which is a function type (and not the return type).
assignability
Enumeration, may be one of the following values:
set
Member may only be set, i.e. it could only be used on the left hand side of an assignment.
get
Member may only be retrieved, i.e. it could only be used on the right hand side of an assignment. This is the default setting for methods.
any
Member may be set or retrieved, i.e. it could only be used on the left or right hand side of an assignment. This is the default setting for fields.
$assignability$ is related but not equal to writable modifiers used for fields. We define a partial order on this enumeration as follows:
$<\left(l,r\right)\phantom{\rule{3.0mm}{0ex}}::=\phantom{\rule{3.0mm}{0ex}}\left\{\left(\text{set},\text{any}\right),\left(\text{get},\text{any}\right)\right\}$
abstract
All members have a flag $abstract$, which is user-defined for methods, getters and setter, but which is always false for fields.
The following pseudo property is set to make fields compatible with properties of an object literal, however it cannot be changed:
configurable
Boolean flag reflecting the property descriptor $configurable$, this is always set to false for members.
#### 5.2.2. Semantics
The members of a given classifier C must be named such that the following constraints are met:
Req. IDE-52: Member Names (ver. 1)
1. The name of a member is given as an identifier, a string literal, a numeric literal, or as a computed property name with a compile-time expression (see Compile-Time Expressions). In particular, string literals, e.g. ['myProp'], built-in symbols, e.g. [Symbol.iterator], and literals of @StringBased enums are all valid computed property names.
2. No two members may have the same name, except one is static and the other is non-static:
$\begin{array}{c}\forall {m}_{1},{m}_{2}\in C.ownedMembers,{m}_{1}\ne {m}_{2}:{m}_{1}.name\ne {m}_{2}.name\vee {m}_{1}.static\ne {m}_{2}.static\end{array}$
3. The member name must be a valid identifier name, see Identifier Grammar.
Thus, overloading of methods is not supported [30] and no field may have the same name as a method. However, overriding of methods, getters, and setters are possible, see Redefinition of Members. Static members may also have the same name as non-static members.[31]
The dollar character $ is not allowed for user-defined member identifiers as the dollar sign is used for rewriting private members. #### 5.2.3. Methods Methods are simply JavaScript functions. They are defined similarly to methods as proposed in [ECMA15a(p.S13.5)] except for the type information and some modifiers. ##### 5.2.3.1. Syntax Syntax Method Declaration N4MethodDeclaration <Yield>: => ({N4MethodDeclaration} annotations+=Annotation* accessModifier=N4JSMemberAccessModifier? (abstract?=’abstract’ | static?=’static’)? TypeVariables? ( generator?='*' LiteralOrComputedPropertyName<Yield> -> MethodParamsReturnAndBody <Generator=true> | AsyncNoTrailingLineBreak LiteralOrComputedPropertyName<Yield> -> MethodParamsReturnAndBody <Generator=false> ) ) ';'? ; fragment MethodParamsAndBody <Generator>*: StrictFormalParameters<Yield=Generator> (body=Block<Yield=Generator>)? ; fragment MethodParamsReturnAndBody <Generator>*: StrictFormalParameters<Yield=Generator> (':' returnTypeRef=TypeRef)? (body=Block<Yield=Generator>)? ; fragment LiteralOrComputedPropertyName <Yield>*: name=IdentifierName | name=STRING | name=NumericLiteralAsString | '[' (=>((name=SymbolLiteralComputedName<Yield> | name=StringLiteralAsName) ']') | computeNameFrom=AssignmentExpression<In=true,Yield> ']') ; SymbolLiteralComputedName <Yield>: BindingIdentifier<Yield> ('.' IdentifierName)? ; BindingIdentifier <Yield>: IDENTIFIER | <!Yield> 'yield' | N4Keyword ; IdentifierName: IDENTIFIER | ReservedWord | N4Keyword; NumericLiteralAsString: DOUBLE | INT | OCTAL_INT | HEX_INT | SCIENTIFIC_INT; StringLiteralAsName: STRING; fragment AsyncNoTrailingLineBreak *: (declaredAsync?='async' NoLineTerminator)?; // See Asynchronous Functions fragment StrictFormalParameters <Yield>*: '(' (fpars+=FormalParameter<Yield> (',' fpars+=FormalParameter<Yield>)*)? ')' ; FormalParameter <Yield>: {FormalParameter} BindingElementFragment<Yield> ; fragment BindingElementFragment <Yield>*: (=> bindingPattern=BindingPattern<Yield> | annotations+=Annotation* ( variadic?='...'? name=BindingIdentifier<Yield> ColonSepTypeRef? ) ) ('=' initializer=AssignmentExpression<In=true, Yield>)? ; fragment ColonSepTypeRef*: ':' declaredTypeRef=TypeRef ; ##### 5.2.3.2. Properties Methods have all the properties of members and the following additional properties can be explicitly defined: abstract Method is declared but not defined. typePars Collection of type parameters of a generic method; empty by default. returnTypeRef Return type of the method, default return type is $Void$. The type of the method as a member of the owning classifier is not the method’s return type but is instead a function type. fpars List of formal parameters, may be left empty. body The body of the method (this is not available in the pure types model) The following pseudo properties are defined via annotations: final Boolean flag set to true if annotation @Final is set. The flag indicates that method must not be overridden in subclasses; see Final Methods. declaresOverride Flag set to true if annotation @Overrides is set. The flag indicates that method must override a method of a superclass; see Overriding of Members. Additionally, we define the following pseudo properties: overrides True if method overrides a super method or implements an interface method, false otherwise. typeRef Type of the method. This is, in fact, a function type (and not the return type). The following pseudo property is set to make methods compatible with properties of an object literal, however it cannot be changed: enumerable Boolean flag reflecting the property descriptor $enumerable$, this is always set to false for methods. ##### 5.2.3.3. Semantics Since methods are ECMAScript functions, all constraints specified in Function Type apply to methods as well. This section describes default values and function type conformance which is required for overriding and implementing methods. In addition, method declarations and definitions have to comply with the constraints for naming members of classifiers (cf. [Req-IDE-52]) and with the constraints detailed in the following sections on final methods (Final Methods), abstract methods (Abstract Methods and method overriding and implementation (Overriding of Members, Implementation of Members). The following constraints are defined for methods in ECMAScript 6 [ECMA15a(p.207)] Req. IDE-53: Method Definition ECMAScript 6 (ver. 1) • It is a Syntax Error if any element of the BoundNames of StrictFormalParameters also occurs in the VarDeclaredNames of FunctionBody. • It is a Syntax Error if any element of the BoundNames of StrictFormalParameters also occurs in the LexicallyDeclaredNames of FunctionBody. Methods – like functions – define a variable execution environment and therefore provide access to the actual passed-in parameters through the implicit arguments variable inside of their bodies (c.f. Arguments Object). Methods are similar to function definitions but they must not be assigned to or from variables. The following code issues an error although the type of the method would be compatible to the type of the variable v: class C { m(): void {} } var v: {function():void} = new C().m; Req. IDE-54: Method Assignment (ver. 1) 1. In contrast to ECMAScript 2015, methods are defined as readonly, that is, it is not possible to dynamically re-assign a property defined as method with a new value. This is because assigning or re-assigning a method breaks encapsulation. Methods are the Acronyms of a class, their implementation is internal to the class. 2. When assigning a method to a variable, a warning is issued since this would lead to an detached this reference inside the method when it is called without explicitly providing the receiver. No warning is issued only if it is guaranteed that no problems will occur: 1. The method’s body can be determined at compile time (i.e., it has been declared @Final) and it lacks usages of this or super. This is true for instance and static methods. 2. The method is the constructor. The following code demonstrates problems arising when methods are assigned to variables in terms of function expressions. Given are two classes and instances of each class as follows: class C { m(): void { } static k(): void {} } class D extends C { @Override m(): void { this.f()} f(): void {} @Override static k(): void { this.f()} static f(): void {} } var c: C = new C(); var d: C = new D(); // d looks like a C Assigning an instance method to a variable could cause problems, as the method assumes this to be bound to the class in which it is defined. This may work in some cases, but will cause problems in particular in combination with method overriding: var v1: {@This(C)function():void} = c.m; var v2: {@This(C)function():void} = d.m; v1.call(c); v2.call(c); Calling c.m indirectly via v1 with c as this object will work. However, it won’t work for v2: the method is overridden in D, and the method in expects other methods available in D but not in C. That is, the last call would lead to a runtime error as method f which is called in D.m won’t be available. The same scenario occurs in case of static methods if they are retrieved polymorphically via the variables of type constructor{C}: var ctor: constructor{C} = C; var dtor: constructor{C} = D; var v3: {@This(constructor{C})function():void} = ctor.k; var v4: {@This(constructor{C})function():void} = dtor.k; In both cases, the problem could be solved by restricting these kinds of assignments to final methods only. In the static case, the problem would also be solved by accessing the static method directly via the class type (and not polymorphically via the constructor). Both restrictions are severe but would be necessary to avoid unexpected runtime problems. The following example shows a problem with breaking the encapsulation of a class. class C { x: any = ""; f(): void { this.g(this); } g(c: C): void { c.h(); } h(): void {} } class D extends C { @Override f(): void { this.g(this.x); } @Override g(c: any) { // do nothing, do not call h()) } } var c = new C(); var d = new D(); var v5: {@This(C)function():void} = c.f; var v6: {@This(C)function():void} = d.f; v5.call(c) v6.call(c) In D, method g is overridden to accept more types as the original method defined in C. Calling this new method with receiver type C (as done in the last line) will cause problems, as in D not only f has been adapted but also g. Eventually, this would lead to a runtime error as well. ##### 5.2.3.4. Final Methods By default, methods can be overridden. To prevent a method from being overridden, it must be annotated with @Final. Of course, a method cannot be declared both abstract and final (cf. [Req-IDE-46]). Private methods are implicitly declared final. Because static methods can be overridden in subclasses (which is different to Java), they also can be marked as final. Default methods in interfaces, cf. Default Methods in Interfaces, may also be declared @Final. Example 36. Final Methods in Interfaces If a method in an interface is provided with a body, it may be declared final. This will ensure that the given method’s body will be in effect for all instances of the interface. Note that this means that; 1. a class implementing that interface must not define a method with the same name and 2. a class inheriting a method of that name cannot implement this interface. The latter case is illustrated here: interface I { @Final m(): void {} } class C1 { m(): void {} } // error at "I": "The method C1.m cannot override final method I.m." class C2 extends C1 implements I { } ##### 5.2.3.5. Abstract Methods A method can be declared without defining it, i.e. without providing a method body, and is then called an abstract method. Such methods must be declared with modifier abstract and have their property $abstract$ set to true. Constraints for abstract methods are covered in [Req-IDE-46] (see Abstract Classes). In interfaces, methods are always abstract by default and they do not have to be marked as abstract. If a method in an interface provides a body, then this is the default implementation. See Implementation of Members about how the default implementation may be mixed in the consumer. ##### 5.2.3.6. Generic Methods Methods of generic classes can, of course, refer to the type variables defined by type parameters of the generic class. These type variables are used similarly to predefined or declared types. Additionally, methods may be declared generic independently from their containing class. That is to say that type parameters (with type variables) can be defined for methods as well, just like for generic functions (see Generic Functions). For a given generic method M of a class C, the following constraint must hold: Since type variables can be used similarly to types in the scope of a generic class, a generic method may refer to a type variable of its containing class. class C { <T> foo(p: T p): T { return p;} }; If a generic type parameter is not used as a formal parameter type or the return type, a warning is generated unless the method overrides a member inherited from a super class or interface. #### 5.2.4. Default Methods in Interfaces If a method declared in an interface defines a body, then this is the so-called default implementation and the method is called a default method. This will be mixed into an implementor of the interface if, and only if, neither the implementing class nor any of its direct or indirect superclasses already provides an implementation for this method; for details see Member Consumption. Since the implementor is not known, some constraints exist for the body. I.e., no access to super is possible, cf. [Req-IDE-124]. In order to declare an interface to provide a default implementation in a definition file, annotation @ProvidesDefaultImplementation can be used, cf. [Req-IDE-167]. When a method in an interface is provided with a default implementation, it may even be declared @Final, see Final Methods. ##### 5.2.4.1. Asynchronous Methods N4JS implements the async/await concept proposed for ECMAScript 7, which provides a more convenient and readable syntax for writing asynchronous code compared to using built-in type Promise directly. This concept can be applied to methods in exactly the same way as to declared functions. See Asynchronous Functions and Asynchronous Arrow Functions for details. #### 5.2.5. Constructors A constructor is a special function defined on a class which returns an instance of that class. The constructor looks like a normal method with name "constructor". The constructor can be defined explicitly or implicitly and every class has an (implicit) constructor. For a given a class C, the constructor is available via two properties: $ownedCtor$ the explicitly defined constructor (if any). $ctor$ the explicit or implicit constructor. If C is provided with an explicit constructor, we have $C.ctor=C.ownedCtor$ and $C.ownedCtor\in C.ownedMembers$. Note that $C.ctor\notin C.ownedMethods$ in all cases. The return type of the constructor of a class C is C. If C has type parameters ${T}_{1},.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}},{T}_{n}$, then the return type is $C<{T}_{1},.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}},{T}_{n}>$. The constructor is called with the operator. Since the return type of a constructor is implicitly defined by the class, it is to be omitted. By this definition, a constructor looks like the following: class C { public constructor(s: string) { // init something } } Constructors define a variable execution environment and therefore provide access to the actual passed-in parameters through the implicit variable inside of their bodies (c.f. Arguments Object). Req. IDE-56: Defining and Calling Constructors (ver. 1) For a constructor $ctor$ of a class C, the following conditions must hold: 1. $ctor$ must neither be abstract nor static nor final and it must not be annotated with @Override. 2. If a class does not explicitly define a constructor then the constructor’s signature of the superclass constructor is assumed. 3. If a class defines a constructor with formal parameters then this constructor has to be called explicitly in constructors defined in subclasses. 4. If a super constructor is called explicitly, this call must be the only expression of an expression statement which has to be the first statement of the body. 5. Constructors may appear in interfaces, but some restrictions apply: 1. constructors in interfaces must not have a body. 2. constructors in interfaces or their containing interface or one of its direct or indirect super interfaces must be annotated with @CovariantConstructor. 6. A constructor must not have an explicit return type declaration. 7. The implicit return type of a constructor is this?. 8. A constructor must not have any type parameters. Properties of object literals may be called constructor. However they are not recognized as constructors in these cases. 1. Required attributes must be initialized: $\forall a\in C.attr:a.required⇒\exists e\in r.elements:a.name=e.name$ Note on syntax: ECMAScript 6 defines constructors similarly, [ECMA15a(p.S13.5)]. In ECMAScript 6 the super constructor is not called automatically as well. The super literal used in order to call super methods is further described in The super Keyword. ##### 5.2.5.1. Structural This Type in Constructor The use of a structural this reference as a formal parameter type is possible only in constructors. This parameter can be annotated with @Spec which causes the compiler to generate initialization code. Simply using this as a type in the constructor causes the constructor to require an object providing all public fields of the class for initialization purposes. The fields have to be set manually as shown in the following code snippet. class A{ public s: string; public constructor(src: ~~this) { this.s = src.s; } } Remarks: • The type of the formal parameter ~~this refers to the structural field type, see Structural Typing for details on structural typing. It contains all public fields of the type. • Subclasses may override the constructor and introduce additional parameters. They have to call the super constructor explicitly, however, providing a parameter with at least all required attributes of the superclass. Usually the type this is replaced with the actual subclass, but in the case of a super() call the this type of structural formal parameters is replaced with the this type of the superclass, hence only required fields of the superclass must be present. As with other structural references, it is possible to add the structural reference with additional structural members, which can be used to initialize private fields which become not automatically part of the structural field type. For example: class A{ public s: string; private myPrivateNumber: number; public constructor(src: ~~this with { x: number; }) { this.s = src.s; this.myPrivateNumber = src.x; } } Defining additional members may become a problem if a subclass defines public fields with the same name, as the ~~this type will contain these fields in the subclass. This is marked as an error in the subclass. If the structural this type is used in a constructor of a class C, and if this structural reference contains an additional structural member $SM$, the following constraints must hold true: 1. For any subclass S of C, with $S.ctor=C.ctor$ (the subclass does not define its own constructor), S must not contain a public member with same name as $SM$: $S<:C,S.ctor=C.ctor$ $\phantom{\rule{3.0em}{0ex}}⇒\nexists M\in S.members:$ $\phantom{\rule{5.0em}{0ex}}M.acc=\text{public}\wedge M.name=SM.name$ 2. C itself must not contain a public member with same name as $SM$: $\nexists M\in C.members:M.acc=\text{public}\wedge M.name=SM.name$ Example 37. Field name conflicts with structural member name The situation described in [Req-IDE-58] is demonstrated in the following code fragment: class A { private myPrivateNumber: number; public constructor(src: ~~this with { x: number; }) { this.myPrivateNumber = src.x; } } class B extends A { public x: number; // will cause an error message } ##### 5.2.5.2. @Spec Constructor The tedious process of copying the members of the parameter to the fields of the class can be automated via the @Spec annotation if the argument has ~i~this structural initializer field typing. More details about this typing can be found in Structural Read-only, Write-only and Initializer Field Typing. This can be used as shown in the following listing: class A { public field: string; public constructor(@Spec spec: ~i~this) {} } let a = new A({field: 'hello'}); console.log(a.field); // prints: hello The code for initializing the public field of A is automatically generated, thanks to the @Spec annotation being given in the constructor. Req. IDE-59: @Spec Constructor (ver. 1) 1. Annotation @Spec may only appear on a formal parameter of a constructor. Such a formal parameter is then called @Spec parameter or simply spec parameter and its owning constructor is referred to as a @Spec constructor or spec constructor. An argument to the spec parameter is called spec object. 2. Only a single formal parameter of a constructor may be annotated with @Spec. 3. If a formal parameter is annotated with @Spec, the parameter’s type must be ~i~this (i.e. a use-site structural initializer field type of this, see Structural Read-only, Write-only and Initializer Field Typing). 4. Using the data provided in the spec object, i.e. in the argument to the spec parameter, a spec constructor will automatically initialize 1. all owned data fields and owned setters of the containing class, and 2. all data fields and setters from interfaces implemented by the containing class if and only if those members are also part of the spec parameter’s structural initializer field type. 5. Fields explicitly added to the spec parameter, e.g. @Spec spec: ~i~this with {name:string}, are used for initialization if a non-public field of the same name exists in the class, either as an owned member or from an implemented interface. The type of such an additional field must be a subtype of the declared type of the field being initialized: $\forall s\in ctor.fpar.structuralMembers,ctor.fpar.spec:$ $\phantom{\rule{2.0em}{0ex}}\exists f\in ctor.owner.ownedFields⇒\Gamma ⊢s\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}f$ 6. Even if the @Spec annotation is used, the super constructor must be invoked explicitly (as usual). It follows from no. 4 above that 1. non-public data fields and setters are never initialized (because they will never be part of the spec parameter’s structural initializer field type), 2. properties provided in the spec object but not defined in the parameter’s structural initializer field type, are not used for initialization, even if a (protected or private) field of the same name exists in the class, 3. data fields and setters inherited from a super class are never initialized by a spec constructor (instead, this will happen in the spec constructor of the super class). The last of these implications will be detailed further at the end of the coming section. @Spec Constructors and Inheritance Spec constructors are inherited by subclasses that do not have a constructor and, when creating instances of the subclass, will then require properties for writable public fields of the subclass in the spec object and include initialization code for them. class A { public fa; public constructor(@Spec spec: ~i~this) {} } class B extends A { public fb; } const b = new B({fa: 'hello', fb: 'world'}); // requires & initializes fb too! console.log(b.fa); // prints: hello console.log(b.fb); // prints: world Public writable fields from implemented interfaces are included as well, i.e. required as property in spec object and initialized by auto-generated code in the @Spec constructor: interface I { public fi; } class B implements I { public fb; public constructor(@Spec spec: ~i~this) {} } const a = new B({fb: 'hello', fi: 'world'}); // requires & initializes fi too! console.log(a.fb); // prints: hello console.log(a.fi); // prints: world When having a spec constructor in a class B that extends a super class A without an owned or inherited spec constructor, it should be noted that the ~i~this type will require properties for public writable fields of A, but the initialization code automatically generated due to the @Spec annotation will not initialize those members. For public writable fields from an interface I implemented by B, however, both a property will be required by ~i~this and initialization code will be generated in the @Spec constructor. This is illustrated in the following code example. class A { public fa; } interface I { public fi; } class B extends A implements I { public fb; public constructor(@Spec spec: ~i~this) { // <- fa, fi, fb required in spec object // Constructor is responsible for initializing fa, fi, fb. // The @Spec annotation will generate initialization code // for fb and fi, but not for fa! } } let b = new B({ fa: 'hello', // <- fa is required (removing it would be a compile error) fi: 'world', fb: '!!' }); console.log(b.fa); // undefined console.log(b.fi); // world console.log(b.fb); // !! The rationale for this different handling of fields from super classes and implemented interfaces is 1. fields from an implemented interface are not seen as inherited but rather implemented by implementing class, so from the @Spec annotation’s perspective the field is a field of the implementing class, and 2. in case of a field inherited from a super class the correct way of initialization may depend on details of the super class and has to be taken care of by custom code in the constructor of the subclass (usually by invoking the non-@Spec constructor of the superclass with super). Special Use Cases The following examples illustrate further details of other use cases of spec constructors. Example 38. Anonymous Interface in Constructor The base class A in the examples redefines the constructor already defined in N4Object. This is not generally necessary and is only used here to make the example legible. class A { public s: string; public constructor(@Spec spec: ~i~this) { // initialization of s is automatically generated } } class B extends A { public t: string; private n: number; public constructor(spec: ~~this with {n: number;}) { super(spec); // only inherited field s is set in super constructor } } Example 39. Spec Object and Subclasses class A1 { public s: string; public n: number; public constructor(@Spec spec: ~i~this) {} } class B extends A1 { public constructor() { super({s:"Hello"}); // <-- error, n must be set in object literal } } class C extends A1 { public constructor() { super({s:"Hello"}); // <-- error, n must be set in object literal this.n = 10; // <-- this has no effect on the super constructor! } } class A2 { public s: string; public n: number?; // now n is optional! public constructor(@Spec spec: ~i~this) {} } class D extends A2 { public constructor() { super({s:"Hello"}); // and this is ok now! this.n = 10; // this explains why it is optional } } class A3 { public s: string; public n: number = 10; // now n is not required in ~~this public constructor(@Spec spec: ~i~this) {} } class E extends A3 { public constructor() { super({s:"Hello"}); // and this is ok now! } } The last case (class E) demonstrates a special feature of the typing strategy modifier in combination with the this type, see Structural Typing for details. The constructor in class B contains an error because the super constructor expects all required attributes in A1 to be set. The additional initialization of the required field A1.n as seen in C does not change that expectation. In this example, the field n should not have been defined as required in the first place. Optional fields like n? in class A2 or fields with default values like n=10 in class A3 are not required to be part of the spec object. Example 40. Superfluous Properties in @Spec Constructors Each non-$\text{public}$ field has to be set in the constructor via the $\text{with}$ to the parameter otherwise properties are not used to set non-$\text{public}$ fields. class C { public s: string; n: number; constructor(@Spec spec: ~i~this) {} } // n is ignored here new C( { s: "Hello", n: 42 }); // but: var ol = { s: "Hello", n: 42 }; // "ol may be used elsewhere, we cannot issue warning here" at "ol" new C(ol) ; // of course this is true for all superfluous properties // weird is not used in constructor new C( { s: "Hello", weird: true } ); Restriction when initializing interface fields via @Spec constructor In most cases, interface definitions in n4jsd files simply declare functions and fields that are supposed to be provided by the runtime environment. As a result, there are restrictions as to whether fields of interfaces defined in n4jsd files can initialized via @Spec constructors or not. In particular, fields of an interface declared in a n4jsd file cannot be initialized via @Spec constructor if the interface 1. is a built-in or 2. does not have an @N4JS annotation The following example illustrates this restriction. Example 41. Interface fields that cannot be initialized via @Spec constructors Inf.n4jsd export external interface I { public m: string; } @N4JS export external interface J { public n: string; } Test.n4js import { I } from "Inf"; // I is an external interface WITHOUT @N4JS annotation class C implements I { constructor(@Spec spec:~i~this) {} } // J is an external interface with @N4JS annotation class D implements J { constructor(@Spec spec:~i~this) {} } // XPECT warnings --> "m is a property of built-in / provided by runtime / external without @N4JS annotation interface I and can not be initialized in Spec constructor." at "m" let c:C = new C({m: "Hello"}); // XPECT nowarnings let d:D = new D({n: "Bye"}); console.log(c.m) console.log(d.n) /* XPECT output --- <== stdout: undefined Bye stderr: ==> --- */ In this example, the interface I is defined in the Inf.n4jsd file without the @N4JS annotation. As a result, its field m cannot be initialized via the @Spec constructor and hence the output of console.log(c.m) is undefined. On the other hand, since the interface J is declared with the annotation @N4JS, it is possible to initialize its field n in the @Spec constructor. That’s why the result of console.log(d.n) is Bye. ##### 5.2.5.3. Callable Constructors ##### 5.2.5.4. Covariant Constructors Usually, the constructor of a subclass need not be override compatible with the constructor of its super class. By way of annotation @CovariantConstructor it is possible to change this default behavior and enforce all subclasses to have constructors with override compatible signatures. A subclass can achieve this by either inheriting the constructor from the super class (which is usually override compatible, with the special case of @Spec constructors) or by defining a new constructor with a signature compatible to the inherited constructor. The same rules as for method overriding apply. The @CovariantConstructor annotation may be applied to the constructor, the containing classifier, or both. It can also be used for interfaces; in fact, constructors are allowed in interfaces only if they themselves or the interface is annotated with @CovariantConstructor (see [Req-IDE-60]). Definition: Covariant Constructor A classifier C is said to have a covariant constructor if and only if one of the following applies: 1. C has a direct super class ${C}^{\text{'}}$ and ${C}^{\text{'}}$ is annotated with @CovariantConstructor or ${C}^{\text{'}}$ has a constructor annotated with @CovariantConstructor. 2. C has a directly implemented interface I and I is annotated with @CovariantConstructor or I has a constructor annotated with @CovariantConstructor. 3. C has a direct super class or directly implemented interface that has a covariant constructor (as defined here). Note that C does not need to have an owned(!) constructor; also a constructor inherited from a super class can be declared covariant. The following rules apply to covariant constructors. Req. IDE-60: Covariant Constructors (ver. 1) 1. Annotation @CovariantConstructor may only be applied to classes, interfaces, and constructors. Annotating a constructor with this annotation, or its containing classifier, or both have all the same effect. 2. Given a class C with an owned constructor $ctor$ and a super class $Sup$ that has a covariant constructor (owned or inherited, see Covariant Constructors), then $Sup.constructor$ must be accessible from C, 1. $ctor$ must be override compatible with $S.constructor$: $overrideCompatible\left(ctor,S.constructor\right)$ This constraint corresponds to [Req-IDE-72] except for the Override annotation which is not required here. 3. Given a classifier C implementing interface I and I has a covariant constructor (owned or inherited, see Covariant Constructors), we require 1. $I.constructor$ must be accessible from C, 2. an implementation-compatible constructor $ctor$ must be defined in C with $overrideCompatible\left(ctor,I.constructor\right)$ This constraint corresponds to [Req-IDE-74] except for the @Override annotation, which is not required, here. 3. Given a classifier C without an owned constructor and an extended class or interface $Sup$ that has a covariant constructor (owned or inherited, see Covariant Constructors), we require the inherited constructor $ctor$ of C within the context of C to be override compatible to itself in the context of $Sup$. Using notation $m\left[T\right]$ to denote that a member M is to be treated as defined in container type T, which means the this-binding is set to T, we can write: $overrideCompatible\left(ctor\left[C\right],ctor\left[Sup\right]\right)$ This constraint does not correspond to any of the constraints for the redefinition of ordinary members. The following example demonstrates a use case for covariant constructors. It shows a small class hierarchy using covariant constructors, Cls and Cls2, together with a helper function createAnother that creates and returns a new instance of the same type as its argument value. Example 42. Covariant Constructors class A {} class B extends A {} @CovariantConstructor class Cls { constructor(p: B) {} } class Cls2 extends Cls { constructor(p: A) { // it's legal to generalize the type of parameter 'p' super(null); } } function <T extends Cls> createAnother(value: T, p: B): T { let ctor = value.constructor; return new ctor(p); } let x = new Cls2(new A()); let y: Cls2; y = createAnother(x, new B()); In the code of Covariant Constructors, we would get an error if we changed the type of parameter p in the constructor of Cls2 to some other type that is not a super type of B, i.e. the type of the corresponding parameter of Cls’s constructor. If we removed the @CovariantConstructor annotation on Cls, we would get an error in the new expression inside function createAnother. The next example illustrates how to use @CovariantConstructor with interfaces and shows a behavior that might be surprising at first sight. Example 43. Covariant Constructors in Interfaces @CovariantConstructor interface I { constructor(p: number) } class C implements I { // no constructor required! } class D extends C { // XPECT errors --> "Signature of constructor of class D does not conform to overridden constructor of class N4Object: {function(number)} is not a subtype of {function()}." at "constructor" constructor(p: number) {} } Interface I declares a covariant constructor expecting a single parameter of type number. Even though class C implements I, it does not need to define an owned constructor with such a parameter. According to [Req-IDE-60], it is enough for C to have a constructor, either owned or inherited, that is override compatible with the one declared by I. Class C inherits the default constructor from N4Object, which does not have any arguments and is thus override compatible to I’s constructor. In addition, subclasses are now required to have constructors which are override compatible with the constructor of class C, i.e. the one inherited from N4Object. Covariant Constructors in Interfaces shows that this is violated even when repeating the exact same constructor signature from interface I, because that constructor now appears on the other side of the subtype test during checking override compatibility. #### 5.2.6. Data Fields A data field is a simple property of a class. There must be no getter or setter defined with the same name as the data field. In ECMAScript 6, a class has no explicit data fields. It is possible, however, to implicitly define a data field by simply assigning a value to a variable of the this element (e.g. this.x = 10 implicitly defines a field x). Data fields in N4JS are similar to these implicit fields in ECMAScript 6 except that they are defined explicitly in order to simplify validation and user assistance. ##### 5.2.6.1. Syntax N4FieldDeclaration <Yield>: {N4FieldDeclaration} FieldDeclarationImpl<Yield> ; fragment FieldDeclarationImpl <Yield>*: (declaredModifiers+=N4Modifier)* BogusTypeRefFragment? declaredName=LiteralOrComputedPropertyName<Yield> (declaredOptional?='?')? ColonSepTypeRef? ('=' expression=Expression<In=true,Yield>)? Semi ; ##### 5.2.6.2. Properties Fields have the following properties which can be explicitly defined: declaredOptional Tells whether the accessor was declared optional. typeRef Type of the field; default value is $Any$. expr Initializer expression, i.e. sets default value. static Boolean flag set to true if field is a static field. const Boolean flag set to true if field cannot be changed. Note that const fields are automatically static. Const fields need an initializer. Also see Assignment Modifiers. $const$ is not the (reversed) value of the property descriptor $writable$ as the latter is checked at runtime while const may or may not be checked at runtime. The following pseudo properties are defined via annotations for setting the values of the property descriptor: enumerable Boolean flag reflecting the property descriptor $enumerable$, set via annotation @Enumerable(true|false). The default value is $\text{true}$.[32] declaredWriteable Boolean flag reflecting the property descriptor $writeable$, set via annotation @Writeable(true|false). The default value is $\text{true}$.[33] final Boolean flag making the field read-only, and it must be set in the constructor. Also see Assignment Modifiers. ###### Derived values for fields readable Always true for fields. abstract Always false for fields. writeable Set to false if field is declared const or final. In the latter case, it may be set in the constructor (cf. Assignment Modifiers). ###### 5.2.6.2.1. Semantics Req. IDE-61: Attributes (ver. 1) For any attribute $a$ if a class C, the following constraints must hold: 1. A required data field must not define an initializer: $a.required⇒a.init=null$ 2. There must be no other member with the same name of a data field f. In particular, there must be no getter or setter defined with the same name: If a subclass should set a different default value, this has to be done in the constructor of the subclass. For the relation of data fields and field accessors in the context of extending classes or implementing interfaces see Redefinition of Members. ###### 5.2.6.2.2. Type Inference The type of a field is the type of its declaration: $\frac{}{\Gamma ⊢f:\Gamma ⊢d}$ The type of a field declaration is either the declared type or the inferred type of the initializer expression: $\phantom{\rule{3.0mm}{0ex}}\frac{d.declaredType\ne \text{null}\phantom{\rule{3.0mm}{0ex}}T=d.declaredType}{\Gamma ⊢d:T}$ $\phantom{\rule{3.0mm}{0ex}}\frac{d.declaredType=\text{null}\phantom{\rule{3.0mm}{0ex}}d.expression\ne \text{null}}{\Gamma ⊢d:T}$ $\phantom{\rule{3.0mm}{0ex}}E=\Gamma ⊢d.expression\phantom{\rule{3.0mm}{0ex}}E\notin \left\{\text{null, undefined}\right\}\phantom{\rule{3.0mm}{0ex}}T=E\right\}$ $\phantom{\rule{3.0mm}{0ex}}\frac{else}{\Gamma ⊢d:\text{any}}$ If the type contains type variables they are substituted according to type parameters which are provided by the reference: ##### 5.2.6.3. Assignment Modifiers Assignment of data fields can be modified by the assignment modifiers const (similar to constant variable declarations, see Const) and @Final. Req. IDE-62: Const Data Fields (ver. 1) For a data field f marked as const, the following constraints must hold: 1. An initializer expression must be provided in the declaration (except in n4jsd files): $f.expr\ne \text{null}$ 2. A constant data field is implicitly static and must be accessed only via the classifier type. It is not possible, therefore, to use the this keyword in the initializer expression of a constant field: $\nexists sub\in f.exp{r}^{*}:sub="this"$ 3. A constant data field must not be annotated with @Final: $f.const\to ¬f.final$ 4. Constant data fields are not writeable (cf. [Req-IDE-68]): $f.const\to ¬f.writeable$ Req. IDE-63: Final Data Fields (ver. 1) For a data field f marked as @Final, the following constraints must hold: 1. A final data field must not be modified with const or static: $f.final\to ¬f.const\wedge ¬f.declaredStatic$ 2. A final data field is not writeable: $f.final\to ¬f.writeable$ A final field may, however, be set in the constructor. See [Req-IDE-68] for details. 3. A final data field must be either initialized by an initializer expression or in the constructor. If the field is initialized in the constructor, this may be done either explicitly or via a spec style constructor. $\begin{array}{c}\phantom{\rule{3.0mm}{0ex}}f.expr\ne \text{null}\\ \phantom{\rule{3.0mm}{0ex}}\vee \left(\exists assignExp:assignExpr.containingFunction=f.owner.constructor\\ \phantom{\rule{3.0mm}{0ex}}\phantom{\rule{3.0em}{0ex}}\wedge assignExpr.left.target=\text{"this"}\\ \phantom{\rule{3.0mm}{0ex}}\phantom{\rule{3.0em}{0ex}}\wedge bind\left(assignExpr.left.property,f\right)\right)\\ \phantom{\rule{3.0mm}{0ex}}\vee \left(f.public\wedge \exists fpar\in f.owner.constructor.fpars:\\ \phantom{\rule{3.0mm}{0ex}}\phantom{\rule{3.0em}{0ex}}fpar.spec\wedge \exists sm\in structuralMembers:sm.name=f.name\right)\end{array}$ ##### 5.2.6.4. Field Accessors (Getter/Setter) Instead of a simple data field, a field can be defined by means of the getter and setter accessor methods. These accessor methods are similar to the accuser methods in object literals: ###### 5.2.6.4.1. Syntax N4GetterDeclaration <Yield>: => ({N4GetterDeclaration} (declaredModifiers+=N4Modifier)* GetterHeader<Yield>) (body=Block<Yield>)? ';'? ; fragment GetterHeader <Yield>*: BogusTypeRefFragment? 'get' -> declaredName=LiteralOrComputedPropertyName<Yield> (declaredOptional?='?')? '(' ')' ColonSepTypeRef? ; N4SetterDeclaration <Yield>: =>({N4SetterDeclaration} (declaredModifiers+=N4Modifier)* 'set' ->declaredName=LiteralOrComputedPropertyName <Yield> ) (declaredOptional?='?')? '(' fpar=FormalParameter<Yield> ')' (body=Block<Yield>)? ';'? ; Notes with regard to syntax: Although ECMAScript 6 does not define fields in classes, it defines getter and setter methods similarly (cf. [ECMA15a(p.S13.3, p.p.209)]). Example 44. Getter and Setter The getter and setter implementations usually reference data fields internally. These are to be declared explicitly (although ECMAScript allows creating fields on the fly on their first usage). The following example demonstrates a typical usage of getter and setter in combination with a data field. The getter lazily initializes the field on demand. The setter performs some notification. Getter Setter class A {} class C { private _data: A = null; public get data(): A { if (this._data==null) { this._data = new A(); } return this._data; } public set data(data: A) { this._data = data; this.notifyListeners(); } notifyListeners(): void { // ... } } ###### 5.2.6.4.2. Properties Properties for field accessors: declaredOptional Tells whether the accessor was declared optional. readable Derived value: true for getters and false for setters. writable Derived value: false for getters and true for setters. ###### 5.2.6.4.3. Semantics There must be no field or method with the same name as a field accessor (follows from [Req-IDE-52]). In addition, the following constraints must hold: Req. IDE-64: Field Accessors (ver. 1) • The return type of a getter must not be void. • The type of the parameter of a setter must not be void. • If a getter $g$ is defined or consumed (from an interface) or merged-in (via static polyfill) in a class C and a setter S with $s.name=g.name\wedge s.static=g.static$ is inherited by C from one of its super classes, then C must define a setter ${s}^{\text{'}}$ with ${s}^{\text{'}}.name=g.name\wedge {s}^{\text{'}}.static=g.static$ [34]. • A setter must have exactly one formal parameter, i.e. variadic or default modifiers are not allowed. The same applies to setters, accordingly. A getter and setter with the same name need not have the same type, i.e. the getter’s return type need not be the same as a subtype of the type of the setter’s parameter (the types can be completely unrelated).[35] Getters and setters – like functions – define a variable execution environment and therefore provide access to the actual passed-in parameters through the implicit arguments variable inside of their bodies (c.f. Arguments Object). ##### 5.2.6.5. Optional Fields Data fields and field accessors of a classifier C can be declared optional, meaning that a structural subtype of C need not provide this field, but if it does, the field must be of correct type. However, to ensure overall type safety, the scope of application of this optionality is limited to a small number of specific use cases, as described in the following. ###### 5.2.6.5.1. Syntax To denote a data field or accessor as optional, a question mark is placed right after the name: Syntax of optional fields class C { public field?: string; public get getter?(): number { return 42; } public set setter?(value: number) {} } The detailed grammar is given in the sections for data fields, cf. Syntax, and field accessors, cf. Syntax. ###### 5.2.6.5.2. Semantics It is important to note that the optionality of a field is, by default and in most cases, ignored and has an effect only in certain special cases. The effect of a field being optional is defined by the following requirement. Req. IDE-240500: Optional Fields (ver. 1) By default, a data field, getter, or setter that is declared optional is handled in the exact same way as if no optionality were involved (i.e. by default, optionality is ignored). Optionality has an effect only in case of structural subtype checks $L<:R$ in which the left-hand side is one of the following: 1. an object literal. 2. a new expression. 3. an instance of a final class, i.e. the type of the value on left-hand side must be nominal and refer to a final class. 4. a reference to a const variable if its initializer expression is one of the following: 1. an object literal. 2. a new expression. 3. an instance of a final class (as explained above). 4. an ternary expression and then • in cases 1 and 4a, both fields and accessors (getters and setters) are optional. That means, an optional data field, getter, or setter of $R$ needs not be present in $L$. • in cases 2, 3, 4b, and 4c, only getters are optional, setters are not optional. That means, an optional getter of $R$ needs not be present in $L$ and an optional field of $R$ requires only a setter in $L$. Note that these cases are more restricted than the cases 1 and 4a. Moreover, optionality has an effect in case of ternary expression $L<:R$ in which the left-hand side is a ternary expression, e.g. l = b? trueExpr : falseExpr whose trueExpr or falseExpr possibly recursively contains an expression of the kind mentioned above. In this case, the optionality effect is the more restricted optinality of trueExpr and falseExpr. If, according to these rules, a data field / getter / setter of $R$ need not be present in $L$ but a member with the same name and access is actually present in $L$, that member in $L$ must be a data field / getter / setter of the same type / a subtype / a super type, respectively. In other words, if a not actually required member is present in the subtype, ordinary rules for member compatibility apply as if no optionality were involved (cf. general subtyping rules for structural types). In other words, in object literals (cases 1 and 4a) neither optional getters, optional setters, nor optional data fields are required. However, in case of new expressions and instances of final classes (cases 2, 3, 4b, 4c) only optional getters are not required in a subtype; optional setters are required as normal (i.e. optionality ignored) and optional data fields require at least a setter. The following table summarizes the most common cases and shows how this relates to the different forms of structural typing. let x: ΔC = {}; let x: ΔC = new D0(); let x: ΔC = new DG(); let x: ΔC = new DS(); Δ Case Comment ~ ~~ ~w~ ~r~ ~i~ may have setter never has setter ✓ ✓ ✓ ✓ ✓ 1 nothing mandatory ✓ ✓ 2 setters mandatory ✓ ✓ 2 setters mandatory ✓ ✓ ✓ ✓ ✓ 2 setters mandatory none D0 not final none fooSF0() not nominal ✓ ✓ 3 setters mandatory In the table, a "✓" means that the particular example is valid; in all other cases an error would be shown in N4JS source code. Here are the classes and functions used in the above table: Classes and functions used in table class C { public field?: string; } class D0 {} class DG { public get field(): string { return "hello"; } } class DS { public set field(value: string) {} } @Final class F0 {} function fooD0(): D0 { return new D0(); } function fooSF0(): ~F0 { return new F0(); } function fooF0(): F0 { return new F0(); } It follows from the above definitions in Requirements [Req-IDE-240500] that cases 4a and 4b are not transitive across a chain of several const variables, whereas case 4c is transitive. For example: Transitivity of the use cases of optional fields class C { public get getter?(): string {return null;} } class D {} @Final class F {} let c: ~C; // no transitivity via several const variables in use case "object literal": const ol1 = {}; const ol2 = ol1; // XPECT errors --> "~Object is not a structural subtype of ~C: missing getter getter." at "ol2" c = ol2; // no transitivity via several const variables in use case "new expression": const new1 = new D(); const new2 = new1; // XPECT errors --> "D is not a structural subtype of ~C: missing getter getter." at "new2" c = new2; // BUT: we do have transitivity via several const variables in use case "final nominal type": const finalNominal1 = new F(); const finalNominal2 = finalNominal1; // XPECT noerrors --> c = finalNominal1; // XPECT noerrors --> "transitivity applies in this case" c = finalNominal2; The following example demonstrates how optionality behaves in ternay expressions. Optional fields in ternay expressions interface ~I { public m?: int; } class ~C { } @Final class F { } let b: boolean; const cc: C = {} let f1 = new F(); let f2: ~F = {}; // True expression is a const object literal, so both fields and accessors in I are optional. // False expression is a new expression, so only getters in I are optionals. // As a result, only getters in I are optional. // XPECT errors --> "C is not a structural subtype of I: missing field m." at "b? cc : new C()" var te1: I = b? cc : new C() // No errors because both true and false expressions are object literal constants and hence // Both fields and accessors in I are optional. // XPECT noerrors var te2: I = b? cc : {} ###### 5.2.6.5.3. Background The following example illustrates why optionality of fields has to be restricted to the few special cases defined above (i.e. object literals, new expressions, etc.). Problem 1 of optional fields class C { public field?: string = "hello"; } class D {} class DD extends D { public field: number = 42; } let c: ~C; let d: D; d = new DD(); c = d; // without the restrictive semantics of optional fields, this assignment would be allowed (but shows compile-time error in N4JS) console.log(c.field); // prints 42 even though the type is string c.field.charAt(0); // exception at runtime: c.field.charAt is not a function In the last line of the above example, c.field is actually 42 but the type systems claims it is of type string and thus allows accessing member charAt of type string which is undefined at runtime the actual value 42. The next example shows why cases 2 and 3 (i.e. new expressions and instances of final classes) have to be handled in a more restrictive manner than case 1 (i.e. object literals). Problem 2 of optional fields class C { public field?: string; } class D {} let c: ~C; c = new D(); // error: new expression but D is missing setter c.field = "hello"; In the previous code, if c = new D() were allowed, we would add a new property field to the instance of class D in the last line, which N4JS aims to avoid in general, unless unsafe language features such as dynamic types are being employed. #### 5.2.7. Static Members Static data fields, field accessors and methods are quite similar to instance members, however they are not members of instances of the type but the type itself. They are defined similarly to instance members except that they are specified with the modifier static. Since they are members of the type, the this keyword is not bound to instances of the class, but again to the type itself. This is similar as in ECMAScript 6 ([ECMA15a(p.14.5.15)]). Since static members are not instance but type members, it is even possible that a static member has the same name as an instance member. Note that static members are not only allowed in classes but also in interfaces, but there are important differences (for example, no inheritance of static members of interfaces, cf. Section Static Members of Interfaces). Req. IDE-65: Static member not abstract (ver. 1) For a static field accessor or method S, the following constraint must hold: • $s.static⇔¬s.abstract$ Like instance methods, static methods of classes are inherited by subclasses and it is possible to override static methods in subclasses. The very same override constraints are valid in this case as well. ##### 5.2.7.1. Access From and To Static Members Req. IDE-66: Accessing Static Members (ver. 1) Let M be a static member of class C. Except for write-access to fields, which will be explained later, you can access M via: 1. The class declaration instance, i.e. the classifier or constructor type, constructor{C}, i.e. C.m 2. The class declaration instance of a subtype, i.e. the classifier or constructor type, i.e. D.m, if D is a subclass of C. 3. v.m, if v is a variable of type C (i.e. classifier type as defined in Constructor and Classifier Type) or a subtype thereof. 4. this.m inside the body of any static method declared in C or any sub-class of C. 5. Via a type variable T which upper bound is a subclassof C e.g., function <T extends C> f(){T.m} Req. IDE-67: Static Member Access (ver. 1) It is not possible to access instance members from static members. This is true in particular for type variables defined by a generic classifier. For static data fields and static setter f the following constraint must hold: • For every assign expression $assignExpr$ with $f.static\wedge assignExpr.left=T.f⇒T=f.owner$. • For every writing unary expression $u$ with $u.op\in \left\{++,--\right\}\wedge f.static\wedge u.expression=T.f⇒T=f.owner$. In the special case of m being a static data field, write-access is only possible via the defining type name C.m. In the list above, only the first line can be used when assigning values to a field. Note that this only applies to fields and set-accessors.[36] It is even possible to call a static field accessor or method of a class using dynamic polymorphism, as demonstrated in the following example: Example 45. Static members of classes, inheritance and polymorphism class A { static m(): void { console.log('A#m'); } static foo(): void { console.log('A#foo'); } static bar(): void { this.foo(); } } class B extends A { @Override static foo(): void { console.log('B#foo'); } } A.m(); // will print "A#m" B.m(); // will print "A#m" (m is inherited by B) var t: type{A} = A; t.foo(); // will print "A#foo" t = B; t.foo(); // will print "B#foo" // using 'this': A.bar(); // will print "A#foo" B.bar(); // will print "B#foo" This is quite different from Java where static methods are not inherited and references to static methods are statically bound at compile time depending on the declared type of the receiver (and not its value): Example 46. Static members in Java // !!! JAVA CODE !!! public class C { static void m() { System.out.println("C#m"); } public static void main(String[] args) { final C c = null; c.m(); // will print "C#m" (no NullPointerException at runtime) } } ##### 5.2.7.2. Generic static methods It is not possible to refer to type variables of a generic class, as these type variables are never bound to any concrete types. A static method can, however, be declared generic. Generic static methods are defined similarly to generic instance methods. Since they cannot refer to type variables of a generic class, the constraint to avoid type variables with equal names (see [Req-IDE-55]) does not need to hold for generic static methods. ##### 5.2.7.3. Static Members of Interfaces Data fields, field accessors and methods of interfaces may be declared static. A few restrictions apply: Req. IDE-69: Static Members of Interfaces (ver. 1) 1. Static members of interfaces may only be accessed directly via the containing interface’s type name (this means, of the four ways of accessing static members of classes defined in [Req-IDE-66] above, only the first one applies to static members of interfaces). 2. The this literal may not be used in static methods or field accessors of interfaces and it may not be used in the initializer expression of static fields of interfaces. See [Req-IDE-173]. 3. The super literal may not be used in static methods or field accessors of interfaces (in fact, it may not be used in interfaces at all, cf. [Req-IDE-123]). Note that the this type as a return type for methods is only allowed for instance methods and as an argument type only in constructors (structurally typed). There is no need to disallow these cases for static interface methods in the constraints above. In general, static members may not be abstract, cf. [Req-IDE-46], which applies here as well. Static methods and field accessors of interfaces, therefore, always have to provide a body. Static members of interfaces are much more restricted than those of classes. Compare the following example to Static Polymorphism for classes above: Example 47. Static members of interfaces interface I { static m(): void { console.log('I#m'); } } interface J extends I {} I.m(); // prints "I#m" J.m(); // ERROR! (m is not inherited by J) var ti: type{I} = I; ti.m(); // ERROR! (access to m only allowed directly via type name I) ti = J; ti.m(); // ERROR! (access to m only allowed directly via type name I) The last line in is the reason why access to static members has to be restricted to direct access via the type name of the containing interfaces. #### 5.2.8. Redefinition of Members Members defined in classes or interfaces can be redefined by means of being overridden or implemented in subclasses, sub-interfaces, or implementing classes. Fields and methods with default implementation defined in interfaces can be consumed by the implementor, but certain restrictions apply. Req. IDE-70: Override Compatible (ver. 1) A member M is override compatible to a member S if and only if the following constraints hold: 1. The name and static modifiers are equal: $M.name=S.name\wedge M.static=S.static$ 2. The metatypes are compatible: $\mu \left(S\right)=\text{Method}\phantom{\rule{3.0mm}{0ex}}\to \phantom{\rule{3.0mm}{0ex}}\mu \left(M\right)=\text{Method}$ $\mu \left(S\right)=\text{Field}\phantom{\rule{3.0mm}{0ex}}\to \phantom{\rule{3.0mm}{0ex}}\mu \left(M\right)\in \text{Field, Getter, Setter}$ $\mu \left(S\right)=\text{Getter}\phantom{\rule{3.0mm}{0ex}}\to \phantom{\rule{3.0mm}{0ex}}\mu \left(M\right)\in \text{Field, Getter}$ $\mu \left(S\right)=\text{Setter}\phantom{\rule{3.0mm}{0ex}}\to \phantom{\rule{3.0mm}{0ex}}\mu \left(M\right)\in \text{Field, Setter}$ 3. The overridden member must not be declared final: $¬S.final$ 4. Overridden member declared const can only be overridden (redefined) by const members: $S.const⇔M.const$ 5. It is not possible to override a non-final / non-const field or a setter with a final / const field: $\left(\mu \left(S\right)=\text{Field}\wedge ¬\left(S.final\vee S.const\right)\right)\vee \mu \left(S\right)=\text{Setter}⇒¬\left(\mu \left(M\right)=\text{Field}\wedge \left(M.final\vee M.const\right)\right)$ 6. It is not possible to override a non-abstract member with an abstract one: $¬M.abstract\vee S.abstract$ 7. The types are compatible: $\left(\mu \left(M\right)\in \text{Method, Getter, Field}\wedge \mu \left(S\right)\ne \text{Setter}\right)\phantom{\rule{3.0mm}{0ex}}\to \Gamma ⊢M\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}S$ $\left(\mu \left(M\right)\in \text{Setter, Field}\wedge \mu \left(S\right)\ne \text{Getter}\wedge ¬S.const\right)\phantom{\rule{3.0mm}{0ex}}\to \Gamma ⊢S\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}M$4 8. The access modifier is compatible: $M.acc\ge S.acc$ We define a relation $overrideCompatible\left(M,S\right)$ accordingly. Members overriding or implementing other members must be declared as override. If a member does not override another, however, it must not be declared as override. Req. IDE-71: Non-Override Declaration (ver. 1) If and only if a member M of a class C (extending a class S and interfaces ${I}_{i}$) does not override or implement another member, then it must not be declared as override. That is the following constraint must hold: $\phantom{\rule{3.0mm}{0ex}}¬M.override$ $\phantom{\rule{3.0mm}{0ex}}\wedge$ $\phantom{\rule{3.0mm}{0ex}}\nexists {M}^{\text{'}}\in C.super.members\cup {\bigcup }_{i=1}^{n}{I}_{i}.members:$ $\phantom{\rule{3.0mm}{0ex}}{M}^{\text{'}}.name=M.name\wedge {M}^{\text{'}}.static=M.static$ $\phantom{\rule{3.0mm}{0ex}}\wedge {M}^{\text{'}}.acc>\text{private}$ ##### 5.2.8.1. Overriding of Members In general, the N4JS platform supports overriding members by redefining them in sub-classes. This definition allows for overriding of static methods, but it does not apply to constructors because $C.ctor\notin C.ownedMethods$. Req. IDE-72: Overriding Members (ver. 1) Given a class C and a superclass $Sup$. If for an instance or static member M defined in C a member S exists with null then we call M the overriding member and S the overridden member. In that case the following constraints must hold: 1. S must be accessible from C 2. M must be override compatible with S: $overrideCompatible\left(M,S\right)$ 3. If S is a field and M is an accessor, then an additional accessor ${M}^{\text{'}}$ must exists so that $M,{M}^{\text{'}}$ are an accessor pair for S: $\phantom{\rule{3.0mm}{0ex}}\mu \left(S\right)=\text{Field}\wedge \mu \left(M\right)=Accessor$ $\phantom{\rule{3.0mm}{0ex}}\to \exists {M}^{\text{'}}\in C.member:$ $\phantom{\rule{3.0mm}{0ex}}\phantom{\rule{4.0em}{0ex}}overrideCompatible\left({M}^{\text{'}},S\right)\wedge \left\{\mu \left(M\right),\mu \left({M}^{\text{'}}\right)\right\}=\text{Getter,Setter}$ 4. M must be declared as override: M.override Remarks: • An overridden method, getter, or setter may called via super. Note that this is not possible for fields. • There is no ’hiding’ of fields as in Java, instead there is field overriding. • It is not possible to override a field with a consumed getter and an overridden setter, because the getter is not consumed if there exists a field in a superclass. In this case, the consuming and extending class needs to define the accessor pair explicitly. The same is true for other combination of accessors and fields. • Overriding a field usually makes only sense if the visibility of the field is to be increased. ##### 5.2.8.2. Implementation of Members For the following constraints, we define two helper sets ${M}_{C}$ and ${M}_{I}$ as follows: Given a C, and interface ${I}_{1},.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}},{I}_{n}$, implemented by C, with ${M}_{C}\phantom{\rule{3.0mm}{0ex}}=C.ownedMembers\cup \left\{m\in C.superType.members|m.acc>\text{private}\right\}$ ${M}_{I}\phantom{\rule{3.0mm}{0ex}}={\bigcup }_{i=1}^{n}{I}_{i}.members$ Note that these sets already contain only non-private data fields. ###### 5.2.8.2.1. Member Consumption A member M defined in an interface I is consumed by an implementor C, if it becomes a member of the class, that is, $M\in C.members$. A member M is consumed if there is no member defined in the implementor with the same name and if there is no non-private, non-abstract member with that name inherited by the implementor from its superclass. [37] If the implementor defines the member itself, then the member is implemented rather than consumed. The concrete rules are described in the following; It is not always possible to directly consume a member. In general, a rather conservative strategy is used: if two implemented interfaces define the same (non-abstract) member then the implementor must redefine the member in order to solve conflicts. Even if the two conflicting members have the same types, the implementor must redefine them as we generally assume semantic differences which the consumer has to be aware of. Data fields defined in interfaces, in particular, are assumed to be concrete. It is not, therefore, possible to consume a field defined in two implemented interfaces. Req. IDE-73: Consumption of Interface Members (ver. 1) Given a classifier C [38], and interfaces ${I}_{1},.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}},{I}_{n}$ implemented (or extended) by C, and sets ${M}_{C}$ and ${M}_{I}$ as defined in [interface_and_class_member_sets]. A non-static member M defined in any interface ${I}_{i}$ is merged into the consumer (C), if for all other (possible) members ${M}^{\text{'}}$ of C $\forall {M}^{\text{'}}\in {M}_{C}\cup {M}_{I}\setminus \left\{M\right\}:M.name={M}^{\text{'}}.name\wedge ¬{M}^{\text{'}}.static$ the following constraints hold: 1. The other member’s meta type matches the meta type of the merge candiate: $\mu \left(M\right)=\text{Method}\phantom{\rule{3.0mm}{0ex}}\to \mu \left({M}^{\text{'}}\right)=\text{Method}$ $\mu \left(M\right)\ne \text{Method}\phantom{\rule{3.0mm}{0ex}}\to \mu \left({M}^{\text{'}}\right)\in \text{Field, FieldAccessor}$ 2. The other member is abstract and not owned by the consumer: $\phantom{\rule{3.0mm}{0ex}}\mu \left(M\right)=\mu \left({M}^{\text{'}}\right)\vee \mu \left(M\right)=\text{Field}$ $\phantom{\rule{3.0mm}{0ex}}\phantom{\rule{2.0em}{0ex}}\to {M}^{\text{'}}.abstract\wedge {M}^{\text{'}}\notin C.ownedMembers$ 3. The merge candidate’s access modifier is not less than the modifier of the other member: $\phantom{\rule{3.0mm}{0ex}}\mu \left(M\right)=\mu \left({M}^{\text{'}}\right)\vee \mu \left(M\right)=\text{Field}$ $\phantom{\rule{3.0mm}{0ex}}\phantom{\rule{2.0em}{0ex}}\to M.acc\ge {M}^{\text{'}}.acc$ 4. The merge candidate’s type compatible with the other member: $\mu \left(M\right)\in \left\{\text{Method, Getter, Field}\right\}\wedge \mu \left({M}^{\text{'}}\right)\ne \text{Setter}\phantom{\rule{3.0mm}{0ex}}\to \Gamma ⊢M\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}{M}^{\text{'}}$ $\mu \left(M\right)\in \left\{\text{Setter, Field}\right\}\wedge \mu \left({M}^{\text{'}}\right)\ne \text{Getter}\phantom{\rule{3.0mm}{0ex}}\to \Gamma ⊢{M}^{\text{'}}\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}M$ ###### 5.2.8.2.2. Member Implementation Req. IDE-74: Implementation of Interface Members (ver. 1) For any non-static abstract member M defined in an interface I implemented (or extended) by a classifier C, M must be accessible from C and one or two member(s) in C must exist which are implementation-compatible with M. The implementing member(s) must be declared as override if they are directly defined in the consumer. 1. M must be accessible from C. 2. An implementation-compatible member ${M}^{\text{'}}$ must exist in C: 1. if M is not a field: $\mu \left(M\right)\ne \text{Field}\phantom{\rule{3.0mm}{0ex}}\to$ $\phantom{\rule{3.0mm}{0ex}}\exists {M}^{\text{'}}\in C.members:$ $\phantom{\rule{3.0mm}{0ex}}\phantom{\rule{3.0em}{0ex}}overrideCompatible\left({M}^{\text{'}},M\right)$ $\phantom{\rule{3.0mm}{0ex}}\phantom{\rule{3.0em}{0ex}}\wedge \left({M}^{\text{'}}\in C.ownedMembers\to {M}^{\text{'}}.override\right)$ 2. if M is a field, then either an implementation-compatible field ${F}^{\text{'}}$ or accessor pair ${G}^{\text{'}},{S}^{\text{'}}$ must exist: $\mu \left(M\right)=\text{Field}\phantom{\rule{3.0mm}{0ex}}\to$ $\phantom{\rule{3.0mm}{0ex}}\exists {F}^{\text{'}}\in C.fields:$ $\phantom{\rule{3.0mm}{0ex}}\phantom{\rule{3.0em}{0ex}}overrideCompatible\left({F}^{\text{'}},M\right)$ $\phantom{\rule{3.0mm}{0ex}}\phantom{\rule{3.0em}{0ex}}\wedge \left({F}^{\text{'}}\in C.ownedMembers\to {F}^{\text{'}}.override\right)$ $\phantom{\rule{3.0mm}{0ex}}\vee$ $\phantom{\rule{3.0mm}{0ex}}\exists {G}^{\text{'}}\in C.getters,{S}^{\text{'}}\in C.setters:$ $\phantom{\rule{3.0mm}{0ex}}\phantom{\rule{3.0em}{0ex}}overrideCompatible\left({G}^{\text{'}},M\right)$ $\phantom{\rule{3.0mm}{0ex}}\phantom{\rule{3.0em}{0ex}}\wedge overrideCompatible\left({S}^{\text{'}},M\right)$ $\phantom{\rule{3.0mm}{0ex}}\phantom{\rule{3.0em}{0ex}}\wedge \left({G}^{\text{'}}\in C.ownedMembers\to {G}^{\text{'}}.override\right)$ $\phantom{\rule{3.0mm}{0ex}}\phantom{\rule{3.0em}{0ex}}\wedge \left({S}^{\text{'}}\in C.ownedMembers\to {S}^{\text{'}}.override\right)$ Methods defined in interfaces are automatically declared abstract if they do not provide a default implementation. This can also be expressed explicitly via adding the abstract modifier. If a class implementing an abstract interface does not implement a method declared in the interface, the class needs to be declared abstract (cf. Abstract Classes). Consequences for method implementation: 1. It may be require the implementor to explicitly define a method in order to solve type conflicts produced by methods of different interfaces with same name but different signatures. 2. Methods in an implementor cannot decrease the accessibility of methods from implemented interfaces, that is $\phantom{\rule{3.0mm}{0ex}}\forall M\in C.methods,{M}^{\text{'}}\in {I}_{i}.methods\left(i=1.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}n\right):$ $\phantom{\rule{3.0mm}{0ex}}\phantom{\rule{2.0em}{0ex}}M.name={M}^{\text{'}}.name\to M.acc\ne private⇒M.acc\ge {M}^{\text{'}}.acc$ 3. Methods in the implementor must be a supertype [39] of methods from implemented interfaces. That is to say the implemented methods are override-compatible. 4. There may be several methods ${M}_{1},.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}}.\phantom{\rule{1.0mm}{0ex}},{M}_{n}$ defined in different implemented interfaces and a single owned method ${M}^{\text{'}}$ in ${M}_{C}$. In this case, the above constraints must hold for all methods. In particular, ${M}^{\text{'}}$’s signature must conform to all conflicting methods’ signatures. This is possible by using union types for the arguments and an intersection type as return type. Such a method ${M}^{\text{'}}$ is said to resolve the conflict between the implemented (and also inherited) methods. 5. Since abstracts methods may become part of the implementor methods, the implementor must either define these methods or it must be declared abstract itself. Since interfaces are abstract by default, responsibility for implementing abstract methods is passed on to any implementor of interfaces. 6. If two implemented interfaces provide (non-abstract) members with the same name, they are not automatically consumed by the implementor even if the types would be similar. In these cases, the implementor has to redefine the members in order to be aware of possible semantic differences. There is currently no separate annotation to indicate that methods are implemented or overridden in order to solve conflicts. We always use the @Override annotation. Example 48. Method Consumption Consumption of methods shows simple examples of above rules. Assuming that class C extends super class S and implements interface I1 and I2: class C extends S implements I1, I2 {...} The columns describe different scenarios in which a method (with same name) is defined in different classifiers. We assume that the defined methods are always non-abstract (i.e. have default implementations), non-private and have the same signature. The last row shows which method will be actually used in class C. If the method is defined in class C, and if this method is printed bold, then this means that the method is required to be defined in C in order to solve conflicts. Interface I1 Interface I2 class S class C $\in C.members$ MI1 MI1 MI1 MI1 MI1 MI1 MI2 MI2 MI2 MS MS MS MC MC MC MI1 MC MC MS MS MC Consuming Field Initializers Aside from the fields themselves, an implementor always consumes the field initialization if the field is consumed – this is how the consumption is noticed at runtime. Example 49. Field and Field Initializer Consumption /* XPECT output ~~~ <== stdout: s: C , t: D ,u: I1 ,v: I2 stderr: ==> ~~~ */ interface I0 { v: string = "I0"; } interface I1 { s: string = "I1"; t: string = "I1"; u: string = "I1"; } interface I2 extends I1, I0 { @Override t: string = "I2"; @Override v: string = "I2"; } class C { s: string = "C"; } class D extends C implements I1, I2 { @Override t: string = "D"; } var d = new D(); console.log( "s:", d.s, ", t:", d.t, ",u:", d.u, ",v:", d.v ) We expect the following output (for each field): • d.s = "C" : s: is inherited from C, so it is not consumed from I1 (or I2). Consequently, the initializer of s in C is used. • d.t = "D": t is defined in D, solving a conflict stemming from the definition of t in I1 and I2. Thus, the initializer of t in D is used. • d.u = "I1" : u is only defined in I1, thus the initializer defined in I1 is used. • d.v = "I2" : v is overridden in I2, so is the field initializer. This is why d.v must be assigned to I2 and not I0. ### 5.3. Structural Typing In general, N4JS uses nominal typing. This is to say that a duck is a duck only if it is declared to be a duck. In particular when working with external APIs, it is more convenient to use structural or duck typing. That is, a thing that can swim and quacks like a duck, is a duck. Interfaces or classes can be used for this purpose with a typing strategy modifier. Given a type $T$, the simple ~ (tilde) can be added to its declaration (on definition-site) or in a reference (on use-site) to indicate that the type system should use structural typing rather than nominal typing.[40] This means that some other type must provide the same members as type $T$ to be deemed a structural subtype. However, the operator cannot be used anymore with the type or reference as this operator relies on the declaration information (or at least the closest thing available at runtime). In this case, $T$ is, therefore, always a structural subtype of ~T. Sometimes it is convenient to refer only to the fields of a classifier, for example when the initial field values are to be provided in a variable passed to the constructor. In that case, the type can be modified with ~~ (two tildes). This is only possible on use-site, i.e. on type references. Furthermore, only on the use-site, it is possible to consider only either readable or writable or fields by using the read-only ~r~ or write-only ~w~ structural field typing. For initialization blocks, it is even possible to use structural initializer field typing via the ~i~ operator. #### 5.3.1. Syntax Structural typing is specified using the typing strategy modifier. There are two modifiers defined; one for definition-site and one for use-site structural typing. Structural Type Operator and References TypingStrategyUseSiteOperator returns TypingStrategy: '~' ('~' | STRUCTMODSUFFIX)?; TypingStrategyDefSiteOperator returns TypingStrategy: '~'; terminal STRUCTMODSUFFIX: ('r' | 'i' | 'w') '~' ; ParameterizedTypeRefStructural returns ParameterizedTypeRefStructural: definedTypingStrategy=TypingStrategyUseSiteOperator declaredType=[Type|TypeReferenceName] (=> '<' typeArgs+=TypeArgument (',' typeArgs+=TypeArgument)* '>')? (=> 'with' '{' astStructuralMembers+=TStructMember* '}')? ; ThisTypeRefStructural returns ThisTypeRefStructural: definedTypingStrategy=TypingStrategyUseSiteOperator 'this' ('with' '{' astStructuralMembers+=TStructMember* '}')? ; #### 5.3.2. Definition Site Structural Typing An interface or class can be defined to be used with structural typing by adding the structural modifier to its definition (or, in case of external classes, to the declaration). This changes the default type system strategy from nominal to structural typing for that type. That means that all types with the same members as the specified type are subtypes of that type, except for subtypes of N4Object. In the latter case, programmers are enforced to nominal declare the type relation. If a type $T$ is declared as structural at its definition, $T.defStructural$ is true. Req. IDE-75: Definition Site Structural Typing (ver. 1) 1. The structurally defined type cannot be used on the right hand side of the instanceof operator: 2. A type $X$ is a subtype of a structurally defined type $T$ either 1. if it is not a subtype of N4Object [41] but it contains all public, non-static members of that type $X\phantom{\rule{1.0mm}{0ex}}\nless \text{:}\phantom{\rule{1.0mm}{0ex}}\left\{\text{N4Object}\right\}\phantom{\rule{3.0em}{0ex}}T.defStructural\right\}$ $\forall m\in T.members,m.acc=\text{public},¬m.static,m\ne T.ctor:\right\}$ $\phantom{\rule{1.0em}{0ex}}\exists {m}^{\text{'}}\in X.members:\right\}$ $\phantom{\rule{2.0em}{0ex}}{m}^{\text{'}}.acc=\text{public}\wedge ¬{m}^{\text{'}}.static\wedge {m}^{\text{'}}.name=m.name\right\}$ $\phantom{\rule{2.0em}{0ex}}\wedge \Gamma ⊢{m}^{\text{'}}\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}m\right\}$ $\frac{\phantom{\rule{2.0em}{0ex}}\wedge \mu \left(m\right)=\text{Field}⇒\Gamma ⊢m\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}{m}^{\text{'}}}{\phantom{\rule{13.0em}{0ex}}\Gamma ⊢X\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}T\phantom{\rule{13.0em}{0ex}}}$ or 2. if it is a subtype of N4Object which explicitly extends or implements the structurally defined type. $\frac{X\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}\left\{\text{N4Object}\right\}\phantom{\rule{3.0mm}{0ex}}T.defStructural\phantom{\rule{3.0mm}{0ex}}T\in X.superType{s}^{*}}{\Gamma ⊢X\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}T}$ 3. A structurally defined type $T$ is implicitly derived from Object if no other type is specified. In particular, a structurally defined type must not be inherited from $\frac{T.defStructural}{\Gamma ⊢T\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}\text{Object}}$ $T.defStructural⇒\Gamma ⊢T\phantom{\rule{1.0mm}{0ex}}\nless \text{:}\phantom{\rule{1.0mm}{0ex}}\text{N4Object}\wedge \text{N4Object}\notin T.superType{s}^{*}$ Example 50. Declaration Site Structural Typing The following snippet demonstrates the effect of definition-site structural types by comparing them to nominal declared types: Declaration Site Structural Typing interface ~Tilde { x; y; } interface Nominal { x; y; } class C { public x; public y;} class D extends C implements Tilde { } function f(p: Tilde) {} function g(p: Nominal) {} f(new C()); // error: nominal typing, C does not implement ~Tilde f(new D()); // ok, D is a nominal subtype (as it implements Tilde) f({x:10,y:10}); // ok: Tilde is used with structural typing for non-N4-classifiers Definition site structural typing may lead to unexpected results. For example; class C{} class ~E extends C{} It may be unexpected, but E is not a subtype of C, i.e. $E\phantom{\rule{1.0mm}{0ex}}\nless \text{:}\phantom{\rule{1.0mm}{0ex}}C$! E.g., instanceof won’t work with E, while it works with C! #### 5.3.3. Use-Site Structural Typing Use-site structural typing offers several typing strategy modifiers to define the accessability of public properties of classes and interfaces. They can be used e.g. on variable declarations like this: var c : ~C. The table Available Fields of Structural Types shows which properties of structural types can be accessed in the different type strategies. For example, when using the write-only structural strategy (i.e. $~\text{w}~\text{X}$), only the writeable fields, can be accessed for writing. In the table, the term field to both, public datafields and accessors of type $X$. Non-public properties are never accessable in use-site structural types. In any field-structural type, methods of the referenced nominal type $X$ are not available. The initializer structural typing provides readable fields for every writeable field of the references type. Table 9. Available Fields of Structural Types Property of $X$ $~\text{X}$ $~~\text{X}$ $~\text{r}~\text{X}$ $~\text{w}~\text{X}$ $~\text{i}~\text{X}$ public method $\varnothing$ $\varnothing$ $\varnothing$ $\varnothing$ public writable field $\varnothing$ $\varnothing$ public readable field $\varnothing$ writable fields Multiple structural typing strategies can be nested when there are multiple use sites, like in the example Nested Structural Typing Strategies below at the locations ST1 and ST2. In the example, the datafield a.field has the nested structural type ~r~ ~i~ A and thus the datafield a.field.df is readable. Nested structural types are evaluated on the fly when doing subtype checks. Example 51. Nested Structural Typing Strategies class A { public df : string; } interface I<T> { public field : ~r~T; // ST1 } var a : ~i~A; // ST2 The following example demonstrates the effect of the structural type modifiers: Example 52. Effect of structural type modifiers on use-site Let’s assume the type defined on the left. The following pseudo code snippets explicitly list the type with its members virtually created by a structural modifier. Note that this is pseudo code, as there are no real or virtual types created. Instead, only the subtype relation is defined accordingly: Effect of structural type modifiers on use-site Effect of structural type modifiers on use-site var c:C class C { private x; public y; public f() private g() public get z():Z public set z(z:Z) } interface I { a; func(); } var cstructural:~C class cstructural { public y; public f() public get z():Z public set z(z:Z) } interface ~I { a; func(); } var cfields:~~C class cfields { public y; public get z():Z public set z(z:Z) } interface ~~I { a; } Type Structural Type Structural Field Type Structural Read-only Field Type Structural Write-only Field Type Structural Initializer Field Type var crofields:~r~C class crofields { public get y():Y public get z():Z } interface ~r~I { public get a():A } var cwofields:~w~C class cwofields { public set y(y:Y) public set z(z:Z) } interface ~w~I { public set a(a:A) } var cinitfields:~i~C class cinitfields { public get y():Y public get z():Z } interface ~i~I { public get a():A } Note that even if a type is defined without the structural modifier, it is not possible to use instanceof for variables declared as structural, as shown in the next example: Type Structural Type Structural Field Type class C {..} interface I {..} foo(c: C, i: I) { c instanceof C; // ok c instanceof I; // ok } class C {..} interface I {..} foo(c: ~C, i: ~I) { c instanceof C; // error c instanceof I; // error } class C {..} interface I {..} foo(c: ~~C, i: ~~I) { c instanceof C; // error C instanceof I; // error } Within this spec, we define the following attributes of a type reference $T$: • If a type is referenced with the structural type modifier ~ , the property $T.refStructural$ is true. • If a type is referenced with the structural field type modifier ~~, the property $T.refStructuralField$ is true. • If a type is referenced with the structural read-only field type modifier ~r~, the property $T.refStructuralReadOnlyField$ is true. • If a type is referenced with the structural write-only field type modifier ~w~, then the property $T.refStructuralWriteOnlyField$ is true. • If a type is referenced with the structural initializer field type modifier ~i~, then the property $T.refStructuralInitField$ is true. • We use $T.isStructural$ to simply refer any structural typing, i.e.+ $T.isStructural=T.refStructural\vee T.refStructuralField\vee$T.refStructuralReadOnlyField \lor T.refStructuralWriteOnlyField || T.refStructuralInitField || T.defStructural$
• We use $T.isNominal$ as the opposite of $T.isStructural$, i.e.
$T.isNominal=¬T.isStructural$
We call the following:
• $T$ the (nominal) type T,
• $~\text{T}$ the structural version of $T$,
• $~~\text{T}$ the structural field version of $T$,
• $~\text{r}~\text{T}$ the structural read-only field,
• $~\text{w}~\text{T}$ the structural write-only field and
• $~\text{i}~\text{T}$ the structural initializer field version of $T$.
Req. IDE-76: Use-Site Structural Typing (ver. 1)
1. The structural version of a type is a supertype of the nominal type:
$T\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}:~\text{T}$
2. The structural field version of a type is a supertype of the structural type:
$~\text{T}\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}:~~\text{T}$
3. The structural read-only field version of a type is a supertype of the structural field type:
$~~\text{T}\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}:~\text{r}~\text{T}$
4. The structural write-only field version of a type is a supertype of the structural field type:
$~~\text{T}\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}:~\text{w}~\text{T}$
5. The structural (field) version of a type cannot be used on the right hand side of the instanceof operator:
$\phantom{\rule{3.0mm}{0ex}}\phantom{\rule{3.0em}{0ex}}⇒¬\left(T.refStructural$
$\phantom{\rule{3.0mm}{0ex}}\phantom{\rule{6.0em}{0ex}}\vee T.refStructuralField$
$\phantom{\rule{3.0mm}{0ex}}\phantom{\rule{6.0em}{0ex}}\vee T.refStructuralReadOnlyField$
$\phantom{\rule{3.0mm}{0ex}}\phantom{\rule{6.0em}{0ex}}\vee T.refStructuralWriteOnlyField$
$\phantom{\rule{3.0mm}{0ex}}\phantom{\rule{6.0em}{0ex}}\vee T.refStructuralInitField\right)$
That is, the following code will always issue an error: x instanceof ~T [42].
6. A type $X$ is a subtype of a structural version of a type $~\text{T}$, if it contains all public, non-static members of the type $T$: [43]
$\forall m\in T.members,m.owner\notin \text{N4Object},m.acc=\text{public},¬m.static,m\ne T.ctor:$
$\phantom{\rule{1.0em}{0ex}}\exists {m}^{\text{'}}\in X.members:$
$\phantom{\rule{2.0em}{0ex}}{m}^{\text{'}}.acc=\text{public}\wedge ¬{m}^{\text{'}}.static\wedge {m}^{\text{'}}.name=m.name$
$\frac{\phantom{\rule{2.0em}{0ex}}\wedge \Gamma ⊢{m}^{\text{'}}\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}\Gamma ⊢m}{\phantom{\rule{13.0em}{0ex}}\Gamma ⊢X\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}\Gamma ⊢~\text{T}\phantom{\rule{13.0em}{0ex}}}$
7. A type $X$ is a subtype of a structural field version of a type $~~\text{T}$, if it contains all public, non-static fields of the type $T$. Special cases regarding optional fields are described in Optional Fields.
$\forall m\in T.fields,m.owner\notin \text{N4Object},m.acc=\text{public},¬m.static$
$\phantom{\rule{3.0em}{0ex}}{m}^{\text{'}}.acc=\text{public}\wedge ¬{m}^{\text{'}}.static\wedge {m}^{\text{'}}.name=m.name$
$\phantom{\rule{3.0em}{0ex}}\wedge \Gamma ⊢{m}^{\text{'}}:{T}_{m}\wedge \Gamma ⊢m:{T}_{{m}^{\text{'}}}\wedge {T}_{m}={T}_{{m}^{\text{'}}}\right\}$
$\frac{\phantom{\rule{3.0em}{0ex}}\wedge {m}^{\text{'}}.assignability\ge m.assignability}{\phantom{\rule{13.0em}{0ex}}\Gamma ⊢X\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}~~\text{T}\phantom{\rule{13.0em}{0ex}}}$
8. A type $X$ is a subtype of a structural read-only field version of a type $~\text{r}~\text{T}$, if it contains all public and non-static readable fields of the type $T$. Special cases regarding optional fields are described in Optional Fields.
$\forall m\in T.fields\cup T.getters,m.owner\notin \text{N4Object},m.acc=\text{public},¬m.static$
$\phantom{\rule{3.0em}{0ex}}{m}^{\text{'}}.acc=\text{public}\wedge ¬{m}^{\text{'}}.static\wedge {m}^{\text{'}}.name=m.name$
$\phantom{\rule{3.0em}{0ex}}\wedge \Gamma ⊢{m}^{\text{'}}:{T}_{m}\wedge \Gamma ⊢m:{T}_{{m}^{\text{'}}}\wedge {T}_{m}={T}_{{m}^{\text{'}}}$
$\frac{\phantom{\rule{3.0em}{0ex}}\wedge {m}^{\text{'}}.assignability\ge m.assignability}{\phantom{\rule{13.0em}{0ex}}\Gamma ⊢X\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}~\text{r}~\text{T}\phantom{\rule{13.0em}{0ex}}}$
9. A type $X$ is a subtype of a structural write-only field version of a type $~\text{w}~\text{T}$, if it contains all public and non-static writable fields of the type $T$. Special cases regarding optional fields are described in Optional Fields.
$\forall m\in T.fields\cup T.setters,m.owner\notin \text{N4Object},m.acc=\text{public},¬m.static,¬m.final$
$\phantom{\rule{3.0em}{0ex}}{m}^{\text{'}}.acc=\text{public}\wedge ¬{m}^{\text{'}}.static\wedge {m}^{\text{'}}.name=m.name$
$\phantom{\rule{3.0em}{0ex}}\wedge \Gamma ⊢{m}^{\text{'}}:{T}_{m}\wedge \Gamma ⊢m:{T}_{{m}^{\text{'}}}\wedge {T}_{m}={T}_{{m}^{\text{'}}}$
$\frac{\phantom{\rule{3.0em}{0ex}}\wedge {m}^{\text{'}}.assignability\ge m.assignability}{\phantom{\rule{13.0em}{0ex}}\Gamma ⊢X\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}~\text{w}~\text{T}\phantom{\rule{13.0em}{0ex}}}$
10. A type $X$ is a subtype of a structural field version of a type $~~\text{this}$, if it contains all public, non-static fields, either defined via data fields or field get accessors, of the inferred type of this. Special cases regarding optional fields are described in Optional Fields.
$\Gamma ⊢this:T$
$\forall m\in T.fields\cup T.setters,m.owner\notin \text{N4Object},m.acc=\text{public},¬m.static$
$\vee m.expr\ne \text{null}$
$\phantom{\rule{3.0em}{0ex}}{m}^{\text{'}}.acc=\text{public}\wedge ¬{m}^{\text{'}}.static\wedge {m}^{\text{'}}.name=m.name$
$\frac{\phantom{\rule{3.0em}{0ex}}\wedge \Gamma ⊢{m}^{\text{'}}\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}m\wedge {m}^{\text{'}}.assignability\ge m.assignability}{\phantom{\rule{13.0em}{0ex}}\Gamma ⊢X\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}~~\text{t}his\phantom{\rule{13.0em}{0ex}}}$
11. A structural field type $~~\text{T}$ is a subtype of a structural type $~\text{X}$, if $~\text{X}$ only contains fields (except methods inherited from Object) and if $~~\text{T}\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}~~\text{X}$.
$\frac{X.methods\setminus \text{Object}.methods=\varnothing \wedge \Gamma ⊢~~\text{T}\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}~~\text{X}}{\Gamma ⊢~~\text{T}\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}~\text{X}}$
12. Use-site structural typing cannot be used for declaring supertypes of classes or interfaces. That is to say that structural types cannot be used after extends, implements or with in type declarations [44].
Note that all members of N4Object are excluded. This implies that extended reflective features (cf. Reflection meta-information ) are not available in the context of structural typing. The instanceof operator is still working as described in Relational Expression, in that it can be used to check the type of an instance.
If a type $X$ is a (nominal) subtype of T, it is, of course, also a subtype of $~\text{T}$:
$\frac{\Gamma ⊢X\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}\Gamma ⊢T}{\Gamma ⊢X\phantom{\rule{1.0mm}{0ex}}<\text{:}\phantom{\rule{1.0mm}{0ex}}\Gamma ⊢~\text{T}}$
This is only a shortcut for the type inference defined above.
Req. IDE-77: Definition and Use-Site Precedence (ver. 1)
If a type is structurally typed on both definition and use-site, the rules for use-site structural typing ([Req-IDE-76]) are applied.
Example 53. Use-Site Structural Typing
The following example demonstrates the effect of the structural (field) modifier, used in this case for function parameters.
interface I { public x: number; public foo()};
class C { public x: number; public foo() {}};
function n(p: I) {}
function f(p: ~I) {}
function g(p: ~~I) {}
n(new C()); // error: nominal typing, C does not implement I
f(new C()); // ok: C is a (structural) subtype of ~I
f({x:10}); // error, the object literal does not provide function foo()
g({x:10}); // ok: object literal provides all fields of I
Example 54. Structural types variable and instanceof operator
It is possible to use a variable typed with a structural version of a type on the left hand side of the instanceof operator, as demonstrated in this example:
class C {
public x;
public betterX() { return this.x||1;}
}
function f(p: ~~C) {
if (p instanceof C) {
console.log((p as C).betterX);
} else {
console.log(p.x||1);
}
}
The following table describes the member availability of X in various typing scenarios. Such as ~~X, ~r~X, ~w~X, ~i~X.
Table 10. Member Availability in various Typing Scenarios
Member of type X ~~X ~r~X ~w~X ~i~X
private m0;
—
—
—
—
public set m1(m) { }
write
—
write
public get m2() {…}
—
public m3;
write
public m4 = 'init.m4';
write
public m5: any?;
write
read$?$
@Final public m6 = 'init.m6';
@Final public m7;
public get m8() {…}
write
public set m8(m) { }
#### 5.3.4. Structural Read-only, Write-only and Initializer Field Typing
Structural read-only, write-only and initializer field typings are extensions of structural field typing. Everything that is defined for the field structural typing must comply with these extension field typings. For the read-only type, readable fields (mutable and @Final ones) and setters are considered, for the write-only type; only the setters and mutable fields are considered. Due to the hybrid nature of the initializer type it can act two different ways. To be more precise, a type $X$ (structural initializer field type) is a supertype of $Y$ (structural initializer field type) if for each public, non-static, non-optional writable field (mutable data field of setter) of $X$, there is a corresponding, public, non-static readable field of $Y$. All public member fields with @Final annotation are considered to be mandatory in the initializer field typing @Spec constructors. The already-initialized @Final fields can be either omitted from, or can be re-initialized via, an initializer field typing @Spec style constructor.
Example 55. Subtype relationship between structural field typing
class A1 {
public s: string;
}
class A2 {
public set s(s: string) { }
public get s(): string { return null; }
}
class A3 {
@Final public s: string = null;
}
class A4 {
public get s(): string { return null; }
}
class A5 {
public set s(s: string) { }
}
A1 ~A1 ~~A1 ~r~A1 ~r~A2 ~r~A3 ~r~A4 ~r~A5 ~w~A1 ~w~A2 ~w~A3 ~w~A4 ~w~A5 ~i~A1 ~i~A2 ~i~A3 ~r~A4 ~r~A5 A1 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ~A1 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ~~A1 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ~r~A1 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ~r~A2 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ~r~A3 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ~r~A4 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ~r~A5 ✓ ✓ ✓ ✓ ✓ ~w~A1 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ~w~A2 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ~w~A3 ✓ ✓ ✓ ✓ ✓ ~w~A4 ✓ ✓ ✓ ✓ ✓ ~w~A5 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ~i~A1 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ~i~A2 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ~i~A3 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ~r~A4 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ~r~A5 ✓ ✓ ✓ ✓ ✓
#### 5.3.5. Public Setter Annotated With ProvidesInitializer
Public setters with ProvidesInitializer annotation can declare optional fields implemented by means of field accessors instead of data fields. Data fields with an initializer are treated as optional in the initializer field type.
It is important to note that it is valid to use the ProvidesInitializer annotation for setters in n4js files and not only definition files.
Example 56. Setters with @ProvidesInitializer treated as optional
class C {
private _y: int = 1;
public get y() { return this._y; }
@ProvidesInitializer
public set y(v: int) { this._y = v; }
public constructor(@Spec spec: ~i~this) { }
}
console.log(new C({}).y); // 1
console.log(new C({y: 42}).y); //24
#### 5.3.6. Structural Types With Optional Fields
Public optional fields become a member of the structural (field) type as well. To ensure the overall type safety, the semantics of optionality (e.g. on or off) depends on the context, in which the optional fields are currently being used (See Optional Fields).
#### 5.3.7. Structural Types With Access Modifier
The access modifier of the subtype have to provide equal or higher visibility.
Example 57. Access modifier in structural typing
class C {
public s: number;
}
class D {
project s: number;
}
function f(c: ~C) {}
f(new D()); // error: D is no (structural) subtype of ~C, as visibility of s in D is lower
function g(d: ~D) {}
g(new C()); // ok: C is a (structural) subtype of ~D, as visibility of s in C is greater-than-or-equal to s in D
#### 5.3.8. Structural Types With Additional Members
It is possible to add additional members when structurally referencing a declared type.
##### 5.3.8.1. Syntax
TStructMember:
TStructGetter | TStructGetterES4 | TStructSetter | TStructMethod | TStructMethodES4 | TStructField;
TStructMethod:
=>
({TStructMethod} ('<' typeVars+=TypeVariable (',' typeVars+=TypeVariable)* '>')?
returnTypeRef=TypeRef name=TypesIdentifier '(')
(fpars+=TAnonymousFormalParameter (',' fpars+=TAnonymousFormalParameter)*)? ')'
';'?;
TStructMethodES4 returns TStructMethod:
=>
({TStructMethod} ('<' typeVars+=TypeVariable (',' typeVars+=TypeVariable)* '>')?
name=TypesIdentifier '(')
(fpars+=TAnonymousFormalParameter (',' fpars+=TAnonymousFormalParameter)*)? ')'
(':' returnTypeRef=TypeRef)?
';'?;
TStructField:
(
typeRef=TypeRef name=TypesIdentifier
| name=TypesIdentifier (':' typeRef=TypeRef)?
)
';';
TStructGetter:
=> ({TStructGetter}
declaredTypeRef=TypeRef
'get'
name=TypesIdentifier)
'(' ')' ';'?;
TStructGetterES4 returns TStructGetter:
=> ({TStructGetter}
'get'
name=TypesIdentifier)
'(' ')' (':' declaredTypeRef=TypeRef)? ';'?;
TStructSetter:
=> ({TStructSetter}
'set'
name=TypesIdentifier)
'(' fpar=TAnonymousFormalParameter ')' ';'?;
TAnonymousFormalParameter:
| variadic?='...'? (=> name=TIdentifier ':') typeRef=TypeRef;
###### 5.3.8.1.1. Semantics
Req. IDE-78: Additional structural members (ver. 1)
It is only possible to add additional members to a type if use-site structural typing is used.
The following constraints must hold:
1. For all additional members defined in a structural type reference, the constraints for member overriding ([Req-IDE-72]) apply as well.
2. All additional members have the access modifier set to public.
3. Type variables must not be augmented with additional structural members.
Additional fields may be declared optional in the same way as fields in classes. The rules given in Structural Types With Optional Fields apply accordingly. Consider the following example:
Example 58. Additional optional members in structural typing
class C {
public f1: number;
}
var c1: ~C with { f3: string; } c1;
var c2: ~C with { f3: string?; } c2;
c1 = { f1:42 }; // error: "~Object with { number f1; } is not a subtype of ~C with { string f3; }."
c2 = { f1:42 }; // ok!!
Augmenting a type variable T with additional structural members can cause collisions with another member of a type argument for T. Hence, type variables must not be augmented with additional structural members like in the following example.
Example 59. Forbidden additional structural members on type variables
interface I<T> {
public field : ~T with {prop : int} // error "No additional structural members allowed on type variables."
}
Using an additional structural member on a type variable T could be seen as a constraint to T. However, constraints like this should be rather stated using an explicit interface like in the example below.
Example 60. Use explicitly defined Interfaces
interface ~J {
prop : int;
}
interface II<T extends J> {
} | 2021-03-03 00:09:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 335, "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": 0.2826981842517853, "perplexity": 4815.037697377394}, "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-2021-10/segments/1614178364932.30/warc/CC-MAIN-20210302221633-20210303011633-00596.warc.gz"} |
http://discretize.simpeg.xyz/en/master/api/generated/discretize.TreeMesh.TreeCell.html | # discretize.TreeMesh.TreeCell¶
class discretize.TreeMesh.TreeCell
Bases: object
A Cell of the TreeMesh
This cannot be created in python, it can only be accessed by indexing the TreeMesh object.
This is also the object that is passed to the user defined refine function when calling TreeMesh.refine(func).
Notes
When called as part of the refine function, only the x0, center, and h properties are valid.
Attributes: center numpy.array of length dim dim “int dimension of cell edges indexes of this cell’s edges faces indexes of this cell’s faces h numpy.array of length dim index integer index of this cell neighbors The indexes of this cell’s neighbors nodes indexes of this cell’s nodes x0 numpy.array of length dim
## Attributes¶
TreeCell.center
numpy.array of length dim
TreeCell.dim
“int dimension of cell
TreeCell.edges
indexes of this cell’s edges
Returns: list of ints
TreeCell.faces
indexes of this cell’s faces :returns: :rtype: list of ints
TreeCell.h
numpy.array of length dim width of this cell
TreeCell.index
integer index of this cell
TreeCell.neighbors
The indexes of this cell’s neighbors
Indexes of this cell’s neighbors. If a cell has more than one neighbor in a certain direction (i.e. when a level changes between adjacent cells), then that entry will also be a list of all of those neighbor indices. The list is order -x, +x, -y, +y, -z, +z. If a cell has no neighbor in that direction, the value will be -1.
Returns: list of ints or list of ints
TreeCell.nodes
indexes of this cell’s nodes
Returns: list of ints
TreeCell.x0
numpy.array of length dim | 2019-10-20 19:29:14 | {"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": 0.3618108928203583, "perplexity": 5038.829179023552}, "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/1570986718918.77/warc/CC-MAIN-20191020183709-20191020211209-00060.warc.gz"} |
http://mathoverflow.net/questions/88364/is-this-a-situation-where-triple-mutual-information-is-always-non-negative/88990 | # Is this a situation where triple mutual information is always non-negative?
Suppose I have three identically-distributed homogeneous continuous-time discrete state space Markov chains $X_1(t), X_2(t), X_3(t)$, $t\geq 0$. They evolve independently but share a common random variable $X_0$ as an initial condition. I let $$X_1=X_1(t_1), \ \ \ X_2=X_2(t_2), \ \ \ X_3=X_3(t_3)$$ for some times $t_1, t_2, t_3\geq 0$.
I want to show that $$I(X_1; X_2; X_3) \geq 0$$ where $I$ is the multivariate mutual information (or information interaction) $$I(A,B,C)= H(A,B,C) -H(A,B) - H(B,C) - H(A,C) + H(A) + H(B) + H(C)$$ where $H$ is the usual Shannon entropy.
## Background/Motivation
There are well-known situations where $I(A;B;C)<0$, a famous one being if $A$ and $B$ are independent random variables, each $\pm 1$ with probability $1/2$, and $C=AB$. But I conjecture that in the case I have described above $I(X_1;X_2;X_3)\geq 0$. I believe that the Markov chains being continuous-time and homogeneous is essential.
The more general motivation is that I want to find very general situations where multivariate mutual information is non-negative. (One well-known example is if $A,B,C$ form a Markov chain.)
-
You are touching a treacherous place that coincidentally intersects with some of my own research. I have several responses.
(1) If at all possible---stay out the synergy/redundancy waters. Instead, see if either of the two known non-negative generalizations of mutual information fit your needs. They are:
(a) "Total Correlation": http://en.wikipedia.org/wiki/Total_correlation
(b) "Dual Total Correlation": http://en.wikipedia.org/wiki/Dual_total_correlation
I personally think the Dual Total Correlation makes a lot more sense, but that's just my opinion.
If you really want to go into the synergy/redundancy waters, here's the deal---
(2) The "triple mutual information" you refer to is actually the redundant information minus the synergistic information. Therefore the triple mutual information will be nonnegative anytime redundancy >= synergy. Here's a paper that describes this, http://arxiv.org/pdf/1004.2515.pdf . Note that the above paper doesn't correctly define the "redundant information" among variables, but it does correctly show that the "triple mutual information" is redundancy minus synergy.
(3) There actually is an alternative (correct) way of measuring the redundant and synergistic information two random variables have about a third. But the "triple mutual information" is not involved. It is based on the "intrinsic conditional mutual information", described here:
http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=748999
The details aren't obvious, so if you really want to go this route email me and I'll LaTeX a paper for you for how to do this.
-
Thanks for these references, and your comments. I will contact you in an e-mail. – Paul Tupper Aug 27 '12 at 20:42
Let $X_i(t)$ be continuous time Markov chains with two states 0 and 1, such that the rate of transition from 0 to 1 and from 1 to 0 is $1$. Let $X_0$ be 0 with probability $0.9$ and 1 with probability $0.1$. Choose $t_1=t_2=t_3$ such that the probability that $X_i=X_0$ is $3/4$ for all $i$.
Then for all $i$, $$H(X_i)=0.88129 \ldots,$$ and for all $i\neq j$, $$H(X_i,X_j)= 1.75448 \ldots,$$ and $$H(X_1,X_2,X_3)=2.61812 \ldots$$ This gives $$I(X_1;X_2;X_3) = -0.00144 \ldots < 0.$$ | 2013-12-10 23:52:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8946129083633423, "perplexity": 442.81415666315297}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164027110/warc/CC-MAIN-20131204133347-00018-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://mathematica.stackexchange.com/questions/161361/merging-one-element-into-a-two-element-list | # Merging one element into a two-element list
I am trying to find how can do this really quick:
list = {{a, b}, c, {d, e}, f, {g, h}, i}
Desired output List is:
{{a,b,c},{d,e,f},{g,h,i}}
Looks like simple but it takes me plenty of time to think... sorry if this one has been asked before, I didn't find any answer yet.
• BTW, you cannot set values to List. – Αλέξανδρος Ζεγγ Dec 5 '17 at 3:44
• Yes, that's a typo, thanks for pointing it out! – leon365 Dec 5 '17 at 14:10
Also:
Append @@@ Partition[list, 2]
ArrayReshape[list, {3, 3}]
SequenceReplace[{{x_, y__}, z_} :> {x, y, z}]@list
Flatten /@ Partition[list, 2] (* thanks: Okkes Dulgerci *)
{## & @@ #, #2} & @@@ Partition[list, 2]
all give
{{a, b, c}, {d, e, f}, {g, h, i}}
Update: You can also use the undocumented 6th argument of Partition:
Partition[list, 2, 2, {1, -1}, {}, Append]
{{a, b, c}, {d, e, f}, {g, h, i}}
• Out of these 3 methods, I'm guessing that ArrayReshape would be the most efficient, right? – Sjoerd Smit Dec 5 '17 at 10:52
• Thank you! Great answers! – leon365 Dec 5 '17 at 14:03
• @SjoerdSmit, sorry for late response. Yes, based on limited tests ArrayReshape and Append+Partition combination are the fastest of the four methods. – kglr Dec 20 '17 at 5:13
• Flatten /@ Partition[list, 2] – Okkes Dulgerci Nov 10 at 2:06
• Thank you @OkkesDulgerci. – kglr Nov 10 at 21:35
BlockMap[Flatten, list, 2]
Of course, the BlockMap also can be DeveloperPartitionMap
• This should be the most concise one. – Αλέξανδρος Ζεγγ Dec 5 '17 at 8:49
• Thank you, sir! – leon365 Dec 5 '17 at 14:08
If the input list is consistently dimensioned, then this would work:
list = {{a, b}, c, {d, e}, f, {g, h}, i};
Partition[ Flatten[list, 1], 3]
I hope that this gets you what you need.
• yes, the pattern is consistent. Thanks for your help, this is work! – leon365 Dec 5 '17 at 3:28
• Partition[Flatten[list], 3] is enogh – yode Dec 5 '17 at 9:11
Method I
Step by step
Cases[list, _List]
List /@ Complement[list, %]
Join[%%, %, 2]
give respectively
{{a, b}, {d, e}, {g, h}}
{{c}, {f}, {i}}
{{a, b, c}, {d, e, f}, {g, h, i}}
Method II
Flatten /@ Transpose @ GatherBy[list, Head]
` | 2018-11-18 10:53:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2085546851158142, "perplexity": 12979.466323051913}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039744348.50/warc/CC-MAIN-20181118093845-20181118115845-00473.warc.gz"} |