content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
How to plot 3d heat map in matlab?
In the provided instructions, the "heatmap" function from the Statistics and Machine Learning Toolbox is not appropriate for creating a 3D heatmap. The "heatmap" function is used for 2D heatmaps.
For creating a 3D heatmap (also known as a 3D surface plot with color representing a third dimension), you would typically use the "surf" function in MATLAB. Here's an example of how to create a 3D
heatmap using the "surf" function:
1 % Generate some sample data for the heatmap
2 x = linspace(-2, 2, 100);
3 y = linspace(-2, 2, 100);
4 [X, Y] = meshgrid(x, y);
5 Z = X.^2 + Y.^2;
7 % Create the 3D heatmap using surf
8 surf(X, Y, Z, 'EdgeColor', 'none');
9 colormap jet;
10 colorbar;
12 % Add labels and title
13 xlabel('X-axis');
14 ylabel('Y-axis');
15 zlabel('Z-axis');
16 title('3D Heatmap');
In this code snippet:
• We first generate some sample data for the heatmap where Z represents the height of the surface at each (X, Y) coordinate.
• We then use the "surf" function to plot the 3D heatmap with the specified data. The 'EdgeColor', 'none' option is used to remove the grid lines for a smoother appearance.
• We set the colormap to 'jet' and add a colorbar for reference.
• Finally, we add labels to the axes and a title to the plot.
You can modify the sample data generation and customize the plot further based on your specific requirements. | {"url":"https://devhubby.com/thread/how-to-plot-3d-heat-map-in-matlab","timestamp":"2024-11-04T16:49:23Z","content_type":"text/html","content_length":"130450","record_id":"<urn:uuid:3513e5c3-f570-4ebe-a07e-8ec20e28924e>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00072.warc.gz"} |
Another tough integral
1 cos^-1(cos^2(pi*x))
Let f(x) = --- + -------------------,
2 3*sin(pi*x)
a fair approximation to the square wave function with an offset of 1/2 below:
______ 1 |______ ______ ______
| | | | | | | |
| | | | | | | |
| | | | | | | |
----------------+------------------------------------- x
-2 -1 0 | 1 2 3 4 5
Let us find the integral of f(x) from x=0 to x=1.
WolframAlpha returns only a numerical result, 0.995555, which suggests there is no known closed-form solution.
Let's see how some calculators handle the task:
WP 34S:
001 LBL 00
002 # pi
003 *
004 COS
005 x^2
006 ACOS
007 x<>y
008 # pi
009 *
010 SIN
012 *
013 /
014 .
016 +
017 RTN
0.9955554977636525 wp34s (v2)
0.9955554973850958 wp34s (v3.0 2458)
HP-15C LE:
0.9955555102 (FIX 7)
0.995555497542 (Fix 9)
It looks like our calculators arsenal cannot give us more than 8 correct decimal places, 0.995555497. (I haven't tried the latest WP 34S release yet).
^Edited for grammar.
Edited: 13 May 2012, 3:22 p.m.
05-13-2012, 03:59 PM
The latest WP 34S release should return the same result but you can try in double precision and compare the figures.
05-13-2012, 06:57 PM
1 cos^-1(cos^2(pi*x))
Let f(x) = --- + -------------------,
2 3*sin(pi*x)
a fair approximation to the square wave function with an offset of 1/2 below:
______ 1 |______ ______ ______
| | | | | | | |
| | | | | | | |
| | | | | | | |
----------------+------------------------------------- x
-2 -1 0 | 1 2 3 4 5
Let us find the integral of f(x) from x=0 to x=1.
Well, it doesn't seem that tough at all, a simple integration routine I wrote for high-precision returns:
essentially instantly (0:00:00, to the second), which is correct to all places shown. A second or so will get you 200 decimal places beginning with:
This might be useful to check for accuracy the results you get from calculators.
Best regards from V.
05-13-2012, 09:30 PM
Hello Valentin,
Quote: A second or so will get you 200 decimal places beginning with:
I've checked the digit frequency of your result and compared it against that of Pi for the first 156 significant digits:
your Pi
I would expect the result of the integral to have a more even digit distribution. Of course there is no solid reason behind this. Often integrals of periodic transcendental functions, depending on
the limits of integration, are not transcendental numbers themselves, much less normal. The repetition of the digit '5' in the beginning might be an indication the result is not normal (as I may have
wrongly assumed). Not that I don't trust it, but is there another integration method that may confirm your result?
Best regards,
05-14-2012, 02:02 AM
Quote: Not that I don't trust it, but is there another integration method that may confirm your result?
It might be the case that the meager 156 digits I gave aren't enough for an statistically significant digit frequency calculation so I let my routine run for a minute and there you are, about 1,000
decimal places for you to fiddle with to your heart's content
For reliable digit frequency results it's frequently the case that many thousands or even millions of decimal digits are necessary, and even more. Consider for instance this simple square root
sqrt(308642) = 555.55557777777733333335111111022222227199999...
Seeing that 44-digit result for its 6-digit integer argument you might be induced to believe that it is anything but normal, with obviously biased digit counts. However, the mirage quickly fades away
the moment you look further:
which surely looks much more "normal" and about as normal as an irrational square root is going to be.
Making judgments based on scarce data is what Wikipedia calls "Hasty generalization, a logical fallacy also known as 'the law of small numbers'"
Best regards from V.
05-14-2012, 06:03 AM
Thank you, Valentin, for taking the time to compute a staggering one thousand digits of the integral and providing yet another interesting example and explanations.
Best regards,
05-14-2012, 06:55 AM
Quote: Thank you, Valentin, for taking the time to compute a staggering one thousand digits of the integral and providing yet another interesting example and explanations.
You're welcome, Gerson, but this does not deserve any thanks, it was just a 1-min computation plus the time to copy & paste the result to post it.
What's weird about it all is that I can perform such computations, i.e., multi-digit integrals, matrix operations, general programming, etc., not just on a desktop PC, not just on a laptop, not just
on a tablet or even a mobile phone, but also on a device as simple as my eReader !
So I think handheld calculators are really, really a thing of the past and can't compete at all with modern technology. Just imagine, doing pretty convoluted math on a humble eReader which weights
even less than a typical advanced graphing calculator, doesn't tax your eyes, consumes extremely little power, and which you can carry with you at all times with no burden whatsoever.
We old-timers are now living in the future ... :D
Best regards from V.
05-14-2012, 12:24 PM
Hi Valentin,
Quote: Just imagine, doing pretty convoluted math on a humble eReader which weights even less than a typical advanced graphing calculator, doesn't tax your eyes, consumes extremely little power,
and which you can carry with you at all times with no burden whatsoever.
Yes, I second that.
Were you extrapolating, or do you have an actual implementation on your eReader? If so, native code or JavaScript?
If latter, I'd be very interested to add your integration routine to my recent BigFloat implementation, if you'd be willing to share it.
05-14-2012, 06:42 PM
Quote: Were you extrapolating, or do you have an actual implementation on your eReader? If so, native code or JavaScript?
No, no, I wasn't extrapolating, my high-precision numerical integration routine actually does run flawlessly on my Sony eReader (10.4 ounces ~ 295 grams) under DosBox for Android, as I wrote it many
years ago for an MS-DOS environment.
Matter of fact lots of pretty useful MS-DOS software runs as well (Turbo Pascal, for instance, DOS-based emulators, etc.), as well as lots of native Android applications. As long as the particular
app isn't heavy with animations (Chess, Sudoku, crosswords, word-search puzzles, comics readers, audio players, etc), it'll run fine on its beautiful E-Ink Pearl 800x600 screen.
Thanks a lot for your interest and
Regards from V.
05-15-2012, 01:58 AM
Ah, I see. Thanks for the clarification.
I thought of JavaScript because it's the one language that comes built-in with these devices, as part of their browsers.
I share your enthusiasm about what could be done with eReaders and calcs.
Edited: 15 May 2012, 2:00 a.m.
05-14-2012, 01:02 PM
So I think handheld calculators are really, really a thing of the past and can't compete at all with modern technology. Just imagine, doing pretty convoluted math on a humble eReader which
weights even less than a typical advanced graphing calculator, doesn't tax your eyes, consumes extremely little power, and which you can carry with you at all times with no burden whatsoever.
We old-timers are now living in the future ... :D
Best regards from V.
I'm not sure why the moral here is "handheld calculators are a thing of the past" rather than "handheld calculators of the future have something to learn from existing ereaders".
05-15-2012, 07:56 AM
It's the "convegence theorem"
05-15-2012, 09:17 PM
I love the idea of a capacitive touch screen for a calculator. For one, I'd like to use it literally like scratch paper for "manual" calculations. I'd also like to see a way of inputting formulas
through handwriting recognition.
And I'd like to see these two capacities work together.
05-14-2012, 06:57 AM
Thank you for your insightfulness. However, are the 1000 digits generated from a different method than to 200 digits? If not, then you have not answered Gerson's question (you have merely refuted the
basis of his query).
Quote: Making judgments based on scarce data is what Wikipedia calls "Hasty generalization, a logical fallacy also known as 'the law of small numbers'"
The same goes for the number of answers from fundamentally different sources to affirm the accuracy. So far, yours is the only one more than 16 digits.
Quote: This might be useful to check for accuracy the results you get from calculators.
It would be very stupid indeed for anyone to take your single source as a definitive answer to check all others against. Perhaps as a comparison, but certainly not a check. I do not even use Wolfram
Alpha answers as definite checks, but merely as comparisons.
So far from this thread, the best we can come up with is the first 6 digits (0.995555) and then only with a very low confidence due to the small sample. I.e. it could all be horribly wrong...
(unlikely, but possible).
05-14-2012, 07:48 PM
I can confirm the result given by Valentin to the first 30 digits of precision, using a high-precision method I programmed a few years back. Adaptive integration using double exponential quadrature.
My method was designed for tough integrals, but doesn't achieve results very quickly in high precision applications. It took about a minute to get (and confirm) 30 digits.
As always I admire Valentin's abilities and the tools he creates.
Edited: 14 May 2012, 7:51 p.m.
05-14-2012, 11:36 PM
Quote: I can confirm the result given by Valentin to the first 30 digits of precision, using a high-precision method I programmed a few years back. Adaptive integration using double exponential
My method was designed for tough integrals, but doesn't achieve results very quickly in high precision applications. It took about a minute to get (and confirm) 30 digits.
As always I admire Valentin's abilities and the tools he creates.
First of all, thank you very much for your appreciation and kind words.
It would be simple to check my 1000+ digit result using Mathematica, Maple, or PARI/GP for instance but I have none of them at hand right now, just my eReader with 894 books, 1200+ (mostly math)
papers, 550+ audio files, and about two dozen assorted applications installed on it which, regrettably, don't include any of those.
However, I have no doubts whatsoever that my results are correct, as I've thoroughly tested the routine innumerable times using much more troublesome cases than Gerson's fairly mild integral, with
reasonable success so far.
Just for instance, this simple, elementary integral is probably a harder nut than Gerson's if dealing with it straightforwardly:
/ 10
| x^2*sin(x^3).dx
To some 750+ decimal places, my quadrature routine produces:
which is demonstrably correct to all digits shown. On the other hand, the naive approach using the HP-71B and asking for full precision gives:
An interval approach fares somewhat better:
>S=0 @ FOR I=0 TO 9 @ S=S+INTEGRAL(I,I+1,0,IVAR^2*SIN(IVAR^3)) @ NEXT I @ S
You might want to try assorted calculators and your own high-precision quadrature to see how they cope with this simple integral.
Best regards from V.
Edited: 14 May 2012, 11:44 p.m.
05-15-2012, 12:19 AM
Quote: However, I have no doubts whatsoever that my results are correct, as I've thoroughly tested the routine innumerable times using much more troublesome cases than Gerson's fairly mild
integral, with reasonable success so far.
I should have known better. Sorry!
I have yet to see a wrong result coming from you. Well, at least this has elicited some interesting discussions and insights.
Best regards,
05-15-2012, 10:10 AM
The Casio fx-115ES Plus ("The Quadra-nator") confirms the accuracy of the first 11 significant digits shown above. :-)
But seriously...when programmed in LineIO mode (using the 115's default integration tolerance value) as S(x^2sin(x^3),0,10) it returns 0.145873641232327 (accurate to 11 digits).
The 10 digits shown in the display are correct. Computation time is 1003 seconds...just making it before the 115's 1050-second time-out halt. Better have some bright light on the solar cell to do
many of these!
The equivalent integral (Ssin(u)du)/3 evaluated from 0 to 1000 produced 0.145873641237629, a result accurate only to 11 digits. It took 737 seconds to compute.
When solved and computed analytically as -(cos(1000)-cos(0))/3 on the 115, the result is 0.145873641236197. This is accurate only to 12 digits. I was expecting a better result.
Edited to add results from 115's analytic solution.
Edited: 16 May 2012, 1:01 a.m.
05-15-2012, 02:47 PM
That is certainly a numerically challenging function to integrate over the suggested range. It increases while oscillating more and more as x increases. And it has the benefit of being analytically
easy to solve with a simple substitution.
My high-precision routine had the grace to give up, since it quickly hit built in limits.
Other double precision integrators in my toolkit had mixed results.
Some gave close results for lower precision demands, but hung (taking too much time) when trying for precision better than 1E-14 or so.
Other integrators gave close answers, but with error estimates that were off by a factor of 10 or more.
A nice test case indeed. And motivation to attempt to make my tools more robust.
05-15-2012, 10:00 PM
Quote: A nice test case indeed. And motivation to attempt to make my tools more robust.
I wish there were at least one HP calculator with a numerical integration function that could even remotely match the performance found in my cheap non-programmable calculator that is sold in
mass-market department stores like Walmart and Target in the USA. It's embarrassing that all of the HP line, even the 50g, performs so poorly when executing its native quadrature firmware.
Like Casio, the TI line has utilized Gauss-Kronrod quadrature for many years. It would be interesting to try this test case on my TI-89 Titanium HW version 4, but I don't have it nearby at present.
It is HP that needs its tools to be more robust.
05-16-2012, 03:33 AM
I confirmed your 750 digit result with the following W|A command-line: NIntegrate[x^2*sin(x^3), {x, 0, 10}] 1000 digits
Strangely enough, this command-line does not work with Gerson's function ("1/2+(acos(cos(pi*x)^2))/(3*sin(pi*x))" or "1/2+(ArcCos[Cos[Pi*x]^2])/(3*Sin[Pi*x])" for Mathematica syntax).
Instead, this NIntegrate[1/2+(acos(cos(pi*x)^2))/(3*sin(pi*x)), {x, 0, 1},WorkingPrecision->100] will work, but be limited to 200 digits.
While NIntegrate[x^2*sin(x^3), {x, 0, 10},WorkingPrecision->100] also works, it doesn't work with WorkingPrecision->1000 (though the first command proves that W|A can compute 1000 digits without
timing out).
If someone can make rhyme or reason out of this, I'd love to hear it.
I'm presently working on a CAS for ND1 that uses W|A to do the hard work. This thread inspired me to sort out high-precision numerical integration, and the result of a little bit of integration work
is that RPL code like this
\<< 200 setBigFPrecision
0 toBig
'x^2*sin(x^3)' 'x' integrate
will numerically integrate a function to a desired number of digits. The "integrate" function, when fed BigFloat values, will produce a BigFloat result. A type in ND1, it can be used for subsequent
The internal implementation of "integrate" looks like this
"integrate": function(a, b, expr, name) { return BigFloat.fromString(calculator.callWA("", "NIntegrate[" + calculator.expressionToWAexpression(expr) + ", {" + calculator.unquote(name) + ", " + a + ", " + b + "},WorkingPrecision->" + BigFloat.precision + "]")); },
The idea is that users can write their own adapter functions and thereby compute anything they like in Mathematica/WolframAlpha, while using a stack-based calc that operates very much like a 50g.
05-16-2012, 05:32 AM
Quote: I confirmed your 750 digit result with the following W|A command-line: NIntegrate[x^2*sin(x^3), {x, 0, 10}] 1000 digits
Thanks, you can get as many digits as you want for checking purposes or otherwise by simply evaluating:
which is the exact result.
Regards from V.
05-16-2012, 07:58 PM
Valentin, I'd be quite interested in learning any details you could share about the workings of your quadrature routine, as well as your extended precision setup.
I've programmed adaptive quadrature using Simpson's and Gauss+Lobatto (which both require endpoint evaluation).
Also programmed Gauss+Kronrod, and Gauss+Bond routines which don't require endpoint evaluation. For these I have fixed weights and abscissa in the code making variable precision hard.
I also programmed double exponential, which works much differently than the above methods, but lends itself to variable precision calculation admirably.
I use MAPM by Michael C. Ring for extended precision since it's freeware and fairly easy to use and quite reliable. It can also be slow, especially if you're not careful with rounding results.
Anyway, if you can share anything more about your methods, I'd appreciate it. If not, I certainly understand.
05-17-2012, 06:15 AM
Hi, Steve:
Quote: Valentin, I'd be quite interested in learning any details you could share about the workings of your quadrature routine, as well as your extended precision setup.
Thanks for your interest, Steve, and congratulations on your many and varied implementations of assorted quadrature algorithms, I see you're really keen on numerical integration. As for details on my
own quadrature routine which I mentioned a few posts ago, I'm sorry but I'll keep them private for the time being, mainly for the following reasons:
1) my routine uses a simple algorithm, nothing fancy at all, and thus I don't think you'd learn anything worthwhile by having a look at it, in fact I think you'd be somewhat disappointed, a likely
case of unfulfilled high expectations.
2) I wrote this routine some years ago for an article I was going to submit to Datafile for publication as part of my "Boldly going ..." regular series of sizable articles featuring novel techniques,
as it was purpose-made to achieve a "bold" goal. After a month or so I eventually finished the article, which includes the routine proper with full comments and, as usual, fully-worked-out examples
galore, with lots of relevant comparisons and tests, caveats, etc.
Alas, things went south with Datafile, regrettably, and the 16-page article remains unpublished to this day so I'd rather not disclose its fine details till I eventually try and publish it somewhere
else, together with about six or seven additional unpublished ones. A pity indeed because the finished article looks awesome, even if I say so myself. As a policy, I always write the kind of articles
I'd love to read myself.
Thanks for your understanding and
Best regards from V.
05-17-2012, 01:32 PM
Valentin wrote:
Quote: As for details on my own quadrature routine which I mentioned a few posts ago, I'm sorry but I'll keep them private for the time being ... I'd rather not disclose its fine details till I
eventually try and publish it somewhere else...
You have got to be kidding!
What sort of reward do you envision for "publishing" your programmable-calculator code (outside of this forum)? And where might you "publish"? Do you really believe that there is a big, receptive
audience outside of this forum?
05-17-2012, 04:27 PM
Since you've gone to a lot of work, and hope to publish, I certainly respect your choice to keep it private for now.
I was kind of hoping it might be appropriate for use in calculator circles, and it appears that this is the case!
In the meantime, my interest in numerical integration has been re-kindled. I've already found ways to speed up my variable precision routine by a factor of 3, so that has been worthwhile.
It's not so much finding test cases that a given routine can't handle that bothers me, but finding cases which the routine claims to handle with a reported error estimate, but is way off.
My code handles cases with singularities (integrated from 0 to 1) like:
without much problem. Also (integrating say from 0.01 to 1):
also fairly readily. I'm always looking for more interesting test cases as have been presented here. Hopefully we can soon have calculator routines at least as good as Casio's available to us on our
HP machines.
As always, I greatly appreciate your input to the forum.
05-17-2012, 05:07 PM
Quote: I'm always looking for more interesting test cases as have been presented here.
Have you tried Kahan's integral already?
I = | (sqrt(x)/(x-1) - 1/ln(x))dx
It has the advantage of having a closed-form solution, as shown here:
I = 2 - ln(4) - EulerGamma
I = 0.036489973978576520559023667001244432806840395339565892952872746128345029282945897851326282715415875401
An alternate form is
I = Psi[3/2]
which can be checked at
05-18-2012, 02:14 PM
At the risk of boring folks with more fx-115ES Plus results, I report that Dr. Kahan's classic HP-34C test integral produces for once slightly mixed results. Adding 1E-12 offsets to both limits of
integration to eliminate singularity at those points during the Gauss-Kronrod process, the 15-digit result after 121 seconds is:
The actual displayed value is 0.03648997399, so only the first 9 significant digits of the result are correct...the last should have been 8.
Still, this is a much better result in far less time than any HP calculator I've used. This includes the HP 50G, which has much faster hardware producing results for other problems about 20 times
faster than does the 115ES Plus. This is a trivial-cost high school student's calculator. I can't help being impressed with the algorithm engineering at Casio, and a little disappointed by HP's in
this area. It's way past time for Gauss-Kronrod quadrature to be an option in HP scientific calculator firmware.
Edited: 18 May 2012, 2:27 p.m.
05-18-2012, 04:30 PM
I'm surprised you had to modify the end-points.
My understanding is that Gauss-Kronrod does not need to evaluate the function at the extremes. That's part of it's usefulness.
My implementation of Gauss-Kronrod also comes up with a reasonable approximation of the integral, but the error estimate isn't as good as I would hope. Overly optimistic I guess I might say.
A good test case.
05-18-2012, 04:19 PM
My double exponential quadrature routine with variable precision handles that one quite nicely too. I incorporated it into my test suite when I read about it in this forum.
Thanks for the suggestion. That function does have interesting behavior at certain spots.
05-17-2012, 06:36 PM
Another simple integral with a closed-form solution is the integral of FRAC(x) from 0 to 6.4 . Every HP handheld calculator to date with a built-in integrator produces a >50% relative error (or >100%
depending on how you define relative error). The Casio fx-115ES Plus gets it exact.
05-17-2012, 07:47 PM
Submit your articles to HP Solve. I know it is PDF, but it would reach a great deal more people than Datafile ever has.
Be good if more people would learn about Valentin!
05-18-2012, 07:22 PM
Quote: Submit your articles to HP Solve. I know it is PDF, but it would reach a great deal more people than Datafile ever has.
Be good if more people would learn about Valentin!
Thanks for your always kind words and your continued appreciation, Gene.
The problem is, I already have the article written as a PDF document formatted per the mandatory Datafile template in use at the time and reformatting it per HP Solve guidelines would surely be a lot
of work and thus a lot of additional time I simply don't have.
Also, I have two versions of the routine, one of them is written for and runs in a non-HP multiprecision environment and thus is likely to be unsuitable for HP Solve. The other version is a
standard-precision HP-71B program but I think that HP Solve will probably be much more interested in publishing materials for newer, current, easily available models than for hard-to-get deprecated
old timers like the HP-71B.
Anyway, thanks again for your interesting, well-meant advice and
Best regards from V.
05-13-2012, 11:41 PM
Quote: Well, it doesn't seem that tough at all
Well, it is tough enough for all HP calculators I tried. For instance, on the ultra fast HP-15C LE it apparently will run forever when FIX mode is set to 8 or 9. Mike Morrow's Casio fx-115ES Plus
below has 15 internal digits, but will get only the first 11 digits right. It is very fast and all 10 digits in the display will be correct, however.
Best regards,
05-14-2012, 01:32 AM
The interesting thing about the speed of the results from the fx-115ES Plus (aka 991ES Plus C) is that the equivalent of a 2500-iteration Savage benchmark takes 23 times longer on the Casio than on
the HP 15C-LE. The magic in the Casio's quadrature result comes from the algorithm, not the hardware.
Edited: 14 May 2012, 1:51 a.m.
05-13-2012, 09:49 PM
The Casio fx-115ES Plus uses a Gauss-Kronrod quadrature method and requires the function to be evaluated at the endpoints. Adding 1E-12 to each endpoint and setting a tolerance value of 1E-14 yields
the following result after 63.0 seconds:
What shows up in the 10-digit display is 0.9955554978, which is accurate for every digit.
That's not too shabby for an $18 calculator.
Edited: 13 May 2012, 10:17 p.m.
05-14-2012, 06:17 AM
Quote: That's not too shabby for an $18 calculator.
Agreed! One of these days I had the band of my wristwatch replaced at a Casio representative and saw one of these on the shelf. Coincidence or not the price tag was exactly R$ 115 (~US$ 57.50), what
made me lose the interest.
05-14-2012, 09:59 AM
www.numberempire.com gives:
05-14-2012, 05:41 PM
Thanks for the link!
Previously I had tried this link, which offers various integration methods, but I wasn't able to make it work with this example.
I tried the WP 34S (v2) again and divided the integration interval into three subintervals, [0,0.1], [0.1,0.9] and [0,9,1], and obtained 0.9955554977624464. Only slightly closer though.
P.S.: The following is better and will give 14 correct digits:
WP 34S v2:
001 LBL 00
002 # pi
003 *
004 COS
005 x^2
006 ACOS
007 x<>y
008 # pi
009 *
010 SIN
011 /
012 RTN
0 ENTER .5 g Integral 00 2 * 3 / .5 + --> 0.9955554977623607
Edited: 14 May 2012, 9:30 p.m.
05-16-2012, 03:24 AM
That is an interesting link, I had not come accross it. I like the choice of different methods too.
05-14-2012, 10:46 PM
Gerson, you seem to have a glitch in the code for your functions program. Once you consume your x with the cos(Pi x) calculation, etc, where are you getting a second x for the denominator?
This is what I used:
# Pi
STO* X
I use my own integrator on the 34S, which is a simple port of the PPC IG routine. My routine requires that only the last two extrapolated estimates agree, whereas the internal routine requires three
and this means things take much longer.
With my routine, on a WP34S with the latest firmware, I get 9.95555497791e-1 at SCI 11. This is off just one ULP in the 12th digit.
The integral isn't that pathological and deserves another look with a routine that computes it correctly.
05-14-2012, 10:59 PM
PS: My batteries are getting sluggish so the WP34S built-in integration routine in the first surge and shuts the calc off and I am too lazy to dig out fresh ones at the moment.
However, in the most recent emulator set to ALL 00 and DP, I not only get the first 12 digits exactly, but internally the routine gives us 6 more, to 18 in totally. And the emulator does so
instantaneously, which means this will not take forever on the real calc. And considering that the calc's routine picks the accuracy according to display mode (ALL 00 is treated as SCI 11), getting 6
more digits is darned impressive.
Edited: 14 May 2012, 11:00 p.m.
05-14-2012, 11:08 PM
Quote: With my routine, on a WP34S with the latest firmware, I get 9.95555497791e-1 at SCI 11. This is off just one ULP in the 12th digit.
Nope, it's 3 units off in the 11th digit, see my 1,000+ decimal places result above which begins with:
05-15-2012, 12:12 AM
Sorry, typo. 11th digit in mine should be a 6, not a 9.
My claim is right. I transcribed my result incorrectly.
05-14-2012, 11:16 PM
Hello Les,
Quote: Once you consume your x with the cos(Pi x) calculation, etc, where are you getting a second x for the denominator?
From the stack, which is automatic filled with the content of the X register every time the function to be integrated is called. From the manual:
"Lower and upper integration limits must
be supplied in Y and X, respectively. Otherwise,
the user interface is as in HP-15C. Please refer to
the HP-15C Owner’s Handbook, Section 14 and
Appendix E, for more information about automatic
integration and some caveats."
Quote: With my routine, on a WP34S with the latest firmware, I get 9.95555497791e-1 at SCI 11. This is off just one ULP in the 12th digit.
By simplifying the integrand prior to integration I was able to obtain 14 correct digits. Please see my reply to Bart above.
Quote: The integral isn't that pathological and deserves another look with a routine that computes it correctly.
I think you're right. The transitions at integer values of x appear to be fast enough to be a problem.
05-15-2012, 03:55 PM
Please, someone give me a hint on how long this takes on the 34s and the 15c LE. It seems to take forever and I don't want to wear out my batteries.
05-15-2012, 04:34 PM
Hello Alexander,
On the WP 34S (v2) it takes only a couple of seconds.
On the HP-15C you should set FIX mode to 7 at most. Alternatively use the following program:
001 LBL A
002 pi
003 *
004 COS
005 x^2
006 ACOS
007 x<>y
008 pi
009 *
010 SIN
011 /
012 RTN
0 ENTER .5 g Integral A 2 * 3 / .5 + f --> 0.995555504
It will take less than 3 seconds on the HP-15C LE, even when FIX mode is set to 9.
05-15-2012, 04:52 PM
Duh! Switching to RAD did it on the WP-34s V2 and the 15c LE, but on the WP-34s V3 2931 I can't get it to work.
EDIT: Solved on the WP-34s V3 by explicitly setting an END at the end instead of terminating only by RTN. I didn't know that.
Edited: 16 May 2012, 2:10 a.m.
05-17-2012, 01:20 AM
This sounds strange. Can you send me an email with details so that I can have a closer look when I find the time?
05-18-2012, 02:02 AM
I can't remember what state the calculator was in when I encountered this, so I took a recently converted machine (VERS 3049, only calc.bin) and did a RESET.
Set it to [g][RAD]
Enter the WP-34s program from posting #1, exactly the 17 steps, no less, no more
0 [ENTER] 1 [g][2] 0 0
For a short time I see 'Running Program' then the integral solving symbol appears and the 9.95555497761E-1 converges. So far so good.
Because the RCL annunciator is still flashing, I [EXIT]. I key in again
0 [ENTER] 1 [g][2] 0 0
Now I see 'Running Program' and the RCL flashing, no integral symbol. After about 2 minutes RCL flashing is off, the matrix display shows 'Reset' and 9.96943604201E-1 is in the lower display.
0 [ENTER] 1 [g][2] 0 0
has the machine running for 5 minutes before I forcibly [EXIT] with 'Stopped' and 4.72450120691E-1 in the display.
Now I do [GTO][.][.] and enter the following program
001 LBL A
002 [g][x^2]
004 -
005 [g][RTN]
Then [f][SLV] brings up -2.2360679775 in the display.
0 [ENTER] 1 [g][2] 0 0
does a fast conversion, with RCL still flashing when I [EXIT]. The next
0 [ENTER] 1 [g][2] 0 0
has to be aborted after 5 minutes with [EXIT].
[SLV]ing program A from above results in 'Running Program'. I end the experiment here.
Marcus, I send you this as an email but post this here so everybody can have fun... ;-)
05-18-2012, 02:19 AM
I'd be guessing that the return stack isn't being cleared each time and that is confusing things. Try a g-shirt RTN in run mode to clear things.
Still, it seems like a bug/annoyance.
- Pauli
05-18-2012, 02:49 AM
Pauli is right, you need to do do g-RTN before you can try again. What happens is that you are effectively nesting calls to he integrator. EXIT stops the process while in your user code routine to
integrate. You can R/S to continue. If, instead, you start a new integration operation this will nest. If you stop it again, it will eat more and more of the available memory.
There is still something wrong. Showing "Running Program" is not intended. "Reset" is most probably the watch dog kicking in.
05-18-2012, 03:34 AM
[g][RTN] does it.
I just wanted to mention, if I enter program mode after an integral run and without [g][RTN], memory is only about 415 steps after the first try and 320 steps after the second try, 225 after 3rd...
05-18-2012, 03:46 AM
About right. The integration code allocates a lot of local registers.
We probably should clear the locals/return stack when solve or integrate is run from the keyboard....
- Pauli
05-18-2012, 01:58 PM
Quote: We probably should clear the locals/return stack when solve or integrate is run from the keyboard....
That's what I came up with on the last kilometers of my latest bike trip. Let me see when I find the time or a clean implementation. The "Running Program" is indeed intentional because the software
detects that it is not run from the top level and is therefore quiet (no display of intermediate results). This leaves the original "Running" message in the display. I'm still thinking about the
"Reset" case. I have an idea how to fix this (if it's the watchdog) but I must check if I'm right.
05-19-2012, 03:21 AM
I did something about the issue. Please report any findings!
05-18-2012, 04:06 AM
Quote: I just wanted to mention, if I enter program mode after an integral run and without [g][RTN], memory is only about 415 steps after the first try and 320 steps after the second try, 225
after 3rd...
I guess this happens only if you abort the integral calculation with [EXIT] each time!?
I've now tried a few integrals with the emulator (last build) but I don't get this problem. Of course I couldn't interrupt the integration because the emulator is just too fast.
Pauli has already posted the solution: clearing the return stack (and local regs) for a manual call should solve the problem. This should also be done for SUM and PROD (not only INTG and SLV), I
don't know if there are other such routines which call a user program.
Edited: 18 May 2012, 4:07 a.m.
05-19-2012, 09:49 PM
Quote: It seems to take forever and I don't want to wear out my batteries.
Same here with versions 3.0 2609 and 3.1 2988. Has it been fixed? Thanks!
05-20-2012, 02:25 AM
The adaptive integrator in WP 34S V3 takes a few seconds even on the emulator so be prepared for some delay. I have no idea what the END/RTN issue is.
05-20-2012, 05:59 PM
After 25 minutes it is showing 9.95555498002^-1 for five or six minutes now, diverging from a previous closer result. A useful feature here would be the possibility of interrupting the calculation
whilst keeping the latest approximation on the display instead of the not meaningful result I get when I eventually abort it, 1.54227330201 (v3.1 2988). | {"url":"https://archived.hpcalc.org/museumforum/thread-221230-post-222314.html#pid222314","timestamp":"2024-11-05T04:25:08Z","content_type":"application/xhtml+xml","content_length":"199297","record_id":"<urn:uuid:933d5b7d-7f2c-4d55-b72c-dc9d3e7a2263>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00044.warc.gz"} |
Efficiency and Betweenness Centrality of Graphs and some Applications
The distance $d_{G}(i,j)$ between any two vertices $i$ and $j$ in a graph $G$ is the minimum number of edges in a path between $i$ and $j$. If there is no path connecting $i$ and $j$, then $d_G(i,j)=
infty$. In 2001, Latora and Marchiori introduced the measure of efficiency between vertices in a graph. The efficiency between two vertices $i$ and $j$ is defined to be $in_{i,j}=frac{1}{d_G(i,j)}$
for all $ineq j$. The textit{global efficiency} of a graph is the average efficiency over all $ineq j$. The {it power of a graph} $G^m$ is defined to be $V(G^m)=V(G)$ and $E(G^m)={(u,v)|d_G(u,v)le m}
$. In this paper we determine the global efficiency for path power graphs $P_n^m$, cycle power graphs $C_n^m$, complete multipartite graphs $K_{m,n}$, star and subdivided star graphs, and the
Cartesian products $K_{n}times P_{m}^{t}$, $K_{n}times C_{m}^{t}$, $K_{m}times K_{n}$, and $P_{m}times P_{n}$.
The concept of global efficiency has been applied to optimization of transportation systems and brain connectivity. We show that star-like networks have a high level of efficiency. We apply these
ideas to an analysis of the Metropolitan Atlanta Rapid Transit Authority (MARTA) Subway system, and show this network is 82% as efficient as a network where there is a direct line between every pair
of stations. From BOLD fMRI scans we are able to partition the brain with consistency in terms of functionality and physical location. We also find that football players who suffer the largest number
of high-energy impacts experience the largest drop in efficiency over a season.
Latora and Marchiori also presented two local properties. The textit{local efficiency} $E_{loc}=frac{1}{n}sumlimits_{iin V(G)}E_{glob}left(G_{i}right) $ is the average of the global efficiencies over
the subgraphs $G_{i}$, the subgraph induced by the neighbors of $i$. The clustering coefficient of a graph $G$ is defined to be $CC(G)=frac{1}{n}sumlimits_{i}C_{i}$ where $C_{i}=|E(G_i)|/binom{|V
(G_i)|}{2}$ is a degree of completeness of $G_{i}$. In this paper, we compare and contrast the two quantities, local efficiency and clustering coefficient.
Betweenness centrality is a measure of the importance of a vertex to the optimal paths in a graph. Betweenness centrality of a vertex is defined as $bc(v)=sum_{x,y}frac{sigma_{xy}(v)}{sigma_{xy}}$
where $sigma_{xy}$ is the number of unique paths of shortest length between vertices $x$ and $y$. $sigma_{xy}(v)$ is the number of optimal paths that include the vertex $v$. In this paper, we
examined betweenness centrality for vertices in $C_n^m$. We also include results for subdivided star graphs and $C_3$ star graphs.
A graph is said to have unique betweenness centrality if $bc(v_i)=bc(v_j)$ implies $i=j$: the betweenness centrality function is injective over the vertices of $G$. We describe the betweenness
centrality for vertices in ladder graphs, $P_2times P_n$. An appended ladder graph $U_n$ is $P_2times P_n$ with a pendant vertex attached to an tbl endtbr. We conjecture that the infinite family of
appended graphs has unique betweenness centrality.
Library of Congress Subject Headings
Paths and cycles (Graph theory); Graph theory--Data processing
Publication Date
Degree Name
Applied and Computational Mathematics (MS)
Department, Program, or Center
School of Mathematical Sciences (COS)
Advisor/Committee Member
Jobby Jacob
Advisor/Committee Member
Paul Wenger
Physical copy available from RIT's Wallace Library at QA166.22 .E4 2014
Recommended Citation
Ek, Bryan, "Efficiency and Betweenness Centrality of Graphs and some Applications" (2015). Thesis. Rochester Institute of Technology. Accessed from | {"url":"https://repository.rit.edu/theses/8585/","timestamp":"2024-11-12T13:00:21Z","content_type":"text/html","content_length":"43049","record_id":"<urn:uuid:62075798-48f3-4b94-938e-6bd9d52ab42b>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00572.warc.gz"} |
Making Math Matter for Students Who Don't Care
Making math matter is a tricky thing. Is there a way to "do math" from a Christian or even a generic moral perspective? Isn't 2+2 the same for everyone?
In undergrad I never really cared about whether math had applications; it was beautiful and that was incentive enough. I would teach my students to value aesthetics of proofs and for them, that too
would be enough. And then came Pre-Algebra in a 95% free and reduced lunch school in rural Northern Michigan. My students hated math. There was no beauty, only previous failure. All of a sudden, I
had to think of some way to motivate my students to engage with the subject. For several years I floundered; Jimmy shovels at a rate of 8 ft every minute, if the driveway is... You know what? It
doesn't really matter how long Jimmy is out there, because he's going to stay out there until he finishes shoveling the driveway. Trite doesn't work. Does math matter or not? If it's just some
intellectual exercise I'm forcing on my students because politicians and textbook makers think its a good idea then we should change the good idea.
There's a long road to my infatuation with problem-based learning and using experiments in my classroom, but developing lessons for the Kuyers Institute
was pivotal. Can teaching a math lesson be about
more than just the math
? Can we talk about social inequality using numbers? What role does math play in convincing people of the rightness or wrongness of an opinion in our society?
One of the few times I've been successful with such a lesson
is in studying pay inequality between men and women in the U.S.A. I've done this lesson in both Algebra 1 and Algebra 2. I think the older students are better able to delve into the underlying
factors (not surprising).
We start by looking at the data from the census bureau: 1955 to 2003. Before beginning group work, we have a discussion about why we use data for median incomes of full-time year round workers.
(Women are more likely than men to work part-time in the US and that would skew the data, hence 'full-time year round'. Also, men are more likely to be top earners in the US (think Bill Gates and the
like), which would also skew the data, hence 'median'.
Then we look for trends in the tables. Almost always students say that men earn more than women and that as the years increase people make more. Then we see if the graphs can help us be more specific
(page 2 of the doc). What do you see? Can you use math to describe that? Students suggest exponential curves. We try them and they don't work. Then we talk about cutting off parts of the data. If you
cut the data at 1975-ish, it becomes approximately linear.
After laying the foundation for the lesson, it's group time. In groups of 3 or 4, students start to work through the questions in the lesson The lesson is divided up into 3 tasks to make for natural
discussion breaks as a class for formalization of content. Here's a quick overview:
In Task 1 students find equations for both men and women incomes using lines of best fit and two points. Students must also explain what each aspect of the function stands for and whether it is
appropriate to predict backwards using the functions.
Task 2 tackles linear regression, correlation coefficients and whether our equations have any merit when comparing with current data (I have 2010 incomes in the lesson).
Task 3 takes a brief foray into rational functions (cents on the dollar). I couldn't help bring up a conceptual problem by asking students to create a line of best fit for this function as well (it
looks linear). However, this yields drastically different results than the findings from the previous tasks. Lovely discussion about why this doesn't work.
All in all, it's an interesting lesson that leads to great conversations about the structure of our society, concepts like 'justice' and how we use math to 'prove' our points.
I'd be interested in your thoughts and/or recommendations for improving it.
2 Comments
Megan Schmidt link
10/7/2013 10:47:42 am
I teach probability and statistics and I REALLY love this activity. I like how you get students interested in a topic that piques their curiosity. I love how your tasks get students to develop a
model for the data before learning a procedure to finding the line of best fit. I am in the middle of teaching this topic to my class and I would love to try this. I'll let you know how it goes and
give you some better feedback.
Andrew Busch link
10/7/2013 09:33:49 pm
Thanks for the feeback Megan. I'm interested to know how it goes and how you would tweak it after teaching it.
Your comment will be posted after it is approved.
Leave a Reply. | {"url":"https://www.andrewbusch.us/home/making-math-matter-for-students-who-dont-care","timestamp":"2024-11-02T02:05:51Z","content_type":"text/html","content_length":"62163","record_id":"<urn:uuid:add49772-257a-4af9-90bd-ab88bebe416c>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00026.warc.gz"} |
Flatten --- Introduction ---
This module is an exercise on the infinitesimals (and its notation by O and/or o). The server generates a function $f\left(x\right)$ of one real variable $x$, containing parameters ${a}_{1}$, ${a}_
{2}$,... Your aim is to find values for these parameters${a}_{i}$ such that $f\left(x\right)$ is infinitesimal to a required order, at the neighborhood of a given point ${x}_{0}$.
Except the trivial levels, one solves the problem by the Taylor expansion of $f\left(x\right)$. In the difficult cases, the resolution of a system of linear equations is also necessary.
Choose the level of difficulty suitable for you: 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 .
Detailed menu for the parameters of this exercise. The most recent version
This page is not in its usual appearance because WIMS is unable to recognize your web browser.
Please take note that WIMS pages are interactively generated; they are not ordinary HTML files. They must be used interactively ONLINE. It is useless for you to gather them through a robot program.
• Description: parametrize a function to make it infinitesimal at a point. interactive exercises, online calculators and plotters, mathematical recreation and games
• Keywords: interactive mathematics, interactive math, server side interactivity, analysis, continuity, derivative, limit | {"url":"https://wims.divingeek.com/wims/wims.cgi?lang=en&+module=U1%2Fanalysis%2Faplat.en","timestamp":"2024-11-07T13:12:55Z","content_type":"text/html","content_length":"6292","record_id":"<urn:uuid:915c503a-7471-408f-836e-9dd6879c7f0f>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00736.warc.gz"} |
M Theory Lesson 198
M Theory Lesson 198
with $(\mathbb{R}, \mathbb{C}, \mathbb{H})$ and the
of Riemann surface moduli $(M(0,6), M(1,3), M(2,0))$, which have Euler characteristics $-6, - \frac{1}{6}, - \frac{1}{120}$ respectively. Observe that 120 is the number of elements in the icosahedral
group, whereas 6 is the number of elements in $S_3$.
The triple of (
orthogonal, unitary, symplectic
) appeared in
Mulase-Waldron T duality
for partition functions over twisted graphs. Here, the unitary case is self dual, just like the Platonic
. The real (orthogonal) case has half the number of matrix dimensions (punctures) as the quaternionic case, suggesting we associate the genus 1 moduli to $\mathbb{R}$ and the genus 0 moduli to $\
mathbb{H}$. The dual graph to the cube is basically the 6 punctured sphere. This leaves the genus 2 moduli for the icosahedron and indeed the 120 in the Euler characteristic suggests a relation.
Observe that without the octonions, one does not naturally encounter
structures in the triples, but such triples are also highly relevant to M Theory.
From a categorical perspective, one views these trinities as models of the category
, the basic triangle, because they naturally form categories with only 3 objects and one natural map between any two objects. The collection of all such
sets of three elements
is the object 3 as an
which counts cardinalities of sets, except that we have
the sets by making them categories! This is why it is not surprising to encounter grouplike cardinalities in the Euler characterstics of these models. (Actually, it is the orbifold structure of the
moduli that gives them a groupoid character).
3 Comments:
lievenlb said...
thanks Kea! ill have another go on all your Mulase-related material and pay special attention to Kneemo-comments...
probably the -1120 must be -120 both in this post and in your former one?
Kea said...
Lieven, it's 1/120 which is the usual form of an orbifold Euler characteristic, or a groupoid characteristic chi = 1/|Aut(g)|.
lievenlb said...
yeah, sorry about that. while i'm spite trying to convince people to use firefox-like browsers to defeat google's imperialism, i'm a total wanker still preferring safari....atb. | {"url":"https://kea-monad.blogspot.com/2008/06/m-theory-lesson-198.html","timestamp":"2024-11-07T02:27:15Z","content_type":"application/xhtml+xml","content_length":"31241","record_id":"<urn:uuid:86e20e33-d8ea-4229-916a-14309f9b69e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00137.warc.gz"} |
2-Sample Poisson
To ensure that your results are valid, consider the following guidelines when you collect data, perform the analysis, and interpret your results.
The sample data should be selected randomly
In statistics, random samples are used to make generalizations, or inferences, about a population. If your data are not collected randomly, your results may not represent the population. For more
information, go to Randomness in samples of data.
The data must be counts per unit, such as the number of calls per hour to a call center or the number of defects per unit in a shipment
If your data classify each observation into one of two categories, such as pass/fail, use 2 Proportions. For more information on data types, go to Data types you can analyze with a hypothesis
Each observation should be independent from all other observations
For observations to be independent, the probability of a particular outcome does not depend on any previous outcome. For example, if you select two parts and record whether they are defective or
not, the outcome of the second part should not depend on the outcome of the first. If your observations are not independent, your results may not be valid.
Determine an appropriate sample size
Your sample should be large enough so that the following are true:
□ The estimates have enough precision.
□ The confidence intervals are narrow enough to be useful.
□ You have adequate protection against type I and type II errors.
To determine the appropriate sample size for your hypothesis test, go to
Power and Sample Size for 2-Sample Poisson Rate | {"url":"https://support.minitab.com/en-us/minitab/help-and-how-to/statistics/basic-statistics/how-to/2-sample-poisson-rate/before-you-start/data-considerations/","timestamp":"2024-11-06T01:45:42Z","content_type":"text/html","content_length":"12098","record_id":"<urn:uuid:36206a0e-f3cc-4ab7-a3bd-580c97b57658>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00801.warc.gz"} |
Fail/Pass tests and the implied failure rate and confidence levels
Let’s say we conduct a fail/pass test. We subject $n_\mathrm{s}$ samples to an accelerated (representative) life test of $m=1$ lifetime equivalents. The test is considered a success if $100\,\%$ of
the $n_\mathrm{s}$ samples survive. Yet, this leaves the important question of how certain can we be that the population as a whole (from which the $n_\mathrm{s}$ samples are a representative
sub-set) will survive the $m=1$ lifetime equivalents? And, apart from the implied confidence level, what fraction of the population is still expected to fail even if $100\,\%$ of the $n_\mathrm{s}$
samples did survive?
We consider a binomial distribution: the samples either survive (pass), or die (fail). Each sample has a probability $p$ of dying. Hence, starting with $n_\mathrm{s}$ samples, the probability of
ending up with $k$ dead (failed) samples is
B(k) = \frac{n_\mathrm{s}!}{k!(n_\mathrm{s}-k)!} p^k (1-p)^{(n_\mathrm{s}-k)}.
The case of interest is with $k=0$ dead samples at the end of the test, i.e. all passed. This leaves us with the special case of
B(0) = (1-p)^{n_\mathrm{s}}.
With $p$ being the probability of dying, $(1-p)=R$ can be said to be a measure of the reliability (in the sense of the probability to survive the foreseen lifetime) of the devices being tested. The
probability $B(0)$ represents a measure of how often we expect to see this survival rate. In other words, it is $B(0)=(1-C)$, with $C$ the confidence level.
In other words, we can rewrite Eq. (\ref{eq:B0}) as
(1-C) = R^{n_\mathrm{s}}.
And from Eq. (\ref{eq:CRn}) follows how $n_\mathrm{s}$ relates to $R$ and $C$:
n_\mathrm{s} = \frac{\ln(1-C)}{\ln(R)}.
As a numerical example, let’s assume we want to be $C=90\%$ confident that the whole population shows a $R=90\%$ reliability (i.e. at least $90\%$ of the whole will survive). In this case, the
minimum number of samples to test is $n_s = 21.85$, i.e. 22 samples.
Another example is that with $n_s=77$ and an assumed reliability of $R=95.0%$, the confidence level is $C = 98.07%$. Meaning, if we test 77 samples, we should find at least one defective sample in
98% of the cases, if the underlying reliability is less than 95%.
We can reuse already tested samples and expose them to the accelerated life test a second, third, … , $m$ time. Up to a certain point this would be equivalent a higher number of samples. Written
n_\mathrm{s} = m \times n_\mathrm{actual\_samples},
and thus
n_\mathrm{s} = \frac{\ln(1-C)}{m \ln(R)}.
Sidenote, it should be clear that a test conducted with only one sample but which was exposed to $m=100$ lifetimes would NOT be statistically equally meaningful as a test with $100$ samples exposed
to $m=1$ lifetime. In what context $m>1$ might be meaningful depends on various circumstances.
The confidence level $C$ is a critical parameter of a fail/pass test. The purpose of a fail/pass test is to learn something (that’s the purpose of any test, of course, not only fail/pass). Often this
tool comes into play to confirm that a new product can be launched. However, testing costs money (for the samples) and time (for the actual testing). Hence, small sample sizes are typically
Yet, confirmation bias aside, what can we possibly learn from a passed test with few samples? Let me assume an extreme case example: testing two samples, which are so inherently flawed that they fail
half of the time. Such a test will fail in $C=75\,\%$ of the time. Upon failure the development team would likely investigate the failed sample and subsequently improve it. This could be considered a
good outcome.
On the other hand, starting off with an only $R=50\,\%$ reliable product left a lot of room for improvement.
And maybe even worse, there are still $B(0)=25\,\%$ of the time in which the two samples do NOT fail, $25\,\%$ of the time in which the team would not investigate the failure mechanisms, and $25\,\%$
of the time in which the product launch would continue according to schedule and being shipped. It should be without saying, no customer is interested in a $R=50\,\%$ reliable product.
If the product was slightly more reliable (say, a still rather bad $R=70\,\%$), the two samples would pass in $B(0)=49\,\%$ of the cases. In other words, the fail/pass test would be only just
slightly more informative than an independent coin-flip!
Numerical examples
Let’s add some images to the discussion.
Given $n_\mathrm{s}$ samples which each have an (assumed) intrinsic reliability $R$, how often would we run a test without any sample failing? This rate of at least one failure represents our
confidence level that we would have encountered at least one failing sample if the underlying reliability was lower than the assumed $R$.
In the following, I’m going to work with python. I encourage anyone reading this post to copy, execute, and modify the code blocks below to reproduce my images: play around with some of the
parameters, get to an understanding where what comes from; that’s the best way to make sense of random processes.
First, I set the stage by defining a basic class called Device. The objects of this class represent the devices under test (DUT). I assume the population of DUTs have a reliability $R$, intrinsic to
the population. Some of the DUTs will fail the test, the others will pass it. Which DUT fails is random (after closer inspection one might find that the failing DUTs happen to be unlucky in how the
the individual manufacturing tolerances accumulated, for example), and the frequency is given by said intrinsic reliability. Numerically speaking, we take a random value between 0 and 1 and compare
it with the set intrinsic reliability. The test() function returns the binary value 1 if the DUT survived the test, 0 if it failed. Note, we’re looking at pass/fail tests: we don’t care how
catastrophically the DUT failed the test, we only care whether it did pass or not.
In a test plan we have multiple DUTs (assume $n$) to work with, numbered as $\mathrm{DUT}_0,\ldots,\mathrm{DUT}_{n-1}$. For simplicity, I call the aggregate of these DUTs a test battery. Concretely,
by conducting a test, we expose these $n$ DUTs of the test battery to certain conditions (such as temperature shocks, high/low temperature storage, etc.) and then check each individually, whether it
is still within the agreed upon specifications — in short, whether it is still alive.
Below a list of parameters I want to explore and visualize.
As a first inspection, we find back the analytical values from above:
• an intrinsic reliability of $R=0.9$, paired with $n_s=22$ samples, gives us a value for the confidence of $C=0.9$ (print_summary(0.9, 22)),
• and with $R=0.95$ and $n_s=77$ samples, we find back the $C=0.98$ (print_summary(0.95, 77)).
All the usual disclaimers regarding statistical explorations apply: The “tested” samples are assumed to be identical and independent. The analytical solution is true only in the limit of infinite
repetitions. And so on.
Mapping out R vs C analytically
In the above sections we have looked at the question: “how many samples are we supposed to test assuming a target $R$ and $C$?” In the present section we approach the problem from the other side.
Assuming we have $n_\mathrm{s}$ samples available, what can we learn from a fail/pass test?
As already mentioned above, if all $n_\mathrm{s}$ samples pass the test, we forego the opportunity to actually learn something (meaning, by opening up a failing sample and learning about the root
causes). The only statement we can make given a $100\,\%$ pass rate is that the underlying population is $R$ reliable with $C$ confidence, related to each other through Eq. (\ref{eq:CRn}) and (\ref
{eq:n_lnCR}), respectively.
As an example let’s assume we have $n_\mathrm{s}=4$ samples available. What is the underlying reliability $R$ of the population supposed to be for us to have a reasonable chance of catching a failing
sample among these $n_\mathrm{s}=4$ samples?
One way to go about answering this question is to flip a coin: tail means the population is impecable and can be shipped, head means the population shows some not-further-specified flaws. This
strategy, stated in this extreme form, does not take the fail/pass test into account in any way whatsoever. However, an unrelated coin flip is not actually worth less than a test with expected
frequency of finding a failing part of $C\leq0.5$.
For example, if none of the $n_\mathrm{s}=4$ samples failed, but, if at the same time we were to expect the underlying population to show a reliability of at least $R\geq0.84$, then the fact that all
$n_\mathrm{s}=4$ samples have passed is actually less informative than a completely unrelated coin flip!
More generally, from Eq. (\ref{eq:CRn}) we find
R = \sqrt[\leftroot{-2}\uproot{2}n_\mathrm{s}]{1-C}.
For a test with $n_\mathrm{s}=4$ to be meaningful — say $C\geq0.9$ — the underlying population has to show a reliability of at most $R=0.56$.
Further reading
This notebook was originally inspired by an Accendo Reliability podcast on the topic, https://accendoreliability.com/podcast/arw/making-use-reliability-statistics/
Regarding what we can learn from a failed vs passed test (equally applicable for software and hardware), recommended
K. Henney “A Test of Knowledge” https://medium.com/@kevlinhenney/a-test-of-knowledge-78f4688dc9cb
You must be logged in to post a comment. | {"url":"https://stefantkeller.com/numerics/sample-size-fail-pass-test","timestamp":"2024-11-03T15:51:16Z","content_type":"text/html","content_length":"51579","record_id":"<urn:uuid:5ba7e7a1-b474-4edf-91af-87d18d737ee7>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00125.warc.gz"} |
Bus number 74
Every evening, Professor Oak walks from his job to the bus stop, waits for a bus 74, and then rides it home. But he has lately noticed that, usually, several 74’s have just passed when he reaches the
bus stop, so he has to wait for a long time for the next 74. Prof. Oak, who is a bit paranoiac, starts thinking that there is a conspiration against him. Funny, isn’t it?
Surprisingly, Prof. Oak is right. The same moment he leaves his job, a geostationary satellite contacts all the 74’s. From that moment on, every 74 chooses its speed so that Prof. Oak has to wait for
the maximum time at the bus stop. During this time, the 74’s don’t attend the rest of users, and do not stop no matter what, red signals included. There is just one exception to this rule: since
buses have just one lane, they cannot overtake (nor overlap) each other.
Let’s formalize more the problem you must solve. Lett be the time for Prof. Oak to reach the bus stop, letm be the minimum speed for a 74, letM be the maximum speed for a 74, letL be the
length of the bus lane, and letn be the number of 74’s. Model the bus lane with the circular interval [0, L). The bus stop is located at [0, 1). Every 74 has length 1 as well. At time0, each 74
numbered i = 1, …, n is at some known interval [p[i], p[i] + 1). From that moment on, every 74 moves to the right (circularly, if the end is reached). From the momentt on, the first instant that
the bus stop is overlapped by one or more 74’s, Prof. Oak rides any of them. Your task is to compute the maximum possible waiting time.
Note: Every moment, every bus can choose any speed (fractional or not) between m and M, both included. Look at the sample input and sample output below for some limiting cases.
Input is all integer numbers, and consists of several cases with t, m, M, L and n in this order, followed by p[1], …, p[n]. Assume t ≥ 0, 1 ≤ m ≤ M, 1 ≤ n ≤ L, that all the p[i]’s are correct and
different, and that no given number is larger than 10000.
For every case, print with four digits after the decimal point the maximum waiting time. | {"url":"https://jutge.org/problems/P55376_en","timestamp":"2024-11-10T12:17:36Z","content_type":"text/html","content_length":"25300","record_id":"<urn:uuid:cedf3f71-ac9c-4ddc-8bb6-b5184de9dfb5>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00897.warc.gz"} |
global optimization problem
This paper presents an outcome-space outer approximation algorithm for solving the problem of minimizing the product of two convex functions over a compact convex set in $\R^n$. The computational
experiences are reported. The proposed algorithm is convergent. Article Download View An Outcome Space Algorithm for Minimizing the Product of Two Convex Functions over a Convex … Read more
Outcome-Space Outer Approximation Algorithm for Linear Multiplicative Programming
This paper presents an outcome-space outer approximation algorithm for globally solving the linear multiplicative programming problem. We prove that the proposed algorithm is finite. To illustrate
the new algorithm, we apply it to solve some sample problems. Citation 10, Hanoi University of Technology, 07/2007 Article Download View Outcome-Space Outer Approximation Algorithm for Linear
Multiplicative Programming | {"url":"https://optimization-online.org/tag/global-optimization-problem/","timestamp":"2024-11-08T03:02:29Z","content_type":"text/html","content_length":"85445","record_id":"<urn:uuid:80be5ed5-2bba-47cd-96be-2adb7a0eab92>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00433.warc.gz"} |
Answers that are equations or inequalities that involve symbols
Suppose the correct answer to a question is the inequality
ad - bc != 0.
Is there a good way to check such an answer in WW? My current solution is to tell the students that their answer must have the form
[some expression in a, b, c, d] [= or != (popup)] [value].
The answer checker I'm currently using allows for two possibilities based on how I expect students to solve the problem: it will accept in the first blank an algebraic expression that is equivalent
to ad - bc or equivalent to d - bc/a (a is assumed nonzero in this problem). But of course 3(ad - bc) would be a valid expression as well ... it is unlikely that students would come up with this, but
it's technically correct, so ...
Any ideas would be appreciated! | {"url":"https://webwork.maa.org/moodle/mod/forum/discuss.php?d=4809&parent=14330","timestamp":"2024-11-08T01:38:01Z","content_type":"text/html","content_length":"66835","record_id":"<urn:uuid:2921ff18-9d65-4129-a765-df8c40fa6294>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00180.warc.gz"} |
HP Prime - problem with Solve App
05-19-2016, 04:57 PM
Post: #1
MarkF Posts: 3
Junior Member Joined: May 2016
HP Prime - problem with Solve App
Hi everybody, I recently updated my hp prime's firmware and have since been having trouble with using Solve app the way I used to. I used it mainly to solve (non-linear) equation systems e.g 4 eqs
involving 4 variables and putting appropriate seed values, then I was given the solution to the system. Now the message "cannot find the solution" keeps appearing no matter what the system is (tried
very simple ones to check if I was mistaken). The only way I can use the app now is to check equations with just one variable. Another thing I noticed that is different is that in the Plot view now
appears a message "check exactly one equation". I wonder if anyone could help with this problem. I thought of going back to the previous firmware version but I don't know how to do that if possible.
I hope you can help me soon, I'm an engineering student and the app used to be really useful to me. Thanks for your time!
05-19-2016, 05:40 PM
Post: #2
DrD Posts: 1,136
Senior Member Joined: Feb 2014
RE: HP Prime - problem with Solve App
Just a short way down the list is this thread:
(May be of some help).
05-19-2016, 06:13 PM
Post: #3
MarkF Posts: 3
Junior Member Joined: May 2016
RE: HP Prime - problem with Solve App
Ok, I get there is a problem with new version (which I find incredible to happen). Until it is corrected I need to come back to the last version. Is that possible? How should i procede?
Thanks for your time,
05-20-2016, 12:45 PM
Post: #4
roadrunner Posts: 449
Senior Member Joined: Jun 2015
RE: HP Prime - problem with Solve App
I don't know if it possible to downgrade to an earlier version, but there are alternatives to the solve app. Solve(), csolve(), and fsolve(), while more cumbersome, are more powerful than the solve
05-20-2016, 06:56 PM
Post: #5
MarkF Posts: 3
Junior Member Joined: May 2016
RE: HP Prime - problem with Solve App
Great piece of advice! I'll go through how to use these functions. Thanks a lot
05-23-2016, 02:06 PM
Post: #6
Tim Wessman Posts: 2,293
Senior Member Joined: Dec 2013
RE: HP Prime - problem with Solve App
Internally, the solve app is just doing a "fsolve([<equation_here>,<equation_here>,...],[<var>=<guess>,<var>=<guess>,...])" when there are multiple equations.
Although I work for HP, the views and opinions I post here are my own.
05-24-2016, 01:23 PM
Post: #7
retoa Posts: 168
Member Joined: Jan 2015
RE: HP Prime - problem with Solve App
Is there a plan to correct this error in a short time?
It is not very useful to have a solver where you can insert 10 equations if you can only solve one at a time. I did it with my 32sII ...
My students had saved the solve app with other names with various equations for electrothecnics, cinematics, dynamics, ...
It does not work anymore. I know you can use solve, fsolve, ..., but you have to rewrite the equations every time...
Go back to the older firmware would not be a solution for me, as the older one can not evaluate correctly the roots of complex numebers.
05-25-2016, 11:26 AM
(This post was last modified: 05-25-2016 11:39 AM by roadrunner.)
Post: #8
roadrunner Posts: 449
Senior Member Joined: Jun 2015
RE: HP Prime - problem with Solve App
I don't know if Tim will answer that question, but using fsolve is only a little more trouble than using the solve app.
You can store E0 thru E9 in a list as needed, then typing fsolve(listname,lname(listname)) in CAS get's you the results.
E1: A+B=12/C
E2: (A/B)=-3
E3: C=A*B
In CAS:
list1:={'E1','E2','E3'} // mind the quotes
returns: [−3.77976314968,1.25992104989,−4.7622031559]
edit: An industrious and benevolent soul could even create and app to hijack the input form of the solve app, perform the calculations, and output the results; then share it with the rest of the
05-25-2016, 12:59 PM
(This post was last modified: 05-25-2016 01:21 PM by DrD.)
Post: #9
DrD Posts: 1,136
Senior Member Joined: Feb 2014
RE: HP Prime - problem with Solve App
Interesting approach, road. This worked also:
fsolve("eval(L0),lname(eval(L0))"); ==> [−3.77976314968,1.25992104989,−4.7622031559]
fsolve(eval(L0),lname(eval(L0))); ==> [−3.77976314968,1.25992104989,−4.7622031559]
Even simpler:
fsolve("L1,lname(L1)"); ==> [−3.77976314968,1.25992104989,−4.7622031559]
fsolve(L1,lname(L1)); ==> [−3.77976314968,1.25992104989,−4.7622031559]
05-25-2016, 02:10 PM
Post: #10
roadrunner Posts: 449
Senior Member Joined: Jun 2015
RE: HP Prime - problem with Solve App
In CAS, the same approach, but with csolve, returns:
Which is the exact real solution from fsolve along with the two exact complex solutions. I haven't checked, but I assume the two complex solutions are correct; making this approach much more powerful
than the original solve app, which only returns real solutions.
User(s) browsing this thread: 1 Guest(s) | {"url":"https://hpmuseum.org/forum/thread-6281-post-56351.html","timestamp":"2024-11-03T12:19:49Z","content_type":"application/xhtml+xml","content_length":"42167","record_id":"<urn:uuid:077cda87-c426-41b5-a554-8de8eb38a4ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00746.warc.gz"} |
CPM Homework Help
Write the inequality that represents the $x$-values highlighted on each number line below.
The boundary point is $6$.
The arrow is pointing in the direction of numbers less than $6$.
The circle is filled in so the inequality includes $6$.
Follow the steps in part (a).
The highlighted $x$-values are greater than or equal to $2$ less than $7$.
Follow the steps in part (c). | {"url":"https://homework.cpm.org/category/CCI_CT/textbook/int1/chapter/9/lesson/9.3.3/problem/9-92","timestamp":"2024-11-04T02:37:52Z","content_type":"text/html","content_length":"43896","record_id":"<urn:uuid:acbad103-eaef-4b32-9f16-c2230dcc8989>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00102.warc.gz"} |
Nilpotent orbits of linear and cyclic quivers and Kazhdan-Lusztig polynomials of type A
The intersection cohomologies of closures of nilpotent orbits of linear (respectively, cyclic) quivers are known to be described by Kazhdan-Lusztig polynomials for the symmetric group (respectively,
the affine symmetric group). We explain how to simplify this description using a combinatorial cancellation procedure, and derive some consequences for representation theory.
arXiv Mathematics e-prints
Pub Date:
January 2005
□ Mathematics - Representation Theory;
□ Mathematics - Combinatorics;
□ 17B37 (Primary) 05E15;
□ 20C08 (Secondary)
34 pages | {"url":"https://ui.adsabs.harvard.edu/abs/2005math......1054H/abstract","timestamp":"2024-11-12T16:35:43Z","content_type":"text/html","content_length":"35243","record_id":"<urn:uuid:d698b953-d3ca-4ef0-ac26-d5c8f4425324>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00578.warc.gz"} |
Truth Table Calculator
This truth table calculator will provide the truth table values for the given propositional logic formulas. The propositional logic statements can only be true or false.
What is Truth Table?
The truth table is a tabular view of all combinations of values for the inputs and their corresponding outputs. It is a mathematical table that shows all possible results that may be occur from all
possible scenarios. It is used for logic tasks such as logic algebra and electronic circuits.
Prepositional Truth Tables Logic
A proposition is a set of declarative statements with a truth value of "true" or a truth value of "false". Propositional expressions are composed of connectives and propositional variables. We use
capital letters to represent the propositional variables (A, B). The connectives connect the propositional variables.
How to Make a Truth Table?
In propositional logic truth table calculator uses the different connectives which are −
• OR (∨)
• AND (∧)
• Negation/ NOT (¬)
• Implication / if-then (→)
• If and only if (⇔)
• Absurdity (#)
• Sheffer Stroke (|)
Propositional Equivalences
Two statements A and B are logically equivalent if any of the following two conditions hold –
• The bi-conditional statement A⇔B is a tautology.
• The truth tables of every statement have the same truth variables.
Example: Prove ~(P ∨ Q) and [(~P) ∧ (~Q)] are equivalent Solution: The truth tables calculator perform testing by matching truth table method
P Q P ∨ Q ¬ (P ∨ Q) ¬ P ¬ Q [(¬ P) ∧ (¬ Q)]
T T T F F F F
T F T F F T F
F T T F T F F
F F F T T T T
Here, we can see the truth values of ~(P ∨ Q) and [(~P) ∧ (~Q)] are same, hence all the statements are equivalent.
How does Truth Table Calculator Works?
An online truth table generator provides the detailed truth table by following steps:
• First, enter a propositional logic equation with symbols.
• Hit the calculate button for results.
• Our calculator construct a truth table for 4 variables of the given expression.
Use this online truth table generator to create the multivariate propositional logic truth tables. Propositional logic deals with statements that can be truth values, "true" and "false". The purpose
is to analyze these statements individually or collectively.
From the source of Wikipedia: Unary operations, Logical true, Logical false, Logical identity, Logical negation, Binary operations, Logical conjunction (AND), Logical disjunction (OR), Logical | {"url":"https://calculator-online.net/truth-table-calculator/","timestamp":"2024-11-03T22:11:25Z","content_type":"text/html","content_length":"60652","record_id":"<urn:uuid:bedcede9-61ea-4494-9a33-736ec0610945>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00010.warc.gz"} |
Where was the President?
canadyjd Well-Known Member
Oct 13, 2005
Likes Received:
I was watching the news this evening. They had footage of Rosa Parks' funeral service. I didn't see George Bush. I don't think I saw a single Republican. Am I wrong, or did no Republicans show
up? Did anyone else notice this?
Joseph_Botwinick <img src=/532.jpg>Banned
Nov 12, 2000
Likes Received:
Nope. I didn't notice.
Joseph Botwinick
LadyEagle <b>Moderator</b> <img src =/israel.gif>
Feb 7, 2002
Likes Received:
I noticed Bill Clinton was there. He was telling how he and two of his friends, when they heard what Rosa Parks did, got up & moved to the back of the bus. I didn't know that Southern white kids
8 or 9 years old were that involved in civil rights back in the 1950s...but maybe it was just me.
Enoch New Member
Mar 12, 2004
Likes Received:
Were there certain Republicans you were looking for? Was this your objective in watching the funeral service? Just curious. Because how can you tell who is a Republican or not?
Bro. Curtis <img src =/curtis.gif>
Site Supporter
Oct 25, 2001
Likes Received:
I couldn't beleive he (Clinton) said that.
BTW, & slightly O/T....Aretha Franklin has still got it.
emeraldctyangel New Member
Jun 11, 2005
Likes Received:
Were there certain Republicans you were looking for? Was this your objective in watching the funeral service? Just curious. Because how can you tell who is a Republican or not? </font>[/QUOTE]Why
dont you know? They (the republicans) all wear special buttons on their clothing!
For those fans of Bush who watch his every move...this was recorded in several news sites. Make sure you put it in your diary now, we wont want to forget where the President was during a funeral.
Parks Remembered for Her Courage, Strength
By KEN THOMAS, Associated Press Writer
And for the record: I dont watch funerals on tv. Kind of wierd if you ask me.
NaasPreacher (C4K) Well-Known Member
Oct 21, 2003
Likes Received:
I do. He will turn any event into an opportunity for self glorification and we all know he has no problem lying.
Mexdeaf New Member
Mar 14, 2005
Likes Received:
It's pretty simple, the Republicans were dis-invited. If they had showed up, Jesse and the "Rev" Al would have pitched a fit. The media would have accused the President of attending with some
ulterior motive.
As to Clinton- typical. Sounds like some preachers whose illustrations are nearly always about themselves.
music4Him New Member
Feb 7, 2004
Likes Received:
I thought the prez was doing lunch with the Prince Charles and Camilla? Then again a royal gala dinner later that night.
carpro Well-Known Member
Site Supporter
Oct 14, 2004
Likes Received:
Well, you know he is, after all, truth challenged.
canadyjd Well-Known Member
Oct 13, 2005
Likes Received:
Were there certain Republicans you were looking for? Was this your objective in watching the funeral service? Just curious. Because how can you tell who is a Republican or not? </font>[/QUOTE]I
didn't have an objective. I was watching the news...which included a segment on the funeral. I simply found it curious that I saw no Republicans, that I recognized anyway, in attendance or
MatthewHenry New Member
Aug 27, 2005
Likes Received:
The thing that really got me, was how Martin Luther Kings Daughter, took Ephesians 2:6, so far out of context and tried to apply that to Rosa Parks!
She needs to read this Scripture:
Also, Did anyone else see the crazy "Pentecostal" outbreak during the service, when that one guy was praying or whatever he was doing...
Bible does say:
Anyhow, It was a LOOOONG funeral, they said they tried to tell everyone, 15 minutes a piece, that worked for like 2 speakers... | {"url":"https://www.baptistboard.com/threads/where-was-the-president.492/","timestamp":"2024-11-08T14:24:18Z","content_type":"text/html","content_length":"97928","record_id":"<urn:uuid:65f49049-90d0-46bd-9e27-dfc591c95bb4>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00642.warc.gz"} |
3.1.3. Problem Definition
Let X be the input (feature) space and Y the target (label) space. A domain is defined as a joint distribution $P X Y$ on $X × Y$. For a specific domain $P X Y$, we refer to $P X$ as the marginal
distribution on X, $P Y | X$ as the posterior distribution of Y given X, and $P X | Y$ as the class-conditional distribution of X given Y.
In the context of Domain Generalization (DG), we have access to K similar but distinct source domains $S = { S k = { ( x ( k ) , y ( k ) ) } } k = 1 K$, each associated with a joint distribution $P X
Y ( k )$. Note that $P X Y ( k ) ≠ P X Y ( k ′ )$ for $k ≠ k ′$ and $k , k ′ ∈ { 1 , . . . , K }$. The goal of DG is to learn a predictive model $f : X → Y$ using only source domain data such that
the prediction error on an unseen target domain $T = { x T }$ is minimized. The corresponding joint distribution of the target domain T is denoted by $P X Y T$. Also, $P X Y T ≠ P X Y ( k )$ for all
$k ∈ { 1 , . . . , K }$.
Two fundamental types of DG scenarios are identified:
• Single-Source DG: In this case, training data stems from a homogeneous source domain, i.e., $K = 1$.
• Multi-Source DG: This setting involves the study of DG across multiple sources. The majority of research is focused on the multi-source DG scenario, where diverse and relevant domains ($K > 1$)
are available.
Lv et al. [
] presented a structural causal model that integrates both causal and non-causal factors, raw inputs, and category labels, as depicted in
Figure 4
(II). This work addresses the limitations of statistical approaches in domain generalization by adopting a causal perspective. The authors introduce the
Causality Inspired Representation Learning (CIRL)
algorithm, which leverages a structural causal model to enforce representations that adhere to essential causal properties and simulate causal factors. The proposed representation learning approach
comprises three key modules: the
causal intervention module
, the
causal factorization module
, and the
adversarial mask module
• Causal Intervention Module: This module focuses on separating causal factors from non-causal factors through do-interventions. By doing so, the causal factors remain unchanged despite non-causal
perturbations. This process generates representations that are independent of non-causal influences.
• Causal Factorization Module: This module promote independence among representation dimensions. It achieves this by minimizing correlations between dimensions. This transformation converts
initially interdependent and noisy representations into independent ones, aligning with the characteristics of ideal causal factors.
• Adversarial Mask Module: In this module, the representations efficacy for the classification task $X → Y$ is enhanced. An adversarial masker identifies dimensions of varying importance. This step
helps distinguish superior dimensions from inferior ones, allowing the former to contribute more significantly. As a result, the representations become more causally informative.
The optimization objective of the proposed Causality Inspired Representation Learning (CIRL) encompasses combining classification losses for both superior and inferior dimensions ($L sup cls$ and $L
inf cls$) with the causal factorization loss ($L F ac$). This optimization process is balanced by a trade-off parameter ($τ$).
With the same structure, a causal regularizer is proposed by Shen et al. [
] with the primary objective was to address the non-i.i.d. problem, characterized by distribution disparities between training and testing image sets that often lead to classification challenges. To
mitigate this issue, they harness causal inference to scrutinize the causal influence of individual image features on corresponding labels, with the aim of identifying pivotal causal factors. By
considering each image feature as an individual variable, they distinguish between treated and control images, thus enabling the estimation of causal effects. This unique methodology uncovers
compelling causal impacts that endure across various distributions. This attemt is formally posed as the Causal Classification Problem, entailing the discovery of causal contributions for image
features and the construction of an image classifier grounded in these contributions. The authors proposed algorithm synergistically combines a causal regularizer inspired by the concept of
confounder balancing.
In the context of observational studies, the rectification of bias stemming from non-random treatment assignments necessitates balancing confounder distributions. However, the authors depart from
direct confounder distribution balancing and introduce a novel approach focused on equilibrating confounder moments by adjusting sample weights. The determination of these weights, represented as
, is accomplished through the optimization problem:
$W = arg min w ∥ X ¯ t − ∑ j : T j = 0 W j · X j ∥ 2 2 .$
Here, $X ¯ t$ and $∑ j : T j = 0 W j · X j$ denote the mean values of confounders in treated and control samples, respectively, in the context of a specific treatment feature T. Notably, this method
concentrates on the first-order moment, but it can be easily extended to incorporate higher-order moments. The authors intend to adapt this technique to simultaneously balance confounder
distributions related to all treatment features.
Expanding upon the concept of confounder balancing, the authors introduce a causal regularizer:
In this equation, W signifies sample weights, and the term $X − j T · ( W ⊙ I j ) W T · I j − X − j T · ( W ⊙ ( 1 − I j ) ) W T · ( 1 − I j )$ characterizes the loss associated with confounder
balancing when considering image feature j as a treatment variable. $X − j$ encompasses all other features (treated as confounders) obtained from X by zeroing its jth column. $I j$ denotes the jth
column of the identity matrix I, and $I i j$ signifies the treatment status of unit i with regard to feature j. The underlying concept of this regularizer is its facilitation of estimating causal
effects related to the treatment variable.
] The author’s objective revolves around comprehensively explaining the phenomena of forgetting and anti-forgetting within the framework of Class-Incremental Learning (CIL) through the lens of causal
inference. This endeavour is undertaken by framing the data, features, and labels within causal graphs for each incremental learning step, effectively elucidating the underlying causal relationships
governing these components. By adopting this approach, the author systematically dissects the mechanisms driving both forgetting and anti-forgetting in CIL.
Beginning with the causal graph, the author delineates the intricate relationships between old data (D), new training samples (I), extracted features (X and Xo), and predicted labels (Y and Yo).
These relationships are represented by distinct links: I → X, X → Y, (D, I) → Xo, (D, Xo) → Yo, D → I, Xo → X, and Yo → Y as shown in (
Figure 5
Building upon the causal graph, the author proceeds to formulate the causal effect between variables through causal intervention, denoted as the do(·) operation. This operation effectively enforces
specific values upon variables, resembling a “surgical” intervention within the graph. The objective is to isolate variables from their causal influences. With this framework in place, the author
delves into explaining the rationale behind forgetting within the new training process. The causal effect of old data (EffectD) on new predictions is quantified, effectively capturing the causal
essence of the previous knowledge. Notably, the author defines EffectD as the difference in predicted labels with and without the presence of old data, shedding light on the pivotal role of this
metric in understanding forgetting.
To contrast forgetting, the author introduces the concept of anti-forgetting and dissects the causal effects that underpin this phenomenon. This exploration encompasses prevailing anti-forgetting
techniques, including data replay, feature distillation, and label distillation. By leveraging the causal relationships depicted in the causal graph, the author meticulously analyzes the effects of
these techniques in mitigating forgetting. The causal relationships enable an in-depth examination of the interactions between these techniques and the system dynamics. Specifically, the impact of
data replay on causal relationships, the effects of feature and label distillation, and the introduction of non-zero effects are all cogently discussed in the context of thwarting forgetting.
Mahajan et al. [
] tackle the challenge of achieving effective generalization across domains in machine learning models. Their focus is on ensuring models can generalize accurately to new domains despite domain
The authors employ causal inference to understand relationships between variables. They create a Structural Causal Model (SCM) to depict data generation and variable interconnections in domain
generalization. This causal approach helps them identify crucial causal features for cross-domain classification, leading to robust generalization.
The key steps of their approach are as follows:
• Recognizing the Confounder and Invariance Condition: The authors introduce the object variable O as a confounder that influences features X and class labels Y. They aim to find invariant
representations across domains that are informative about O.
• Introducing the Matching Function $Ω$: They propose a matching function $Ω$ to assess if pairs of inputs from different domains correspond to the same object. This function enforces consistency
of representation across different domains but with the same object.
• Defining the Invariance Condition: An average pairwise distance condition between representations of the same object from different domains is stipulated. This condition ensures close
representations for the same object across various domains.
• Learning Invariant Representations: To learn invariant representations, the authors introduce the “perfect-match” invariant, combining classification loss and the invariance condition. This loss
function encourages representations that are invariant to domain shifts while preserving object-related information.
Liu et al. [
] address the challenge of out-of-distribution (OOD) generalization by mitigating the confounding effects of mixed semantic and variation factors in learned representations. The authors introduce a
Causal Semantic Generative (CSG) model that explicitly models separate causal relationships between semantic and variation factors, enhancing the model’s performance on OOD examples. The key
components of their approach include:
• Introducing the CSG model to represent causal relationships between semantic (s), variation (v), and observed data (x, y).
• Disentangling semantic and variation factors using latent variables s and v, ensuring accurate modeling of causal relations.
• Addressing confounding by attributing x-y relationships to latent factor z and accounting for interrelation between semantic and variation factors.
Sun et al. [
] tackled the challenge of degraded prediction accuracy due to distributional shifts and spurious correlation. They introduced a Latent Causal Invariance Model (LaCIM) to handle this confounding.
LaCIM incorporates causal structure and a domain variable to address confounding issues. The authors identified a spurious correlation between latent factors
, arising from an unobserved confounder
. This correlation can negatively impact model performance across domains. To address this, LaCIM employs a structural causal model framework, representing relationships between latent factors. A
domain variable accounts for distributional shifts, while Disentangled Causal Mechanisms (DCMs) capture attribute-specific shifts. By leveraging DCMs, LaCIM replaces unobserved attributes with proxy
variables, aiding model adaptation. The authors achieve this by implementing the transportability theory.
The paper by [
] et al. addresses the challenge of adapting models from a source to a target domain in the context of Unsupervised Domain Adaptation (UDA). UDA seeks to enhance model performance on a target domain
without target domain labels. The authors introduce a novel approach called Transporting Causal Mechanisms (TCM) to bridge the domain gap. The authors identify a confounding factor, the domain
selection variable
, as the key challenge in UDA. They propose using disentangled causal mechanisms as proxies for unobserved domain-specific attributes. The TCM approach involves three main steps: (1) identifying
disentangled causal mechanisms that correspond to attribute-specific shifts, (2) using transformed outputs as proxy variables for unobserved attributes, and (3) transporting these mechanisms to align
feature representations between domains. By substituting unobserved attributes with proxy variables derived from disentangled mechanisms, the authors effectively mitigate the confounding effect. The
core intervention equation
$P ( Y ∣ do ( X ) , S ) = P ( Y ∣ X , U = u ) P ( U = u ∣ S )$
captures domain-specific shifts in a principled manner.
In their work [
], the authors introduce the Contrastive Causal Model (CCM) as a solution to the domain generalization problem. CCM employs contrastive similarity to convert new images into prior knowledge and
amplify the causal effects from images to labels. The CCM framework comprises crucial components, including a teacher-student backbone, a classifier, and a knowledge queue. The approach encompasses
domain-conditioned supervised learning, causal effect learning, and contrastive similarity learning. The authors identify a significant confounder in the domain generalization challenge, manifested
as the spurious correlation introduced by domain shifts. This confounding factor can undermine genuine causal relationships among variables. To mitigate this confounding effect, the authors introduce
a structural causal model (SCM) that explicitly delineates the causal paths from images to labels. This strategy serves to disentangle the authentic causal effects from the spurious correlations,
thereby allowing the model to concentrate on the true causal relationships. The training of CCM encompasses a sequence of essential steps:
• Domain-Conditioned Supervised Learning: The model optimizes cross-entropy loss while conditioning on the domain. This strategy captures the correlation between images and labels across diverse
• Causal Effect Learning: Leveraging the front-door criterion, the authors measure and enhance causal effects. A knowledge queue is leveraged to retain historical features and labels, aiding the
translation of new images into acquired knowledge.
• Contrastive Similarity Learning: The application of contrastive similarity serves to cluster features sharing the same category. This process quantifies feature similarity, facilitating the
separation of features.
Wang et al. [
] authors address the challenge of domain generalization building a classifier that performs well in unseen environments. The fluctuating correlation between features and labels across different
environments hinders effective generalization. To tackle this, they propose a two-phased approach. First, a Variational Autoencoder (VAE) learns the data distribution for each environment, capturing
latent variables responsible for inter-environment variations. Then, balanced mini-batch sampling is introduced. It pairs examples with similar balancing scores from VAE’s conditional prior, forming
balanced mini-batches. This counters unstable correlations between features and labels across environments. The main confounder addressed is the variable correlation between label
and latent
across environments. This variance affects the causal relationship between features and labels, challenging robust generalization. A Variational Autoencoder (VAE) is employed to learn data
distribution within each environment. Model parameters and latent variable
are learned from training data. The model captures causal connections among
, and observed feature
. To manage varying
correlations across environments, the authors propose balanced mini-batch sampling. Dissimilar examples with matching balancing scores are paired, forming balanced mini-batches. This cultivates a
balanced distribution that emphasizes stable
causal relations. It mitigates adverse effects of unstable correlations in training data.
Wang et al. [
] addresses domain generalization by leveraging invariance in causal mechanisms to enhance model generalization across distributions while preserving semantic relevance for downstream tasks. The
authors defined domains as distributions over input (
) and label (
) spaces. They proposed aggregating instances from diverse domains into the training dataset
and employing Empirical Risk Minimization (ERM) loss to optimize encoding (
$f θ$
) and decoding (
$g ϕ$
) models. The paper connects neural networks and causal models, treating datasets as Structural Causal Models (SCMs). This interpretation enhances confidence in the neural model’s ability to capture
causal mechanisms from observational data. The average Causal Effect (ACE) quantifies the influence of each input feature on the output. Neural network (
$g ϕ$
) serves as a tool for causal quantification, with a clear procedure for ACE computation. Inspired by contrastive representation learning, they introduced contrastive ACE loss. It identifies
differences in ACE values across domains, optimizing interclass dissimilarity and intra-class similarity via positive and negative sets. Their training involves encoding and decoding model
initialization, positive and negative set construction, and loss optimization using gradient descent. The innovative contrastive ACE loss, coupled with ERM, stabilizes performance against domain
The core problem addressed by Chen et al. [
] was the challenge of domain generalization, where a model trained on a single source domain must be capable of handling various unseen target domains. This scenario is characterized by the need to
account for the domain shift between the source and target domains. The proposed solution introduced the novel “simulate-analyze-reduce” paradigm, which represents a comprehensive approach to
mitigate the challenges posed by domain shifts in single-domain generalization. This paradigm comprises three primary stages: domain shift simulation, causal analysis, and domain shift reduction. The
underlying idea was to first simulate domain shifts by generating an auxiliary domain through the transformation of source data. These shifts are based on variant factors, representing extrinsic
attributes causing domain shifts. Subsequently, a meta-causal learning method was introduced, enabling the model to infer causal relationships between domain shifts in the auxiliary and source
domains during the training phase. This inferred meta-knowledge is then utilized for analyzing the domain shifts between target and source domains during testing. | {"url":"https://www.preprints.org/manuscript/202312.2087/v1","timestamp":"2024-11-13T04:21:28Z","content_type":"text/html","content_length":"903134","record_id":"<urn:uuid:73372bbe-3733-4a42-96c2-b3317a89d76c>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00297.warc.gz"} |
Arun Sharma
• Queensland University of Technology, Brisbane, Australia
• University of New South Wales, School of Computer Science and Engineering, Sydney, Australia
According to our database
, Arun Sharma authored at least 60 papers between 1989 and 2007.
Collaborative distances:
Book In proceedings Article PhD thesis Dataset Other
Online presence:
On csauthors.net:
Deduction, Induction, and beyond in Parametric Logic.
Proceedings of the Induction, Algorithmic Learning Theory, and Philosophy, 2007
On the data consumption benefits of accepting increased uncertainty.
Theor. Comput. Sci., 2007
On ordinal VC-dimension and some notions of complexity.
Theor. Comput. Sci., 2006
Unifying logic, topology and learning in Parametric logic.
Theor. Comput. Sci., 2006
Identifying Clusters from Positive Data.
SIAM J. Comput., 2006
On a Syntactic Characterization of Classification with a Mind Change Bound.
Proceedings of the Learning Theory, 18th Annual Conference on Learning Theory, 2005
Some applications of logic to feasibility in higher types.
ACM Trans. Comput. Log., 2004
Generalized notions of mind change complexity.
Inf. Comput., 2004
On the classification of recursive languages.
Inf. Comput., 2004
Web Searching and S 2 Queries.
Proceedings of the Advanced Web Technologies and Applications, 2004
Learning power and language expressiveness.
Theor. Comput. Sci., 2003
Theor. Comput. Sci., 2002
Mind change complexity of learning logic programs.
Theor. Comput. Sci., 2002
Learning to Win Process-Control Games Watching Game-Masters.
Inf. Comput., 2002
Learning in Logic with RichProlog.
Proceedings of the Logic Programming, 18th International Conference, 2002
Learning, Logic, and Topology in a Common Framework.
Proceedings of the Algorithmic Learning Theory, 13th International Conference, 2002
Synthesizing noise-tolerant language learners.
Theor. Comput. Sci., 2001
Predictive learning models for concept drift.
Theor. Comput. Sci., 2001
On a Generalized Notion of Mistake Bounds.
Inf. Comput., 2001
A General Theory of Deduction, Induction, and Learning.
Proceedings of the Discovery Science, 4th International Conference, DS 2001, Washington, 2001
Team Learning of Computable Languages.
Theory Comput. Syst., 2000
Robust Learning Aided by Context.
J. Comput. Syst. Sci., 2000
Ordinal Mind Change Complexity of Language Identification.
Theor. Comput. Sci., 1999
On Sufficient Conditions for Learnability of Logic Programs from Positive Data.
Proceedings of the Inductive Logic Programming, 9th International Workshop, 1999
The VC-Dimension of Subclasses of Pattern.
Proceedings of the Algorithmic Learning Theory, 10th International Conference, 1999
A Note on Batch and Incremental Learnability.
J. Comput. Syst. Sci., 1998
Generalization and Specialization Strategies for Learning r.e. Languages.
Ann. Math. Artif. Intell., 1998
LIME: A System for Learning Relations.
Proceedings of the Algorithmic Learning Theory, 9th International Conference, 1998
Learning from Multiple Sources of Inaccurate Data.
SIAM J. Comput., 1997
The Structure of Intrinsic Complexity of Learning.
J. Symb. Log., 1997
Elementary Formal Systems, Intrinsic Complexity, and Procrastination.
Inf. Comput., 1997
Characterizing Language Identification in Terms of Computable Numberings.
Ann. Pure Appl. Log., 1997
On the Classification of Computable Languages.
Proceedings of the STACS 97, 14th Annual Symposium on Theoretical Aspects of Computer Science, Lübeck, Germany, February 27, 1997
ILP with Noise and Fixed Example Size: A Bayesian Approach.
Proceedings of the Fifteenth International Joint Conference on Artificial Intelligence, 1997
Anomalous Learning Helps Succinctness.
Theor. Comput. Sci., 1996
The Intrinsic Complexity of Language Identification.
J. Comput. Syst. Sci., 1996
Computational Limits on Team Identification of Languages.
Inf. Comput., 1996
Machine Induction Without Revolutionary Changes in Hypothesis Size.
Inf. Comput., 1996
Team Learning of Recursive Languages.
Proceedings of the PRICAI'96: Topics in Artificial Intelligence, 1996
Finite Identification of Functions by Teams with Success Ratio 1\over2 and Above
Inf. Comput., September, 1995
Complexity Issues for Vacillatory Function Identification
Inf. Comput., February, 1995
On Aggregating Teams of Learning Machines.
Theor. Comput. Sci., 1995
Prudence in Vacillatory Language Identification.
Math. Syst. Theory, 1995
On Identification by Teams and Probabilistic Machines.
Proceedings of the Algorithmic Learning for Knowledge-Based Systems, GOSLER Final Report, 1995
Machine Induction Without Revolutionary Paradigm Shifts.
Proceedings of the Algorithmic Learning Theory, 6th International Conference, 1995
Program Size Restrictions in Computational Learning.
Theor. Comput. Sci., 1994
Characterizing Language Identification by Standardizing Operations.
J. Comput. Syst. Sci., 1994
Vacillatory Learning of Nearly Minimal Size Grammars.
J. Comput. Syst. Sci., 1994
On Monotonic Strategies for Learning r.e. Languages.
Proceedings of the Algorithmic Learning Theory, 1994
Learning with the Knowledge of an Upper Bound on Program Size
Inf. Comput., January, 1993
On the Non-Existence of Maximal Inference Degrees for Language Identification.
Inf. Process. Lett., 1993
Probability is More Powerful Than Team for Language Identification from Positive Data.
Proceedings of the Sixth Annual ACM Conference on Computational Learning Theory, 1993
On Learning Limiting Programs.
Int. J. Found. Comput. Sci., 1992
Prudence in Vacillatory Language Identification (Extended Abstract).
Proceedings of the Algorithmic Learning Theory, Third Workshop, 1992
Learning in the Presence of Partial Explanations
Inf. Comput., December, 1991
Hypothesis Formation and Language Acquisition with an Infinitely-Often Correct Teacher.
Proceedings of the 3rd Conference on Theoretical Aspects of Reasoning about Knowledge, 1990
Language Learning by a "Team" (Extended Abstract).
Proceedings of the Automata, Languages and Programming, 17th International Colloquium, 1990
Finite Learning by a "Team".
Proceedings of the Third Annual Workshop on Computational Learning Theory, 1990
Anomalous Learning Helps Succinctness (Extended Abstract).
Proceedings of the Algorithmic Learning Theory, First International Workshop, 1990
Convergence to Nearly Minimal Size Grammars by Vacillating Learning Machines.
Proceedings of the Second Annual Workshop on Computational Learning Theory, 1989 | {"url":"https://www.csauthors.net/arun-sharma-001/","timestamp":"2024-11-10T17:19:19Z","content_type":"text/html","content_length":"67603","record_id":"<urn:uuid:7769053e-6e94-40d4-ba81-dcd69687734e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00497.warc.gz"} |
Manually Convert a Floating-Point MATLAB Algorithm to Fixed Point
This example shows how to convert a floating-point algorithm to fixed point and then generate C code for the algorithm. The example uses the following best practices:
• Separate your algorithm from the test file.
• Prepare your algorithm for instrumentation and code generation.
• Manage data types and control bit growth.
• Separate data type definitions from algorithmic code by creating a table of data definitions.
For a complete list of best practices, see Manual Fixed-Point Conversion Best Practices.
Separate Your Algorithm from the Test File
Write a MATLAB^® function, mysum, that sums the elements of a vector.
function y = mysum(x)
y = 0;
for n = 1:length(x)
y = y + x(n);
Since you only need to convert the algorithmic portion to fixed-point, it is more efficient to structure your code so that the algorithm, in which you do the core processing, is separate from the
test file.
Write a Test Script
In the test file, create your inputs, call the algorithm, and plot the results.
1. Write a MATLAB script, mysum_test, that verifies the behavior of your algorithm using double data types.
n = 10;
rng default
x = 2*rand(n,1)-1;
% Algorithm
y = mysum(x);
% Verify results
y_expected = sum(double(x));
err = double(y) - y_expected
rng default puts the settings of the random number generator used by the rand function to its default value so that it produces the same random numbers as if you restarted MATLAB.
2. Run the test script.
The results obtained using mysum match those obtained using the MATLAB sum function.
For more information, see Create a Test File.
Prepare Algorithm for Instrumentation and Code Generation
In your algorithm, after the function signature, add the %#codegen compilation directive to indicate that you intend to instrument the algorithm and generate C code for it. Adding this directive
instructs the MATLAB code analyzer to help you diagnose and fix violations that would result in errors during instrumentation and code generation.
function y = mysum(x) %#codegen
y = 0;
for n = 1:length(x)
y = y + x(n);
For this algorithm, the code analyzer indicator in the top right corner of the editor window remains green telling you that it has not detected any issues.
For more information, see Prepare Your Algorithm for Code Acceleration or Code Generation.
Generate C Code for Your Original Algorithm
Generate C code for the original algorithm to verify that the algorithm is suitable for code generation and to see the floating-point C code. Use the codegen (MATLAB Coder) function (requires MATLAB
Coder™) to generate a C library.
1. Add the following line to the end of your test script to generate C code for mysum.
codegen mysum -args {x} -config:lib -report
2. Run the test script again.
MATLAB Coder generates C code for mysum function and provides a link to the code generation report.
3. Click the link to open the code generation report and view the generated C code for mysum.
/* Function Definitions */
double mysum(const double x[10])
double y;
int n;
y = 0.0;
for (n = 0; n < 10; n++) {
y += x[n];
return y;
Because C does not allow floating-point indices, the loop counter, n, is automatically declared as an integer type. You do not need to convert n to fixed point.
Input x and output y are declared as double.
Manage Data Types and Control Bit Growth
Test Your Algorithm With Singles to Check for Type Mismatches
1. Modify your test file so that the data type of x is single.
n = 10;
rng default
x = single(2*rand(n,1)-1);
% Algorithm
y = mysum(x);
% Verify results
y_expected = sum(double(x));
err = double(y) - y_expected
codegen mysum -args {x} -config:lib -report
2. Run the test script again.
err =
Attempt to write a value of type 'single' into a variable defined
as type 'double'. Code generation does not support changing types
through assignment. To investigate the cause of the type mismatch,
check preceding assignments or input type specifications.
Code generation fails, reporting a data type mismatch on line y = y + x(n);.
3. To view the error, open the report.
In the report, on the line y = y + x(n), the report highlights the y on the left side of the assignment in red to indicate that there is an error. The issue is that y is declared as a double but
is being assigned to a single. y + x(n) is the sum of a double and a single which is a single. If you place your cursor over variables and expressions in the report, you can see information about
their types. Here, you can see that the expression, y + x(n) is a single.
4. To fix the type mismatch, update your algorithm to use subscripted assignment for the sum of elements. Change y = y + x(n) to y(:) = y + x(n).
function y = mysum(x) %#codegen
y = 0;
for n = 1:length(x)
y(:) = y + x(n);
Using subscripted assignment, you also prevent the bit growth which is the default behavior when you add fixed-point numbers. For more information, see Bit Growth. Preventing bit growth is
important because you want to maintain your fixed-point types throughout your code. For more information, see Controlling Bit Growth.
5. Regenerate C code and open the code generation report. In the C code, the result is now cast to double to resolve the type mismatch.
Build Instrumented Mex
Use the buildInstrumentedMex function to instrument your algorithm for logging minimum and maximum values of all named and intermediate variables. Use the showInstrumentationResults function to
propose fixed-point data types based on these logged values. Later, you use these proposed fixed-point types to test your algorithm.
1. Update the test script:
1. After you declare n, add buildInstrumentedMex mySum —args {zeros(n,1)} -histogram.
2. Change x back to double. Replace x = single(2*rand(n,1)-1); with x = 2*rand(n,1)-1;
3. Instead of calling the original algorithm, call the generated MEX function. Change y = mysum(x) to y=mysum_mex(x).
4. After calling the MEX function, add showInstrumentationResults mysum_mex -defaultDT numerictype(1,16) -proposeFL. The -defaultDT numerictype(1,16) -proposeFL flags indicate that you want to
propose fraction lengths for a 16-bit word length.
Here is an updated test script.
%% Build instrumented mex
n = 10;
buildInstrumentedMex mysum -args {zeros(n,1)} -histogram
%% Test inputs
rng default
x = 2*rand(n,1)-1;
% Algorithm
y = mysum_mex(x);
% Verify results
showInstrumentationResults mysum_mex ...
-defaultDT numerictype(1,16) -proposeFL
y_expected = sum(double(x));
err = double(y) - y_expected
%% Generate C code
codegen mysum -args {x} -config:lib -report
2. Run the test script again.
The showInstrumentationResults function proposes data types and opens a report to display the results.
3. In the report, click the Variables tab. showInstrumentationResults proposes a fraction length of 13 for y and 15 for x.
In the report, you can:
• View the simulation minimum and maximum values for the input x and output y.
• View the proposed data types for x and y.
• View information for all variables, intermediate results, and expressions in your code.
To view this information, place your cursor over the variable or expression in the report.
• View the histogram data for x and y to help you identify any values that are outside range or below precision based on the current data type.
To view the histogram for a particular variable, click its histogram icon, .
Separate Data Type Definitions from Algorithmic Code
Rather than manually modifying the algorithm to examine the behavior for each data type, separate the data type definitions from the algorithm.
Modify mysum so that it takes an input parameter, T, which is a structure that defines the data types of the input and output data. When y is first defined, use the cast function like syntax — cast
(x,'like',y) — to cast x to the desired data type.
function y = mysum(x,T) %#codegen
y = cast(0,'like',T.y);
for n = 1:length(x)
y(:) = y + x(n);
Create a Table of Data Type Definitions
Write a function, mytypes, that defines the different data types that you want to use to test your algorithm. In your data types table, include double, single, and scaled double data types as well as
the fixed-point data types proposed earlier. Before converting your algorithm to fixed point, it is good practice to:
• Test the connection between the data type definition table and your algorithm using doubles.
• Test the algorithm with singles to find data type mismatches and other problems.
• Run the algorithm using scaled doubles to check for overflows.
function T = mytypes(dt)
switch dt
case 'double'
T.x = double([]);
T.y = double([]);
case 'single'
T.x = single([]);
T.y = single([]);
case 'fixed'
T.x = fi([],true,16,15);
T.y = fi([],true,16,13);
case 'scaled'
T.x = fi([],true,16,15,...
T.y = fi([],true,16,13,...
For more information, see Separate Data Type Definitions from Algorithm.
Update Test Script to Use Types Table
Update the test script, mysum_test, to use the types table.
1. For the first run, check the connection between table and algorithm using doubles. Before you declare n, add T = mytypes('double');
2. Update the call to buildInstrumentedMex to use the type of T.x specified in the data types table: buildInstrumentedMex mysum -args {zeros(n,1,'like',T.x),T} -histogram
3. Cast x to use the type of T.x specified in the table: x = cast(2*rand(n,1)-1,'like',T.x);
4. Call the MEX function passing in T: y = mysum_mex(x,T);
5. Call codegen passing in T: codegen mysum -args {x,T} -config:lib -report
Here is the updated test script.
%% Build instrumented mex
T = mytypes('double');
n = 10;
buildInstrumentedMex mysum ...
-args {zeros(n,1,'like',T.x),T} -histogram
%% Test inputs
rng default
x = cast(2*rand(n,1)-1,'like',T.x);
% Algorithm
y = mysum_mex(x,T);
% Verify results
showInstrumentationResults mysum_mex ...
-defaultDT numerictype(1,16) -proposeFL
y_expected = sum(double(x));
err = double(y) - y_expected
%% Generate C code
codegen mysum -args {x,T} -config:lib -report
6. Run the test script and click the link to open the code generation report.
The generated C code is the same as the code generated for the original algorithm. Because the variable T is used to specify the types and these types are constant at code generation time; T is
not used at run time and does not appear in the generated code.
Generate Fixed-Point Code
Update the test script to use the fixed-point types proposed earlier and view the generated C code.
1. Update the test script to use fixed-point types. Replace T = mytypes('double'); with T = mytypes('fixed'); and then save the script.
2. Run the test script and view the generated C code.
This version of C code is not very efficient; it contains a lot of overflow handling. The next step is to optimize the data types to avoid overflows.
Optimize Data Types
Use Scaled Doubles to Detect Overflow
Scaled doubles are a hybrid between floating-point and fixed-point numbers. Fixed-Point Designer™ stores them as doubles with the scaling, sign, and word length information retained. Because all the
arithmetic is performed in double-precision, you can see any overflows that occur.
1. Update the test script to use scaled doubles. Replace T = mytypes('fixed'); with T = mytypes('scaled');
2. Run the test script again.
The test runs using scaled doubles and displays the report. No overflows are detected.
So far, you’ve run the test script using random inputs which means that it is unlikely that the test has exercised the full operating range of the algorithm.
3. Find the full range of the input.
-1.000000000000000 0.999969482421875
DataTypeMode: Fixed-point: binary point scaling
Signedness: Signed
WordLength: 16
FractionLength: 15
4. Update the script to test the negative edge case. Run mysum_mex with the original random input and with an input that tests the full range and aggregate the results.
%% Build instrumented mex
T = mytypes('scaled');
n = 10;
buildInstrumentedMex mysum ...
-args {zeros(n,1,'like',T.x),T} -histogram
%% Test inputs
rng default
x = cast(2*rand(n,1)-1,'like',T.x);
y = mysum_mex(x,T);
% Run once with this set of inputs
y_expected = sum(double(x));
err = double(y) - y_expected
% Run again with this set of inputs. The logs will aggregate.
x = -ones(n,1,'like',T.x);
y = mysum_mex(x,T);
y_expected = sum(double(x));
err = double(y) - y_expected
% Verify results
showInstrumentationResults mysum_mex ...
-defaultDT numerictype(1,16) -proposeFL
y_expected = sum(double(x));
err = double(y) - y_expected
%% Generate C code
codegen mysum -args {x,T} -config:lib -report
5. Run the test script again.
The test runs and y overflows the range of the fixed-point data type. showInstrumentationResults proposes a new fraction length of 11 for y.
6. Update the test script to use scaled doubles with the new proposed type for y. In myTypes.m, for the 'scaled' case, T.y = fi([],true,16,11,'DataType','ScaledDouble')
7. Rerun the test script.
There are now no overflows.
Generate Code for the Proposed Fixed-Point Type
Update the data types table to use the proposed fixed-point type and generate code.
1. In myTypes.m, for the 'fixed' case, T.y = fi([],true,16,11)
2. Update the test script, mysum_test, to use T = mytypes('fixed');
3. Run the test script and then click the View Report link to view the generated C code.
short mysum(const short x[10])
short y;
int n;
int i;
int i1;
int i2;
int i3;
y = 0;
for (n = 0; n < 10; n++) {
i = y << 4;
i1 = x[n];
if ((i & 1048576) != 0) {
i2 = i | -1048576;
} else {
i2 = i & 1048575;
if ((i1 & 1048576) != 0) {
i3 = i1 | -1048576;
} else {
i3 = i1 & 1048575;
i = i2 + i3;
if ((i & 1048576) != 0) {
i |= -1048576;
} else {
i &= 1048575;
i = (i + 8) >> 4;
if (i > 32767) {
i = 32767;
} else {
if (i < -32768) {
i = -32768;
y = (short)i;
return y;
By default, fi arithmetic uses saturation on overflow and nearest rounding which results in inefficient code.
Modify fimath Settings
To make the generated code more efficient, use fixed-point math (fimath) settings that are more appropriate for C code generation: wrap on overflow and floor rounding.
1. In myTypes.m, add a 'fixed2' case:
case 'fixed2'
F = fimath('RoundingMethod', 'Floor', ...
'OverflowAction', 'Wrap', ...
'ProductMode', 'FullPrecision', ...
'SumMode', 'KeepLSB', ...
'SumWordLength', 32, ...
'CastBeforeSum', true);
T.x = fi([],true,16,15,F);
T.y = fi([],true,16,11,F);
2. Update the test script to use 'fixed2', run the script, and then view the generated C code.
short mysum(const short x[10])
short y;
int n;
y = 0;
for (n = 0; n < 10; n++) {
y = (short)(((y << 4) + x[n]) >> 4);
return y;
The generated code is more efficient, but y is shifted to align with x and loses 4 bits of precision.
3. To fix this precision loss, update the word length of y to 32 bits and keep 15 bits of precision to align with x.
In myTypes.m, add a 'fixed32' case:
case 'fixed32'
F = fimath('RoundingMethod', 'Floor', ...
'OverflowAction', 'Wrap', ...
'ProductMode', 'FullPrecision', ...
'SumMode', 'KeepLSB', ...
'SumWordLength', 32, ...
'CastBeforeSum', true);
T.x = fi([],true,16,15,F);
T.y = fi([],true,32,15,F);
4. Update the test script to use 'fixed32' and run the script to generate code again.
Now, the generated code is very efficient.
int mysum(const short x[10])
int y;
int n;
y = 0;
for (n = 0; n < 10; n++) {
y += x[n];
return y;
For more information, see Optimize Your Algorithm.
External Websites | {"url":"https://de.mathworks.com/help/fixedpoint/gs/manually-convert-a-floating-point-matlab-algorithm-to-fixed-point.html","timestamp":"2024-11-04T14:38:58Z","content_type":"text/html","content_length":"100156","record_id":"<urn:uuid:062ac478-905c-4824-9494-b39321fe5283>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00013.warc.gz"} |
Help with a bundle of vintage Seamaster inserts
Hey guys
I’ve recently bought a little bundle of inserts.
I don’t really need all of them and want to sell these therefore. I need a little help from you in determining the watches to which these belong and maybe a little help with the pricing. If any other
or more detailed pictures are needed please let me know.
1. (Top left): Omega Ploprof B1
2. (Top right): Omega Speedmaster 125
3. (Bottom left): Omega Seamaster (y. 67-69)
4. (Bottom right): Omega Seamaster 120
Kind regards
Ok, I will give it a try, maybe you guys can tell me if Iam totally out of range or not 😀
1. Seamaster Ploprof : 600€
2. Speedmaster 125: ??? (I cannot find any comparable auction)
3. Seamaster 300: 1000€
4. Seamaster 120: 500€
Absolutely nobody? 😕
Dan S
Why are you changing the numbering of the bezels in different posts. That makes it hard to follow your thread.
Oh sorry. Didn’t recognized that.
Fixed 😀
All garbage, I’ll send you a self addressed, stamped envelope to dispose of them.
All garbage, I’ll send you a self addressed, stamped envelope to dispose of them.
If he sends them, I’ll pay double the cost of shipping and split them with ya
· ·Oooo subtitles!
I’m not being funny but if you are trying to make a buck don’t expect people to help for gratis. Such things a have sold before, do your own homework. EBay is surely your friend. A lazy half arsed
sales post looks exactly like that. And it’s in the wrong place.
Just check your pm's and you know what they are worth!
I’m not being funny but if you are trying to make a buck don’t expect people to help for gratis. Such things a have sold before, do your own homework. EBay is surely your friend. A lazy half
arsed sales post looks exactly like that. And it’s in the wrong place.
Why so offending? Here are several threads in which people asking about the value of sth they have bought, gonna buy or want to sell. If you have a problem which such threads just stay out of it.
Nobody want to read your unsolicited opinion.
An in addition I’ve not only asked about the worth of the inserts only but also to which watches they belong.
Why so offending? Here are several threads in which people asking about the value of sth they have bought, gonna buy or want to sell. If you have a problem which such threads just stay out of it.
Nobody want to read your unsolicited opinion.
An in addition I’ve not only asked about the worth of the inserts only but also to which watches they belong.
well, to be fair, I was into reading his unsolicited opinion.... as I agree with it.
he has a point. research should be done first, then asking about what you couldn't find yourself.
you asked what they belong to, but clearly already know what they fit as you've described them by name on the OP.
here's my $0.02 (since you asked)...
in my extensive (13 min on google) research, I am seeing these SELL (SOLD< NOT SELLING FOR) for the ballpark between 82 and 1200 each with the Ploprof going for between 120-1250, selling in the last
year. some for more.
I've not sold or bought any of these specifically, (with the exception of a ploprof bezel, which I sold a couple years ago, it took 11 months and I got half of what I expected out of it.)
there are a few for sale at the numbers you have above just sitting for sale for months upon months.
yours aren't very rare, a couple have damage and the market will only bear what people will pay.
anyways, thanks for listening to my unsolicited (albeit solicited) advice. | {"url":"https://omegaforums.net/threads/help-with-a-bundle-of-vintage-seamaster-inserts.175561/","timestamp":"2024-11-09T06:11:49Z","content_type":"text/html","content_length":"97580","record_id":"<urn:uuid:d4c5e5ef-96fd-415a-a1fc-b5fe6668ed08>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00462.warc.gz"} |
Physics - Online Tutor, Practice Problems & Exam Prep
Hey guys. In this video, we're going to talk about inductors and the role that they play in AC circuits. Alright? Let's get to it. Now remember that the current in an AC circuit at any time is going
to be given by the equation that by now we've seen a bunch of times. Okay? Okay, this is going to tell us that the current is simply oscillating with an angular frequency of omega between some value
positive i[max] and some value negative i[max]. Now the question is, how does the voltage across the inductor look? Well, remember that the voltage across an inductor, which we saw during our
discussion of Faraday's law, is the inductance times delta I over delta t, which is the rate at which the current is changing. Okay. Now I can't show you how but using calculus you guys can arrive at
the answer that the rate at which the current is changing looks like this.
So if I multiply this by the inductance, then I get the voltage across an inductor in an AC circuit at any time. This is going to be i[max] times omega times L times cosine of omega t + π/2. Okay, so
once again, remember the voltage across a resistor. The voltage across a resistor looks like i[max] times R times cosine of omega t. So the angle that it operates at omega t is different than the
angle that the voltage across the inductor operates at. This is some other angle theta prime which is omega t + π/2. Okay? So the current and the voltage across the resistor are in phase, right,
their plots line up, but the current and the voltage across an inductor are not going to line up. They're going to be out of phase. If we plot the current across an inductor and the voltage across an
inductor, you can see that the voltage across an inductor actually leads the current by 90 degrees. Okay? What's happening here is that the voltage is deciding to go up at a time when the current is
0. Okay? Then at a future time, the current starts to go up. But at this point, the voltage has already peaked. Then at a future time, the current peaks. Right? It's trying to match what the voltage
is doing. But at this point, the voltage is already decreasing. So then at a future time, the current decreases but the voltage has already bottomed out. So you see the current is trying and trying
and trying to match the voltage, but it's lagging behind, or we can say that the voltage leads the current. Either one is fine. Okay? Now, the maximum voltage across an inductor is going to look like
i[max] times omega L. This looks a lot like Ohm's Law where we said that the voltage across the resistor was I times the resistance. There appears to be a resistance-like quantity of omega L. That
resistance-like quantity for capacitors we called the capacitive reactance. Now we are calling it the inductive reactance for inductors. The units are still ohms. Right? The same unit as for
resistance. Alright. Let's do an example. An AC power source delivers a maximum voltage of 120 volts at 60 hertz. If an unknown inductor is connected to the source and the maximum current in the
circuit is found to be 5 amps, what is the inductance of the inductor? Okay? What is the maximum current in an inductor circuit? This is just going to be the maximum voltage across the inductor
divided by what they both have to share the maximum voltage. That's what Kirchhoff's loop rule says. So this is just going to be V[max], the maximum voltage by the source, divided by XL. And what is
that inductive capacitance? Well, that is just going to be omega L. Okay so plugging this into our equation, we can say that i[max] is simply V[max]/omega L. And our unknown is the inductance. So
what I want to do is multiply the inductance up and divide i[max] over, and then I have inductance as V[max] / omega i[max]. Before I can continue though, we need to know what omega is. We're told
the linear frequency is 60 hertz. Okay. Remember, this is linear frequency because the units are hertz. So the angular frequency, which is 2πf, is going to be 2π times 60 hertz, which is going to be
about 377 inverse seconds. Now we can solve for the inductance. And the inductance is just going to be V[max] / omega i[max]. V[max] is 120, right? Always look at and make sure that's a maximum
voltage not an RMS voltage, divided by 377, which was our angular frequency. The maximum current in the circuit was 5 amps. So that's 5 and this whole thing is 0.064 henrys, the unit of inductance.
Alright guys, that wraps up our discussion on inductors and AC circuits. Thanks for watching. | {"url":"https://www.pearson.com/channels/physics/learn/patrick/alternating-current/inductors-in-ac-circuits?chapterId=8fc5c6a5","timestamp":"2024-11-12T02:37:52Z","content_type":"text/html","content_length":"502146","record_id":"<urn:uuid:2e07b3db-6363-4099-9dcd-a76524bc3fcc>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00678.warc.gz"} |
Mechanical Engineering Calculators
Mechanical engineering formulas & calculators
- step by step calculations on strength of materials, fluid dynamics, statics, solid mechanics, thermodynamics, heat transfer, fuels, internal combustion engines etc. Perform, understand & verify the
mechanical engineering calculations as quick as possible. These online calculators along with the formulas not only provides the answers for the given input values but also provides the complete
step by step calculations
for each calculations users do by using these calculators. The main objective of these mechanical engineering calculators is to assist students to perform, understand & verify the mechanical
engineering calculations; professionals and researchers may quickly perform or verify the results of such calculations to save their time. | {"url":"https://dev.ncalculators.com/mechanical/","timestamp":"2024-11-12T19:19:58Z","content_type":"text/html","content_length":"36003","record_id":"<urn:uuid:a7f6eca4-585c-40ed-b75a-5124b5c02f6b>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00520.warc.gz"} |
Wave Induced Oscillations in Harbors of Arbitrary Shape
Theoretical and experimental studies were conducted to investigate the wave induced oscillations in an arbitrary shaped harbor with constant depth which is connected to the open-sea.
A theory termed the “arbitrary shaped harbor” theory is developed. The solution of the Helmholtz equation, ∇^2f + k^2f = 0, is formulated as an integral equation; an approximate method is employed to
solve the integral equation by converting it to a matrix equation. The final solution is obtained by equating, at the harbor entrance, the wave amplitude and its normal derivative obtained from the
solutions for the regions outside and inside the harbor.
Two special theories called the circular harbor theory and the rectangular harbor theory are also developed. The coordinates inside a circular and a rectangular harbor are separable; therefore, the
solution for the region inside these harbors is obtained by the method of separation of variables. For the solution in the open-sea region, the same method is used as that employed for the arbitrary
shaped harbor theory. The final solution is also obtained by a matching procedure similar to that used for the arbitrary shaped harbor theory. These two special theories provide a useful analytical
check on the arbitrary shaped harbor theory.
Experiments were conducted to verify the theories in a wave basin 15 ft wide by 31 ft long with an effective system of wave energy dissipators mounted along the boundary to simulate the open-sea
Four harbors were investigated theoretically and experimentally: circular harbors with a 10° opening and a 60° opening, a rectangular harbor, and a model of the East and West Basins of Long Beach
Harbor located in Long Beach, California.
Theoretical solutions for these four harbors using the arbitrary shaped harbor theory were obtained. In addition, the theoretical solutions for the circular harbors and the rectangular harbor using
the two special theories were also obtained. In each case, the theories have proven to agree well with the experimental data.
It is found that: (1) the resonant frequencies for a specific harbor are predicted correctly by the theory, although the amplification factors at resonance are somewhat larger than those found
experimentally,(2) for the circular harbors, as the width of the harbor entrance increases, the amplification at resonance decreases, but the wave number bandwidth at resonance increases, (3) each
peak in the curve of entrance velocity vs incident wave period corresponds to a distinct mode of resonant oscillation inside the harbor, thus the velocity at the harbor entrance appears to be a good
indicator for resonance in harbors of complicated shape, (4) the results show that the present theory can be applied with confidence to prototype harbors with relatively uniform depth and reflective
interior boundaries.
Item Type: Thesis (Dissertation (Ph.D.))
Subject Keywords: (Civil Engineering)
Degree Grantor: California Institute of Technology
Division: Engineering and Applied Science
Major Option: Civil Engineering
Thesis Availability: Public (worldwide access)
Research Advisor(s): • Raichlen, Fredric
Thesis Committee: • Unknown, Unknown
Defense Date: 11 November 1969
Funders: ┌──────────────────────────┬────────────────────────┐
│ Funding Agency │ Grant Number │
│ Army Corps of Engineers │ DA-22-079-CIVENG-64-11 │
Record Number: CaltechTHESIS:12072015-163443717
Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:12072015-163443717
DOI: 10.7907/SEBX-ZE52
Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code: 9313
Collection: CaltechTHESIS
Deposited By: Bianca Rios
Deposited On: 08 Dec 2015 15:49
Last Modified: 16 May 2024 23:20
Thesis Files
PDF (Final) - Final Version
See Usage Policy.
Repository Staff Only: item control page | {"url":"https://thesis.library.caltech.edu/9313/","timestamp":"2024-11-03T05:56:27Z","content_type":"application/xhtml+xml","content_length":"30955","record_id":"<urn:uuid:9dad2285-99c9-497c-99a8-468d38100d46>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00663.warc.gz"} |
class compas.geometry.Translation(matrix=None, check=True)[source]
Bases: Transformation
Class representing a translation transformation.
matrix (list[list[float]], optional) – A 4x4 matrix (or similar) representing a translation.
translation_vector (Vector) – The translation vector.
ValueError – If the default constructor is used, and the provided transformation matrix is not a translation.
>>> T = Translation.from_vector([1, 2, 3])
>>> T[0, 3] == 1
>>> T[1, 3] == 2
>>> T[2, 3] == 3
>>> from compas.geometry import Vector
>>> T = Translation.from_vector(Vector(1, 2, 3))
>>> T[0, 3] == 1
>>> T[1, 3] == 2
>>> T[2, 3] == 3
>>> T = Translation([[1, 0, 0, 1], [0, 1, 0, 2], [0, 0, 1, 3], [0, 0, 0, 1]])
>>> T[0, 3] == 1
>>> T[1, 3] == 2
>>> T[2, 3] == 3
from_vector Create a translation transformation from a translation vector.
Inherited Methods
ToString Converts the instance to a string.
concatenate Concatenate another transformation to this transformation.
concatenated Concatenate two transformations into one Transformation.
copy Returns a copy of the transformation.
decomposed Decompose the Transformation into its components.
from_change_of_basis Construct a change of basis transformation between two frames.
from_data Construct an object of this type from the provided data.
from_euler_angles Construct a transformation from a rotation represented by Euler angles.
from_frame Construct a transformation from world XY to frame.
from_frame_to_frame Construct a transformation between two frames.
from_json Construct an object from serialized data contained in a JSON file.
from_jsonstring Construct an object from serialized data contained in a JSON string.
from_list Creates a transformation from a list of 16 numbers.
from_matrix Creates a transformation from a list[list[float]] object.
inverse Returns the inverse transformation.
invert Invert this transformation.
inverted Returns the inverse transformation.
sha256 Compute a hash of the data for comparison during version control using the sha256 algorithm.
to_data Convert an object to its native data representation.
to_json Serialize the data representation of an object to a JSON file.
to_jsonstring Serialize the data representation of an object to a JSON string.
transpose Transpose the matrix of this transformation.
transposed Create a transposed copy of this transformation.
validate_data Validate the object's data against its data schema.
validate_json Validate the object's data against its json schema. | {"url":"https://compas.dev/compas/1.16.0/api/generated/compas.geometry.Translation.html","timestamp":"2024-11-04T10:40:04Z","content_type":"text/html","content_length":"34316","record_id":"<urn:uuid:7fdb74b3-d53f-4904-8777-27295e706fe4>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00224.warc.gz"} |
Utility class for triangle cells. More...
#include <MeshVizXLM/extractors/MxTriangleCellExtract.h>
static MbVec3d getIsoParametricCoord (const MiGeometryI &meshGeometry, const MiSurfaceCell *triangleCell, const MbVec3d &point)
Computes the iso parametric coordinates of the given point in the given cell with the given geometry.
static MbVec3d getIsoParametricCoord (size_t nodeIndex)
Returns the iso parametric coordinates of one of the 3 nodes of a triangle cell.
static void getWeight (const MiGeometryI &meshGeometry, const MiSurfaceCell *triangleCell, const MbVec3d &point, std::vector< double > &weights)
Gets the weights in the given cell of the given point.
static void getWeight (const MbVec3d &ipcoord, std::vector< double > &weights)
Gets the weights of a point defined by its iso parametric coordinates.
static bool isPointInsideCell (const MiGeometryI &meshGeometry, const MiSurfaceCell *triangleCell, const MbVec3d &point, std::vector< double > &weights)
Checks if a point is inside or outside a triangle cell.
static double getLongestEdgeLength (const MiGeometryI &meshGeometry, const MiSurfaceCell *cell)
Gets the longest edge of a triangle cell.
static double getShortestEdgeLength (const MiGeometryI &meshGeometry, const MiSurfaceCell *cell)
Gets the shortest edge of a triangle cell.
Utility class for triangle cells.
Utility class that provides a static implementation of the MiSurfaceCell's methods in a triangle cell. This class is provided to make easier the creation of a class in the application that implements
the MiSurfaceCell
interface for a triangle cell.
• value of iso parametric coordinates in a linear triangle cell. (see getIsoParametricCoord())
• value of the 3 weights (aka shape functions) in a triangle cell. (see getWeight())
• localization test (see isPointInsideCell())
The following image shows the nodes numbering used by this class. The weights (see getWeight()), and parametric coordinates (see getIsoParametricCoord()) are defined according to this node numbering.
Definition at line 56 of file MxTriangleCellExtract.h.
◆ getIsoParametricCoord() [1/2]
static MbVec3d MxTriangleCellExtract::getIsoParametricCoord ( const MiGeometryI & meshGeometry,
const MiSurfaceCell * triangleCell, static
const MbVec3d & point
Computes the iso parametric coordinates of the given point in the given cell with the given geometry.
As computing the iso parametric coordinates of a point needs the coordinates of the cell's nodes, the given triangleCell is assumed to contain 3 nodes.
Each node coordinate of the given cell is retrieved in the following way:
for each i in the range [0-2].
[in] meshGeometry The geometry of the mesh.
[in] triangleCell The input cell.
[in] point The input point given in the same space coordinate as meshGeometry.
◆ getIsoParametricCoord() [2/2]
static MbVec3d MxTriangleCellExtract::getIsoParametricCoord ( size_t nodeIndex ) static
Returns the iso parametric coordinates of one of the 3 nodes of a triangle cell.
This static method helps to implement the method MiCell::getIsoParametricCoord().
[in] nodeIndex Must be defined in the range [0-2]
◆ getLongestEdgeLength()
static double MxTriangleCellExtract::getLongestEdgeLength ( const MiGeometryI & meshGeometry,
const MiSurfaceCell * cell static
Gets the longest edge of a triangle cell.
◆ getShortestEdgeLength()
static double MxTriangleCellExtract::getShortestEdgeLength ( const MiGeometryI & meshGeometry,
const MiSurfaceCell * cell static
Gets the shortest edge of a triangle cell.
◆ getWeight() [1/2]
static void MxTriangleCellExtract::getWeight ( const MbVec3d & ipcoord,
std::vector< double > & weights inlinestatic
Gets the weights of a point defined by its iso parametric coordinates.
This static method helps to implement the method MiCell::getWeight(ipcoord,weights) for a triangle cell.
[in] ipcoord The iso parametric coordinates of the input point. The reference space for the iso parametric coordinates is assumed to be [0-1]. Thus any point inside the cell has iso parametric
coordinates in the interval [0-1].
[out] weights This method computes the 3 values weights[0-2].
It assumes the weights vector array has been already allocated. Its size must be set to 3 (at least) before calling this method, using for instance the weights::resize(3)
Definition at line 118 of file MxTriangleCellExtract.h.
◆ getWeight() [2/2]
static void MxTriangleCellExtract::getWeight ( const MiGeometryI & meshGeometry,
const MiSurfaceCell * triangleCell,
const MbVec3d & point, inlinestatic
std::vector< double > & weights
Gets the weights in the given cell of the given point.
This static method helps to implement the method MiCell::getWeight(meshGeometry,point,weights) for a triangle cell. As computing the weights of a point needs the coordinates of the cell's nodes, the
given triangleCell is assumed to contain 3 nodes.
Each node coordinate of the given cell is retrieved in the following way:
for each i in the range [0-2].
[in] meshGeometry The geometry of the mesh.
[in] triangleCell The input cell.
[in] point The input point given in the same space coordinate as meshGeometry.
[out] weights This method computes the 3 values weights[0-2]. It assumes the weights vector array has been already allocated. Its size must be set to 3 (at least) before calling this method,
using for instance the weights.resize(3)
Definition at line 101 of file MxTriangleCellExtract.h.
◆ isPointInsideCell()
static bool MxTriangleCellExtract::isPointInsideCell ( const MiGeometryI & meshGeometry,
const MiSurfaceCell * triangleCell,
const MbVec3d & point, inlinestatic
std::vector< double > & weights
Checks if a point is inside or outside a triangle cell.
This static method helps to implement the method MiCell::isPointInsideCell(meshGeometry,point,weights) for a triangle cell.
[in] meshGeometry The geometry of the mesh.
[in] triangleCell The input cell.
[in] point The input point given in the same space coordinate as meshGeometry.
[out] weights This method computes the 3 values weights[0-2] if the point is inside the cell. It assumes the weights vector array has been already allocated. Its size must be set to 3 (at
least) before calling this method, using for instance the weights::resize(3)
true if the point is inside the cell.
Definition at line 138 of file MxTriangleCellExtract.h.
The documentation for this class was generated from the following file: | {"url":"https://developer.openinventor.com/refmans/latest/RefManCpp/class_mx_triangle_cell_extract.html","timestamp":"2024-11-07T04:15:14Z","content_type":"application/xhtml+xml","content_length":"30367","record_id":"<urn:uuid:a579d883-99f5-4117-91fb-a1a8625d365b>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00488.warc.gz"} |
Large-Signal Approach Yields Low-Noise VHF/UHF Oscillators
Oscillator design for veryhigh- frequency (VHF) and ultra-high-frequency (UHF) applications has been well documented in books and journals. Most early work focused on frequency stability and, to a
lesser extent, on efficiency and output signal quality. But with increasing use of advanced modulation formats in communications systems, and the growing need for oscillators with extremely low phase
noise, greater design emphasis is now placed on achieving oscillator designs with low phase noise. Fortunately, with the availability of accurate phase-noise measurement equipment and improving
computer-aidedengineering (CAE) tools for predicting and simulating phase-noise performance, the gap between oscillator simulations and measured results has narrowed. Still, many early oscillator
design strategies were based on small-signal approaches that yielded less-than-accurate predictions for output frequency, output power, and phase noise. As an alternative, largesignal, time-domain
calculations will be applied to the design of a groundedbase oscillator (rather than a Colpitts oscillator) to validate the effectiveness of this design approach. Part 1 of this three-part article
series will explore the use of large-signal design techniques for a grounded-base 144-MHz oscillator.
By presenting the use of the largesignal, time-domain approach with nonlinear software simulation tools in the design of VHF/UHF groundedbased oscillators, the goals include (1) accurately predicting
oscillator phase noise and deriving a set of algebraic equations for the noise calculations (many CAE tools provide incorrect answers about the phase noise) and (2) developing a set of empirical
equations that will guide in the synthesis of VHF/UHF oscillators. The approach yields oscillators with the best possible combination of output power and phase noise.
As a point of reference, the traditional small-signal design approach will first be used to create an oscillator for comparison to a more optimized design developed with the novel large-signal
approach. Using a mix of linear equations and one large signal parameter, the device transconductance (g[m]), the important noise parameters will be calculated and validated. Finally, based on this
procedure, a simple but scalable and accurate set of formulas for oscillator synthesis will be presented. The novel large-signal design principles shown here for fixed or narrowband oscillators can
also be applied to broadband voltage-controlled-oscillator (VCO) design. The methodology has been shown to work well even with multi-octave-band (1:3 frequency tuning range) tunable oscillators.^
The grounded base configuration (Fig. 1) is a popular circuit for VHF/UHF oscillators. It is simple and can be made with very low phase noise, since the RF voltage swing at the active-device's
collector can be close to that of the supply voltage. Oscillation is based on the principle that power from the output is fed back to the emitter. This feedback arrangement generates a negative
resistance at the output, compensating for the losses of the output-tuned circuit, and starts oscillating and then stabilizing the oscillation amplitude.^1-4 A complete survey of grounded-base
oscillator configurations and applications can be found in references 5 to 19.
These references include some of the more popular texts recently published on oscillators. Many of the authors have attempted to predict oscillator performance based on a set of linear calculations,
including use of the Leeson model or similar methods to determine phase noise. For accurate predictions of phase noise, however, several key input parameters are needed, including the large-signal
noise figure of the active device, the output power, and the operating quality factor (Q). The values of these parameters are not often known and more typically approximated (or guessed). The first
successful attempts at determining the large-signal phase noise were reported in references 6 and 7. But these approaches are not useful without an accompanying CAE tool, and they don't provide
design guidelines. Another problem with the linear approach is inaccuracy in predicting the actual oscillating frequency, with predicted results at higher frequencies often differing widely from
actual performance.
Well-known for his work on amplifiers, Guillermo Gonzalez recently published a text on oscillators that provides an interesting overview of design based on linear calculations and CAE tools, although
his approach does not provide optimum solutions.8 To demonstrate this, his methods will first be applied to the design of a 144-MHz oscillator. The resulting circuit neither provides the best output
power nor the lowest phase noise and, at high frequencies, requires capacitor values that cannot be easily realized because of parasitic effects.
Figure 1 shows the typical configuration of the grounded base oscillator circuit. This type of oscillator works effectively from about 10 to above 1000 MHz. Following the procedures of ref. 8, and
the large-signal conditions of ref. 11, it is possible to analyze this oscillator circuit. Kenneth Clarke was probably the first to publish the effect of the collector current conducting angle of an
oscillator, but makes no mention of the relationship of it on phase noise, as done in ref. 10.
The oscillator circuit is based on a model BFR193 silicon bipolar transistor from Infineon Technologies (www.infineon.com). Designed for low-noise, high-gain amplifiers to 2 GHz, the transistor
features a transition frequency (fT) of 8 GHz for +8 VDC collectoremitter voltage and 50 mA collector current. The first step in designing the oscillator circuit for this transistor is to determine
the small-signal parameters for the transistor at 144 MHz and under the operating conditions of +8.8 VDC collector-emitter voltage (Vce, 10 mA collector current (Ic), 24 A base current (IB), and
+0.64 VDC base-emitter voltage (Vbe). The 10-mA collector current was selected for stable transistor cut-off transition frequency. For more output power, a collector current of 30 mA is a better
Figure 2 shows a circuit for generating the oscillator's small-signal parameters using Ansoft Designer CAE software from Ansoft (www.ansoft.com) and the time-domain model. The process is based on the
configuration shown in Fig. 3 and the following definition:
I[1] > Y[11] Y[12] > V[1] > (1)
I[2] Y[21] Y[22] V[2]
Once Ansoft Designer is armed with the circuit parameters for the oscillator circuit, it uses the matrix to generate the Y-parameters:
Y11=G[11] + jB[11]= (279.08-j95.07mS (2)
Y21=G[21] + jB[21]= (-271.32 + j100.77)mS (3)
Y12=G[12] + jB[12]= (-1030 + j78.06) μS (4)
Y[22]=G[22] + jB[22]= (1020 + j536.14) μS (5)
Figure 4 shows a standard feedback oscillator topology using parallel circuit elements. In theory, the grounded- base configuration can be rotated into a Colpitts circuit, which is often referenced
in the technical literature and based on the black-box theory (ref. 5). In terms of performance, however, it cannot be said that a mathematical rotation yields the same performance. In the case of
the Colpitts oscillator, the RF voltage swing is now limited by the base-to-emitter and emitter-to-ground voltages. As a result, there is less energy stored in the circuit and, because of loading,
the operational Q can be degraded for the grounded-base oscillator. For the Colpitts oscillator configuration, the collector-to-base voltage (V[cb]) is about 12 V. Also, parameter Y[22cb] is less
than parameter Y[22ce], resulting in less loading than the grounded-base configuration. The Colpitts configuration is popular because of its simplicity and its perceived high isolation since the
output power is extracted from the collector, although this is nothing more than perception due to the strong Miller effect at very high frequencies. In terms of configurations other than the
Colpitts, the general time-domain approach presented here is valid not only for the Colpitts configuration but for other derivative configurations.
Conditions necessary for oscillation for the parallel feedback oscillator configuration of Fig. 4 can be described by
Yout + Y3 =>0 (6)
This condition can be expressed as:
Det Y[11] + Y[1] + Y[2]Y[12]-Y[2] > =0 (7)
Y[21]-Y[2]Y[22] + Y[2] + Y[3]
Y[3]= -Y[22]+Y[2]> + Y[12]-Y[2]{Y[21]-Y[2]} (8)
Y11 + Y1 + Y2>
where Y[ij] (i,j = 1, 2) are the smallsignal parameters of the bipolar or FET model.
Continue on Page 2
Page Title
As shown in Fig. 4, the active two-port network, together with feedback elements Y[1] and Y[2], can be considered as a one-port negative-resistance oscillator circuit. The following is an example of
an oscillator design using the small-signal parameters determined above for 8.8 V and 10 mA at 144 MHz. The resulting output admittance (Y[out]) is shown in Eq. 9 (page 76).
The optimum values of feedback element are calculated from the expression of B*[1] and B*[2] and for 10 mA are shown in Eq. 10 (page 76).
The optimum values of the real and imaginary part of the output admittance are:
are values for conjugate matching needed to compensate for resonator losses in Eq. 19 (on p. 76) and in:
for the case of a center frequency (f[0] of 144 MHz and inductor L[3] of approximately 2.59 nH).
Figure 5 shows the 144-MHz oscillator circuit using the small-signal parameters for establishing oscillation conditions. The required values for this parallel feedback topology are 478 pF for the
feedback capacitor, 459 pF for the emitter-to-ground capacitance, 3.2 nH for the inductor, and 186 pF for capacitors C3A and C3B. Bypass capacitors C[b] and C[c] should be about 220 pF. Because of
the difficulty of producing capacitors above 200 pF at these frequencies, it may be more reasonable to use several, up to 10, capacitors in parallel to achieve these values.
For the higher-output oscillator case operating at 30 mA, the values of the parameters are:
For a center frequency (f[0]) of 144 MHz and 30 mA operation, the component values for the oscillator are L[3] of 3.77 nH, C[1] of 518 pF, C[2] of 503 pF, and C[3] of 324 pF. As mentioned previously,
because of the high values of C[1] and C[2], their values can only be achieved using multiple parallel capacitors of about 100 pF each.
Figure 6 shows the simulated plot of phase noise for the 144-MHz oscillator. The "linear" calculation indicates a resonant frequency of 143.2 MHz, while nonlinear harmonicbalance (HB) analysis
provides the correct frequency of 144.2 MHz (a relatively large difference in percent frequency). Figure 7 shows the output power to be +5.1 dBm. The value was determined using the HB software
program Nexxim from Ansoft Designer, although the Advanced Design System (ADS) software suite from Agilent Technologies (www.agilent.com) provided the same answers. The predictions from both CAE
tools deviate less than 1 dB from measured results for the oscillator, assuming that the input SPICE type parameters for the transistor are accurate.
A variety of efforts have been made to deal with large-signal conditions for oscillator design, such as the timedomain approach. Reference 10 is a first successful attempt to calculate output power
with reasonable effort, notably Eq. 10 within this reference.
There are many problems associated with both the large-signal analysis and noise analysis. From an experimental point of view, it is almost impossible to consider all possible variations. During the
creation of the Ansoft Designer CAE program, for example, it was necessary for the developers to validate the accuracy of that software's large-signal noise analysis. As part of that validation,
several critical circuits were used to compare CAE predictions with measured results, from crystal oscillators to voltage-controlled oscillators (VCOs). Measurements were made with well-known test
equipment, including the R&S FSUP 26 spectrum analyzer with all necessary options from Rohde & Schwarz (www.rohdeschwarz. com). References 12 and 13 showed that the accuracy of this software's
large-signal predictions is high, within 0.5 dB of measured results. This evaluation involved extensive analysis of noise in oscillators using a set of equations with a minimum number of CAE tools.
The equations, derived in ref. 9, will be used for the current analysis.
The search for low oscillator phase noise has been well documented in the technical literature. Designers have published many different recipes, such as those based on the use of certain low-noise
transistors, high-Q circuits, and various other things. In all of these approaches, however, the consequences of device large-signal operation and its effects on phase noise had not been well
understood. To help gain a better grasp on the relationship between device large-signal behavior and phase noise, a complete mathematical analysis of a 144-MHz oscillator follows.
The design steps for achieving a 144-MHz oscillator by means of the large-signal approach include:
1. Calculation of the output power for the selected DC operating conditions. For this example, the same circuit as used above for the small-signal approach will be applied for a center frequency of
144 MHz. From ref. 9, the RF output current can be found from Eq. 27 (above) where:
V[1] = the drive signal and x = the normalized drive level with x = qV[1]/kT with
Considering a 50-Ohm load, the RF output power can be calculated by means of
V[RF](f[0] = I[RF] 50 = 60 10-^3 50 = 1 V (peak amplitude) (with no V[ce] saturation assumed).
The oscillator output power at 144 MHz is then P[out](f[0] = V[RF]^2(f[0]/2R[L] = 1/(2 50) = 10 mW = +10 dBm
2. Calculation of the large-signal transconductance for a normalized drive level can be performed by means of Eq. 28 (above), while the largesignal transconductance (G[m]) can be found from Eq. 29
Continue on Page 3
Page Title
This assumes an ideal intrinsic transistor. To perform the transition from an intrinsic to an extrinsic transistor, parasitics (package effects, lead inductance, and bond wires) are added by
correcting the final results for capacitances and inductances. The transition frequency (f[t]) of the transistor used is high enough so a phase shift correction for the small-signal transconductance
(g[m]) is not necessary at these frequencies (VHF).
3. Values of the feedback capacitors can be calculated in the following way. The value of n can be in the range of n , where n1 is 2 and n2 is 5 for a drive level of x = 15 (in pursuit of
low-phase-noise performance). Assuming n = 5, the values of capacitors C[1] and C[2] can be calculated as
The ratio of C1 to C2 is 4.
The final design step for the 144-MHz oscillator using the largesignal approach involves finding the value of inductor L, which can be performed by knowing the relationship of the oscillator's
operating frequency to the inverse of the square root of the oscillator's inductance and capacitance and selecting a value of L for optimum phase noise.
Next month, Part 2 of this threepart article series will show the details of calculating the value of L for the 144-MHz oscillator, and how to apply the large-signal approach and commercial software
to compute the oscillator's phase noise.
1. F. E. Terman , Radio Engineers' Handbook, McGraw- Hill, New York, 1943, p. 498.
2. F. Langford-Smith, Ed., Radiotron Designers Handbook, Electron Tube Division, RCA, Harrison, NJ, 1954, p. 947.
3. L. J. Giacoletto, Electronics Designers' Handbook, McGraw-Hill, New York, 1977, p. 16-1.
4. F. Vilbig, Lehrbuch der Hochfrequenztechnik, vol. 2, Akademische Verlagsgesellschaft Becker & Erler Kom.- Ges., 1937/1942, p. 235.
5. K. L. Kotzebue and W. J. Parrish, "The Use of Large Signal S-parameters in Microwave Oscillator Design," in Proceedings of the 1975 International Microwave Symposium On Circuits and Systems.
6. V. Rizzoli, A. Neri, A. Costanzo, and F. Mastri, "Modern Harmonic-Balance Techniques for Oscillator Analysis and Optimization," in RF and Microwave Oscillator Design, M. Odyniec, Ed. Artech House,
Norwood, MA, 2002.
7. W. Anzill, F.X. Kaertner, and P. Russer, "Simulation of the Single-Sideband Phase Noise of Oscillators," Second International Workshop of Integrated Nonlinear Microwave and Millimeterwave
Circuits, 1992.
8. Guillermo Gonzalez, Foundations of Oscillator Circuit Design, Artech House, Norwood, MA, 2007.
9. Ulrich L. Rohde, Ajay K. Poddar, and Georg Boeck, The Design of Modern Microwave Oscillator for Wireless Applications Theory and Optimization, Wiley, New York, 2005.
10. Kenneth M. Johnson, "Large Signal GaAs MESFET Oscillator Design," IEEE Transactions on Microwave Theory & Techniques, vol. MTT-27, No. 3, March 1979.
11. Kenneth. K. Clarke, "Design of Self-Limiting Transistor Sine-Wave Oscillator," Circuit and Systems, IEEE Transaction on , vol. 13, issue 1, March 1966, pp. 58-63.
12. U. L. Rohde and J. Whitaker, Communication Receivers: DSP, Software Radios, and Design, 3rd ed., McGraw- Hill, New York, 2001, pp. 413-422.
13. U. L. Rohde and D. P. Newkirk, RF/Microwave Circuit Design For Wireless Applications, Wiley-Interscience, New York, 2000, pp. 798-812, 413-422.
Sponsored Recommendations | {"url":"https://www.mwrf.com/technologies/components/article/21845678/large-signal-approach-yields-low-noise-vhf-uhf-oscillators","timestamp":"2024-11-14T04:35:44Z","content_type":"text/html","content_length":"286081","record_id":"<urn:uuid:ac964c6f-155c-493d-99f9-0e0dc83a564b>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00121.warc.gz"} |
How do you find the domain of the function g(x) = 2/3-5x? | HIX Tutor
How do you find the domain of the function #g(x) = 2/3-5x#?
Answer 1
Domain: all the real $x$.
This is a Linear function that accepts all the real values of #x#. So, the domain is all the real #x#. You can have a look at your function graphically as: graph{(2/3)-5x [-10, 10, -5, 5]}
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer 2
To find the domain of the function ( g(x) = \frac{2}{3-5x} ), we need to identify any values of ( x ) that would make the denominator ( 3-5x ) equal to zero. Set the denominator equal to zero and
solve for ( x ):
[ 3 - 5x = 0 ]
[ 5x = 3 ]
[ x = \frac{3}{5} ]
The domain of the function is all real numbers except ( x = \frac{3}{5} ). So, the domain can be expressed as:
[ \text{Domain of } g(x) = { x \in \mathbb{R} \ | \ x \neq \frac{3}{5} } ]
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
Not the question you need?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-find-the-domain-of-the-function-g-x-2-3-5x-8f9af8dba4","timestamp":"2024-11-05T22:00:10Z","content_type":"text/html","content_length":"568327","record_id":"<urn:uuid:a472b325-9ab3-429b-b41f-c6a60b735017>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00721.warc.gz"} |
How do you define a spiral? - Vypros.COM
How do you define a spiral?
How do you define a spiral?
1a : the path of a point in a plane moving around a central point while continuously receding from or approaching it. b : a three-dimensional curve (such as a helix) with one or more turns about an
axis. 2 : a single turn or coil in a spiral object.
What is a spiral shape?
A spiral is a coil or curl, like the shape of a piece of hair wound around your finger, a Slinky toy, or a corkscrew. A curve forming a series of circles that become gradually larger or smaller is
one kind of spiral.
What is a spiral in math?
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.
What is the pitch of a spiral?
The pitch of a helix is the height of one complete helix turn, measured parallel to the axis of the helix. A double helix consists of two (typically congruent) helices with the same axis, differing
by a translation along the axis.
What is spiral and example?
Spirals. A spiral is a curved pattern that focuses on a center point and a series of circular shapes that revolve around it. Examples of spirals are pine cones, pineapples, hurricanes.
Is a helix a spiral?
helix Add to list Share. A helix is a twisted, spiral shape, like a corkscrew. In math, a helix is defined as “a curve in three-dimensional space.” If you have ever seen a spiral staircase, you can
envision the shape of a helix.
Do spirals end?
Does Spiral: From the Book of Saw have a post-credits scene? Again, it does not. However, the film ends on a pretty big cliffhanger, so we don’t blame you for wondering if there’s any more resolution
heading your way.
What is spirals and the golden ratio?
In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter
turn it makes.
What are examples of spirals?
A spiral is a curved pattern that focuses on a center point and a series of circular shapes that revolve around it. Examples of spirals are pine cones, pineapples, hurricanes.
Are spirals geometric?
Like all other geometric shapes, a spiral has certain characteristics which help define it. The center, or starting point, of a spiral is known as its origin or nucleus. The line winding away from
the nucleus is called the tail. Most spirals are also infinite, that is they do not have a finite ending point.
Why is the Archimedean spiral called an arithmetic spiral?
Characteristics. The Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance (equal to 2 πb if θ is
measured in radians ), hence the name “arithmetic spiral”. In contrast to this, in a logarithmic spiral these distances,…
What makes up the center of a spiral galaxy?
Most spiral galaxies contain a central bulge surrounded by a flat, rotating disk of stars. The bulge in the center is made up of older, dimmer stars, and is thought to contain a supermassive black
hole. Approximately two-thirds of spiral galaxies also contain a bar structure through their center, as does the Milky Way.
How are spirals used to squaring the circle?
One method of squaring the circle, due to Archimedes, makes use of an Archimedean spiral. Archimedes also showed how the spiral can be used to trisect an angle. Both approaches relax the traditional
limitation on the use of straightedge and compass.
What was the purpose of the spiral curve?
Spiral Curves Made Simple. HISTORY. Spiral curves were originally designed for the Railroads to smooth the transition from a tangent line into simple curves. They helped to minimize the wear and
tear on the tracks. Spiral curves were implemented at a later date on highways to provide a smooth transition from the tangent line into simple curves. | {"url":"https://vypros.com/how-do-you-define-a-spiral/","timestamp":"2024-11-14T12:02:23Z","content_type":"text/html","content_length":"61530","record_id":"<urn:uuid:04a8eea1-c289-4d4c-b433-9b7ca2d0db79>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00740.warc.gz"} |
Applying Algebra: Jia Huang sees beauty of mathematics, student research - UNK NewsApplying Algebra: Jia Huang sees beauty of mathematics, student research
UNK Communications
There’s an intrinsic beauty Jia Huang sees in mathematics.
The University of Nebraska at Kearney associate professor enjoys the process of analyzing a problem, then developing ways to attack it.
“And if I’m lucky enough, I have the answer in the end,” he said.
That’s another draw. In math, there’s usually a definite answer. You’re either right or wrong.
“Even without any application, sometimes the result itself already looks very beautiful,” Huang said. “That’s something important for mathematics.”
Huang’s fascination with the subject started at a young age while growing up in Jiujiang, a city in eastern China located along the Yangtze River.
“In grade school, I was pretty good at math,” he said. “It’s interesting to me and I can solve a lot of interesting problems.”
Huang earned a bachelor’s degree in math from the University of Science and Technology of China before pursuing a doctorate at the University of Minnesota. He didn’t know much about the Twin Cities
at that time, but the university’s strong reputation in discrete mathematics research was enough to bring him to the Midwest.
“It turned out to be a pretty good decision. Everything worked out pretty well for me and my family,” said Huang, who married his wife Ting Zou shortly before the move to the U.S.
Huang’s doctorate research focused on discrete mathematics, which deals with objects that have distinct, separated values – think individual points instead of continuous lines.
“We’re especially interested in counting problems,” Huang explained.
For example, if there are 10 teams in a sports league and every team plays each other twice, how many total games does the season include? That’s a problem answered by discrete mathematics.
Huang likes discrete mathematics because it often has “an experimental flavor.” Researchers can look for patterns, develop a theory, then prove or disprove that theory using math.
“The methods we use can be very different from the methods we use in other branches of mathematics,” Huang said.
A famous problem in discrete math is the four-color theorem, which states that only four colors are needed to color any planar map while ensuring that no adjacent, contiguous regions share the same
color. That theorem took decades to prove and there are still some doubters.
At UNK, Huang’s research looks at the connections between discrete mathematics and algebra, including algebraic and enumerative combinatorics, combinatorial representation theory and graph theory.
“Algebra is a branch of mathematics with a very long history and having great significance in contemporary science, technology and engineering. On the other hand, discrete mathematics is relatively
young and recently flourishing, with many applications in STEM fields, as well,” he said. “The connections between these two areas are often interesting and also useful. For instance, molecular
symmetry in chemistry benefits from group theory, enumerative combinatorics and graph theory, as well as their connections with each other.”
One project that also included UNK students studied binary operations, which are widely used in science, technology, engineering and math. An example of this work is calculating the number of
possible results when five different numbers are subtracted consecutively, but the subtractions can start anywhere and move in either direction.
Another area of research is domination, for example, determining the fewest number of servers needed to sufficiently cover a communication network and how that number changes if a connection is
“This field has a lot of possible applications in the real world,” Huang said of discrete mathematics. “It has connections to many interesting real-life problems.”
That’s a message he repeats over and over to his students. Huang uses these real-world connections to motivate students and encourage them to take on their own research projects.
“At UNK, we are focused on students. We are student-centered,” he said. “I think research should be an important component of that.”
Undergraduate research, he added, enhances the learning experience by allowing students to apply their knowledge from math courses in a new way while discovering new things.
“It is important to not only teach them mathematical knowledge and skills, but also help them develop their own mathematical thinking and encourage them to apply mathematics in other fields,” Huang
said. “One good way to achieve this goal is to engage them in research and provide them opportunities to independently solve problems by themselves, and this will further consolidate their
understanding of multiple subjects in mathematics.”
In his classroom, Huang welcomes questions and collaboration among students. That’s a major benefit of being at UNK, where he’s worked since 2014.
“The class sizes are pretty small. That’s something really nice to me because I can interact with my students and understand their needs,” he said.
It’s a stark contrast to his time as a postdoctoral associate at the University of Minnesota, when he taught a class with about 200 students in the room.
“Most of the time I only looked at the people in the first couple of rows,” Huang said. “I don’t know how much those students got from my lecture.”
During his job interview at UNK, Huang was asked to teach a class with about 10 students who asked questions and interacted with the professor candidate. That was a selling point, along with
Kearney’s size and the easy commute compared to the one-hour bus rides he endured in Minneapolis.
“I think we have a very nice work environment. It just feels very comfortable,” said Huang, noting the friendliness of faculty and students and his department’s willingness to work together.
He expects to see even more teamwork when UNK’s new STEM building opens in fall 2019. The 90,000-square-foot building, part of the Otto C. Olsen replacement project, will bring the university’s
science, technology, engineering and math programs together inside a state-of-the-art facility that promotes collaboration and innovation among students and faculty.
“That will be a good platform for us to find new collaboration opportunities,” said Huang, who is looking forward to sharing ideas with faculty from other departments and further exploring the
connections between science, technology, engineering and math.
“I also think that will be beneficial for our students,” he said.
Outside the classroom, Huang enjoys reading, particularly science fiction and history books, as well as soccer, although there’s a lot more time for the former with two sons, ages 6 and 2.
Huang did play intramural soccer at the University of Minnesota and pick-up games for a couple years at Ted Baldwin Park in Kearney before his schedule got too hectic. He likes the teamwork and
strategy the sport requires.
“It’s not just one or two players,” Huang said. “It’s more of a team game.”
Huang’s wife Ting Zou teaches English language learners at Central Elementary School in Kearney, giving him a different kind of teammate to bounce ideas off.
“It’s a good job for her,” he said. “She likes teaching kids and building those relationships.”
With two educators in the household, teaching is a common topic of conversation. But does she like talking discrete mathematics as much as he does?
“That’s a different story,” Huang said with a smile. “I don’t think it’s her favorite subject.”
Title: Associate Professor, Math and Statistics
College: Arts and Sciences
Education: Ph.D., mathematics, University of Minnesota, 2013.
Years at UNK: 5
Career: Postdoctoral associate, University of Minnesota, 2013-14; Assistant professor, University of Nebraska at Kearney Department of Mathematics and Statistics, 2014-18; Associate professor, UNK
Department of Mathematics and Statistics, 2018-present.
Family: Wife, Ting Zou; Sons, Yifan, 6, and Yifei, 2.
Hobbies/Interests: Reading (historical and science fiction), soccer.
Honors/Awards: UNK College of Natural and Social Sciences Travel Awards; University of Nebraska Collaboration Initiative Planning and Proposal Generation Grant.
Areas of research/specialization: Discrete mathematics and its connections with algebra, including algebraic and enumerative combinatorics, combinatorial representation theory and graph theory.
Courses taught: Applied Calculus; Calculus I, II and III; Foundations of Math; College Geometry; Abstract Algebra; Complex Analysis and Theory of Numbers.
Recent Published Articles:
“Ordered Set Partitions and the 0-Hecke Algebra,” Algebraic Combinatorics, 2018.
“The Nonassociativity of the Double Minus Operation,” Journal of Integer Sequences, 2017.
“A Uniform Generalization of Some Combinatorial Hopf Algebras,” Algebra and Representation Theory, 2017.
“Absolute Order in General Linear Groups,” Journal of the London Mathematical Society, 2017.
“Modular Catalan Numbers,” European Journal of Combinatorics, 2017. | {"url":"https://unknews.unk.edu/2019/03/05/applying-algebra-jia-huang-sees-beauty-of-mathematics-student-research/","timestamp":"2024-11-06T23:32:15Z","content_type":"text/html","content_length":"78648","record_id":"<urn:uuid:24288e6c-9795-490e-b3d0-823e1f96924b>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00781.warc.gz"} |
Themes, topics - math
Themes, topics - math word problems - last page
This page contains thematic word problems. We have about 20 main topics, such as motion problems, mixtures, work problems etc. Please choose a specific topic that interests you from the menu.
Examples within a topic usually train similar knowledge. For example, in movement tasks, terms such as distance, speed, and time often occur.
Number of problems found: 3110
We apologize, but in this category are not a lot of examples.
Do you have homework that you need help solving?
Ask a question
, and we will try to solve it. | {"url":"https://www.hackmath.net/en/word-math-problems/themes-topics?page=156","timestamp":"2024-11-05T17:06:42Z","content_type":"text/html","content_length":"33839","record_id":"<urn:uuid:99cdcc83-a23e-43c6-bcfd-d6456d6d54b2>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00303.warc.gz"} |
Slope Intercept Form
In geometry, the equation of a line can be written in different forms and each of these representations is useful in different ways. The equation of a straight line is written in either of the
following methods:
1. Slope intercept form
In this article, you will learn about one of the most common forms of the equation of lines called slope-intercept form along with derivation, graph and examples.
Learn what is the intercept of a line here.
Let’s have a look at the slope-intercept form definition.
What is the Slope Intercept Form of a Line?
The graph of the linear equation y = mx + c is a line with m as slope, m and c as the y-intercept. This form of the linear equation is called the slope-intercept form, and the values of m and c are
real numbers.
The slope, m, represents the steepness of a line. The slope of the line is also termed as gradient, sometimes. The y-intercept, b, of a line, represents the y-coordinate of the point where the graph
of the line intersects the y-axis.
Slope Intercept Form Equation
In this section, you will learn the derivation of the equation of a line in the slope-intercept form.
Consider a line L with slope m cuts the y-axis at a distance of c units from the origin.
Here, the distance c is called the y-intercept of the given line L.
So, the coordinate of a point where the line L meets the y-axis will be (0, c).
That means, line L passes through a fixed point (0, c) with slope m.
We know that, the equation of a line in point slope form, where (x[1], y[1]) is the point and slope m is:
(y – y[1]) = m(x – x[1])
Here, (x[1], y[1]) = (0, c)
Substituting these values, we get;
y – c = m(x – 0)
y – c = mx
y = mx + c
Therefore, the point (x, y) on the line with slope m and y-intercept c lies on the line if and only if y = mx + c
Note: The value of c can be positive or negative based on the intercept is made on the positive or negative side of the y-axis, respectively.
Slope Intercept Form Formula
As derived above, the equation of the line in slope-intercept form is given by:
y = mx + c
(x, y) = Every point on the line
m = Slope of the line
c = y-intercept of the line
Usually, x and y have to be kept as the variables while using the above formula.
Slope Intercept Form x Intercept
We can write the formula for the slope-intercept form of the equation of line L whose slope is m and x-intercept d as:
y = m(x – d)
m = Slope of the line
d = x-intercept of the line
Sometimes, the slope of a line may be expressed in terms of tangent angle such as:
m = tan θ
Also, try: Slope Intercept Form Calculator
Derivation of Slope-Intercept Form from Standard Form Equation
We can derive the slope-intercept form of the line equation from the equation of a straight line in the standard form as given below:
As we know, the standard form of the equation of a straight line is:
Ax + By + C = 0
Rearranging the terms as:
By = -Ax – C
⇒y = (-A/B)x + (-C/B)
This is of the form y = mx + c
Here, (-A/B) represents the slope of the line and (-C/B) is the y-intercept.
Slope Intercept Form Graph
When we plot the graph for slope-intercept form equation we get a straight line. Slope-intercept is the best form. Since it is in the form “y=”, hence it is easy to graph it or solve word problems
based on it. We just have to put the x-values and the equation is solved for y.
The best part of the slope-intercept form is that we can get the value of slope and the intercept directly from the equation.
Solved Examples
Example 1:
Find the equation of the straight line that has slope m = 3 and passes through the point (–2, –5).
By the slope-intercept form we know;
y = mx+c
m = 3
As per the given point, we have;
y = -5 and x = -2
Hence, putting the values in the above equation, we get;
-5 = 3(-2) + c
-5 = -6+c
c = -5 + 6 = 1
Hence, the required equation will be;
y = 3x+1
Example 2:
Find the equation of the straight line that has slope m = -1 and passes through the point (2, -3).
By the slope-intercept form we know;
y = mx+c
m = -1
As per the given point, we have;
y = -3 and x = 2
Hence, putting the values in the above equation, we get;
-3 = -1(2) + c
-3 = -2 + c
c = -3+2 = -1
Hence, the required equation will be;
y = -x-1
Example 3:
Find the equation of the lines for which tan θ = 1/2, where θ is the inclination of the line such that:
(i) y-intercept is -5
(ii) x-intercept is 7/3
Given, tan θ = 1/2
So, slope = m = tan θ = 1/2
(i) y-intercept = c = -5
Equation of the line using slope intercept form is:
y = mx + c
y = (1/2)x + (-5)
2y = x – 10
x – 2y – 10 = 0
(ii) x-intercept = d = 7/3
Equation of slope intercept form with x-intercept is:
y = m(x – d)
y = (1/2)[x – (7/3)]
2y = (3x – 7)/3
6y = 3x – 7
3x – 6y – 7 = 0
Practice Problems
1. Find the slope of the line y = 5x + 2.
2. Find the slope of the line which crosses the line at point (-2,6) and have an intercept of 3.
3. What is the equation of the line whose angle of inclination is 45 degrees and x-intercept is -⅗?
4. Write the equation of the line passing through the point (0, 0) with slope -4. | {"url":"https://mathlake.com/Slope-Intercept-Form","timestamp":"2024-11-06T23:36:48Z","content_type":"text/html","content_length":"17716","record_id":"<urn:uuid:460fb800-ebd7-4616-8e84-278c1bd176b3>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00222.warc.gz"} |
Understanding Measurement Units in Chemistry - Measuring Expert
Chemistry, the central science, hinges on the understanding and correct utilization of units of measurement. Ensuring consistency in scientific calculations worldwide, units of measurement are the
fundamental building blocks in the field of chemistry. Spanning from basic concepts to complex applications, this exploration of units of measurement will guide you through their significance, use,
conversion, and problem-solving applications in chemistry. Gain insights on the International System of Units (SI Units) – the standard system used by chemists worldwide – and familiarize yourself
with common units like the mole, meters, kilograms, seconds, and Kelvin. By engaging with this, you’ll acquire the knowledge to convert units and apply them effectively in chemistry-related
Basic Concepts of Units of Measurement in Chemistry
Understanding Units of Measurement in Chemistry
Units of measurement are an elemental aspect of science, and the field of Chemistry is not an exception. These units are necessary for accurate descriptive and predictive scientific work.
Measurements provide quantitative data, which aid in comparing observations and improving the precision of theoretical predictions.
Absolute measurements refer to quantities like time, temperature, and mass, which do not require to be related with other quantities for their understanding. For instance, ten grams is a definitive
weight, regardless of what substance is being weighed. Relative measurements, on the other hand, measure one quantity in relevance to another. An example is density, which requires both mass and
volume to be defined.
Units of measurement also allow for standardization across the field of Chemistry. This standardization makes it possible for different scientists to repeat or modify experiments conducted by others.
Also, creating a standard universal system prevents discrepancies that could arise from using different unit systems.
The Adoption of SI Units in Chemistry
Science, including Chemistry, has accepted the International System of Units (SI Units) as the standard system for measurements. The system, developed by the Bureau International des Poids et Mesures
(BIPM) in France in the 1960s, is based on seven base units. These units include the meter (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, kelvin (K) for
temperature, mole (mol) for amount of substance, and candela (cd) for luminous intensity.
Chemistry often utilizes derived units, which are combinations of these base units. For example, velocity is defined in meters per second (m/s), which are derived from the base units of meter (m) and
second (s).
Chemical Quantities and SI Units
In Chemistry, the mole (mol) is a crucial base unit as it provides a way to express amounts of a chemical substance. It allows chemists to count entities at the atomic and molecular level. One mole
of any substance contains the same number of entities (atoms, molecules, ions, or other particles), known as Avogadro’s number (approximately 6.022 x 10^23).
Other commonly used chemical units derive from SI base and derived units, such as liters for volume and grams for mass.
An Overview
The understanding and standardization of units of measurement are of paramount importance in the scientific discipline of Chemistry. The standardized measures facilitate a universal comprehension of
results, breaking down barriers created by different countries, cultures, and languages. This unanimity accelerates the global advancement of scientific knowledge.
Common Units of Measurement in Chemistry
Exploring Units of Measurement in Chemistry
Chemistry, like all other science domains, is heavily dependent on standardized units of measurement to articulate the quantities of various elements or compounds. A fundamental measurement unit in
chemistry is the mole (mol), the basic unit for quantity in the International System of Units (SI). Essentially, a mole corresponds to the number of atoms present in exactly 0.012 kilograms of
Carbon-12, equating to approximately 6.022 x 10^23, a value famously known as Avogadro’s number.
For length, the metric used generally in laboratory settings is the meter (m). However, when describing the minute dimensions of atoms and molecules, subunits like nanometers (10^-9 meters) or
picometers (10^-12 meters) are often utilized. These units make these incredibly small values manageable for scientific research.
Mass, an inherent property of matter, is typically quantified in kilograms (kg) in the SI system. However, in chemistry, smaller measures like grams (g) and milligrams (mg) are more common due to the
typically lesser quantities used.
Time, a universal physical quantity, is employed across various areas within chemistry, such as in tracking reaction rates or determining the half-lives of unstable isotopes. The second (s) is the SI
system’s primary unit of time.
Temperature is another crucial variable in many chemical processes. Scientists generally use the Kelvin (K) scale to measure temperature while Celsius (°C) is commonly used in laboratories. The
Kelvin scale begins from absolute zero, the lowest achievable temperature, hence making it suitable for rigorous thermodynamic calculations.
Another unit specific to chemistry is concentration, typically presented in the form of molarity (mol/L). Molarity conveys the number of solute moles within a liter of solution. This unit proves
particularly handy when undertaking calculations related to solution reactions.
Lastly, the energy unit in the SI system is the joule (J). In chemistry, it is also frequent to express energy in terms of calories (cal) or kilocalories (kcal), particularly when referring to heat
Familiarizing oneself with these units and their usage is crucial for accurately understanding chemical data, carrying out calculations, and validating theories. Each unit plays a critical role in
deciphering and interpreting the vast realm of Chemistry.
Conversion of Units in Chemistry
Expanding the Scope: Units of Measurement in Chemistry
In chemistry, units of measurement encompass an extensive range of categories, including distance, mass, volume, energy, and significantly, number of particles. Elements such as the intensity of
interactions, rate of chemical reactions, and quantification of distinct physical properties are all expressed through these units. A thorough understanding of these units forms an essential
cornerstone in the mastery of chemical science.
Significance of Conversion Factors
Conversion factors play an essential role in chemistry. Essentially, a conversion factor is a ratio that expresses how one unit of measurement equates to another. It’s used to convert a given unit in
one form to another equivalent form without adjusting the quantity’s value. The conversion factor is derived by taking the relationship between two units and expressing this relationship as a
fraction. Conversion factors have a value of one, since they express a relationship between two equivalent measurements.
Application of Conversion Factors in Chemistry
Conversion factors are commonly used in chemistry for various calculations. For instance, chemists use Avogadro’s number (6.02 x 10^23), a conversion factor, to convert between number of particles
such as atoms, ions, molecules, and moles. The mole is a standard unit in chemistry and can be defined as the amount of substance that contains as many entities (atoms, molecules or other particles)
as there are in 12 grams of pure carbon-12.
A specific example would be converting grams to moles using the molar mass of a substance. If you know the molar mass, you could use it as a conversion factor to convert grams of the substance to
moles. For instance, in carbon with a molar mass of approximately 12g/mol, 1 mole of carbon is equivalent to 12 g. So if you had 24 g of carbon, you could use the conversion factor of (1 mol/12g) to
determine that you have 2 moles of carbon.
Conversion in Chemical Equations and Reactions
Conversion factors are also used in the stoichiometry of chemical reactions. Stoichiometry is the relationship between reactants and products in a chemical reaction. One mole of a substance produces
or consumes a specific number of moles of another substance according to the balanced chemical equation.
For example, in the combustion of hydrogen gas to form water (2H2 + O2 → 2H2O), two moles of hydrogen gas react with one mole of oxygen gas to produce two moles of water. Hence, using conversion
factors, you can determine the amount of product (in moles) produced from a given amount of reactants or the amount of reactants needed to produce a certain amount of product.
Conversion factors and measurement units lay the foundation of every scientific computation in the field of chemistry. Expertise in these tools enables chemists to measure and interpret the natural
phenomena precisely, enhancing their communication. Their usage spans from rudimentary laboratory calculations to advanced theoretical modeling of chemical reactions. Grasping these foundational
concepts opens doors to a wider exploration and research in chemistry.
Problem Solving Using Units of Measurement
Grasping the Fundamentals of Measurement Units in Chemistry
Chemistry acts as the linking bridge between physical sciences, for instance, physics and geology, and life sciences such as biology and medicine. A fundamental underpinning that strengthens the
understanding of chemistry is its consistency in the application of measurement units. The most prevalently used units in chemistry within the International System of Units (SI), include meters (m)
for length, kilograms (kg) for mass, and seconds (s) for time.
For quantifying the amounts of substances, chemists rely on the mole (mol), which encompasses approximately 6.02214076 x 10^23 representative particles, also known as Avogadro’s number. Temperature
in the chemical context is generally measured in Kelvin (K) or degrees Celsius (°C), and the quantification of energy is often in Joules (J) or kilojoules (kJ).
Applying Units of Measurement in Chemistry Calculations
Accurate use and conversion of units of measurements are critical to problem solving in chemistry. For instance, if you’re asked to convert 250 grams of a substance to moles, you need to know the
molar mass of the substance, which can be found on the periodic table. Then, you use the formula moles = mass (grams) / molar mass (grams/mole) to get the required conversion.
Another everyday example is determining the concentration of a solution, measured in moles per liter (M). If you dissolve 0.1 moles of sodium chloride (NaCl) in a liter of water, you’ll get a 0.1 M
NaCl solution.
Interpreting Results and Units in Chemical Contexts
Understanding the meaning of units in a chemical context is as important as knowing how to use them. For instance, when we say the density of a substance is 1 g/cm^3, it means that a cubic centimeter
of the substance weighs one gram. Similarly, a heat capacity listed as 4.18 J/g°C indicates that 4.18 joules of energy are needed to raise the temperature of a gram of the substance by one degree
In interpreting units like M (moles per liter), it means that for every liter of solution, there is an equivalent mole of solute present.
Examples and Problems
Here are some simple problems:
1. Calculate the number of moles in 50g of water (H2O). (Hint: The molar mass of water is approximately 18g/mol).
2. If you have a 0.25 M solution of NaCl and you have 2 liters of it, how many moles of NaCl do you have?
3. If 4.18 J of energy is required to heat up a gram of water by 1 degree Celsius and you have 100g of water, how much energy do you need to increase the water’s temperature by 10 degrees Celsius?
Understanding the units of measurement in chemistry is imperative in various aspects of the science and in our daily lives. From medical dosages to cooking recipes to industrial applications,
measuring substances correctly ensures desired results and safety. Further, understanding units helps in academic success in chemistry, and in professional applications in the field.
By now, you should have a strong understanding of the pivotal role units of measurement play in the field of chemistry. From providing standardization with the SI Units to the utility of common units
such as the mole, meters, kilograms, seconds, and Kelvin, these tools act as the backbone of every calculation and scientific exploration in the discipline. With the ability to convert units, you can
navigate through varied expressions, equations, and reactions, enhancing your problem-solving skills in chemistry. It’s through the mastery of these fundamental elements that one can truly appreciate
and excel in their chemistry endeavors, be it academic, professional, or personal interests. Always remember, every significant chemical breakthrough began with an understanding and application of
these seemingly simple units of measurement. | {"url":"https://www.measuringexpert.com/understanding-measurement-units-in-chemistry/","timestamp":"2024-11-04T08:23:14Z","content_type":"text/html","content_length":"127785","record_id":"<urn:uuid:55e84aa8-e9b6-4297-a8c8-c0f1af7b8663>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00885.warc.gz"} |
inertia for objects in motion. It helps quantify how difficult it is to stop something. Momentum has both magnitude and direction, making it a vector quantity. The change in momentum is called
The law of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. This means that in the absence of external forces, the total
momentum of a system before an event must be equal to the total momentum after the event. This principle is often applied in the analysis of collisions and explosions.
Momentum Types
Linear Momentum - This is the most common type of momentum and is associated with the motion of an object in a straight line. It is defined as the product of an object's mass and its linear velocity.
Angular Momentum - This type of momentum is associated with the rotation or spinning of an object. It depends on the object's moment of inertia and its angular velocity.
Impulse Momentum - Impulse is the change in momentum of an object. It is the product of the force applied to an object and the time over which the force is applied.
Relativistic Momentum - In the context of special relativity, momentum is modified due to the relativistic effects at high velocities.
Momentum formula
\( p \;=\; m \; v \) (Momentum)
\( m \;=\; p \;/\; v \)
\( v \;=\; p \;/\; m \)
Symbol English Metric
\( p \) = Momentum \(lbm-ft\;/\;sec\) \(kg-m\;/\;s\)
\( m \) = Mass \(lbm\) \(kg\)
\( v \) = Velocity \(ft\;/\;sec\) \(m\;/\;s\)
Tags: Motion Momentum Inertia | {"url":"https://www.piping-designer.com/index.php/properties/classical-mechanics/692-momentum","timestamp":"2024-11-13T16:16:48Z","content_type":"text/html","content_length":"29687","record_id":"<urn:uuid:bb39f8d4-d1d7-4096-88d7-e5c8ba893e1b>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00593.warc.gz"} |
Unlocking the Mysteries of Algebra – A Guide to the Glencoe Algebra 1 Workbook Answer Key PDF
Remember the frustration of staring at a math problem, feeling like the solution was just out of reach? The feeling of relief when you finally understood the concept and the answer clicked? That’s
the power of learning, and for many students, the Glencoe Algebra 1 workbook has been a trusted companion on the journey to mastering algebra. But what about those tricky problems that leave you
scratching your head? That’s where the Glencoe Algebra 1 workbook answer key PDF can come to the rescue.
Image: www.amazon.com
The answer key is more than just a cheat sheet; it’s a learning tool. It provides insights into the problem-solving process, revealing the steps and strategies used to arrive at the correct answer.
It’s like having a private tutor who can guide you through any roadblock, allowing you to grasp the underlying principles and build confidence in your algebraic abilities.
The Glencoe Algebra 1 Workbook Answer Key: Your Secret Weapon
The Glencoe Algebra 1 workbook is a staple in many classrooms, offering a comprehensive approach to algebra with plenty of practice problems. However, even the most dedicated student can benefit from
the occasional guidance. The answer key PDF provides a detailed breakdown of every solution, illuminating the logic behind each step. It’s like having a step-by-step walkthrough that demystifies even
the most complex equation.
The key is to use the answer key strategically. Don’t just blindly copy the solutions. Instead, work through the problems first, then use the answer key to verify your answers and analyze your
approach. This active learning approach helps you to:
• Identify your strengths and weaknesses. See where you excel and where you need to focus more of your energy.
• Deepen your understanding. By examining the solution process, you’ll gain a more profound understanding of the underlying concepts.
• Build confidence in your problem-solving abilities. Each solved problem strengthens your belief that you can conquer any mathematical challenge.
Navigating the Answer Key for Maximum Learning
The Glencoe Algebra 1 workbook answer key PDF is your guide, but it’s up to you to make the most of it. Here are some tips to unlock its full potential:
• Start with the basics. Don’t jump into the more complex problems before mastering the fundamental concepts.
• Work through the problems systematically. Follow the order of the workbook, tackling the concepts one at a time.
• Don’t be afraid to ask for help. If you’re stuck on a problem, seek clarification from your teacher or a tutor. They can provide additional insights and guidance.
• Use the answer key as a tool, not a crutch. It’s designed to support your learning, not replace it.
Beyond the Answers: Mastering Algebra
The Glencoe Algebra 1 workbook answer key PDF is a valuable resource, but it’s just one piece of the puzzle. True mastery of algebra requires a multi-faceted approach that goes beyond simply
memorizing solutions. Here are some strategies for achieving lasting success:
• Practice, practice, practice. The more you practice, the more comfortable you’ll become with the concepts.
• Visualize the concepts. Use diagrams, graphs, and real-world examples to help you visualize the abstract principles of algebra.
• Collaborate with your classmates. Work together to discuss problems, share strategies, and help each other learn.
• Embrace the challenges. Don’t get discouraged by difficult problems. They’re an opportunity for growth and deeper understanding.
Image: www.pdfprof.com
Glencoe Algebra 1 Workbook Answer Key Pdf
The Journey to Mathematical Mastery
The quest to master algebra is often compared to a journey with its twists, turns, and moments of uncertainty. But with the right tools and the right mindset, you can achieve your goals. The Glencoe
Algebra 1 workbook answer key PDF is a powerful ally on your journey, offering valuable insights and support along the way. It’s a testament to the power of resources and the commitment to learning
that can unlock the mysteries of math and empower you to achieve academic success.
Take charge of your learning, use the answer key strategically, and enjoy the satisfaction of conquering algebraic challenges. The journey is yours to own, and with perseverance and the right
guidance, you can reach new heights in your mathematical understanding. | {"url":"https://www.johnsonpaper.com/2705.html","timestamp":"2024-11-04T04:48:07Z","content_type":"text/html","content_length":"107383","record_id":"<urn:uuid:05587692-7cd0-4f7a-89d0-144107419779>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00059.warc.gz"} |
Convert km to m
Please provide values below to convert kilometer [km] to meter [m], or vice versa.
Definition: A kilometer (symbol: km) is a unit of length in the International System of Units (SI). One kilometer is equivalent to 0.6214 miles.
History/origin: The prefix kilo- is a metric prefix indicating one thousand. One kilometer is therefore one thousand meters. The origin of the kilometer is linked to that of the meter, and its
current definition as the distance traveled by light in 1/299 792 458 second. This definition is subject to change, but the relationship between the meter and the kilometer will remain constant.
Current use: It is currently the official unit of measurement for expressing distances between geographical places on land in most of the world. However, there still remain a number of countries that
primarily use the mile instead of the kilometer including the United States and the United Kingdom (UK). Unlike the United States, the UK has adopted the metric system; while the metric system is
widely used in government, commerce, and industry, remnants of the imperial system can still be seen in the UK's use of miles in its road systems.
Definition: A meter, or metre (symbol: m), is the base unit of length and distance in the International System of Units (SI). The meter is defined as the distance traveled by light in 1/299 792 458
of a second. This definition was slightly modified in 2019 to reflect changes in the definition of the second.
History/origin: Originally, in 1793, the meter was defined as one ten-millionth of the distance from the equator to the North Pole. This changed in 1889, when the International prototype metre was
established as the length of a prototype meter bar (made of an alloy of 90% platinum and 10% iridium) measured at the melting point of ice. In 1960, the meter was again redefined, this time in terms
of a certain number of wavelengths of a certain emission line of krypton-86. The current definition of the meter is effectively the same as the definition that was adopted in 1983, with slight
modifications due to the change in definition of the second.
Current use: Being the SI unit of length, the meter is used worldwide in many applications such as measuring distance, height, length, width, etc. The United States is one notable exception in that
it largely uses US customary units such as yards, inches, feet, and miles instead of meters in everyday use.
Kilometer to Meter Conversion Table
Kilometer [km] Meter [m]
0.01 km 10 m
0.1 km 100 m
1 km 1000 m
2 km 2000 m
3 km 3000 m
5 km 5000 m
10 km 10000 m
20 km 20000 m
50 km 50000 m
100 km 100000 m
1000 km 1000000 m
How to Convert Kilometer to Meter
1 km = 1000 m
1 m = 0.001 km
Example: convert 15 km to m:
15 km = 15 × 1000 m = 15000 m
Popular Length Unit Conversions
Convert Kilometer to Other Length Units | {"url":"https://www.tbarnyc.com/length/km-to-m.html","timestamp":"2024-11-08T18:11:31Z","content_type":"text/html","content_length":"18269","record_id":"<urn:uuid:3293097d-051c-4b6f-85ed-6c37deea5c79>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00126.warc.gz"} |
Conductor Facts: Lay Direction and Length - Fisk Alloy
Stranded conductors are manufactured by twisting strands of non-insulated wire. The direction of twisting is designated as the “lay direction.” The degree of twist per unit length defines the “lay
Lay Direction
The lay direction is determined by the direction the machine is turning during the stranding operation. The conventional method to determine the lay direction is to observe the upper surface of the
stranded conductor with one end pointing toward you and the wire leading away from you.
If the strands turn left leading away from the observer and have the same slant as the middle of the letter “S”, the convention denotes an “S” lay direction.
If the strands turn right leading away from the observer and have the same slant as the middle of the letter “Z”, the convention denotes a “Z” lay direction.
Lay Length
Lay length is defined as the distance required to complete one revolution of the strand around the diameter of the conductor.
When a conductor has more than one layer, it usually refers to the lay length of the outer layer. In the case of Unilay, Equilay and bunch, the lay length of all layers is equal. In True Concentric
and Unidirectional, the lay lengths of the inner layers are less, this also holds true for rope constructions.
Generally Accepted Practices
There are some general practices that pertain to the lay direction and lengths of conductor as specified by industry standards such as ASTM, NEMA and military, however, requirements for specific
applications vary.
Direction of the Outer Layer:
The direction away from the outer layer of strands or members is usually S. Inner layer directions depend upon the construction (True concentric, Unilay, etc).
Length of the Outer Layer:
The lay length of the outer layer of strands or members varies with different applications.
• For most conductor applications, lay lengths of between 8 – 16 times the outer diameter of a given layer are specified in ASTM B 286. In general, lay lengths in the range of 12 – 15 times the
outer diameter are used for tighter tolerance and geometric pattern control. Shorter lay lengths of 12 times or less have the disadvantage of slightly higher weight per unit length.
For 7 strand and bunch applications, where tight diameter tolerance is less of a concern, lay lengths in excess of 30 times the outer diameter are common. Longer lay lengths are sometimes preferred
by customers for cost, yield and weight considerations.
Stranding Factors
The increase in weight and resistance due to stranding can be calculated mathematically. ASTM refers to this increase as the stranding or “k-factor”, defined as “incremental percentage (increase) of
weight and electrical resistance.” ASTM B 8, B 229, B 231, and others give a method of calculating the “k“:
k = 100 (m – 1)
Where k is the incremental (increase) in mass and electrical resistance, the factor m is the ratio of the mass or electrical resistance of a unit length of the stranded conductor to that of a
conductor monofilament of the same section or that of the stranded conductor with an infinite length of lay (all the strands run parallel to the axis). The factor m of the strand is the average of
the factors for each of the individual wires in the conductor including the straight wire core, if any (for which the lay factor is unity).
The lay factor mind for any given wire in a concentric stranded conductor is calculated as follows:
Where n = (length of lay) + (diameter of helical path of wire).
Example: the lay factor for a 19 strand conductor is the numerical average of the 19 individual strands:
m = (1 + 6m[6] = 12m[12]) ÷ 19
Where m[6] = m[ind] calculated for each of the 6 strands of the inner layer and m[12] = mind calculated for each of the 12 strands of the outer layer. | {"url":"https://fiskalloy.com/conductor-facts/lay-direction-and-length/","timestamp":"2024-11-10T15:55:51Z","content_type":"text/html","content_length":"56241","record_id":"<urn:uuid:bd1cdd66-d41f-47a5-9c9d-7f5723386924>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00062.warc.gz"} |
Dimitris G. Giovanis
Oct 29, 2021
Abstract:Computational models of the human head are promising tools for estimating the impact-induced response of brain, and thus play an important role in the prediction of traumatic brain injury.
Modern biofidelic head model simulations are associated with very high computational cost, and high-dimensional inputs and outputs, which limits the applicability of traditional uncertainty
quantification (UQ) methods on these systems. In this study, a two-stage, data-driven manifold learning-based framework is proposed for UQ of computational head models. This framework is demonstrated
on a 2D subject-specific head model, where the goal is to quantify uncertainty in the simulated strain fields (i.e., output), given variability in the material properties of different brain
substructures (i.e., input). In the first stage, a data-driven method based on multi-dimensional Gaussian kernel-density estimation and diffusion maps is used to generate realizations of the input
random vector directly from the available data. Computational simulations of a small number of realizations provide input-output pairs for training data-driven surrogate models in the second stage.
The surrogate models employ nonlinear dimensionality reduction using Grassmannian diffusion maps, Gaussian process regression to create a low-cost mapping between the input random vector and the
reduced solution space, and geometric harmonics models for mapping between the reduced space and the Grassmann manifold. It is demonstrated that the surrogate models provide highly accurate
approximations of the computational model while significantly reducing the computational cost. Monte Carlo simulations of the surrogate models are used for uncertainty propagation. UQ of strain
fields highlight significant spatial variation in model uncertainty, and reveal key differences in uncertainty among commonly used strain-based brain injury predictor variables. | {"url":"https://www.catalyzex.com/author/Dimitris%20G.%20Giovanis","timestamp":"2024-11-04T15:44:31Z","content_type":"text/html","content_length":"68156","record_id":"<urn:uuid:fc465fd0-ef42-4c9f-8493-cfe33dadf630>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00164.warc.gz"} |
by finding LCA
Solve RMQ (Range Minimum Query) by finding LCA (Lowest Common Ancestor)¶
Given an array A[0..N-1]. For each query of the form [L, R] we want to find the minimum in the array A starting from position L and ending with position R. We will assume that the array A doesn't
change in the process, i.e. this article describes a solution to the static RMQ problem
Here is a description of an asymptotically optimal solution. It stands apart from other solutions for the RMQ problem, since it is very different from them: it reduces the RMQ problem to the LCA
problem, and then uses the Farach-Colton and Bender algorithm, which reduces the LCA problem back to a specialized RMQ problem and solves that.
We construct a Cartesian tree from the array A. A Cartesian tree of an array A is a binary tree with the min-heap property (the value of parent node has to be smaller or equal than the value of its
children) such that the in-order traversal of the tree visits the nodes in the same order as they are in the array A.
In other words, a Cartesian tree is a recursive data structure. The array A will be partitioned into 3 parts: the prefix of the array up to the minimum, the minimum, and the remaining suffix. The
root of the tree will be a node corresponding to the minimum element of the array A, the left subtree will be the Cartesian tree of the prefix, and the right subtree will be a Cartesian tree of the
In the following image you can see one array of length 10 and the corresponding Cartesian tree.
The range minimum query [l, r] is equivalent to the lowest common ancestor query [l', r'], where l' is the node corresponding to the element A[l] and r' the node corresponding to the element A[r].
Indeed the node corresponding to the smallest element in the range has to be an ancestor of all nodes in the range, therefor also from l' and r'. This automatically follows from the min-heap
property. And is also has to be the lowest ancestor, because otherwise l' and r' would be both in the left or in the right subtree, which generates a contradiction since in such a case the minimum
wouldn't even be in the range.
In the following image you can see the LCA queries for the RMQ queries [1, 3] and [5, 9]. In the first query the LCA of the nodes A[1] and A[3] is the node corresponding to A[2] which has the value
2, and in the second query the LCA of A[5] and A[9] is the node corresponding to A[8] which has the value 3.
Such a tree can be built in $O(N)$ time and the Farach-Colton and Benders algorithm can preprocess the tree in $O(N)$ and find the LCA in $O(1)$.
Construction of a Cartesian tree¶
We will build the Cartesian tree by adding the elements one after another. In each step we maintain a valid Cartesian tree of all the processed elements. It is easy to see, that adding an element s
[i] can only change the nodes in the most right path - starting at the root and repeatedly taking the right child - of the tree. The subtree of the node with the smallest, but greater or equal than s
[i], value becomes the left subtree of s[i], and the tree with root s[i] will become the new right subtree of the node with the biggest, but smaller than s[i] value.
This can be implemented by using a stack to store the indices of the most right nodes.
vector<int> parent(n, -1);
stack<int> s;
for (int i = 0; i < n; i++) {
int last = -1;
while (!s.empty() && A[s.top()] >= A[i]) {
last = s.top();
if (!s.empty())
parent[i] = s.top();
if (last >= 0)
parent[last] = i; | {"url":"https://cp-algorithms.com/graph/rmq_linear.html","timestamp":"2024-11-08T05:52:56Z","content_type":"text/html","content_length":"127482","record_id":"<urn:uuid:8076eca8-9be6-423f-bb2f-09dad64ec9ca>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00890.warc.gz"} |
What is the slope of y=3? | Socratic
What is the slope of #y=3#?
1 Answer
The slope of $y = 3$ is $0$.
Slope is defined as the change in $y$ for a given change in $x$.
Since the value of $y$ doesn't change, its change is $0$.
Impact of this question
1861 views around the world | {"url":"https://api-project-1022638073839.appspot.com/questions/what-is-the-slope-of-y-3","timestamp":"2024-11-07T06:04:26Z","content_type":"text/html","content_length":"32002","record_id":"<urn:uuid:4539090a-3038-4f9b-8a24-1d490bfc4c46>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00171.warc.gz"} |
Institute for Gravitation and the Cosmos | Murat Gunaydin
Research Interests:
M/Superstring theory, supergravity, AdS/CFT dualities, exceptional groups and related algebraic and geometric structures, representations of noncompact groups and supergroups and their applications,
algebraic foundations of Quantum Mechanics, generalized spacetimes defined by Jordan algebras, double-copy constructions of the amplitudes of supergravity theories in terms of gauge theory
amplitudes, higher spin theories, the classification of higher spin algebras in various dimensions and their unitary realizations.
I have done extensive work on the construction and classification of novel supergravity theories and their gaugings in various dimensions and AdS/CFT dualities. Of particular interest from physics
point of view are gauged supergravity theories that describe the low energy effective theories of flux compactifications from M/Superstring theory that are relevant for AdS/CFT dualities in M/
Superstring theory. Another focus of my research has been on the problem of understanding how the spectra of various superstring theories or M-theory are related to the unitary representations of
their non-perturbative symmetry groups or supergroups. Towards this goal I have done extensive work on the unitary representations of continuous U-duality groups of supergravity theories, some of
which arise as low energy effective theories of compactified M-theory or superstring theories. I have also been studying the unitary representations of spectrum generating symmetry groups in five and
four dimensional supergravity theories and their applications to the BPS black hole spectra. More recently I have been trying to extend these results to discrete arithmetic subgroups of U-duality
groups which lead to some fascinating connections with number theory.
U-duality groups of five dimensional supergravity theories with homogeneous scalar manifolds admit extensions to spectrum generating generalized conformal groups. Similarly, U-duality groups of
corresponding four dimensional theories admit extensions to spectrum generating quasiconformal groups. Quasiconformal realization of the spectrum generating symmetry group E8(8) of the maximal
supergravity in four dimensions, constructed in 2000, was the first known geometric realization of E8. Quasiconformal realizations exist for different real forms of all Lie groups and they leave
invariant a generalized light-cone defined by a quartic distance function. The quantization of geometric quasiconformal realization of a noncompact group leads directly to its minimal unitary
More recent work on quasiconformal realizations of noncompact groups have established a one-to-one correspondence between massless conformal fields in all spacetime dimensions and their minimal
unitary representations and their deformations. This correspondence extends also to conformal superalgebras that exist in space-time dimensions six or less. The enveloping algebras of the minimal
unitary representation and its deformations describe the higher spin algebras and their deformations in the respective dimensions.
My current research focuses on the construction of the scattering amplitudes of matter coupled supergravity theories as double copies of gauge theory amplitudes using BCJ relations. Another area of
my current research focuses on the deep connections between discrete arithmetic U-duality groups and number theory.
Publications: See my papers listed in Spires: http://inspirehep.net/search?p=find+a+gunaydin,+m.
1. Marco Chiodaroli, Murat Gunaydin, Radu Roiban, "Superconformal symmetry and maximal supergravity in various dimensions." JHEP 03 (2012)
2. Marco Chiodaroli, Murat Gunaydin, Henrik Johansson, Radu Roiban, "Complete construction of magical, symmetric and homogeneous N=2 supergravities as double copies of gauge theories." Phys. Rev.
Lett. 117 1 (2016)
3. Marco Chiodaroli, Murat Gunaydin, Henrik Johansson, Radu Roiban, "Gauged Supergravities and Spontaneous Supersymmetry Breaking from the Double Copy Construction." Phys. Rev. Lett. 120 17 (2018)
4. M. Gunaydin, G. Sierra, P. Townsend, "The Geometry of N=2 Maxwell-Einstein Supergravity and Jordan Algebras." Nucl. Phys. B 242 (1984)
5. M. Gunaydin, N. Marcus, "The Spectrum of the s**5 Compactification of the Chiral N=2, D=10 Supergravity and the Unitary Supermultiplets of U(2, 2/4)." Class. Quant. Grav. 2 (1985)
6. M. Gunaydin, L. Romans, N. Warner, "Compact and Noncompact Gauged Supergravity Theories in Five-Dimensions." Nucl. Phys. B 272 (1986)
7. M. Gunaydin, K. Koepsell, H. Nicolai, "Conformal and quasiconformal realizations of exceptional Lie groups." Commun. Math. Phys. 221 (2001)
8. Murat Gunaydin, Andrew Neitzke, Boris Pioline, Andrew Waldron, "BPS black holes, quantum attractor flows and automorphic forms." Phys. Rev. D 73 (2006)
9. Murat Günaydin, E. Skvortsov, Tung Tran, "Exceptional F(4) higher-spin theory in AdS_6 at one-loop and other tests of duality." JHEP 11 (2016)
10. Murat Günaydin, Dmytro Volin, "The Complete Unitary Dual of Non-compact Lie Superalgebra 𝔰𝔲( p, q| m) via the Generalised Oscillator Formalism, and Non-compact Young Diagrams." Commun. Math.
Phys. 367 3 (2019) | {"url":"https://igc.psu.edu/people/bio/gxt/","timestamp":"2024-11-09T00:50:02Z","content_type":"text/html","content_length":"63073","record_id":"<urn:uuid:dd725052-71f1-448f-9d24-4db43fa962e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00637.warc.gz"} |
Parents' Review
The Parents' Review
A Monthly Magazine of Home-Training and Culture
Edited by Charlotte Mason.
"Education is an atmosphere, a discipline, a life."
The First Stage in Arithmetic
by the late Rev. R. H. Quick
Volume 7, 1896, pgs. 186-194
[Robert Hebert Quick, 1831-1891, graduated from Trinity College, Cambridge, and was assistant master at Cranleigh School, and Harrow School (both near London). He lectured on the history of
education, wrote about Froebel, and edited John Locke's "Some Thoughts Concerning Education."]
A lecture given at the College of Preceptors.
(Continued from page 17.)
But at this proposal, I know many experienced teachers would stand aghast. "What!" they would say, "turn one of the easiest lessons (easiest for the teacher) into one of the most difficult!" What can
be easier than to say to a class: "Take out your slates and work the sums I write on the board?" Well, that is easy; but, when I was a lad, the master had a plan still easier. He gave us--or, rather,
made our parents buy for us from him--a book with rules and examples. These examples we had to copy on our slates, and see if we could work them and get out the right answer. If we could do this in
any way, however absurd, or with any expenditure of time, however great, the teacher had no trouble at all; but, if we did not get the right answer, we took our slates and joined a queue of boys that
filed up to the teacher's desk. He looked at each slate in turn, examined the sum, and did it, and so proved to us that the book was right after all. This is the way in which I was not taught
arithmetic! But even my old master had not attained to the easiest method of all of giving an arithmetic lesson. This method, however, has been discovered, and, I may say, brought to perfection, in a
girls' school I once knew about, a school not belonging to the Girls' Public School Company. The teacher, like my old master, made every pupil get a book of sums with answers to them. So furnished,
the pupils sat for an hour getting out as many answers like the book as they conveniently could with reference to their individual knowledge, skill, and industry. At the end of the hour, each pupil
announced to the teacher the number of right answers she had obtained, and the number as given by the pupil was recorded without inspection in the teacher's book. If anybody happens to know an easier
method of "giving a lesson" in arithmetic, he will oblige me by mentioning it!
And here I come with my theoretical notions, and want to take away from the teacher the great practical advantage of having a very simple, straightforward occupation, or even of taking an
after-dinner nap. Vivâ voce or vocal arithmetic with a class is a severe strain on the teacher--not the least doubt about it. The relaxations, well known to youth, whereby the tedious hours of school
are beguiled of some of their tedium, are of various kinds, and, from the master's point of view, those with the pen or pencil are by far the least objectionable. Boys who are playing
noughts-and-crosses [tic-tac-toe] are very quiet. A boy who is producing a likeness of the master invariably behaves in a most exemplary manner as long as the work of art is in progress. Even
correspondence by notes passed from one part of the room to the other seldom obtrudes itself on the notice of the master, and, unless he can attend to two or three things at once, he is probably
totally unconscious of it. So masters who wish to keep everybody quietly occupied mostly set something to be written. Take away from juvenile energy such unobtrusive outlets as I have mentioned, it
it sure to find other outlets by no means unobtrusive. Pinching, e.g., though not in itself a noisy operation, often occasions noise; and then follows inquiry, and the master has to estimate the
value of conflicting statements; the order of the class is threatened, its time is obviously wasted. Even whispering among the boys makes it only too evident that the attention of the class is by no
means concentrated on what is supposed to be the occupation of the hour. Hypocrisy has been called a tribute to virtue. In like manner, the decorum of the schoolboy, though only skin-deep, is a
tribute to the supremacy of the master. So long as everything is quiet, the master may fancy himself monarch of all he surveys; but, directly whispering begins, he feels that he is dethroned.
So, when the practical man points out the advantages of giving a form something to write, I am bound to say: Yes, my friend, I know these advantages of old, and have often jumped at them, though
painfully conscious of the attendant disadvantage, that all my time out of school would go in correcting what had been written in school. This, however, by the way. All I wish you to observe is, that
I know it is a very difficult task to keep up the attention or even the discipline of a large class, if all the work is taken vivâ voce. But this difficulty may be got over by ordinary mortals when
the subject of the hour admits of an infinite number of questions, easily found and easily answered, and where the pupils can be arranged for ready place-taking.
As for easy questions, there is no subject better than arithmetic. Inexperienced masters waste a vast amount of time by making their questions too hard. I do not mean to say that hard questions
should never be asked. They have their uses at times; but the great mass of questions--99 per cent. of the questions--should be easy. This is true in most subjects, for the main duty of the teacher
is to get the minds of his pupils to deal readily with the notions already stored; but it is most of all true in arithmetic, for the art of arithmetic consists chiefly in performing, accurately and
readily, different series of operations, each operation in itself being of the simplest and easiest kind. If a difficult arithmetical problem is set before us, we may not know what to do, so we
cannot get on at all; but, the instant we see what to do, the whole thing resolves itself into a series of small operations, each one of which we could have performed when we were ten years old. Thus
we see that, with reference to the art, the main difference between the skilled arithmetician and the ordinary schoolboy is that the one can do readily and accurately what the other can do indeed,
but only with effort and with frequent mistakes.
The young, then, must be trained to perform these simple operations quite accurately and quite easily. And to get this accuracy and care, an immense amount of brisk practice is needed. The mind works
by means of established sequences or trains, and these trains, whether natural or conventional, are fixed by repetition. The mind goes easily along the accustomed path. If I were to ask you what
letter came after L, you would have not the smallest effort to make in saying M; but suppose I ask what letter comes before L, probably many of you would not be able to answer without running along
the sequence H, I, J, K, L. Now, in arithmetic all our power depends on the perfect ease with which we run along certain short trains, and our children as a rule fail because we do not practise them
in these trains till effort is no longer needed.
We are to children what the professional gymnast is to the ordinary man. The gymnast performs what he considers a very simple and easy feat, explains how it is done, and calls on the untrained man to
imitate him; but the untrained muscles cannot do it. At length, by repeated efforts, the pupil after a fashion succeeds; but his muscles are tired, and he cannot then and there do it again. The
teacher must have patience. He must remember how many years he has been practising, and must not suppose that what is now so easy to him is easy for the beginner. No doubt everybody will readily
assent and say, "Oh, yes, we know teaching requires a great deal of patience;" but I am convinced that only those who have tried to teach young children properly have anything like an adequate
conception of the amount of patience required. In practise the gymnast is so apt to throw a somersault and then scold his pupils for not following him. I have been a schoolmaster most of my adult
like, and I have been intimately associated with all kings of schoolmasters. This has led me to form a very high estimate of their honesty, of their devotion to their work, and of their interest in
their pupils; but, as a rule, they do not seem even to themselves very successful as teachers. You hear them almost universally grumble at the stupidity of their pupils. This, I own, proves to me the
existence of stupidity somewhere. Is this stupidity shown in the construction of the mind of our average pupil, or in the way in which that mind is dealt with? Suppose we came across a music-master,
say a teacher of the piano, who bemoaned the stupidity and clumsiness of his pupils; suppose he complained to us, "I don't know how it is, but it's very rarely indeed that I get a pupil to play the
piano. They are always stumbling and bungling, and in spite of all my showing how, they never play right. And then they have no memories. They never know for certain what time they are playing in,
and constantly take the lines in the bass for the lines in the treble." We should probably say in our minds, though we might be too polite, as we often are, to tell the truth: "My good sir, it is
quite possible for a teacher to stumble and bungle as well as a pupil; and if you tell me that almost all your pupils drive you crazy by their stumbling and bungling, I think I may tell you with some
confidence that you don't understand your business. You should exercise them thoroughly in things they can play without stumbling; and, as they acquire power, you should employ this power on tasks of
gradually increasing difficulty."
Now we often hear the loud complaints of the stumbling and bungling of our children in arithmetic. This raises at least, a presumption that we don't know our business. Let us reckon the school-time
of a child as going on 40 weeks in the year, and during those weeks, for 5 days in the week. If the instruction of the child begins at 5 years old, and he is practised each of the 5 days for 10
minutes in arithmetic, this will give 33 hours in the first year. In the next year we will allow 15 minutes a day, and from that time 30 minutes a day. At this rate the child, on reaching the ago of
10, would have practised arithmetic 380 hours. Now a good deal may be done in 380 hours, or even in 300. But as things go, much of this time will have been spent, not in establishing the necessary
trains and practising the use of them, but in inventing and applying devices for doing without them. I have know a child attain to great accuracy, and (considering the method) remarkable speed in
doing addition sums by counting taps with his slate pencil. Subtraction sums are often performed by making a number of marks on the slate, then rubbing out the number to be subtracted and counting
the remainder. All this sort of thing must be entirely swept away. Those who are inclined to take my advice will abolish the slate altogether. Neatness and accuracy of column should be rigorously
insisted on whenever notation is used, and all the work should be shown up; no scribbling figures and rubbing them out again, no scraps of paper that may be at once destroyed. But what I wish to
insist on, over and over again, is numeration before notation; don't allow any written record of calculations till the short trains are established in the mind, and the power of writing down neat
figures in good upright and transverse columns has been established in the hand.
There has been much discussion at what point the children begin to think of numbers as abstractions. The Germans, as we have seen, insist that numbers should be taught sensuously. When this is done
carefully, without haste, the power and habit of abstraction will come in due time; and when it has come the short trains must be established by daily vivâ voce practice carried on very smartly.
[*As a preparations for written arithmetic, the pupil should be practiced in getting the numbers, not by the ear, but by the eye. Such exercises do not belong strictly to the first stage; but, as I
have spoken of vivâ voce arithmetic, I must draw attention to this form of it. What is generally called "mental arithmetic" is properly arithmetic in which the voice and ear are employed, as
distinguished from that with the eye and hand. The mind may be said to take part more or less in every calculation. As a training, then, for written arithmetic, the eye should be employed. Every
pupil should be furnished with a paper or card on which the same set of figures should be written or printed. It will be a good practice to dictate the figures before the vivâ voce begins; then the
boys will benefit by their own neatness or suffer by their want of it.
A 3 8 5
B 2 4 7
C 9 3 6
D 7 5 1
E 3 7 6
F 8 2 7
G 0 1 6
H 3 8 3
I 6 2 8
J 7 5 9
A set of figures such as these will give a good number of questions. First, for addition the master may demand the sum of say D, or H, or F, the digits being added crosswise. Or he may say "A+C,"
"B+A," &c., &c. Or he may say, "The sum of three lines from F upwards," "The sum of three lines from B downwards," &c. For subtraction, he may direct the boys to add the left-hand and middle digit of
the named line and subtract the right-hand digit from that sum, or to find the sum, say, of F and of G and subtract the less from the greater. Or a fixed subtrahend [number to be taken away] may be
agreed upon, say 7, and the sum of any line or lines may be required -7. (The last exercise, by the way, has the drawback that a sharp boy will drop a 7 at first.) For multiplication, is it very
important that the eye should, as it were, by a glance at two numbers, suggest the product. It is an excellent exercise to take three digits and add the third to the product of the first and second.
If the master first requires the digits to be taken from left to right, he can get ten questions by naming the letters; next he can have the digits taken from right to left and get ten more. He can
then get almost any number of questions by having the digits taken upwards or downwards as he may direct; he then gives the digit to start with by naming it, as, say, "E left" or "E right" or "E
middle." For division, the left-hand digit may be taken as divisor, and the two right-hand digits with their topical value may be taken for dividend or a fixed divisor may be taken and any pair of
figures may be taken for dividend. The details may vary to any extent so long as the main object is kept in view, which is to give an immense amount of practise to the ordinary trains. The work, if
done as it should be, very smartly, is tiring, and 15 or 20 minutes (even less at first) will probably give enough of it.
The practice I said is tiring. Of course, the master is the trained gymnast and should not get tired as soon as the pupils. In teaching very young children, I can get on well enough answering from my
own head all the questions I ask; but directly my boys show any skill (and I have by such practices as I mentioned made some boys very skilful and ready with their trains), directly my boys get
pretty sharp I prepare my own paper with answers in red ink, or I ask the questions from such a book as Henry Hopkins' Teacher's Manual. So fortified, I face my class prepared for rapid place-taking.
To the device (the purely English device, as far as I know) of place-taking, I owe many pleasant hours of my life in the schoolroom, hours in which we none of us ever got drowsy. I must not linger on
the topic, but in passing, I cannot help saying that the ordinary method of place-taking is a very clumsy one, for it gives every advantage to the cleverest boys and enables them to distance the rest
without exertion. A better plan than this is to number up every 10 minutes and then give each boy not the number of his place, but 2 if he is up, 1 in his place, and 0 if he is down; i.e., reckoning
from his place when the class numbered up before. In this method, by the way, the top boy is always up and the bottom boy down. But a still better plan is to mark by the sum of two place-takings, and
after recording the numbers of the first place-taking, start with the order reversed. After a brisk fire of questions, this is the fairest method possible. Of course the question must always be
asked, and sufficient pause allowed for the work, before the master names a boy to answer, and the question must then be passed as rapidly as can be.
Part of the vivâ voce lesson should be spent in securing accurate knowledge of the meaning of the words used in arithmetic. Incredible as some may think it is, we often find boys who can do sums in
G.C.F. or L.C.M. [greatest common factor; least common multiple], and yet don't know what a factor or a multiple is. Further, the boys should be exercised in analysing the numbers greater than 100 so
as to recognize the prime numbers, or to resolve a number not prime into its prime factors. It will also be convenient to teach powers and indices very early.]
[In the expression 5^4, 5 is the base, ^4 is the index (plural is indices) or exponent, and the entire expression is called a power.]
As for the science of arithmetic, it must be taught very gradually. If numeration based on collections of tens is once thoroughly understood, and if every fresh operation is first of all made out
vivâ voce with a number of examples dealing only with low numbers, the difficulties of the science will seldom be found insuperable and rules may be dispensed with. We are perhaps familiar with the
questions, "Please, sir, what rule am I to do it by?" "Please, sir, where am I to put the decimal point?" I am sorry to say anything that may sound harsh, but if you want to know what I think, I must
express a strong opinion that any pupil capable of asking these questions has been badly taught.
My lecture goes on the assumption that our failures in this subject are mostly owing to rotten foundations. I have, therefore, endeavoured to show how sound foundations may be laid. I have not time
this evening to talk about the building that should be erected when the foundations are well laid and have had time to settle; and, although as teachers you are professors of patience, I am aware
that I have put you to a severe test already. I will conclude, then, with a remark of very general application.
We may divide all conscientious teachers into two great classes: first, those whose attention is taken up mainly with what they themselves think and do, and second, those whose attention is taken up
mainly with what is thought and done by their pupils. Perhaps, I shall make myself best understood by telling you of a specimen of the first class of whom I learnt, or at least tried to learn, when I
was young. He was a man of some distinction, and was most conscientious and painstaking. He had a thorough mastery of the subjects, and he was ready to give any amount of time to me. But,
unfortunately, he was like the gymnast who should suppose that he was teaching when he performed his feats before his pupils. I witnessed a feat, performed no doubt slowly, and with every
explanation. I tried to do it after him, but stuck. The gymnast, engrossed by his own performances, never watched mine, and when I announced that I could not get on, all he did was to perform the
feat over again. He was a very good man, and I wish to speak of him kindly and with all respect, but a good teacher he was not. The second class of teachers, a class to which I hope everyone present
belongs, takes pains like the others to get their own notions clear; but when they come to teach, their attention is concentrated on the minds of their pupils. They carefully watch every operation of
those minds. Whatever the mental feat may be, they have analysed the necessary movements, and they see their pupils perform them in the right way and in the right order. The first movement may,
perhaps, be got through indeed, but clumsily and with difficulty. The teacher observes this, and practises the pupil in such movements till the first step can be taken neatly and easily; he does the
same with the following movements, watching the mind of the pupil throughout. To such a teacher, the pupil's mistakes and misconceptions cease to be irritating. As Professor Meiklejohn has well said,
the teacher's business is not simply to correct mistakes, but to trace them to their source; and the teacher should no more be vexed by a blunder than a doctor by a furred tongue. He should be
interested by it, and by means of it should endeavor to diagnose the complaint.
What a vast subject is teaching, boundless as its own subject, which is the human mind! The treatment of the mind, one would think, should not be undertaken by the prentice hand any more than the
treatment of the body; but a man has to go through a course of five years' training before he is thought competent to deal with our bodies, and from the teacher less preparation is required than even
from H.M. Inspectors of Schools, for a Minister of Education has informed us that these have to be in training for "at least a fortnight." But, though we move slowly, teachers are by degrees getting
to see that, whether they are required to learn it or not, there really is something for them to learn. When their minds are in this receptive condition, they may learn from very different people. A
wise doctor will profit by what he hears from a professor lecturing on embryology and from an old woman who can cure warts. So, too, a wise teacher will listen to the results arrived at by the
psychologist, and to the last tip that is suggested for collecting pens. Nothing will be too great or too small for his consideration. All theory concerns him if it is true, all practise if it is
right. There is a sense in which the familiar words "While there is life there is hope" should come home to us all. So long as we are alive, i.e., so long as we have not become mere machines for
performing certain teaching or other functions, so long there is hope. Hope shews the life of the mind, as breath shows the life of the body. The living teacher has hope, the hope of doing better;
his great desire is to know the best that has been thought and done; his great object is to bring his own thought and practice nearer to it.
The paragraphs in small type were omitted in giving the lecture.
Proofread by LNL, Apr. 2021 | {"url":"https://www.amblesideonline.org/PR/PR07p186FirstStageinArithmetic.shtml","timestamp":"2024-11-05T07:23:02Z","content_type":"text/html","content_length":"28484","record_id":"<urn:uuid:8ba9408d-d074-4cdc-a449-b096db7678d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00715.warc.gz"} |
My fist scientific paper - Research management course
My fist scientific paper
From Research management course
News and announcements
The Art of Scientific Research
See you this Saturday at 14:40 m1p.org/go_zoom
The goal is to prepare and select the research topic of your dream. We must be sure that the problem statement and project planning lead you to successful delivery.
m1p Course progress
This course produces student research papers. It gathers research teams. Each team joins a student, a consultant, and an expert. The student is a project driver who wants to plunge into scientific
research activities. The graduate student consultant conducts the research and helps the student. The expert, a professor, states the problem and enlightens the road to the goal. The projects start
in February and end in May, according to the schedule.
Mathematical forecasting, 2024
This course delivers methods of model selection in machine learning and forecasting. The modeling data are videos, audio, encephalograms, fMRIs, and other measurements in natural science. The models
are linear, tensor, deep neural networks, and neural ODEs. The practical examples are brain-computer interfaces, weather forecasting, and various spatial-time series forecasting. The lab works are
organized as paper-with-code reports. See the course page
Functional Data Analysis, 2024
The statistical analysis of spatial time series requires additional methods of data analysis. First, we suppose time is continuous, put to the state space changes \(\frac{d\mathbf{x}}{dt}\) and use
neural ordinary and stochastic differential equations. Second, we analyze a multivariate and multidimensional time series and use the tensor representation and tensor analysis. Third, since the time
series have significant cross-correlation we model them in the Riemannian space. Fourth, medical time series are periodic, the base model is the pendulum model, \(\frac{d^2x}{dt^2}=-c\sin{x}\). We
use physics-informed neural networks to approximate data. Fifth, the practical experiments involve multiple data sources. We use canonical correlation analysis with latent state space. This space
aligns the source and target spaces and generates data in source and target manifolds. See the course page. | {"url":"https://m1p.org/index.php?title=My_fist_scientific_paper&oldid=1806","timestamp":"2024-11-12T09:25:17Z","content_type":"text/html","content_length":"21661","record_id":"<urn:uuid:ec29077a-57ff-4857-87fc-ea1f5767ec33>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00899.warc.gz"} |
Active Analytics
Actuarial GLM Pricing using Metropolis MCMC
This week I will present some material on using the Metropolis-Hastings algorithm to carry out GLM-like actuarial pricing in R. We use the motorins data set from the faraway package and compare the
output with using a standard glm() function in R.
Monte Carlo methods are pretty popular nowadays particularly for analytically intractable problems however we will present a simple model. The paper by Chib and Greenberg outlines the
Metropolis-Hastings method. In this as assume a poisson likelihood and a normal prior for the beta coefficients N(0, 10), we use a multivariate normal distribution as our proposal distribution and we
sample from this using the mvrnorm() function from the MASS package.
An initial variance for the proposal distribution is given by:
mPopVariance <- as.numeric(var(log(dObs/dExposure + .5)))*solve(t(dDesign) %*% dDesign)
There are a few things to note here. Firstly, we have to adjust the observed counts dObs with the exposure dExposure, and the design matrix is dDesign. Note the design matrix can be obtained using
the model.matrix() function. This is appropriate as a “first pass” but as we will later see, variance adaptation methods are required to properly explore the posterior distributions.
Note also that when calculating the density for the poisson distribution, you need to adjust for the exposure:
dpois(dObs, dExposure*exp(dDesign %*% betaProposal))
I will be running the equivalent of this model:
glm(Claims ~ Kilometres + Zone + Bonus + Make, data = motorins, family = poisson(link = "log"), offset = log(motorins$Insured))
The Bonus variable needs to be converted to a factor before the analysis is carried out.
In this analysis we allow for 4000 iterations for burn-in and collect a further 10,000 iterations. The raw outputs show that the burn-in used is appropriate.
A closer look at the chain output shows that the posterior distribution is not being properly explored. At this point our post-burning acceptance probability is 0.055, much lower than the rule of
thumb for multivariate distributions 0.234 {Roberts et al}.
We can easily create an adaptive scheme for the sampling variance that keeps the acceptance probability hovering around where it should be. Doing this gives the full burn-in chart below:
And the post burn in chart below.
Now we bump the number of simulations to 100,000. The full and post burn in charts are given below. The acceptance probability was 0.257 after setting a target of 0.25 on the variance adapting
Below we show the histogram and density line (grey) for the post burn-in chain and compare this with the output from the glm() function summary for the coefficient distribution (red).
MCMC methods applied to generalized regression model schemes give us the opportunity to analyse analytically intractable problems, the simple one tackled here shows that there can be many challenges
while using this technique. The proposal variance algorithm was applied post burn-in. It is possible to obtain faster convergence by investigating different burn-in variance modification schemes. If
we are interested in reducing the dependence withing samples it may be useful to considering “thinning” obtained sample and use the acf() function as a guide. The great disadvantage here of course is
that > 70% of the proposed samples are thrown away post burn-in, which seems wasteful from a resource point of view. Properly formulated Gibbs sampling schemes can over come this. I hope that this
has been useful.
• S. Chib and E. Greenberg, Understanding the Metropolis-Hastings Algorithm, The American Statistician, 1995, Vol. 49, No. 4, p 327-335
• G. O. Roberts, A. Gelman, W. R. Gilks, Weak Convergence and Optimal Scaling of Random Walk Metropolis Algorithms, The Annals of Applied Probability, Vol. 7, No. 1 (Feb., 1997), pp. 110-120. | {"url":"https://active-analytics.com/blog/actuarialpricingusingmetropolis-hastingsmcmc/","timestamp":"2024-11-10T16:04:30Z","content_type":"text/html","content_length":"11913","record_id":"<urn:uuid:66aaa370-ebcf-4b61-9627-db2dddbeaa66>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00867.warc.gz"} |
How do you simplify sqrt(6x^4) times sqrt(12x^3) ? | HIX Tutor
How do you simplify #sqrt(6x^4) times sqrt(12x^3) #?
Answer 1
#=# #sqrt(6)xxx^2xxsqrt(6)xxsqrt(2)xxsqrt(x^2xxx)#
#=# #6xxx^2xxsqrt(2)xxxsqrt(x)#
#=# #6x^3sqrt(2x)#
Check this exhaustively. All care taken, but no responsibility assumed!
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer 2
To simplify sqrt(6x^4) times sqrt(12x^3), we can combine the square roots and simplify the expression.
First, we can simplify the numbers under the square roots. The square root of 6 can be simplified to 2sqrt(3), and the square root of 12 can be simplified to 2sqrt(3).
Next, we can simplify the variables. The x^4 and x^3 can be combined by adding their exponents, resulting in x^7.
Combining the simplified numbers and variables, we have 2sqrt(3) times 2sqrt(3) times x^7.
Multiplying the numbers, we get 4 times 3, which is 12.
Therefore, the simplified expression is 12x^7.
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer 3
To simplify sqrt(6x^4) times sqrt(12x^3), you can use the properties of square roots. First, you multiply the terms inside the square roots together and then simplify if possible.
sqrt(6x^4) times sqrt(12x^3) = sqrt(6x^4 * 12x^3)
= sqrt(72x^7)
= sqrt(36x^6 * 2x)
= sqrt(36x^6) * sqrt(2x)
= 6x^3 * sqrt(2x)
So, the simplified expression is 6x^3 * sqrt(2x).
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
Not the question you need?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-simplify-sqrt-6x-4-times-sqrt-12x-3-8f9af9b081","timestamp":"2024-11-03T10:22:50Z","content_type":"text/html","content_length":"573991","record_id":"<urn:uuid:755df97c-be45-46ea-a44b-9ca8af664b14>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00117.warc.gz"} |
On Markov Processes Charecterised Via Martingale Problem.
Date of Submission
Institute Name (Publisher)
Indian Statistical Institute
Document Type
Doctoral Thesis
Degree Name
Doctor of Philosophy
Subject Name
Computer Science
Theoretical Statistics and Mathematics Unit (TSMU-Delhi)
Karandikar, Rajeeva L. (TSMU-Delhi; ISI)
Abstract (Summary of the Work)
Martingale approach to the study of finite dimensional diffusions was initiated by Stroock-Varadhan, who coined the term martingale problem. Their success led to a similar approach being used to
study Markov processes occuring in other areas such as infinite particle systems, branching processes, genetic models, density dependent population processes, random evolutions etc.Suppose X is a
Markov process corresponding to a semigroup (T)e20 with generator L. Then all the information about X is contained in L. We also have thatMf(t) := f(X(t)) – ∫t0 Lf(X(s))dsis a martingale for
every f ∈ D(L). i.e. X is a solution to the martingale problem for L. Now instead of the generator L, if we start with an operator A, such that there exists a unique solution to the martingale
problem for A, then under some further conditions the solution is a Markov process corresponding to a semigroup which is given by a transition probability function. Hence the operator A determines
the semigroup (Tt)t≥ 0. A is then a restriction of the generator of (T)20 to D(A).
ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843752
Control Number
Recommended Citation
Bhatt, Abhay G. Dr., "On Markov Processes Charecterised Via Martingale Problem." (1993). Doctoral Theses. 393. | {"url":"https://digitalcommons.isical.ac.in/doctoral-theses/393/","timestamp":"2024-11-04T07:10:55Z","content_type":"text/html","content_length":"36369","record_id":"<urn:uuid:6273edd4-3785-4dd6-b01f-0ff915e5080c>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00713.warc.gz"} |
What Is 36 Celsius In Fahrenheit? | WhyDo
What Is 36 Celsius In Fahrenheit?
So, what is 36 celsius in fahrenheit? We call it temperature when we measure the degree of hotness or coldness on any pre-determined scale. It indicates the flow of heat energy of the surface
temperature of an object, which means how fast and how much heat travels from a hotter thing to a colder object.
The US measurement technique for temperature is degrees Fahrenheit while other areas use the metric system, known as degrees Celsius. Both the skills are widely popular temperature units and differ
from each other. To calculate the conversion between Celsius to Fahrenheit, you need mathematical formula. But before that, how did the temperature scale emerge?
What Are The Celsius And Fahrenheit Scales?
What Are The Celsius And Fahrenheit Scales?
The Celsius scale is an invention by Swedish astronomer Anders Celsius dating back to 1742. The 100 degrees interval between specific points in this measurement unit generates the name Celsius.
The freezing point of water in this scale is zero degrees Celsius while the boiling point is 100 degrees Celsius. There is the presence of an absolute zero. Comparing temperatures in Celsius with
other temperature units, such as Kelvin, is more manageable.
The Prussian scientist Daniel Gabriel Fahrenheit proposed using the degree Fahrenheit as a temperature unit in 1724. In this scale, water boils at 212 degrees Fahrenheit, while its freezing point is
32 degrees Fahrenheit.
Daniel Gabriel Fahrenheit proposed that there is no absolute zero in this temperature scale. The interval between the temperature of boiling and the one where water freezes have 180 divisions. This
unit of measurement is much older than degrees Celsius.
How Do We Perform Celsius To Fahrenheit Conversions?
How Do We Perform Celsius To Fahrenheit Conversions?
Temperature conversion of degrees Celsius into Fahrenheit temperature scale is not troublesome. You need to follow the appropriate formula that will help you do the math and calculate the conversion.
This formula is called the C to F formula.
A crucial piece of information here is that the 100 divisions on the Celsius equal 180 divisions in Fahrenheit. Similarly, ten units Celsius will equal 18 units Fahrenheit.
The Celsius to Fahrenheit conversion involves two steps. First, you have to multiply Celsius by 1.8. Then, in the second step, add 32 to the result, which will give you the final answer on the
Fahrenheit temperature scale.
What Is 36 Celsius In Fahrenheit?
What Is 36 Celsius In Fahrenheit?
When you convert the 36 Celsius to Fahrenheit temperature scale, you get the result of 96.8 degrees Fahrenheit. This answer is verifiable through the mathematical formula. Hence, we will be
multiplying 1.8 to 36 degrees Celsius, and adding 32 gives 96.8 degrees Fahrenheit as the product.
Is 36 Celsius Cold Or Hot?
The answer to this question is simple. 36 Celsius is the optimum temperature for the human body. It is neither too hot nor too cold.
This heat energy is adequate for the body to maintain its homeostasis.
What Is 37 Celsius Mean In Fahrenheit?
Following the formula mentioned above, 37 Celsius converts to 98.6 Fahrenheit. Multiply 37 with 1.8 and add 32. The answer will be 98.6 Fahrenheit.
Final Words
Converting Celsius To Fahrenheit can be manageable. It takes much less time if one knows the appropriate formula for the calculation; even if you don’t know, there is nothing to worry about because
multiple online calculator websites can help you with the degree Celsius to Fahrenheit conversion. Some converter websites, such as Kelvin or Rankine, may also work with other temperature units. | {"url":"https://why.do/what-is-36-celsius-in-fahrenheit/","timestamp":"2024-11-08T01:40:42Z","content_type":"text/html","content_length":"47584","record_id":"<urn:uuid:25e71823-d331-41de-86a4-4636780932c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00526.warc.gz"} |
How do you differentiate y=xcosy^2-xy? | HIX Tutor
How do you differentiate #y=xcosy^2-xy#?
Answer 1
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\cos {y}^{2} - y}{1 + 2 x y \sin {y}^{2} + x}$
We can differentiate each component and then put it together
The next one is a chain differentiation and differentiation of a product
#(xcosy^2)'=1*cosy^2 +x(-siny^2)2ydy/dx# #= cosy^2-(2xysiny^2)dy/dx#
The next one is #(xy)'=1*y+xdy/dx#
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer 2
To differentiate ( y = x \cos(y^2) - xy ), you would use the product rule and chain rule. The derivative with respect to ( x ) is:
[ \frac{dy}{dx} = \cos(y^2) - 2xy \sin(y^2) - y ]
This comes from differentiating ( x \cos(y^2) ) and ( -xy ) separately.
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
Not the question you need?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-differentiate-y-xcosy-2-xy-8f9af9ea1b","timestamp":"2024-11-05T19:24:10Z","content_type":"text/html","content_length":"569023","record_id":"<urn:uuid:a7b308b5-acfe-4cf7-9d71-c8fe9e017b61>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00547.warc.gz"} |
CGWAVE Math Details: Numerical Solution
Equation (5) is generally solved using the boundary element method, the finite-difference method, or the finite element method. In general, finite-difference discretizations are not well-suited to
represent the complex domain shapes described, for example, in Fig. 1. Not only are the boundaries distorted, but the number of uniformly spaced grids may also be excessively large. (Adequate
resolution, typically 10 points per wavelength, demands that the spacing be determined from the smallest wavelength.) Most studies with the finite-difference method have been limited to largely
rectangular domains (e.g. Li 1994a, 1994b; Panchang et al. 1991; Li and Anatasiou 1992). Boundary element models can handle arbitrary shapes and require minimal storage since only the boundaries are
discretized; however, they are limited to subdomains with constant depths only (e.g. Isaacson and Qu 1990; Lee and Raichlen 1972; Lennon et al. 1982). Finite element models, on the other hand, allow
the construction of grids with variable sizes (based on the local wavelength) and give a good reproduction of the boundary shapes. Most finite element models (e.g. Tsay and Liu 1983; Tsay et al.
1989; Kostense et al. 1988; Demirbilek and Panchang 1998; Panchang et al. 2000) have used triangular elements, and modern graphical grid generating software permits efficient and accurate
representation of harbors with complex shapes. The Surface Water Modeling System can be used to conveniently generate as many as 1,000,000 elements of varying size, based on the desired
(user-specified) resolution, and to specify the desired reflection coefficients on various segments of the closed boundary. The solution of (1) by the finite element method is described in detail by
Mei (1983) and by Demirbilek and Panchang (1998) when different types of open boundary conditions are used.
Whether one uses finite differences or finite elements for discretization, the numerical treatment of (1) with appropriately chosen boundary conditions leads to system of linear equations:
${\displaystyle [A][\phi ]=[B]\ }$ ^(28)
where [φ] represents the vector of all the unknown potentials. For solving (5), a similar system results as long as W is prespecified. The matrix [A] is usually extremely large. In earlier models
(e.g. Tsay and Liu 1983; Tsay et al. 1989; Chen, 1990; Chen and Houston, 1987) the solution of (28) was accomplished by Gaussian Elimination, which requires enormous memory and is prohibitive when
the number of wavelengths in the domain is large (i.e. short waves or a large domain). Pos and Kilner (1987) were able to alleviate this difficulty somewhat by using the frontal solution method of
Irons (1970).
In recent years, the solution of (28) has been obtained with minimal storage requirements for [A]. This is due to the development by Panchang et al. (1991) and Li (1994a) of iterative techniques
especially suited for (1). These techniques, based on the conjugate gradient method, guarantee convergence and have been found to be extremely robust in a wide variety of applications involving both
finite differences and finite elements for several kinds of boundary conditions. For a review of other methods, see Panchang and Demirbilek (2000). Options based on the work of both papers, viz.
Panchang et al. (1991) and Li (1994a), are available in CGWAVE. It is found that the latter often leads to faster convergence, but in an oscillating fashion. The former leads to a monotonically
decreasing error which can be more reassuring while the iterations are in progress.
Related Topics | {"url":"https://www.xmswiki.com/wiki/SMS:CGWAVE_Math_Details:_Numerical_Solution","timestamp":"2024-11-05T02:46:09Z","content_type":"text/html","content_length":"26080","record_id":"<urn:uuid:022d03cc-a9fd-45d1-8dd9-a974cd80912a>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00075.warc.gz"} |
Notions of work and kinetic energy: The work-energy theorem
We already discussed in the previous article (link here) that there is some relation between work done and energy. Now we will see the theorem that relates them. According to this theorem, the net
work done on a body is equal to the change in kinetic energy of the body. This is known as Work-Energy Theorem. It can be represented as:
Harmony in Motion: Demystifying Notions of Work and Kinetic Energy with the Work-Energy Theorem.
Kf – Ki = W
Where Kf = Final kinetic energy
Ki = Initial kinetic energy
W = net work done
Work (W): In physics, work is done when a force is applied to an object, causing it to move a certain distance in the direction of the force.
Kinetic Energy (KE): This is the energy associated with the motion of an object. It depends on the object's mass and its velocity.
Mathematical Expression of Work:
The work done (W) is mathematically expressed as W=F⋅d⋅cos(θ), where F is the force applied, d is the displacement, and θ is the angle between the force and displacement vectors.
Kinetic Energy Formula:
Kinetic energy ( KE) is given by KE=12mv2 m is the mass of the object, and v is its velocity.
Statement of the Work-Energy Theorem:
The Work-Energy Theorem establishes a direct relationship between work done on an object and the change in its kinetic energy.
Mathematically, it is expressed as W=ΔKE, signifying that the net work done is equal to the change in kinetic energy.
Interpreting the Theorem:
Positive Work: When the net work is positive, it increases the kinetic energy of the object, resulting in acceleration.
Negative Work: Conversely, negative work decreases the kinetic energy, causing deceleration or bringing the object to a stop.
Applications in Real-world Scenarios:
Understanding the Work-Energy Theorem is crucial for analysing motion in various scenarios, including the trajectory of projectiles, the performance of vehicles, and the operation of machines.
Conservation of Energy:
The Work-Energy Theorem aligns with the principle of conservation of energy.
In a closed system with no external forces, the total mechanical energy (kinetic + potential) remains constant.
According to Work energy theorem,
Work done by all the forces = Change in Kinetic Energy
Wg + WN + Wf =Kf – Ki
Where Wg = work done by gravity
WN = work done by a normal force
Wf = work done by friction
Kf = final kinetic energy
Ki = initial kinetic energy
Work done by a constant force
A constant force will produce constant acceleration. Let the acceleration be ‘a’.
From the equation of motion,
v2 = u2 + 2as
2as = v2 – u2
Multiplying both sides with mass ‘m’ (ma).s = (mv2–mu2)2 F.s = (mv2–mu2)2 Comparing the above equation, we get, Work done by force (F) = F.s Where ‘s’ is the displacement of the body.
Work done by Non-Uniform Force
Now the equation,
W = F.ds
This is only valid when force remains constant throughout the displacement. Suppose we have a force represented below,
For these kinds of forces, we can assume that force remains constant for a very small displacement and then integrate that from the initial position to the final position.
W = ∫xf.xi f(x)dx
This is work done by a variable force. A graphical approach to this would be finding the area between F(x) and x from xi to xf.
CBSE Class 11th Downloadable Resources:
│1. CBSE Class 11th Topic Wise Summary │View Page / Download│
│2. CBSE Class 11th NCERT Books │View Page / Download│
│3. CBSE Class 11th NCERT Solutions │View Page / Download│
│4. CBSE Class 11th Exemplar │View Page / Download│
│5. CBSE Class 11th Previous Year Papers │View Page / Download│
│6. CBSE Class 11th Sample Papers │View Page / Download│
│7. CBSE Class 11th Question Bank │View Page / Download│
│8. CBSE Class 11th Topic Wise Revision Notes │View Page / Download│
│9. CBSE Class 11th Last Minutes Preparation Resources │View Page / Download│
│10. CBSE Class 11th Best Reference Books │View Page / Download│
│11. CBSE Class 11th Formula Booklet │View Page / Download│
Being in CBSE class 11th and considering the board examinations you must be needing resources to excel in your examinations. At TestprepKart we take great pride in providing CBSE class 11th all study
resources in downloadable form for you to keep you going.
Below is the list of all CBSE class 11th Downloads available on TestprepKart for both Indian and NRI students preparing for CBSE class 11th in UAE, Oman, Qatar, Kuwait & Bahrain.
Q1. What is the Work-Energy Theorem?
Answer: The Work-Energy Theorem states that the work done on an object is equal to the change in its kinetic energy. Mathematically, it is expressed as W=ΔKE, where W is the work done, and ΔKE is the
change in kinetic energy.
Q2. How is Work Defined in Physics?
Answer: In physics, work is defined as the product of the force applied to an object and the distance over which the force is applied. Mathematically, it is represented as W=F⋅d⋅cos(θ), where F is
the force, d is the displacement, and θ is the angle between the force and displacement vectors.
Q3. What is Kinetic Energy?
Answer: Kinetic energy is the energy an object possesses due to its motion. The kinetic energy (KE) of an object is given by the formula 2KE=21mv2, where m is the mass of the object, and v is its
Q4. How Does the Work-Energy Theorem Relate to Kinetic Energy?
Answer: The Work-Energy Theorem states that the work done on an object is equal to the change in its kinetic energy. Essentially, the energy transferred to or from an object as a result of work
manifests as a change in its kinetic energy.
Q5. What Happens if No Work is Done on an Object?
Answer: If no work is done on an object, according to the Work-Energy Theorem, there is no change in its kinetic energy. The object's velocity remains constant, and there is no acceleration.
│Class 11th CBSE Physics Chapters │
│Chapter1: UNITS AND MEASUREMENTS │
│Chapter2: MOTION IN A STRAIGHT LINE │
│Chapter3: MOTION IN A PLANE │
│Chapter4: LAWS OF MOTION │
│Chapter5: WORK, ENERGY AND POWER │
│> Introduction │
│> Work │
│> Kinetic energy │
│> Work done by a variable force │
│> The concept of potential energy │
│> The conservation of mechanical energy │
│> The potential energy of a spring │
│> Power │
│> Collisions │
│Chapter6: SYSTEM OF PARTICLES AND ROTATIONAL MOTION │
│Chapter7: GRAVITATION │
│Chapter8: MECHANICAL PROPERTIES OF SOLIDS │
│Chapter9: MECHANICAL PROPERTIES OF FLUIDS │
│Chapter10: THERMAL PROPERTIES OF MATTER │
│Chapter12: KINETIC THEORY │
│Chapter13: OSCILLATIONS │
│Chapter14: WAVES │
│Class 11th CBSE Chemistry Chapters │
│Chapter1: SOME BASIC CONCEPTS OF CHEMISTRY │
│Chapter2: STRUCTURE OF ATOMS │
│Chapter3: CLASSIFICATION OF ELEMENTS AND PERIODICITY IN PROPERTIES│
│Chapter4: CHEMICAL BONDING AND MOLECULAR STRUCTURE │
│Chapter5: THERMODYNAMICS │
│Chapter6: EQUILIBRIUM │
│Chapter7: REDOX REACTIONS │
│Chapter8: ORGANIC CHEMISTRY - SOME BASIC PRINCIPLE AND TECHNIQUES │
│Chapter9: Hydrocarbons HYDROCARBONS │
│Class 11th CBSE Mathematics chapter │
│Chapter1: SETS │
│Chapter2: RELATIONS AND FUNCTIONS │
│Chapter3: TRIGONOMETRIC FUNCTIONS │
│Chapter4: COMPLEX NUMBER AND QUADRATIC EQUATIONS │
│Chapter5: LINEAR INEQUALITIES │
│Chapter6: PERMUTATIONS AND COMBINATIONS │
│Chapter7: BINOMIAL THEOREM │
│Chapter8: SEQUENCES AND SERIES │
│Chapter9: STRAIGHT LINES │
│Chapter10: CONIC SECTIONS │
│Chapter11: INTRODUCTION TO THREE-DIMENSIONAL GEOMETRY │
│Chapter12: LIMITS AND DERIVATIVES │
│Chapter13: STATISTICS │
│Chapter14: PROBABILITY │
│Class 8 Link soon │
│Class 9 Link soon │
│Class 10 Link soon │
│Class 12 Link soon │ | {"url":"https://www.testprepkart.com/jee/blog-single.php?id=2719/cbse-class-11th-notions-of-work-and-kinetic-energy:-the-work-energy-theorem-details---preparations-downloads","timestamp":"2024-11-12T00:38:23Z","content_type":"text/html","content_length":"133303","record_id":"<urn:uuid:5c0463f6-e43c-4b1a-97b6-583d013d16a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00042.warc.gz"} |
High School SMILE Meeting
07 March 2006
• Write-ups for academic year SMILE meetings for the current semester may be found at http://www.iit.edu/~johnsonp/acysmile.html, and write-ups from previous semesters are permanently located at
• Your hang tag for parking at IIT is valid in the parking lot by the Keating Gymnasium on Tuesday afternoons, 4-6:30 pm, until the end of the Spring semester 2006. Be sure to put this hang tag on
your inside rear view mirror when you park here.
• The Spring 2006 meeting of the AAPT [American Association of Physics Teachers] will be held on Saturday, 01 April 2006 at Chicago State University. A focus topic at the meeting will be Using
Physics Education Research to Guide Instruction. Further program details are available at the CSAAPT-06 website: http://www.neiu.edu/~csaapt/
• Earl Zwicker received this information concerning workshops to be given in the near future from Laura Nickerson:
"As part of my duties as a Wright Fellow, I am giving workshops for physics teachers on my 'stuff' that I've been working on, and I am giving a set in Chicago. The place is IMSA, and the
dates are April 5th and June 23rd. I know April 5th is a Wednesday, but it was necessary for scheduling. The workshops are absolutely free (and you even get fed!), so all the school district
has to do is pay for a sub for the day. Participants are expected to attend both sessions, but let me know if there are special cases and we'll see what we can do. The title is "Relatively
Physics", and I'll be sharing curricula in relativity, quantum mechanics, and cosmology for use with high school students. No previous background is required, and freshman physics teachers or
physical science teachers are very welcome! Participants will also receive a CD of the workshop materials as well as some demonstrations to take home with them. Did I mention this is all for
FREE? The workshop website is http://www.tufts.edu/as/wright_center/ and you can register here and find out more information."
• Earl also called attention to two articles on the Physical Review Focus webpage.
1. Why the Inner Ear is Snail-shaped: http://focus.aps.org/story/v17/st8. Apparently, to boost sensitivity at low frequencies.
2. Spiked Ice: http://focus.aps.org/story/v17/st7. Fields of spiky ice appear on high-altitude glaciers, and may help preserve them.
• Bryan Noon, a physics teacher at Argo Community High School [7329 W 63rd Street, Summit IL 60501], informed us of a workshop on the Physics First curriculum, to be held there on Saturday, 18
March 2006, 9 am to 1 pm. You may obtain additional information and register for the conference by contacting Jill Alexander at (708) 728-3200 ext 360 or jalexander@argo217.k12.il.us.
• For details on future meetings of the Illinois State Physics Project [ISPP] see their website: http://ispp.info. For details concerning Physics Northwest meetings see their website: https://
Bud Schultz (Aurora Middle School) Gonzo Gizmos
Bud shared a book he bought at American Science and Surplus -- Gonzo Gizmos: Projects & Devices to Channel Your Inner Geek by Simon Field (http://www.kk.org/cooltools/archives/000667.php). This book
is full of great ideas, explanations, and projects -- it contains a lot of interesting activities involving electricity. Thanks for the Info, Bud!
Don Kanner (Lane Tech HS, physics) The Sound of Physics
Don suggested modifying the simple discussion of open ended and closed ended organ pipes, as made at the last meeting by Larry Alofs. The equations l = 4L (for a pipe with one end closed) and l = 2 L
(for a pipe with both ends open) gives the fundamental frequency only for pipes under certain geometrical restrictions. In fact, it is an oversimplified way of describing the vibrations. It is not
just the length of the pipe that determines the pitch; we tested this for various pipes, across which we blew air to try to make standing waves. Another exercise involves a Florence flask and an
Erlenmeyer flask of equal heights and volumes. They produce sounds of rather different pitch when air is blown across them. The size of the neck and opening of the vessel is also important in
determining what tone is made in this way. For additional information see The Resonance of Common bottles and Jugs by Don Kanner: http://www.iit.edu/~smart/kanndon/lessonb.htm. Hermann Helmholtz
actually found out that the ideal shape for a resonating volume is a sphere. For additional discussion see the comments at the 25 February 2003 HS Math-Physics SMILE meeting: http://www.iit.edu/
Sounds good! Thanks, Don.
Fred Schaal (Lane Tech HS, math) Parabolic Points
In an extension of his presentation at the last meeting, Fred used a similar procedure to trace out the points on a parabola using only his (chalk) compass, a meter stick and the blackboard. He chose
a focal point (focus) at random above a horizontal line (directrix). He used the compass to draw a portion of a circular arc with an arbitrary radius, with the center at the focus . Two arcs are then
made with the compass held at the same radius, with their centers on the line. A tangent to these two arcs intersects the first arc at two points, which lie on the parabola. The process is repeated
using the same focal point but different radii, generating points to trace out a parabola. For additional information see the interactive webpage The Parabola by Alex Bogomolny: http://
Neat, Fred!
Earl Zwicker (IIT Physics) Mr Angry is on the left, and Mrs Calm is on the right
Earl had gotten an e-mail from Rudy Keil, including the remarkable image shown here. There are two images of a face, one with a calm look and one with an angry look. The two images seem to switch
depending on whether they are viewed from close in (about 1 foot away) or far away (about 8 feet). It works completely!! But no one knew the reason for this! We will have to look for one!! One way to
investigate it would be to try to find out if there is a consistent distance (for the members for the class) at which the transition occurs. We tried this. Fred tried it and the transition (where the
images looked roughly the same) occurred at a distance of about 6 floor tile widths and the switch was complete at about 10 tiles. Don tried it and got 8 and 10 for the same figures. Walter got 8 and
10; Ed got 6 and 9. Fairly consistent results which did not seem to depend upon whether or not the observer was wearing glasses. For additional discussion see the website http://cvcl.mit.edu/
gallery.htm#hsflsf, from which the following has been excerpted:
"This impressive illusion created by Dr. Aude Oliva and Dr. Philippe G. Schyns, illustrates the ability of the visual system to separate information coming from different spatial frequency
channels. In the right image, high Spatial Frequencies (HSF) represent a woman with a neutral facial expression, mixed with the low spatial frequency (LSF) information from the face of an angry
man. On the left, the face of the angry man is represented in fine details whereas the underlying female face is made of blur only."
Thanks, Earl!
Porter Johnson (IIT, Physics) Sangaku-Followup
Porter continued the discussion of the “Circle Inscribing Sangaku”, which was introduced at the last class by Walter McDonald. This problem is discussed on the Mathworld Website on the web page http:
//mathworld.wolfram.com/CircleInscribing.html. However, that discussion is incomplete, in that it does not prove that the inscribed circle centered at O[3] is tangent to the isosceles triangle ACB.
According to the statement of the problem, the large circle of diameter 1 (unity) is centered at point O, and a smaller circle of diameter r is centered at O[2]. The smallest circle, which is of
radius a and centered at O[3], is tangent to the other two circles, and its center lies on the line O[3]A that is perpendicular to the major diameter XB. The Mathworld website uses the fact that the
right triangles OO[3]A and O[2]O[3]A have a common side, O[3]A, to determine the length y of that side O[2]A, as well as the radius a of the inscribing circle. Their results are
(1 + r) a = r (1 - r) ;
(1 + r) y = r Ö[2 (1 - r)] .
Let the symbol j represent the angle ACD. Because the point C lies on the largest circle, its distance to the center O is 1/2. Furthermore, the right triangle ADC, has these side lengths:
[AD, DC, CA] = Ö(1 - r) / 2 ´ [ Ö(1 - r) , Ö(1 + r) , Ö 2 ] .
Thus, we can show that
sin j = Ö[(1 - r) / 2] .
Because the alternate interior angles O[3]AC and ACD are equal, we can compute the distance from the center O[3 ]of the inscribed circle to the straight line AC:
y sin j = a
Consequently, the inscribed circle, with radius a and centered at O[3], is tangent to the straight line AC. The result is thus established.
Porter then told us about Morley’s Theorem. Start with any triangle and trisect all three angles. Pairs of the trisecting lines from adjacent angles will intersect to make three points inside the
original triangle. Connection of these three points will always produce an equilateral triangle!! Fred then illustrated this by laying out a carefully drawn figure on the board. For more details see
the website http://www.cut-the-knot.org/Curriculum/Geometry/Morley.shtml, which contains an adjustable triangle showing the result. See also http://www.jimloy.com/geometry/morley.htm, which contains
the following comment:
"One of the interesting side results of some of the proofs is that the side of the equilateral triangle is equal to 8R sin(A/3) sin(B/3) sin(C/3), where A, B, and C are the angles of the larger
triangle, and R is the radius of the circumcircle."
Fascinating, Porter.
Lee Slick (Morgan Park HS, retired) Cricket Temperature
Lee described how to estimate the temperature (in degrees Fahrenheit) from the frequency of cricket chirps. (handout by Tom Skilling, Chicago Tribune, February 5, 2006. http://wgntv.trb.com/news/
weather/weblog/wgnweather/archives/ATW020506SUN.jpg ) Count the number of chirps a cricket makes during a 15 second interval, and add 39 to that number. You get a remarkably accurate reading. This
works because crickets are “cold blooded” (poikilothermic), so that their metabolism (and thus frequency of chirping) will increase as the temperature increases. For more details see the website
Oecanthus: Pulse Distribution and Temperature Effects: http://facstaff.unca.edu/tforrest/ASA 98 Seattle/sld005.htm.
Keep on chirping! Thanks, Lee!
Bill Colson (Morgan Park HS, math) Assorted Literature
Bill shared several items with us:
• First he passed around an official Course Planning Assessment Rubric from the CPS. Bill challenged us to try to figure out use this rather complicated rubric!
• Bill then posed a problem recently discussed in the Straight Dope column by Cecil Adams in the 03 February 2006 issue of the Chicago Reader (see the website http://www.straightdope.com/columns/
060203.html. The question has a jet plane sitting on a conveyor belt that is moving as fast as the plane is going, but in the opposite direction, so that the plane is stationary to an outside
observer. Does the jet plane take off? Apparently so! It is true that a jet plane accelerates because of thrust produced by the engines, whereas there must be friction at the wheels for an
automobile to accelerate. But there was some skepticism! We seem not to have reached a consensus on this problem.
• Bill showed a newspaper article about the Society of Physics Students at Purdue University. They built MOJO, a motorized couch with which to ride around town. For details see http://
• Bill passed around the new Hammacher Schlemmer Catalog: http://hammacher.com. Among other items of potential use ,it touted the Turbo-Thruster Remote Control Car: item CC-72943: http://
www.hammacher.com/publish/72943.asp?promo=QSearch .
• Bill also shared a recent article "Is America Flunking Science?" the cover article in the 13 February 2006 issue of Time Magazine®: http://www.time.com/time/archive/preview/
0,10987,1156575,00.html. Is the USA falling behind other countries in scientific research? It is suggested that we need interesting and knowledgeable science teachers in schools in the earliest
grades in order to produce scientifically interested and literate students.
Thanks, Bill!
Our next SMILE meeting will be on Tuesday March 21, 2006. See you there!
Notes prepared by Ben Stark and Porter Johnson. | {"url":"https://smileprogram.info/weekly/hs030706.html","timestamp":"2024-11-15T00:10:00Z","content_type":"text/html","content_length":"17712","record_id":"<urn:uuid:bbac81c0-a3ce-4f1a-bdef-fd47bd57714a>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00343.warc.gz"} |
Cumulative frequency Example The frequency table shows the examination marks of 80 students. MarkFrequency ppt download
Presentation is loading. Please wait.
To make this website work, we log user data and share it with processors. To use this website, you must agree to our
Privacy Policy
, including cookie policy.
Ads by Google | {"url":"http://slideplayer.com/slide/5926574/","timestamp":"2024-11-08T08:50:04Z","content_type":"text/html","content_length":"139354","record_id":"<urn:uuid:235fcb3b-aa90-43e4-8cd2-47e7daff39bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00202.warc.gz"} |
MEMS Gyroscope Performance Parameters - ArticleTed - News and Articles
MEMS Gyroscope Performance Parameters
Tue, Dec 5, 23, 10:17, 12 Months ago
The basic info of MEMS sensor
To judge whether the performance of a MEMS gyroscope is excellent, it needs to be judged by its parameters, such as measuring range, bias instability, angle random walk, resolution, etc. These
parameters are important indicators for judging the performance of MEMS gyroscope, and also determine the application environment of the gyroscope.
Resolution: It refers to the minimum angular velocity that the gyroscope can detect. This parameter and the zero angular velocity output are actually determined by the white noise of the gyroscope.
Measuring range: Usually expressed by the maximum value of the input forward and reverse angular rates. The larger the value, the more sensitive the gyroscope is to angular rate. Within this input
angular rate range, the nonlinearity of the gyroscope scale factor can meet the specified requirements, and the gyroscope’s range can usually be configured. For example, , the measuring ranges of
ER-MG2-50/100 are 50 and 100 respectively, so they are named accordingly.
Bias instability: It is the drift amount of the gyroscope output that changes with time at a stable temperature. Gyroscopes are subject to bias instability, where the gyroscope’s initial zero reading
drifts over time due to the integration of inherent imperfections and noise within the device.
Angular random walk: If you integrate a noisy output signal from a sensor, such as integrating an angular rate signal to determine the angle, the integral will drift in time due to noise. This drift
is called a random walk because the integral appears to take random steps from one sample to the next. The standard deviation of the drift caused by noise can be recovered by multiplying the random
walk by the square root of time.
Bias stability (1σ 10s): This is what we usually call Allen’s variance, which is the most commonly used measurement. The measurement tells you how stable the gyroscope bias is over a specific
specified period of time. In general, the lower the bias stability, the smaller the error when integrating the gyroscope output over time.
Temperature range: The general temperature range of MEMS gyroscope is −45°C to +85°C, which represents the temperature range in which the gyroscope can operate. However, the requirements for the
petroleum logging industry will be higher. Many tools may require the gyroscope to reach 125 degrees or even 175 degrees. Most MEMS gyroscopes can reach a maximum temperature of 85 degrees, but
ER-MG2-022 can reach 125 degrees. This is a MEMS gyroscope specially designed for gyro tools.
Contact me:
Email: info@ericcointernational.com
Whatsapp: 139 9288 4879 | {"url":"https://www.articleted.com/article/705919/239257/MEMS-Gyroscope-Performance-Parameters","timestamp":"2024-11-03T12:14:24Z","content_type":"text/html","content_length":"48910","record_id":"<urn:uuid:d3635a10-c0c2-4ee6-92f8-9e1990af5209>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00572.warc.gz"} |
Keep Science in the Practice of M&V
By Eric Mazzi *
In Issue #10 of the M&V Focus, an article published by John Avina presents an analysis concluding that Cv(RMSE) requirements should be relaxed for Option C regression models. Avina’s analysis of Cv
(RMSE) is introduced and framed using several egregious claims and significant omissions. The effect of accepting these claims, omissions, and conclusions is nothing short of a rejection of basic,
well-proven scientific methods and principles. In this article I describe these egregious claims and omissions. My purpose is to encourage M&V stakeholders, such as facility owners, M&V
practitioners, and financing organizations to maintain the integrity of the practice of M&V by continuing to accept and utilize science-based practices embedded in all M&V protocols.
Before listing the egregious claims and omissions in Avina’s M&V Focus article, it’s important to understand this is only his latest article promoting a thesis to discard the use of proven
statistical tools and scientific uncertainty in the practice of M&V. In 2021, Avina published an article^1 “Why do we calculate uncertainty?” where he concludes about uncertainty that “It is an
arbitrary concept, which really should not be applied at this time.” Then in 2022, he published another article^2 dismissing R^2, Cv(RMSE) and fractional savings uncertainty (FSU) as being
unscientific. Avina wrote that these three parameters “…are not based in reality … inconsistent, unscientific, arbitrary …” He goes on to say “The entire concept of using these indicators to prove
the soundness of a regression model is unscientific…” and “They are invented unscientific concepts that can never be tested against reality.” More than ten times in his article he describes these
three parameters as unscientific or having no basis in reality. He did not stop there, and went further to state “They are agreed upon figments of our collective imagination composed for us by
authorities such as ASHRAE and EVO.” He stated that EVO and ASHRAE, by incorporating these three parameters in their protocols are acting “…akin to the early Church fathers (politicians in practice),
who, in the 4th century Council of Nicaea selected the Christian canon and the gospels that everyone in the Roman Empire had to believe in.”
It is important to highlight that Cv(RMSE) as it is used in M&V is the ratio of the standard error of an OLS regression model divided by the mean of measured energy values. Avina accepts that metered
numbers are scientific, but he argues forcefully that Cv(RMSE) is not scientific. This means he argues that standard errors are unscientific. Moreover, if standard errors are unscientific, then
p-values must be as well because p-values and standard errors are interrelated in statistics. However, the use of standard errors (as well as R^2 and p-values) for statistical models are ubiquitous
in both research and applied sciences such as econometrics, hydrology, epidemiology, biostatistics, modern data science, finance, and countless other scientific disciplines. Avina’s unsubstantiated
dismissal of standard errors as being unscientific is wrong. Similarly, the concept of uncertainty is central to the practice of all sciences, irrespective as to whether statistical tools are used or
not. I summarized the scientific basis of uncertainty and cited several sources in my reply to Avina’s 2022 article^3. Declaring the use of uncertainty as unscientific is an astonishing claim that is
greatly misguided.
In the context of declaring R^2, Cv(RMSE), and uncertainty as being unscientific in his previous articles, Avina makes the following egregious claims and omissions in his M&V Focus article:
1. In his Background material, Avina begins by stating “In the past I have questioned using Cv(RMSE) as a means of deciding whether a linear regression model is acceptable to use or not.” To state
that he only “questioned” the use of Cv(RMSE) is a reframing that grossly misrepresents what he actually wrote, as cited and quoted above. The reality is that he dismissed Cv(RMSE), R^2 and FSU as
unscientific figments of imagination with no basis in real world systems.
2. Next, Avina claimed that I am in agreement with his thesis. He stated that “Recently, Professor Eric Mazzi wrote that many in the statistics community are starting to question the value of R^2 and
Cv(RMSE) as measures to determine whether to use a regression model or not. ‘Statistically significant’ is becoming an outmoded term. So, perhaps I am not alone on this point after all. It appears
others are realizing this as well.” This is not true. Readers are encouraged to actually read my article^3. The article carefully describes the scientific foundations of R^2, Cv(RMSE), and
uncertainty. What I concluded was “Parameters such as R^2 and Cv(RMSE) are essential elements of the process to relate the statistical model to a real-world system. Put another way, using a
statistical model [e.g., Electricity kWh/day = α + β * (CDD/day)] without considering any parameters to assess the model acceptability would be scientifically unsound. As such, I disagree with
Avina’s assertion …” I stated that these parameters are “essential,” based on “sound science,” and I explicitly stated that “I disagree” with Avina’s thesis. It is a false statement to say that we
are in agreement.
3. In the same quote as point #2 above, Avina implied that statisticians were also in agreement with him because some leading statisticians have argued to discontinue the use of the phrase
“statistically significant.” Avina referred to an article published in the American Statistician^4 which I had cited in my article. What Wasserstein and colleagues actually argued is that the
practice of using rigid thresholds for statistical parameters to accept or reject models, such as p < 0.05, should be avoided. For example, they stated that p = 0.051 should not automatically
invalidate a statistical model. They did not argue that p-values are unscientific with no basis in reality. Wasserstein highlighted that using the phrase “statistically significant” has some
drawbacks, and recommended alternative ways of describing the use of statistical models to draw scientific conclusions. The Wasserstein article explicitly described the use of standard errors and
uncertainty as sound science, while Avina has unambiguously declared both of these parameters to be unscientific. In fact, there is an entire section in the Wassertein article titled “Accept
Uncertainty” which readers are encouraged to read. Avina’s implied claim of agreement with statisticians is not based on reality.
4. In his M&V Focus article, Avina claimed that “The general consensus of the experts is that the R^2 value should be ignored…” This is yet another egregious claim. Avina did not identify these
experts or provide any citations. M&V stakeholders should be aware that all commonly used M&V protocols including ASHRAE^5, IPMVP^6, ISO^7, FEMP^8, and DOE^9 all specify the use of R^2. Avina omits
mentioning that all of these protocols specify the use of R^2, and none state to ignore R^2. Avina may be referring to one past article in M&V Focus^10. However, it should be noted that this article
argued for the use of Cv(RMSE) and uncertainty in lieu of R^2 (e.g. “.. the importance of CvRMSE in assessing savings uncertainty…”), while Avina claimed that Cv(RMSE) and uncertainty are
unscientific. There is no evidence of a consensus to ignore R^2. In fact, the opposite is true.
5. The stated purpose of Avina’s article is to examine the case for relaxing the value of Cv(RMSE) used to accept or reject a regression model. However, Avina omitted mentioning that there are
widely-used and proven practices for addressing situations where regression model criteria are not met, such as low values for R^2 or high values of Cv(RMSE)^11. These practices include seeking
additional independent variables, eliminating poor variables, shifting or extending measurement periods, considering different model forms, and collecting additional data. Omitting any mention of
these practices indicates that Avina’s article is biased.
Avina’s erroneous claims and omissions in the introduction of the analysis of Cv(RMSE) represent an inaccurate and biased framing of the utility of Cv(RMSE), as well as an unsupported dismissal of
the validity of statistical parameters and uncertainty in general. It is not surprising that Avina’s analysis concludes there is evidence to relax Cv(RMSE) criteria. He frames Cv(RMSE) as being an
unscientific figment of imagination that is not based on reality, and infers that it is used because it is imposed by EVO and ASHRAE as authorities behaving akin to 4th century religious politicians.
With this framing, what other conclusion could be drawn? His article is a clear case of an analysis conducted to support a pre-determined conclusion.
M&V of energy projects is a critical function with real-world importance, such as climate change mitigation and responsible financing of green projects. Stakeholders such as facility owners, M&V
practitioners, and financing organizations are encouraged to maintain the technical integrity of M&V. Integrity will be preserved by utilizing proven, science-based practices which are used in
countless, practical scientific applications as well as essentially all M&V protocols.
1. Avina J. (2021) “Why Do We Calculate Uncertainty” International Journal of Energy Management, Vol 3, Issue #3.
2. Avina J. (2022) “Statistics and Reality— Addressing the Inherent Flaws of Statistical Methods Used in Measurement and Verification” IJEM Vol 4, Issue #1.
3. Mazzi E. (2022) “Commentary on Article “Statistics and Reality—Addressing the Inherent Flaws of Statistical Methods Used in Measurement and Verification” IJEM Vol 4, Issue #2.
4. Wasserstein et al (2019). “Moving to a World Beyond ‘p < 0.05’ ” The American Statistician, 73:sup1, 1-19.
5. ASHRAE 14-2014 “Measurement of Energy, Demand, and Water Savings” explicitly describes the use of R2.
6. “International Performance Measurement & Verification Protocol Core Concepts” EVO 10000 – 1:2022.
7. ISO 17741 “General technical rules for measurement, calculation and verification of energy savings of projects” 2016-05-01. This protocol explicitly cites the use of EVO’s IPMVP Uncertainty Guide,
which includes R2.
8. FEMP is “M&V Guidelines: Measurement and Verification for Performance-Based Contracts Version 4.0” Prepared for the U.S. Department of Energy Federal Energy Management Program. November, 2015.
This protocol explicitly cites the use of EVO’s IPMVP Uncertainty Guide, which includes R2.
9. U.S. Department of Energy and Cadmus (2018) “The Uniform Methods Project: Methods for Determining Energy Efficiency Savings for Specific Measures” explicitly includes R2. Also see: DOE’s “50001
Ready Measurement & Verification Protocol” (19 April, 2017).
10. Stetz, M. (2019) “Why r2 Doesn’t Matter” M&V Focus Issue #5.
11. See these three sources: 1) “Uncertainty Assessment for IPMVP” EVO 101000 – 1:2019, section 1.8; 2) Bonneville Power Authority (2018) “Regression for M&V: Reference Guide” (Contract Number
00077045); and 3) DOE (2018) “The Uniform Methods Project: Methods for Determining Energy Efficiency Savings for Specific Measures” Chapter 14, p.16.
(*) Eric Mazzi is a member of EVO's IPMVP Committee. | {"url":"https://evo-world.org/en/news-media/m-v-focus/904-may-2023-m-v-focus-issue-11/1378-keep-science-in-the-practice-of-m-v","timestamp":"2024-11-07T04:36:21Z","content_type":"text/html","content_length":"45358","record_id":"<urn:uuid:c5a7ae4d-a7f2-43fc-8579-33e405b37100>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00805.warc.gz"} |
Understanding Rate Groups
Rate Groups Explained
Rate Group – A Rate Group defines how various rates relate to each other, and how that rate will be applied to construct the Grand Total, or amount the customer is to pay. The rates in System Rate
Manager must be placed into a Rate Group. Rate Groups are defined in Limo Anywhere and cannot be altered. The primary reason Rate Groups are used is allow the percentage rate types calculate off the
other fields in a correct manor and to give you an exact calculation.
NOTE: The letters following the Rate Group name indicate if it can be applied as a Fixed(F), Multiplier(M), or Percentage(P).
• Base Rate (F, M): These rates when present in a reservation are added together to create the Base Rate Total. For example a fixed rate to the airport of $100 plus a $20 Waiting Time would result
in a $120 Base Rate Total as part of the Grand Total.
• Gratuities (F,M,P): These rates are applied to the Offset Base Rates Total (See Discount 5 definition below). For example a 15% Standard Gratuity applied to a $100 Base Rate Total will result in
a $15 Gratuity being part of the Grand Total.
• Taxes (P): These rates are applied to the Offset Base Rates Total (See Discount 5 definition below). For example an 8% Sales Tax applied to a $100 Base Rate Total would result in $8 being part of
the Grand Total.
• Miscellaneous (F, M): These rates are applied to the Base Rate Total. For example a Tolls charge of $5.00 would be added to the Base Rate Total and become part of the Grand Total.
• Surcharge 1 (F, M, P): These rates apply to the Offset Base Rates Total (See Discount 5 definition below) plus Gratuity Total, plus Taxes Total plus Miscellaneous Total. For example an airport
trip with a Base Rate Total of $100, a Gratuity of $15, Tolls of $5, and Taxes of 8% ($8) that has a Fuel Surcharge (Surcharge1) of 10% applied to it would mean that $12.80 would be part of the
Grand Total of $140.80
• Surcharge 2 (F, M, P): These rates apply to the Offset Base Rates Total (See Discount 5 definition below), plus Taxes Total plus Miscellaneous Total. For example an airport trip with a Base Rate
Total of $100, a Gratuity of $15 (not included in calculation for Surcharge 2), Tolls of $5, and Taxes of 8% ($8) that has a Fuel Surcharge (Surcharge2) of 10% applied to it would mean that
$11.30 would be part of the Grand Total of $139.30
• Surcharge 3 (F, M, P): These rates apply to the Offset Base Rates Total (See Discount 5 definition below). For example an airport trip with a Base Rate Total of $100, a Gratuity of $15, Tolls of
$5, and Taxes of 8% ($8) that has a Fuel Surcharge (Surcharge3) of 10% applied to it would mean that $10.00 would be part of the Grand Total of $138.00.
• Surcharge 4(F, M, P): These rates apply to Offset Base Rates Total (See Discount 5 definition below) plus Gratuity Total, plus Taxes Total plus Miscellaneous Total, plus Surcharges 1, 2, and 3
less all Discounts. For example an airport trip with a Base Rate Total of $100, a Gratuity of $15, Tolls of $5, Taxes of 8% ($7.6), and an Airport Surcharge (Surcharge 1) of $5 less a Base Rate
discount (Discount 5) of 5% ($5); that has a Fuel Surcharge (Surcharge4) of 10% applied to it would mean that $12.76 would be part of the Grand Total of $140.36
• Surcharge 5 (F, M, P): These rates apply to Offset Base Rates Total (See Discount 5 definition below), plus Taxes Total plus Miscellaneous Total, plus All Surcharges less all Discounts. For
example an airport trip with a Base Rate Total of $100, a Gratuity of $15 (not included in calculation for Surcharge 5), Tolls of $5, Taxes of 8% ($7.6), and an Airport Surcharge (Surcharge 1) of
$5 less a Base Rate discount (Discount 5) of 5% ($5); that has a Fuel Surcharge (Surcharge5) of 10% applied to it would mean that $11.26 would be part of the Grand Total of $138.86.
• Discount 1 (F, M): This discount applies to the Base Rate Total and are a fixed amount deducted as part of the Grand Total. For example an airport trip with a Base Rate Total of $100 and a $5
Discount 1 would result in a $5 deduction being part of the Grand Total.
• Discount 2 (P): This discount applies to the Offset Base Rates Total (See Discount 5 definition below) and are deducted as part of the Grand Total. For example an airport trip with a Base Rate
Total of $100 and a 5% Discount 2 would result in a $5 deduction being part of the Grand Total of $95.00.
• Discount 3 (P): This discount applies to Offset Base Rates Total (See Discount 5 definition below), plus Gratuity Total, plus Taxes Total plus Miscellaneous Total, plus Surcharges 1, 2, and 3 and
less Discounts 1, 2, and 4. For example an airport trip with a Base Rate Total of $100, a Gratuity of $15, Tolls of $5, and Taxes of 8% ($8), with a 5% Discount 3 applied would result in a $6.40
deduction included in the Grand Total of $121.60.
• Discount 4 (P): This discount applies to Offset Base Rates Total (See Discount 5 definition below), plus Taxes Total plus Miscellaneous Total, plus Surcharge 1, 2, and 3 and less Discounts 1 and
2. For example an airport trip with a Base Rate Total of $100, a Gratuity of $15 (not included in calculation for Discount 4), Tolls of $5, and Taxes of 8% ($7.2), and a Fuel Surcharge of 10%
($9) with a 5% Discount 4 applied would result in a $5.56 deduction included in the Grand Total of $120.64.
• Discount 5 (F, M, P): This discount applies to Base Rates Total and using this creates a new Offset Base Rates Total that the Gratuity, Taxes, and other Percentage Rates are calculated from. For
example if an Airport trip had a Base Rate Total of $100 with a 10% Discount 5 the new Offset Base Rates Total would be $90. Then if a 15% Gratuity ($13.50), and 8% taxes ($7.20) were in effect
the Grand Total would be $110.70.
If you need further assistance setting this section up please feel free to call Technical Support Monday – Friday from 8am -7pm (Central Standard Time) 972-701-8887 ext 2. | {"url":"https://kb.limoanywhere.com/docs/understanding-rate-groups/","timestamp":"2024-11-05T15:41:02Z","content_type":"text/html","content_length":"327265","record_id":"<urn:uuid:c679be99-acfa-48aa-b42a-7cbf24e2e8b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00284.warc.gz"} |
Black Hole Tidal Forces Calculator - GEGCalculators
Black Hole Tidal Forces Calculator
Black Hole Tidal Forces Calculator
Drag the object near the black hole to visualize tidal forces.
What is the formula for the tidal forces of a black hole? Tidal forces near a black hole can be calculated using the tidal force formula: F_tidal = (2 * G * M * m * r) / R^3, where F_tidal is the
tidal force, G is the gravitational constant, M is the mass of the black hole, m is the mass of the object experiencing tidal forces, r is the distance from the center of the black hole to the
object, and R is the size of the object.
How much force can a black hole exert? The force exerted by a black hole depends on its mass and the distance from the black hole. Black holes can exert extremely strong gravitational forces due to
their immense mass, capable of bending light and trapping objects within their event horizon.
What is the formula for Spaghettification? Spaghettification is the stretching and elongation of objects near a black hole due to tidal forces. There isn’t a specific formula for spaghettification,
but it is a consequence of the tidal force formula mentioned earlier.
What is the acceleration due to gravity of a black hole? The acceleration due to gravity near a black hole can be calculated using Newton’s law of universal gravitation: g = (G * M) / r^2, where g is
the gravitational acceleration, G is the gravitational constant, M is the mass of the black hole, and r is the distance from the center of the black hole.
How do you calculate tidal force? Tidal force can be calculated using the formula mentioned earlier: F_tidal = (2 * G * M * m * r) / R^3, where G is the gravitational constant, M is the mass of the
black hole, m is the mass of the object, r is the distance, and R is the size of the object.
How do you calculate tidal generating forces? Tidal generating forces are typically calculated using the same formula for tidal forces, taking into account the masses and distances of the interacting
Can anything be stronger than a black hole? In terms of gravitational forces and their ability to trap light and matter, black holes are among the most powerful objects in the universe. It is
challenging to conceive of anything more gravitationally powerful.
How powerful is the pull of a black hole? The pull of a black hole is incredibly powerful, strong enough to trap even light itself within its event horizon. The strength of the pull depends on the
mass of the black hole and the distance from it.
Which force is the greatest in a black hole? Inside a black hole, gravitational forces become dominant. The gravitational force is the greatest force in a black hole, ultimately leading to the
singularity at its core.
Is spaghettification survivable? Spaghettification near a black hole is not survivable for macroscopic objects like humans or spacecraft. The extreme tidal forces would tear apart anything
approaching a black hole.
Is black hole spaghettification painful? Black hole spaghettification is a purely gravitational process, and it does not involve sensory experiences or pain as we understand it. It would be an
extremely rapid and violent process, but it wouldn’t be painful in a conventional sense.
Is it possible to avoid spaghettification? Avoiding spaghettification near a black hole would require staying far outside its event horizon and maintaining a safe distance from its strong
gravitational pull.
Is the gravity of a black hole infinite? No, the gravity of a black hole is not infinite, but it becomes extremely strong as you approach the singularity at its core. However, the concept of
“infinite gravity” does not apply in the context of general relativity.
Is black hole gravity faster than light? Gravity itself does not have a “speed” in the way that light does. The gravitational effects of a black hole, including its bending of light, occur at the
speed of light.
Does light accelerate in a black hole? Light does not accelerate in a black hole, but its path is bent due to the strong gravitational field, as predicted by Einstein’s theory of general relativity.
Where is the tidal force strongest? The tidal force is strongest closer to the massive object causing it. Near a black hole, the tidal force is most pronounced as you approach the event horizon.
What causes the largest tidal force? The largest tidal forces are caused by massive objects, such as black holes, when smaller objects come too close to them.
How strong is tidal force? The strength of tidal forces depends on various factors, including the masses of the objects involved and their distances from each other. Near a black hole, tidal forces
can be extremely strong and destructive.
What is the tidal force in space? Tidal forces exist throughout the universe, wherever there are massive objects exerting gravitational influence on each other. They are not limited to specific
locations in space.
What is the formula of tidal energy? Tidal energy is not directly related to tidal forces near black holes. Tidal energy on Earth is typically harnessed using the formula E_tidal = 0.5 * μ * A * Δh^2
* g, where μ is the density of the fluid, A is the area of the tidal basin, Δh is the height difference between high and low tides, and g is the acceleration due to gravity.
What are the two major tidal generating forces? The two major tidal generating forces on Earth are the gravitational pull of the Moon and the Sun. These forces result in the rise and fall of ocean
Could you theoretically destroy a black hole? Theoretical concepts like Hawking radiation suggest that black holes can slowly lose mass and eventually evaporate over an extremely long period of time.
However, this process is incredibly slow and has not been observed.
What can destroy a black hole? As of current scientific understanding, black holes are incredibly stable and are not easily destroyed by conventional means. Their eventual fate may involve Hawking
radiation, but this is a very slow process.
What is the most powerful thing in the universe? Black holes are among the most powerful objects in the universe in terms of gravitational forces and their ability to warp space and time.
What can overpower a black hole? Black holes are among the most gravitationally powerful objects known, and there is nothing that can easily overpower them in terms of gravitational force.
Is a smaller black hole stronger than a bigger black hole? In terms of gravitational force at a given distance, a larger black hole will exert a stronger force than a smaller one. This is because
gravitational force is directly proportional to mass.
What happens if two black holes collide? When two black holes collide, they can merge into a larger black hole in a violent event called a black hole merger. This process releases energy in the form
of gravitational waves.
What is at the center of a black hole? At the center of a black hole is believed to be a singularity, a point where gravitational forces become infinitely strong and classical physics breaks down.
Is a neutron star more powerful than a black hole? In terms of gravitational forces, black holes are more powerful than neutron stars due to their ability to trap even light within their event
Is a black hole a solid, liquid, or gas? A black hole is not made up of solid, liquid, or gas as we understand them. It is a region in space where the gravitational field is so intense that it warps
space and time to the point where nothing, not even light, can escape.
How long would you stay alive in a black hole? If you were to fall into a black hole, your experience would depend on your trajectory and the size of the black hole. However, once you cross the event
horizon, you would likely be on an irreversible path towards the singularity, and your time would be limited.
Has any star survived a black hole? As of my last knowledge update in September 2021, no star can survive once it crosses the event horizon of a black hole. The intense gravitational forces inside a
black hole would tear apart any star.
How much is 1 minute in a black hole? Time dilation near a black hole means that 1 minute experienced by an observer far from the black hole can correspond to a significantly longer or shorter time
for an observer close to the event horizon. The exact time dilation factor depends on the black hole’s mass and the observer’s position.
What happens to your body during spaghettification? During spaghettification, the tidal forces from the black hole stretch and elongate your body, causing it to become stretched out like spaghetti.
This process is extremely painful and lethal.
What happens after you get Spaghettified? After spaghettification, if you continue to fall into the black hole, you will ultimately reach the singularity at the center, where gravitational forces
become infinitely strong, and your matter is crushed to an unknown state.
What is worse, a black hole or a white hole? Black holes and white holes are theoretical constructs. Black holes are known to exist in the universe and are incredibly powerful, while white holes, if
they exist, are not well understood. Neither has been observed directly.
Has spaghettification been observed? Spaghettification has not been directly observed because it would be lethal to any observer experiencing it. However, its effects are predicted by the laws of
physics near extremely massive objects like black holes.
Has spaghettification been proven? Spaghettification is a theoretical concept based on our understanding of gravitational forces near massive objects like black holes. While it has not been directly
observed, it is a well-established prediction of general relativity.
Will black holes eventually consume everything? Black holes do not “consume” everything in the universe. They have a finite mass, and their gravitational influence is limited to objects that come
within their gravitational reach. Over extremely long timescales, they can slowly lose mass through Hawking radiation.
What cannot escape the gravity of a black hole? Anything that crosses the event horizon of a black hole cannot escape its gravity, not even light. Anything outside the event horizon can potentially
escape if it has enough velocity.
How big is a singularity in a black hole? The singularity at the center of a black hole is a mathematical point with zero volume but infinite density. It is not a “size” in the conventional sense.
Does a black hole have a core? The singularity at the center of a black hole can be considered the “core,” but it is a point of infinite density, not a solid or physical structure.
What is the fastest thing in the universe? In the context of the laws of physics as we understand them, the speed of light in a vacuum, approximately 299,792,458 meters per second, is the fastest
known speed in the universe.
Is there anything in the universe that has stronger gravity than a black hole? Black holes have some of the strongest gravitational forces in the universe. While there may be hypothetical objects
with similar or stronger gravitational forces, none have been observed or confirmed.
How fast is the black hole moving in mph? Black holes do not typically have a “speed” in mph because their motion is relative to other objects in the universe. Their motion would depend on their
interaction with nearby celestial objects.
What happens if you shine a light at a black hole? When you shine a light at a black hole, the light will be gravitationally bent by the black hole’s intense gravitational field. If the light comes
too close to the event horizon, it will not escape.
What if a black hole moves at the speed of light? It is not possible for any massive object, including a black hole, to move at the speed of light according to our current understanding of physics.
As an object with mass approaches the speed of light, its relativistic mass increases, making it require more and more energy to accelerate further.
What happens if a black hole travels at the speed of light? Hypothetically, if a black hole were to travel at or near the speed of light, it would exhibit relativistic effects such as time dilation
and length contraction. However, such a scenario is purely theoretical and not observed in nature.
Why don’t we feel tidal force? We typically don’t feel tidal forces from celestial objects like the Moon and the Sun because the human body is relatively small compared to the Earth, and the tidal
forces on our planet are relatively weak. Tidal forces become more pronounced when you are closer to massive objects like black holes.
What is the fastest tidal flow in the world? The Bay of Fundy in Canada is known to have some of the fastest tidal flows in the world, with tidal ranges that can exceed 50 feet (15 meters).
Where is the fastest tidal flow in the world? As mentioned, the Bay of Fundy in Canada is famous for its fast tidal flows and extreme tidal ranges.
GEG Calculators is a comprehensive online platform that offers a wide range of calculators to cater to various needs. With over 300 calculators covering finance, health, science, mathematics, and
more, GEG Calculators provides users with accurate and convenient tools for everyday calculations. The website’s user-friendly interface ensures easy navigation and accessibility, making it suitable
for people from all walks of life. Whether it’s financial planning, health assessments, or educational purposes, GEG Calculators has a calculator to suit every requirement. With its reliable and
up-to-date calculations, GEG Calculators has become a go-to resource for individuals, professionals, and students seeking quick and precise results for their calculations.
Leave a Comment | {"url":"https://gegcalculators.com/black-hole-tidal-forces-calculator/","timestamp":"2024-11-05T10:56:43Z","content_type":"text/html","content_length":"181215","record_id":"<urn:uuid:2072e7af-8ba4-4e56-b3ba-be56688d463b>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00230.warc.gz"} |
Math, Grade 6, Distributions and Variability, Self Check Exercise
Statistics and Probability
Material Type:
Lesson Plan
Middle School
Media Formats:
Adding to the Rubric
Review the Rubric
Self Check Exercise
Students analyze the data they have collected to answer their question for the unit project. They will also complete a short Self Check.
Students are given class time to work on their projects. Students should use the time to analyze their data, finding the different measures and/or graphing their data. If necessary, students may
choose to use the time to collect data. Students also complete a short pre-assessment (Self Check problem).
Key Concepts
Students will look at all of the tools that they have to analyze data. These include:
• Graphic representations: line plots, box plots, and histograms
• Measures of center and spread: mean, median, mode, range, and the five-number summary
Students will use these tools to work on their project and to complete an assessment exercise.
Goals and Learning Objectives
• Complete the project, or progress far enough to complete it outside of class.
• Review measures of center and spread and the three types of graphs explored in the unit.
• Check knowledge of box plots and measures of center and spread.
Project Rubric
Lesson Guide
Have students look at the rubric and talk with a partner to describe it as completely as possible in their own words.
Tell students they will have today to analyze their data and prepare for the project presentations.
SWD: Students with disabilities may have a more challenging time identifying areas of improvement to target in their projects. Teach your students how to review a project using the rubric and a
sample project.
ELL: If ELLs are still unsure about what a rubric is, show one and explain how it is used. Allow ELLs to use a dictionary if they wish.
Project Rubric
Work with a partner and review the project rubric.
• Take two minutes to study the rubric on your own.
• Then have one partner (without looking at the rubric) take one minute to describe the rubric as completely as possible to the other partner (who can see it). This partner should listen carefully
to the description.
• Briefly look at the rubric again together. The partner who was previously the listener should now take 30 seconds to add to the description—without repeating any of it.
Math Mission
Lesson Guide
Discuss the Math Mission. Students will analyze their project data in order to answer their statistical question.
Organize and analyze your project data in order to answer your statistical question.
Organize and Analyze Project Data
Lesson Guide
Students will work in their groups to make decisions about their project. Explain that they need to be clear about what is left to do on their project by the end of Work Time. Check with students as
they work.
SWD: Students with disabilities may need review and reinforcement of the major skills and concepts covered in this unit. Make sure that students are in the best partnerships for their success and
Make sure students understand the measures of center (mean and median) and how to calculate them. These should be calculated even if they choose to analyze their data with a graph. The measures
cannot be seen from the graphs, but students will have the data sets to calculate the measures.
As students work, ask them which type of graph (or graphs) would be most useful for their data sets and why.
Students have difficulty getting started.
• Try making a line plot of your data to see what it looks like.
• What is the overall shape?
• Are there one or two values that appear more than others?
• Are there clusters or gaps?
• Are the data skewed to one side or another?
• Does the plot give you a sense of what might be typical about your data?
Students have trouble choosing the best measure of what is typical.
• Did you calculate the MAD for your data? How can the MAD help you figure out if the mean is typical?
• In your line plot, are the values close to the mean or are they spread out?
• Are there outliers that affect the mean? If so, consider the median. Does it seem typical?
• Are there one or two values that occur much more than others? Would the mode be a typical value?
Students choose an inappropriate plot or bin width.
Questions depend on a group's data set. Here are some examples:
• Your data have some big gaps, but I don't see this in your plot. What type of plot (or bin width) will show this feature?
• Your data have a value that is much greater than all the others. I don't see this in your plot. Is there a different type of display that will make this clear?
• Your data have two big clusters, but the other values are spread out. Is this clear in your plot? Is there a different type of plot that might show this better?
• Answers will vary.
• Answers will vary.
Work Time
Organize and Analyze Project Data
Organize and analyze the data you collected for your project. Consider these questions:
• What measures can you calculate to help you draw conclusions and to provide support for your conclusions? Examples: mean, median, mode, range, outliers, lower and upper extremes, lower and upper
• What graph(s) will best represent your data? Examples: line plot, box plot, histogram, or more than one graph
Keep the rubric in mind as you work on your project.
Prepare a Presentation
Lesson Guide
Students will work with their groups to prepare their project presentations.
Preparing for Ways of Thinking
Highlight the Mathematical Practices during the Ways of Thinking discussion.
Mathematical Practices
This project provides an opportunity for students to engage in many of the Mathematical Practices.
Mathematical Practice 1: Make sense of problems and persevere in solving them.
• Students must make sense of their statistical question and the data they have collected to help them answer it. They must determine the best way to organize the data and determine the measures
that best summarize and describe what the data show.
Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.
• Students must construct viable arguments to justify their conclusions based on the shape and distribution of the data and the statistical measures they have calculated.
Mathematical Practice 5: Use appropriate tools strategically.
• Students must choose the most appropriate measures and displays for their particular set of data. Students might also use spreadsheet and plotting tools to explore various data displays and help
them choose the best one.
Mathematical Practice 6: Attend to precision.
• Students must communicate their conclusions and reasoning precisely in their project summaries.
Mathematical Practice 7: Look for and make use of structure.
• Students look for structure and patterns in their line plots and other data displays that can help them explain what the data show.
Challenge Problem
• Answers will vary.
• Answers will vary.
Work Time
Prepare a Presentation
Work on preparing your project presentation.
• Make sure your completed project presentation includes the following:
□ Your data set
□ At least one graph and an explanation of why the type of graph you used best represents your data
□ Measure of center and/or spread
□ A summary of what the graph and measures tell you about your data
□ Your conclusion about what is typical for your set of data
Challenge Problem
• Suppose you had collected twice as much data. Do you think having twice the amount of data would change your conclusion, or reinforce it?
• Which of your measures would change? How would your graph(s) change?
Make Connections
Lesson Guide
Have students share their thoughts about the project and discuss possible additions to the rubric.
Consider these questions to elicit discussion about the rubric:
• What specific mathematical representations should we see used in the unit project?
• What should we consider an appropriate level of effort?
• How will we know if the conclusions are correct?
• What should we evaluate that is specific to a statistics unit?
Performance Task
Ways of Thinking: Make Connections
Look at the rubric again.
• Notice the blank column with the heading Specific to This Project. Is there anything that you think should be added to this column?
• Next look at the blank row at the bottom of the rubric. Is there any aspect of the statistics project that you think should be added here?
• Take a few minutes to discuss these questions with your partner. Write down any ideas you have.
• Be prepared to discuss your ideas as a class. As you propose an idea, make sure to say why you think it is important. After all ideas are discussed, the class will decide as a group whether to
adopt any of the suggestions.
Measures of Center and Spread
Lesson Guide
This task allows you to assess students’ work and determine what difficulties they are having. The results of the Self Check will help you determine which students should work on the Gallery and
which students would benefit from review before the assessment. Have students work on the Self Check individually.
Have students submit their work to you. Make notes on what their work reveals about their current levels of understanding and their different problem-solving approaches.
Do not score students’ work. Share with each student the most appropriate Interventions to guide their thought process. Also note students with a particular issue so that you can work with them in
the Putting It Together lesson that follows.
SWD: Some students with disabilities may struggle with self-assessment; use your knowledge of student strengths and vulnerabilities to inform and create interventions you will put into place for this
period of class time.
Student does not see that 0 and 28 must be data points.
• What is the range of the data?
• What are the extreme values?
• What do these two extreme values tell you about the data set?
Student does not see that the two data values around the median must add up to 23 (such as 11 and 12).
• What does the median represent?
• Is the number of data values even or odd?
• Since the median is 11.5, what does that tell you about the 2 data values on either side of it?
Student does not see that 7 and 22 must be data values.
• How many data values are in the first two quartiles and the second two quartiles?
• What is the median data value for the lower half?
• What is the median data value for the upper half?
Student does not connect the mean with the sum of the data.
• If the mean is 14 and there are 10 data values, what is the sum of the data values?
• What is the sum of the known data values?
• What do the rest of the data need to add up to?
Student provides a poor explanation.
• How can you convince a student in another class that your answer is correct?
Student provides adequate solution to all questions.
• Find a different way of solving the problem.
• Answers will vary.
• Answers will vary.
Formative Assessment
Measures of Center and Spread
Complete this Self Check by yourself.
This box plot shows a summary of a data set of 10 measurements, each measured to the nearest inch.
• Create a possible data set so that the mean of the data set is 15 inches.
• Show your work and explain how you created the data set.
Reflect on Your Work
Lesson Guide
Have each student write a brief reflection before the end of class. Review the reflections to find out what students' groups accomplished.
Work Time
Write a reflection about the ideas discussed in class today. Use these sentence starters below if you find them to be helpful. Share your reflections with your group.
Today my group accomplished …
Our next steps are … | {"url":"https://openspace.infohio.org/courseware/lesson/2177/overview","timestamp":"2024-11-13T09:19:40Z","content_type":"text/html","content_length":"70479","record_id":"<urn:uuid:94c0f9d1-a073-414e-b780-94fa1c601e7e>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00674.warc.gz"} |
Algebra (NCTM)
Represent and analyze mathematical situations and structures using algebraic symbols.
Recognize and generate equivalent forms for simple algebraic expressions and solve linear equations
Grade 6 Curriculum Focal Points (NCTM)
Algebra: Writing, interpreting, and using mathematical expressions and equations
Students write mathematical expressions and equations that correspond to given situations, they evaluate expressions, and they use expressions and formulas to solve problems. They understand that
variables represent numbers whose exact values are not yet specified, and they use variables appropriately. Students understand that expressions in different forms can be equivalent, and they can
rewrite an expression to represent a quantity in a different way (e.g., to make it more compact or to feature different information). Students know that the solutions of an equation are the values of
the variables that make the equation true. They solve simple one-step equations by using number sense, properties of operations, and the idea of maintaining equality on both sides of an equation.
They construct and analyze tables (e.g., to show quantities that are in equivalent ratios), and they use equations to describe simple relationships (such as 3x = y) shown in a table. | {"url":"https://newpathworksheets.com/math/grade-6/algebraic-equations-1?dictionary=algebra&did=395","timestamp":"2024-11-09T23:02:35Z","content_type":"text/html","content_length":"53949","record_id":"<urn:uuid:77dfbdad-dee4-40e4-9616-d0d0f127ac6b>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00443.warc.gz"} |
Kelly McCown
Is your school going virtual this year?
Do you want to assign digital math activities that help your students?
How to Assign Digital Math Activities
Assigning digital math activities doesn't have to be difficult. Students will do work in small chunks. Think about how you will assign the math activities beforehand. After setting up the
expectations for the assignment, then give your students time to complete and turn it in.
Assign Digital Math Activities Using Google Apps
The best way to assign digital math activities is with Google Apps. There are a plethora of math activities for students to complete with Google Apps. The possibilities are endless.
Students can access Google Slides with moveable pieces. Create different math equations and solutions with number tiles that they move (as pictured above). Students enjoy being able to have an
interactive part with their learning.
Students also can create different presentations and math projects with Google Slides. This is great for students looking to make their own learning. Google Forms are an awesome tool for self
grading. Students receive instant results based on how they answer questions.
Types of Digital Math Activities
• Vocabulary math activities help students practice terms in context. Students gain conceptual understanding and master key words for math.
• Practice math activities aide students in developing skills. Giving students ten problems or less of practice ensures that they will complete their assignments.
• Assessment math activities allows teachers to see where the students are. Did they master the skill or are the kids still developing?
Try a Digital Math Activity today!
The following digital math activities are all free for use with Google Apps.
Happy Teaching!
Have you made some mistakes in teaching?
I'm sure you've heard the saying "Teachers don't make mistakes".
I've made more mistakes being a Teacher than I can count, but the failures have made the successes that much sweeter.
I want to share 5 mistakes I've made teaching over the years (trust me, there are plenty more).
5 Mistakes I've Made as a Teacher
Mistake #1 Make Every Lesson Epic
I think for a lot of us we were told in teacher college that we had to have an epic hook for every lesson in our classroom. Make every opening as engaging as possible for students wanting to learn
Mistake number one for me was trying to out do the previous lesson. I was overwhelmed from the pressure I created to make it epic. Every. Time.
Here's what I learned for sure: your classroom does not have to be perfect for learning to occur. You can be amazing just the way you are.
I created some resources that would make students have fun in math (without knowing it). These printables helped students work on key skills in a different way.
Practicing key skills daily or weekly should be fun.
Kids really enjoy doing puzzles and don't even realize it's math.
Mistake #2 Teacher Created Rooms are the Better
I thought that I should have everything created and made beautifully. My room would look amazing. What I soon found out was the opposite. My students didn't connect as well with the room. It wasn't
theirs it was mine. They wanted to to be included in the room design too.
How I fixed it: I changed my mindset. The classroom became a student centered classroom. The visuals on the walls were student created. The word wall was student created. The displays were made by
students. Was it still beautiful? Yes. Did the students take ownership and behaviors towards math improve? Yes.
Students write on sticky printed notes about each grade level term.
Mistake #3 Forgetting to Communicate
This is one that I cannot take back. I learned the hard way. Communicate, communicate, and communicate. Don't assume, just call, email or send home a million notes (okay maybe not a million).
The single best thing I ever created for my classroom and I share it for here, was a parent homework assignment. Kids LOVE this! They giggle when you say, "Here's your first homework assignment and
your parent is going to do it for you!" Say what?
Go get the assignment and your students will enjoy giving their parents homework too. The best part is you will have a working email and phone number for EVERY PARENT and GUARDIAN. Yes. Best first
homework assignment.
The first homework assignment is a quick win for communication with parents and guardians.
Mistake #4 Teaching Focused Classroom
A common mistake is forgetting the why or who you're doing something for. I was getting caught up in teaching all the time. EVERY. MINUTE. EVERY. DAY. I was exhausted and spent many hours focused on
teaching activities and not independent student activities.
Once the shift was made from teacher focused to student focused, I was no longer exhausted, fatigued, or tired. Students enjoyed the independent activities. Students loved to be doing something on
"their terms" and not being micro-managed.
Allowing students to work with pattern blocks helped challenge their critical thinking skills.
Students gained independence and worked cooperatively with partners.
They were proud of their notebooks and the learning they created inside of them.
Mistake #5 Not Focusing on Vocabulary
I thought the focus of math was answering the problems correctly. Practicing fluency skills and doing textbook activities all the time. It wasn't until word problems became an issue that I realized
what was missing.
How I fixed it: Implementing weekly vocabulary notebooks. Students had to write out 4-6 vocabulary terms every week in their notebooks. Their confidence went through the roof! Practicing vocabulary
transcended into students being able to read, write, and speak math better. It is the secret sauce my students were missing.
Students would write 4-6 words in their vocabulary notebooks every week.
Students would increase their knowledge of math terms.
By the end of the year they knew all grade level vocabulary words.
I hope my mistakes become your successes. It's okay to make mistakes. We do the most learning from them.
Happy Teaching! | {"url":"https://www.kellymccown.com/2020/08/","timestamp":"2024-11-04T08:26:10Z","content_type":"application/xhtml+xml","content_length":"173816","record_id":"<urn:uuid:031b08b6-3d4f-4256-98a3-562e17ab3409>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00743.warc.gz"} |
Crowd Simulation with Arrival Time Constraints
Department of Game & Mobile, Keimyung University, Daegu 42601, Korea
Division of SW Convergence, Sangmyung University, Seoul 03016, Korea
Author to whom correspondence should be addressed.
Submission received: 12 October 2020 / Revised: 29 October 2020 / Accepted: 29 October 2020 / Published: 31 October 2020
Finding collision-free paths for crowd simulation has been a core technique in video games and the film industry; it has drawn a great deal of attention from computer animation researchers for
several decades. Additionally, theoretical modeling of pedestrian has been a hot topic in physics as well because it allows us to predict any architectural failure of buildings and many city planning
problems. However, the existing studies for path planning cannot guarantee the arrival order, which is critical in many cases, such as arrival symmetry of the characters within video games or films.
To resolve this issue, a path planning algorithm has been developed with a novel method for satisfying the arrival-order constraints. The time constraint we suggest is the temporal duration for each
character, specifying the order in which they arrive at their target positions. In addition to the algorithm that guarantees the arrival order of objects, a new user interface is suggested for
setting up the arrival order. Through several experiments, the proposed algorithm was verified, and can successfully find collision-free paths, while satisfying the time constraint set by the new
user interface. Given the available literature, the suggested algorithm and the interface are the first that support arrival order, and their usability is proven by user studies.
1. Introduction
Crowd simulation has been one of the core techniques in many industries including entertainment, transportation and architecture for many years. Particularly, computer games and feature films have
many scenes where numerous agents move together. To create those crowd scenes, we need a technique to obtain collision-free paths for each individual. Various approaches have been proposed till now,
under the category of multi-agent path planning techniques. One of the most popular methods is the velocity obstacles (VO) method. It can predict where the other moving objects might be in the future
within the pre-defined time duration by extrapolating their velocities; hence, collisions are avoided accordingly [
]. These methods formulate a set of collision regions in the velocity domain from all neighboring characters and static obstacles at each time frame and use an optimization technique to find an
optimal velocity that is as close as possible to the preferred velocity and does not intersect with the collision region. This continuous optimal speed selection creates collision-free paths for
every individual. Extensions of the original idea have been proposed, such as HRVO (hybrid reciprocal velocity obstacle) [
], ORCA (optimal reciprocal collision avoidance) [
], and AVO (acceleration velocity obstacle) [
], which have their own advantages and disadvantages. Because these methods can produce collision-free paths for many three-dimensional (3D) characters successfully, they have been integrated into
current commercial game engines, such as Unity or Unreal.
However, in many cases, collision-free paths alone do not suffice. If game designers or movie directors were able to control crowds so that they arrive at a particular position in a particular order,
it would be possible to express various effects. For example, an entire crowd can be moved from an initial position to a target position at the same time, and could be made to engage with enemies.
However, none of the previous studies [
] consider these critical time constraints when simulating multi-agents. Instead, they are concerned with ensuring that no collisions occur while the agents move to their target positions. In this
study, we try to obtain the collision-free paths for crowds, subject to arrival-time constraints. Since we assume that agents have zero knowledge about the environment, they do not know how long it
takes to get to the intended position. Therefore, it is very difficult to set the absolute arrival time. Instead, our algorithm focuses on the relative time difference among the agents’ arrivals.
This would give the arrival-order constraints over the crowds. For example, all agents could be made to arrive at their intended position simultaneously, or given an initial line formation, we can
make them arrive at their target positions from left to right. As another example, we can make moving objects symmetrically arrive at the targeted positions at the video games, or the films.
Furthermore, we can control the arrival times of drones to give more impressive effects to attendances. In summary, supporting the relative arrival times of objects to specific positions can further
increase the controllability of objects.
To solve the ordered arrival-time constraint, this study proposes algorithms featuring three main contributions. First, to meet the time constraints, we put a velocity-adjustment layer on top of the
existing ORCA method. This layer adjusts the velocities of all agents so that they arrive at their target positions in a particular order. Whereas the original ORCA method has a fixed range of speed,
the proposed method allows the maximum speed parameter to change, to meet the time constraints. Although the original ORCA algorithm creates collision-free paths for multiple agents, it cannot be
used to control their relative arrival times. That is, it cannot control the order in which the agents arrive. However, the proposed methods not only produce the collision-free paths, but also
satisfy the arrival-time constraints.
Second, to find a set of waypoints for a highly complicated environment, a modified PRM (probabilistic roadmap) method is proposed, which can reduce the potential future collisions for multiple
agents. The modification scatters the shared waypoints of multiple agents slightly so that the agents can have different waypoints; this has the effect of reducing future collisions.
Third, a novel user interface is proposed for setting the time constraints. Usually, the timeline interface has been widely used for time-domain multimedia applications. However, it is not efficient
for multi-agent simulation because a timeline must be given for each agent. If hundreds of agents are to be simulated, then the same number of timelines must be created, which is not easy to control.
The novel interface instead integrates the time constraints with the environment. Specifically, it is assumed that the y-axis (up vector) of the environment is a time domain. When the target position
of each agent is set up, another time point is allowed for setting their arrival time. The farther the point is from the ground, the later the agent arrives at the goal. A key point is that the time
point does not represent the absolute time of arrival. Instead, the gap between two time points is of significance, as it represents the time difference between two arrival times. Through a series of
experiments, it was verified that the proposed algorithm could exactly create paths for many agents satisfying diverse time constraints in real-time. Furthermore, a new user interface is proposed
that can be used to set constraints around 30 percent faster than the timeline interface.
The remainder of this paper is organized as follows:
Section 2
lists and overviews all the related work.
Section 3
illustrates the proposed algorithm in detail. In
Section 4
, we show experimental results, including performance graphs. Finally,
Section 5
concludes the paper with discussion and future work. In this paper, the terms “crowds” and “multi-agents” are used interchangeably.
2. Related Work
For the past few decades, crowd simulation has been one of the more active research areas in the computer animation research community, and many different approaches have been proposed [
]. Excellent module-based software architecture for pedestrians, including motion synthesis and path planning has also been proposed [
]. Crowd simulation techniques can be divided into two categories: macroscopic models and microscopic models [
]. Macroscopic models usually view the entire crowd as a single group and focus on the natural flow of crowd movement. Continuum crowds and aggregate dynamics [
] are two popular macroscopic crowd simulation models. As these approaches do not factor in individual behavior, detailed quality of motion in simulation is not their top priority. Recently,
Karamouzas et al. analyzed a large corpus of real crowd data and found that interaction energy between two pedestrians follows a power law as a function of their projected time to collision [
]. Using this law, they were able to simulate a more realistic flow of movement, including the self-forming lane phenomena. They further proposed an implicit crowd simulation [
] method. This method was based on an energy-based formulation in a physics-based simulation. To update the agent’s position, they proposed an optimization-based integration scheme. In their formula,
the entire crowd was considered to solve the optimization; this made it hard to implement the method for large crowds in real-time, although the overall movement of crowds was quite realistic.
Conversely, microscopic models focus on individual behaviors and interactions between agents. The classical social force model and Boids model [
] are two widely used models, for simulating flocks based on interactions between individuals. Another algorithm integrates the popular A* path planning with crowd density information so that agents
could avoid high-density areas when they plan a path [
]. In [
], the algorithm applies PLE (principle of least effort), a general principle of crowds, to find a biomechanically energy-efficient collision-free trajectory by performing an optimization on the
total amount of metabolic energy used when traveling to the goal.
Some other methods have been proposed for multi-agent path planning to meet constraints. For example, the MAPF-POST (multi-agent path finding) algorithm suggested in [
] took into account kinematic constraints such as the maximum translational and rotational velocities of agents, and built a simple temporal network to post-process the output of an MAPF solver using
artificial intelligence (AI) to create a plan-execution schedule. This method is similar to our proposed method, in that both methods targets creating collision-free paths for a multi-agent system,
while satisfying constraints. However, the constraints in [
] include only kinematic constraints. The proposed study aims to solve the problem of temporal constraints, which gives more controllability to the agents.
The proposed method solves the time constraint problem by augmenting a new functional layer that adjusts velocities to meet the given time constraints on the existing velocity-based models. Among the
velocity-based models, ORCA (optimal reciprocal collision avoidance) has an advantage in obtaining a collision-free path for n-body moving objects [
]. In this formulation, each agent considers a neighboring moving object and constructs a half-plane defined in velocity space that is selected to guarantee collision avoidance. The agent then keeps
selecting their optimal velocity from the intersection of all half-planes, which can be done efficiently using a simple optimization technique [
]. This method enables agents to avoid collisions while they are moving to their respective target positions. However, there are still some cases when they get stuck, and it takes time to extract
themselves from the situation. In addition, the original ORCA method does not focus on how long it takes for agents to arrive at the target positions. Instead, it only emphasizes obtaining only
collision-free paths. The proposed method modified the existing ORCA to satisfy time constraints. First, the algorithm checks the visibility to the current waypoint, all the time. If the waypoint is
visible, it uses the time-constrained maximum speed parameter rather than the fixed one, which can expand the range of speeds, to support constrained arrival times. Second, once the ORCA decides the
collision-free final velocity through the optimization process, the algorithm adjusts it automatically to meet the time duration constraints.
Meanwhile, the robotics research community has been working on computing collision-free, feasible motion paths for an object from a given starting position to a given target, in a complicated
workspace. To this end, the probabilistic roadmap (PRM) approach is a commonly used motion planning technique [
]. The core part of this approach is a sampling method that samples the configuration space of the moving objects. In a simple two-dimensional (2D) environment where agents are moving on a flat
ground, the sampling involves obtaining a set of 2D points distributed in the environment that do not collide with static obstacles. Further, those 2D points are connected to generate a roadmap. The
roadmap is then used as a set of waypoints to find a path. To find a path for the given initial and target points, the Dijkstra algorithm is applied on the roadmap [
]. One of the problems of the method is that it is not made for multi-agents. Although the method can produce a path in a highly complicated environment where a lot of obstacles exist, for a finite
number of samples, if the initial and final positions of two agents are similar, then there is a high probability that these two agents keep colliding during the simulation. To solve this problem, a
method is proposed to perturb waypoints in the paths so that the agents have slightly different waypoints in their respective paths; this can reduce the possibility of collision.
In terms of setting time constraints, the timeline interface has been used for multimedia-related authoring tools [
]. In the timeline interface, a horizontal bar represents an event in the time domain. Users can put a marker on the bars, cut them out or concatenate them together. This is quite efficient for a
simple time-related manipulation task. However, it is not efficient for crowd simulation, because a timeline must be created for each agent. As the number of individuals increases, the number of
timelines must increase as well. In this paper, the timeline interface is integrated with the environment directly. Through this interface, users can conveniently set the time duration constraints
interactively. In addition, users can use a grouping tool for setting time constraints to multiple agents at the same time.
3. Proposed Algorithms
3.1. Problem Statement
The problem consists of path planning for N agents, with arrival time constraints between agent pairs. Each agent is assigned a target point, which is denoted as $G i ∈ R 2$, where $0 ≤ i ≤ N$ and
each G[i] consists of a set of waypoints ${ W j ∈ R 2 }$ to get to the target G[i], where $0 ≤ j ≤ m$. The arrival-time constraints are represented as $T i ∈ R$, and ∆T[i] is the difference of
adjacent two T[i] and their average $T ^$. Given these configurations, collision-free paths must be found for all agents such that for each agent, the difference between the actual arrival time and
the estimated average arrival time, $∆ A j = A ^ − A i$ matches $∆ T i$.
3.2. Overview
The simulation follows the steps illustrated in
Figure 1
. In the pre-processing step within
Figure 1
, given a set of static obstacles, the algorithm constructs a roadmap through the PRM planning method. At run time, the user first specifies the target position and time duration constraints for all
agents. Then, the waypoints are obtained through a roadmap query to the PRM. The waypoints behave like sub-goals, as the agent moves to the target. The current waypoint is set to the starting
waypoint in the beginning, and moves to the next waypoint as the agent passes by. The preferred velocity can be computed from the current waypoint and the agent’s position. Further, the visibility is
checked to see if there are any obstacles up to the current waypoint. If there is no obstacle, then the maximum speed parameter is adjusted so that the agent can move through the current waypoint,
while satisfying the time constraints.
As illustrated in [
], given the preferred velocity and maximum speed parameter, the algorithm constructs a set of ORCA half-planes from neighboring agents and static obstacles, and applies linear programming to obtain
the optimal velocity. At the final step, the optimal velocity is adjusted again to satisfy the time duration constraint. Once the final velocity is computed, an Euler integration is applied to update
the agents’ positions.
3.3. Pre-Processing Step
This section explains the roadmap construction using the PRM method. To navigate through the highly complicated environment where many static obstacles exist, the agents must have a rough picture of
the paths that they move along. For this purpose, the algorithm constructs a roadmap that guides agents to the target positions. There are many sophisticated high-level motion planning algorithms
available in robotics research, including PRM, RRT (rapidly-exploring random tree), and others [
]. Among them, we use the PRM owing to its simplicity in implementation.
Figure 2
shows an example of a roadmap constructed by the PRM.
One issue of the original PRM is that it does not support multiple agents, due to which two agents can sometimes have similar paths if their initial and target positions are not separated far enough.
To solve this problem, the proposed algorithm randomly
the path when the two agents share the same waypoint. Assume that there are two paths
as shown in
Figure 3
. In
Figure 3
a, it happens that both
have the same waypoint
. In this case, if those two agents have a similar speed, then there is a high chance that the two agents would bump into each other at some point in time. To prevent this collision, the algorithm
maintains an internal count variable for each waypoint. If a certain path uses a waypoint, then the count variable increases its value. If another path lies on the same waypoint, then instead of
using the original waypoint, the algorithm finds a new position that is close to the original waypoint. The new position should not lie on any static obstacles and should be a within a threshold
distance range from the original waypoint.
Figure 3
b explains the new position, perturbed from the original waypoint.
3.4. ORCA with Modified Preferred Velocity
The velocity obstacle (VO) is defined as a set of velocities that lead to a collision for moving an agent
, if the agent maintains its current velocity for a short period [
], given other dynamically moving agents playing the role of moving obstacles. In general, VO is defined in velocity space. Specifically, assume that
$V O a$
is a VO induced by agent
. Then, VO is constructed by translating a collision cone, a set of velocities that lead to a collision eventually, given the velocity of agent
, denoted by
$v a$
Figure 4
illustrates the VO for agent a given another agent
. Although we discuss the method only for agent
given another agent
, the method can be applied to the agent
given agent
, because the method is reciprocal.
To avoid collision, the agent should find a new optimal velocity outside VO[a]. The meaning of optimal in our approach requires the input of a preferred velocity. The preferred velocity is a vector
obtained by multiplying a speed parameter with a direction vector. The direction vector is defined a unit vector from the agent’s position to the current waypoint. The optimal velocity must then be
chosen to stay outside of $V O a$ and to be as close to the preferred velocity as possible. In our approach, rather than using fixed maximum speed $S m a x$, it is allowed change, depending on the
visibility. This approach makes the agent arrive at the waypoint a little early, which gives some extra time to adjust the velocity to meet the time constraint.
The visibility check decides if there are obstacles to the current waypoint. Clearly, all waypoints are sampled so that they are not inside static obstacles. Therefore, the visibility checking here
only accounts for other moving agents in the straight line to the current waypoint assuming that current waypoint is visible from the current agent unless there are other agents in between.
Mathematically, assume that we have agent
at the position
$p a$
and their current waypoint is
$w i$
. Then, if the distance
, the length of the perpendicular vector to
$w i − p a$
as indicated in
Figure 5
, is smaller than the agent’s size
$r a$
, it shows that the waypoint
$w i$
is not visible from
$p a$
. Otherwise, the agent can go to the waypoint directly. The distances
are calculated by following Equations (1) and (2):
$d = ‖ p b p a → ‖ 2 − l 2$
$l = p b p a → · w i − p a ‖ w i − p a ‖$
Depending on the visibility, the preferred velocity can be calculated as follows:
$v a p r e f = { S * m a x w i − p a ‖ w i − p a ‖ : i f v i s i b l e S m a x w i − p a ‖ w i − p a ‖ : i f n o t v i s i b l e$
$w i$
is the current waypoint,
$p a$
is the current position of agent
$S * m a x = S m a x ∗ l ∗ k . S m a x *$
is the modified maximum speed of the agent, which is a scalar value, where
$S m a x$
is the original maximum speed parameter and
is the distance constant value, which works as a control parameter on how imminent the collisions are. The scalar
represents the length of the projection of the vector
$p b p a →$
on to the vector
$w i p a →$
. The value of
controls the magnitude of
$S m a x .$
Then, the optimal velocity
$v a o p t$
can be defined as follows:
$v a o p t = arg m i n ‖ v − v a p r e f ‖ where v ∈ { v ∈ R 2 | ‖ v ‖ < S m a x , v ∉ V O a } .$
To choose an optimal velocity, it must be outside of the
$V O a$
and be as close to the preferred velocity as possible. The optimal velocity can be found through a mathematical optimization technique such as a linear programming [
The problem of the original VO is that sometimes oscillations may occur during simulation because there is no guarantee that the desired velocity chosen by the agent
lies outside the VO when another agent chooses their velocity simultaneously. To fix this issue, the algorithm is upgraded so that each agent takes half of the responsibility of avoiding collision.
It is called the RVO (reciprocal velocity obstacle). To implement the RVO efficiently while minimizing unnatural movement, the ORCA algorithm was proposed [
]. The ORCA argues that the velocity obstacles, along with the so-called half-plane that contains a set of collision-free velocities, generate smooth trajectories. Let
$v a$
represent the current velocity. In ORCA, a new parameter called the finite time horizon
is introduced. The finite time horizon
is the period that guarantees collision-free movement, which means that future collisions are ignored beyond
. With
, the
$V O a$
is often truncated into
$V O a τ$
, which would reshape the collision cone (to be rounder). Let
be the nearest point on the border of
$V O a τ$
be the vector connecting
$v a$
, and
be the outward normal vector of
$V O a τ$
. Then,
can be calculated as follows:
$q = arg m i n ‖ u − v a ‖ where v ∉ V O a τ , u = q − v a$
To calculate the optimal vector for n neighboring agents, ORCA defines the half-plane as a plane perpendicular to
at the point
$v a + 1 2 u$
Figure 6
shows the ORCA and the truncated
$V O a τ$
. More details on ORCA algorithm can be found in [
3.5. Adjustment of Velocity for Time Constraints
Given arrival-time constraints, the velocity of the agent must be adjusted to satisfy the constraints. First, the algorithm calculates the arrival-time to the target of each agent. To calculate it,
the total length of the path for a given set of waypoints is needed. For this, the distances between adjacent waypoints are summed. Assume that the set of waypoints for agent
${ w i }$
. Then the total length of the path,
$D a$
, can be calculated as follows:
$D a = ∑ ‖ w i − w i − 1 ‖$
Since the
$D a$
does not take collision avoidance into account, the actual trajectory that the agent moves along may be different from this path. Assuming that the agent
maintains its current velocity, and that
$D a$
is the total length of the path it traverses, the estimated arrival time
$A a$
can be calculated as follows:
$A a = D a ( ‖ v a ‖ + ε ) ,$
$v a$
is the current velocity, and
is the smallest floating point value to prevent division by zero.
Since $A a$ is estimated at every time frame, $ε$ is required, as sometimes the agent may not move during the simulation, which generates a zero velocity. Given the estimated value $A a$ for agent a,
the velocity can be recalculated for given time constraints $T a$. The time constraint $T a ∈ R$, is a scalar value that indicates the arrival time of agent a. The key point to note is that $T a$
does not mean the absolute time at which the agent a arrives at the target. In the proposed simulation, the difference of this value with time constraints of other agents is of more importance.
Mathematically, assume that there is a set of constraints $T i$ where $0 ≤ i ≤ n$. For example, $T 0$ is the time constraint for agent 0, $T 1$ is the time constraint for agent 1, and so on.
Succinctly, the value of $T i$ is meaningful only when it is compared with some other $T j$. That is, if $T i$ is smaller than $T j ,$ agent i arrives at its target position faster than agent j. The
extent by which the agent i arrives before agent j depends on the difference between $T i$ and $T j$. For example, let three agents and their $T i$ be 0.1, 0.5, and 0.3, respectively. Then, the order
of arrival is the first, the third, and the second.
The difference between two
$T i$
is the difference in arrival times. By changing values, the arrival time can be controlled. For instance, if
$T i$
is the same for all agents, they must arrive at their targets simultaneously. For the three-agent example above, the suggested algorithm does not distinguish between the cases when all
$T i$
are 0.1, or when all
$T i$
are 0.2. To adjust the speed of agents based on
$T i$
, a standard arrival-time is needed. In this approach, the standard arrival time is set as the average of
$T i$
, denoted as
$T ^$
. Controlling the speed of particular agents, now depends on the difference between
$T i$
$T ^ .$
Similarly, the algorithm calculates the average arrival time,
$A ^$
, for all agents:
$A ^$
$T ^$
, the adjusted velocity
$v i ′$
can be calculated for agent
$v i ′ = ( A ^ − m i n ( ∆ t i , ∆ t m a x ) ) + A i D i v ^ i$
$∆ t i = T ^ − T i$
$∆ t m a x$
is the allowable maximum
$∆ t ,$
$v ^ i$
is the normalized velocity.
In the proposed algorithm, velocity is allowed to change only when the current waypoint is visible. This basic strategy is to make each agent adjust their speed as often as possible whenever the
current waypoint is visible. The maximum $∆ t m a x$ is set so that $∆ t i$ cannot be larger than a threshold, which is needed to prevent an unnecessarily large arrival-time difference.
3.6. User Interface for Setting Time Constraints
To set a time constraint for a specific agent, an efficient user interface is needed. The timeline interface has been used for multimedia applications that require time-domain data. However, it is
not easy to apply it for crowd simulations, where a lot of agents move together. If a timeline is to be put for each agent, the number of timelines can be too many, which needs a lot of scrolling for
a given fixed window size. In the proposed approach, a simple interface is designed, integrated with the environment directly.
Figure 7
shows the user interface. It is assumed that the y-axis (up vector) of the environment is the time domain. Then, the target points are duplicated into so-called time points, and each time point is
allowed to move along the y-axis. The farther the time point is from the ground, the longer the corresponding agent takes to arrive. By changing this target position on the y-axis, the order of
arrivals can be controlled. Simply selecting with a mouse and dragging enable control over the time points. Group selection can also be allowed so that multiple constraints can be moved up and down
at the same time.
4. Experiments
A crowd simulation system was built on Microsoft Windows 10. The hardware specifications included the following: a single Intel i7 CPU with a 16 GB main memory. The graphics card was NVidia’s GeForce
GTX 1060. OpenGL was used for 3D rendering. The library for the original ORCA algorithm was provided by [
]. For all the experiments in this section, the number of samples of the PRM planning was set to 1000, the perturbation value of r was set to 1.0, and the distance constant
for calculating
$S * m a x$
in Equation (3) was set to 5.0. For the ORCA, the maximum speed
$S m a x$
was set to 2.0 and the agent’s size,
was heuristically set to 0.5. For fast neighbor searching, a simple grid-bin method was used on a KD (K-dimension) tree structure. To increase the understandability of the figures in this section, a
result video was created. Please refer to the accompanying video and
Supplementary Materials
at online
Figure 8
shows the simulation result for 10 agents. In the first scenario, the simulation without time constraints is compared with the one with time constraints. The initial positions of agents are arranged
vertically. The time constraints are set such that all agents arrive at their targets simultaneously. If the time constraints were not set, all agents would make their way to the target
independently. However, when the time constraints are set, the agents adjust their velocities considering all other agents, and finally arrive at the targets at the same time. Fast-moving agents
decrease their speed, and slow-moving agents increase their speed. The total amount of time taken to reach their targets is similar in both cases (13 and 12 s, respectively).
Figure 9
shows the proposed user interface for time constraints integrated with the environment. Users can adjust the relative time gap between two agents by moving the time points up and down, i.e., by
conveniently dragging the mouse. In this scenario, agents were required to arrive at their target positions in the left-to-right order. The simulation result shows that the constraints are exactly
satisfied. The total simulation time was around 14 s. The time difference between the first and last arrivals was around 6 s. The value of
$∆ t$
between two adjacent time constraints was 1.5–2.0 s.
Figure 10
shows another simulation example, where all agents formed a circle at the beginning and tried to get to the opposite target positions. The time constraints were set to make them arrive at the targets
simultaneously, which is a harsh condition, as there is heavy traffic at the center. The result shows that all agents were able to solve the situation, and eventually maneuver to their targets at the
same time. The total simulation time was around 24 s, which took longer than other scenarios due to the heavy traffic in the middle.
Figure 11
shows the scenario where the environment has static obstacles. The algorithm applies the modified PRM to obtain the waypoints for each agent. The arrival order was set in an arch and symmetry style
so that agents gradually arrive at their targets, from the center to both ends. The result shows that the agents avoided all obstacles and made their way to their targets in a specific order. The
total simulation time was around 14 s. In addition, to test the scalability of the proposed algorithm, it was tested with more agents and in more complicated environments.
Figure 12
shows the simulation result of 30 agents in the environment with relatively more obstacles. To make the scenario more complex, all agents were required to arrive at their target positions
simultaneously. The result shows that the proposed algorithm satisfies the arrival-time constraints for all the agents.
Figure 13
shows how the algorithm makes the agents satisfy the arrival-time constraints. The x-axis represents the particular agents and y-axis indicates the time domain. The time constraint and the actual
arrival times are put in the same time domain together for comparison, although the two are conceptually different. The value of time constraints shown in blue lines indicate y, the time points of
the environment, and orange lines indicate absolute arrival times. As can be seen in the figure, the two lines have almost identical shapes, meaning that the agents’ arrival times satisfy the
The graph in
Figure 14
represents the performance. On the same complicated environment as in the final scenario, the number of agents were increased from 5 to 100 and the frame rate was checked. The result shows that the
proposed algorithm has linear performance as the number of agents is increased, and maintains a stable real-time performance. Even when the number of agents was increased to 100, the framerate did
not go below 40 frames/s, which substantiates the real-time performance. Note that the framerate takes the simulation and rendering times into account.
In summary, the proposed algorithm was tested in several scenarios. Given a set of targets, the targets were specified to the agents randomly. This setting generates a different result for each
simulation. It was found that the simulation time ranges between 10–20 s, when the Euclidean distance between agent and target is around 10 units. During the simulation, no collisions were observed
between the agents. This demonstrates that the proposed algorithm is reliable and has a stable real-time performance.
Since the suggested algorithm is the first for path planning with time constraints, it was compared with the A* path planning algorithm in
Table 1
, in terms of the total amount of time taken for construction. The amount of time includes the pre-processing steps, in both cases. More details on the A* algorithm and its efficiency can be found in
To this end, 3000 samples were used for the PRM and a grid size of 0.1 was set in the A* algorithm, in the environment of size 1000 × 1000 units. The result shows that the modified PRM is 2.6 times
faster than the A* algorithm, even while supporting the time constraint that the A* algorithm does not. The speculation behind this, is that the A* algorithm requires a significant amount of time to
calculate all-pair-paths. Once the pre-processing is done, actual path planning for the given initial and target positions can be done in real-time, for both cases.
To check the efficiency of the proposed user interface, a simple experiment was performed, in which 30 agents were created and the time taken to set the time-arrival constraints was checked. For
comparison, a timeline interface was also built with FLTK (fast light toolkit).
Figure 15
shows the two user interfaces together. Three students were asked to randomly set generated time constraints with both interfaces, and the amount of time taken to set the constraints was checked.
Table 2
shows the comparison result. The proposed interface enabled the users to set the time constraint around 30% faster than the generic timeline interface. The reason for this is that the users spent a
considerable amount of time scrolling across the screen to find the timeline for a particular agent. This would be worse when the number of agents is beyond the display-limit of the widget. In the
proposed user interface, however, the user can change the position of the camera, and the perspective so that many time points can be seen at the same time, which helps the user find the constraints
5. Discussions
In this study, a time-constrained crowd path planning algorithm is proposed. The constraint that was enforced over the agents is a time gap. By doing so, it is possible to control the arrival time of
each agent, and symmetrical arrival orders of objects can be specified for various effects in video games, films, and robots. The basic collision avoidance structure of the proposed algorithm is
based on the popular ORCA algorithm. However, to use it for a highly complicated environment, the PRM method is applied to obtain the waypoints up to the target position. The preferred velocity,
which is the important parameter in the ORCA, is also adjusted automatically, rather than keeping it fixed during the simulation. The final velocity is also altered to satisfy the time duration
constraints at each frame. Through experiments, it was verified that the proposed algorithm can simulate the agents to meet the time constraints in real-time, and the paths were planned 2.6 times
faster than the A* algorithm. In addition to proposing the algorithm for supporting time constraints, a novel user interface was also suggested, where target points are represented by relative
positions in the y-axis. It is assumed that the y-axis of the environment is the time domain. By dragging a mouse, users can conveniently change the position along the y-axis and control the timeline
30% faster than a general interface.
However, the proposed algorithm and the user interface have limitations. Since we assume that the y-axis of the environment represents time, the algorithm is not applicable to non-flat environments.
However, this can be resolved as the height of the environment can be taken into account when the time constraints are set.
Figure 16
shows an example. Although A and B have different heights along the y-axis, their arrival times should be the same because their distances from the ground are the same. We would like to extend the
proposed interface for this kind of uneven terrain. Another limitation is that there may be cases where the time constraint cannot be satisfied. In particular, when agents are occupied with avoiding
collisions, there is no time to adjust their velocities to satisfy the time constraints. In that case, our algorithm may not work. For example, when a large number of agents are densely packed in a
highly complicated environment they get stuck and barely move to their targets. In this case, adjustment of velocities cannot be carried out as all agents are engaged in avoiding collisions. In the
future, we would like to find a density threshold parameter within which the proposed algorithm would continue to work. Furthermore, all experiments are carried out assuming that all the constraints
are set reasonably. However, arbitrarily set constraints may cause the algorithm to fail. On the user interface level, these kinds of constraints can be prohibited by limiting the position of time
points. This can be taken up as a future study.
As another potential investigation for the future, we would like to implement the proposed algorithm on a GPU (graphics processing unit). To animate a large number of agents, the high performance of
a GPU can be exploited. Using a GPU is quite appropriate for crowd simulation because each agent can independently decide its velocity-warranting the use of the parallel computational power of the
GPU. As another prospective study, rather than representing the agent as a sphere, we would like to use a real 3D character and utilize its motion data instead. This would pose a problem in
synthesizing motion because the agents may change their speed continuously. In the future, we would like to integrate a motion synthesizing algorithm with the current path planning method.
Author Contributions
Methodology, M.S.; software, M.S.; investigation, M.S. and S.K.; writing—original draft preparation, M.S. and S.K.; writing—review and editing, M.S. and S.K.; supervision, M.S.; project
administration, M.S.; funding acquisition, M.S. All authors have read and agreed to the published version of the manuscript.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (2018R1D1A1B07048414).
Conflicts of Interest
The authors declare no conflict of interest.
Figure 2. Roadmap constructed by the PRM. The red rectangles represent the static obstacles. The red circles are the initial positions whereas the blue circles are target positions. The yellow lines
represent the paths obtained from the PRM query. Note that given maximum number of connections, two nearly nodes are connected as an edge when they are visible each other, otherwise, the nodes are
not connected. The nodes are sampled in the environment to avoid all static obstacles.
Figure 3. Left: The two paths (p[1] and p[2]) have a common waypoint $w i$. Right: The waypoint $w i$ is changed to $w i ′$ for the second path for a given range parameter r, so that the two
resulting paths have slightly different waypoints. Note that the waypoints represented as circles are the part of PRM road maps. Additionally, the $w i ′$ is a random point inside the circle center
at $w i$ with radius r.
Figure 4. Left: Two agents ($a$ and $b$ ) are at $p a$ and $p b$, respectively. Their current velocities are $v a$ and $v b$, and their radii are $r a$ and $r b$. Middle: A velocity obstacle of agent
a ($V O a$) induced by agent b, shown in velocity space. Right: A reciprocal velocity obstacle of agent a ($R V O a$) induced by agent b, shown in velocity space.
Figure 5. Given two agents a and b, the visibility checks whether the current waypoint $w i$ is visible to the agent a. This means that there should be no other agents (for example, agent b) in
sight, towards the waypoint.
Figure 6.
Two agents
are at the positions
$p a$
$p a$
, respectively. Their current velocities are
$v a$
$v b$
, and their radii are
$r a$
$r b$
. For time horizon
, the
$V O a τ$
of agent
induced by agent
is the truncated cone (gray cone), in the velocity space. Then, ORCA, which is built by augmenting
$V O a τ$
with an additional linear constraint, provides a half-plane of velocity of collisions, oscillations, and reciprocal dances [
Figure 7. Proposed user interface for time constraints: the user can control the time points (green circles) to change their time (y) coordinates.
Figure 8. Line formation scenario: (a) Simulation without time constraints (b) Simulation with time constraints (all agents arrive at their targets at the same time) (Refer to the video). Note that
all sequences are from left to right in each row, and then from top to bottom to the next row.
Figure 9. Linear shape ordered scenario: Top: Blue points represent the time constraints set in the user interface. All other pictures show the simulation over time (refer to the video). Note that
the sequences are from left to right in each row, and then from top to bottom to the next row.
Figure 10. Circle formation scenario: An example of circle formation-agents are in a circular formation in the beginning, and their target positions are set opposite to the circle (Refer to the
video). Note that the sequences are from left to right in each row, and then from top to bottom to the next row.
Figure 11. Arch-shape ordered scenario: a complex environment example-red blocks represent fixed obstacles. Yellow lines indicate the connection of waypoints for each agent. The time constraints are
set in a way that all agents can arrive at the target positions in a specific order (refer to the video).
Figure 12. Complex environment example where 30 agents are moving to arrive at their target positions (refer to the video).
Figure 13. Orange lines indicate the time constraints for 10 agents. Blue lines show the arrival time of the agents. Note that the two lines must have the same shape to meet the constraints.
Table 1. Comparison between the proposed modified PRM and the A* algorithm in terms of construction time.
Modified PRM A*
Time 1.2 s 3.2 s
Age/Gender Timeline Interface Proposed Interface
A 24/F 196 s 152 s
B 28/M 224 s 175 s
C 27/M 222 s 165 s
Average 214 s 164 s
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http:/
Share and Cite
MDPI and ACS Style
Sung, M.; Kim, S. Crowd Simulation with Arrival Time Constraints. Symmetry 2020, 12, 1804. https://doi.org/10.3390/sym12111804
AMA Style
Sung M, Kim S. Crowd Simulation with Arrival Time Constraints. Symmetry. 2020; 12(11):1804. https://doi.org/10.3390/sym12111804
Chicago/Turabian Style
Sung, Mankyu, and SeongKi Kim. 2020. "Crowd Simulation with Arrival Time Constraints" Symmetry 12, no. 11: 1804. https://doi.org/10.3390/sym12111804
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
Article Metrics | {"url":"https://www.mdpi.com/2073-8994/12/11/1804","timestamp":"2024-11-13T02:02:46Z","content_type":"text/html","content_length":"476846","record_id":"<urn:uuid:5ac682c4-13df-45a7-b873-733ceec9a813>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00761.warc.gz"} |
Why Take a Math Course? | Colby College
Because math is beautiful!
Mathematics is one of the greatest human discoveries
Calculus revolutionized humankind’s ability to precisely describe and shape the world we live in. Linear algebra gives a common framework for understanding both elementary transformations (like
rotations) and the long-term behavior of complicated systems.
Each part of mathematics exhibits its own form of beauty
In courses such as real and complex analysis, abstract algebra, and geometry much of the beauty arises from pristine arguments which make precise intuitive notions of infinity, continuity, symmetry,
and space. The beauty showcased in these classes is similar to the beauty exhibited by a haiku or abstract piece of art.
Mathematics is an intricate set of interlocking hierarchies
Viewed from the outside, the language and methods of formal mathematics may seem intimidating, but the discipline, by its very nature, slowly builds on and develops basic concepts which are
accessible to every person. Each of our courses takes students, with whatever mathematical knowledge and ability they bring, and moves them along on the path to greater mathematical understanding and
ability. As they do so, they connect the main ideas of the course to concepts and themes found in other mathematics, statistics, and science courses.
Mathematics has deep affinities with the arts and humanities
It has inspired paintings (like this one, for example, as explained here.), made a profound impact on art history, provides important examples and problems for philosophy (as in the work of Immanuel
Kant, for example), was responsible for a revolution in economic history and is intimately connected to the modern revolution in physics history. Even mathematics and music are intertwined in many
interesting and complex ways.
Because math is useful!
Abstract mathematical ideas have unplanned impact on the world
• Number theory, once thought of as the “most useless” of abstract mathematics, is now the basis of modern cryptography – every time your credit card number or password is transmitted over a secure
connection, you’re using number theory.
• Google’s algorithm for ranking search results is an application of linear algebra.
• Persistent homology is a new data analysis tool arising from a branch of mathematics, formerly considered to be one of the most abstract subdisciplines of mathematics.
Mathematics and statistics provide the quantitative language of the natural and social sciences
• Biology: Mathematical biology is an upcoming field that uses sophisticated mathematics and statistics to understand biological systems. Indeed, biologists of all sorts will find mathematics and
statistics extremely useful in the modern world. Mathematics can even explain why there are no 3-headed monsters.
Economics: From fixed point theorems (of the sort studied in topology and real analysis) to the Black-Scholes Formula, mathematics and statistics are everywhere in economics. Classes in calculus,
linear algebra, real analysis, and statistics will get you off to a good start in understanding the mathematical foundations.
• Physics: Calculus is the language of classical physics; modern physics makes extensive use of differential equations, topology, and geometry (among other mathematical subjects!)
• Chemistry: The mathematics of tilings (studied in geometry courses) governs the formation of crystals and quasi-crystals. Group theory (studied in our Abstract Algebra course) is the language for
describing the symmetries of molecules. In concert with group theory, topology and knot theory provide techniques for understanding chirality.
• Computer Science: Data visualization relies on a web of concepts from linear algebra, statistics, and geometry. Quaternions (studied in Abstract Algebra) are fundamental to computer graphics.
Turing machines (the theoretical foundation of computing) are formal mathematical constructs like those studied in Mathematical Reasoning (MA 274). Since computers have finite precision, any time
real numbers are used in computing the techniques of numerical analysis become extremely important. Quantum computers will radically change the nature of computing in the 21st century — topology,
geometry, and abstract algebra (in addition to the old standbys of calculus and linear algebra) are essential tools in quantum computing.
• Environmental Science: Mathematical models are the central tool for describing and predicting climate change. Population migrations and predator-prey relationships are studied with differential
equations. To learn more about these (or related) topics take our courses in differential equations or mathematical modeling.
• Engineering: All disciplines of mathematics have contributed to the various engineering disciplines. Since we haven’t mentioned it yet, we should point to Fourier analysis as a beautiful and
highly applicable mathematical subject. One of its many applications is in sound engineering: it explains why musical instruments sound the way they do! Fourier analysis is studied on occasion in
our courses in real analysis, topics in analysis, or partial differential equations.
Because you can be wrong! (and right!)
Mathematics is one of the few human endeavors where there is almost total agreement about what constitutes a correct argument. In our classes, you’ll learn how to construct rigorous arguments, detect
flawed arguments, and spot vague and malformed statements.
Because math is power!
Providing a solid, appropriate mathematical education to elementary and secondary school children empowers them to lead confident, productive lives. We love helping students who want to teach – it’s
a noble calling. If you’re interested, talk to us about particular programs which might be relevant to your interests.
Because they’re fun!
• Solving problems is fun. The moment of sudden enlightenment makes the struggle worth it.
• Connecting disparate mathematical ideas is fun. Topology shows up in real analysis, and real analysis in geometry, and geometry in number theory. Mathematics is one discipline, but with many
rooms. Discovering which rooms adjoin to which is the adventure of a lifetime.
• Applying mathematical and statistical ideas to the real world is fun. Connecting the abstract to the concrete is what science and the liberal arts, more generally, are all about. It’s fascinating
work and a great deal of fun.
Even if you are majoring in something else, taking more math classes may be worthwhile. Math tends to open doors that if left closed can be avoided, but when opened offer opportunities that are worth
having. In almost any field, (and particularly in medicine, the sciences, and economics) more math always gives one an edge over those who have taken less. | {"url":"https://www.colby.edu/academics/departments-and-programs/mathematics/why-math/","timestamp":"2024-11-02T21:25:13Z","content_type":"text/html","content_length":"167049","record_id":"<urn:uuid:4e93bfff-0c79-4746-b3cc-dd1826cb8915>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00089.warc.gz"} |
Generalized Linear Regression
Analysis using GeoAnalytics Tools
Analysis using GeoAnalytics Tools is run using distributed processing across multiple ArcGIS GeoAnalytics Server machines and cores. GeoAnalytics Tools and standard feature analysis tools in ArcGIS
Enterprise have different parameters and capabilities. To learn more about these differences, see Feature analysis tool differences.
• As a GIS analyst at a utility company, you have a dataset of power outages, as well as extreme weather data. You enrich your outage data using the Build Multi-Variable Grid and Enrich from
Multi-Variable Grid tools to create a dataset with extreme weather information for the outages. You use Generalized Linear Regression to determine what event led to the power outages. Now that
you have this information, you can predict outages and allocate resources.
• As an analyst for a large city, you have historic 911 call records, as well as demographic information. You need to answer the following questions: Which variables effectively predict 911 call
volume? Given future projections, what is the expected demand for emergency response resources?
Usage notes
This tool can be used in two operation modes. The Fit a model to assess model performance option can be used to evaluate the performance of different models as you explore different explanatory
variables and tool settings. Once a good model has been found, you can use the Fit a model and predict values option.
Use the Choose a layer to generate a model from parameter with a field representing the phenomena you are modeling (Choose the field to model) and one or more fields representing the explanatory
variables. These fields must be numeric and have a range of values. Features that contain missing values in the dependent or explanatory variable will be excluded from the analysis. If you want to
modify null values, use the Calculate Field tool first to create a new layer with updated values.
The Generalized Linear Regression tool also produces output features and diagnostics. Output feature layers are automatically added to the map with a rendering scheme applied to model residuals. A
full explanation of each output is provided below.
It is important to use the correct model (Continuous, Binary, or Count) for your analysis to obtain accurate results of your regression analysis.
Model summary results and diagnostics are written to the messages window and charts will be created below the output feature class. The diagnostics reported depend on the Model Type. The three
options for model type are as follows:
• Use the Continuous (Gaussian) model type if your dependent variable can take on a wide range of values such as temperature or total sales. Ideally, your dependent variable will be normally
• Use a Binary (logistic) model type if your dependent variable can take on one of two possible values, such as success and failure or presence and absence. The field containing your dependent
variable must be numeric and contain only ones and zeros. There must be variation of the ones and zeros in your data.
• Consider using a Count (Poisson) model type if your dependent variable is discrete and represents the number of occurrences of an event such as a count of crimes. Count models can also be used if
your dependent variable represents a rate and the denominator of the rate is a fixed value such as sales per month or number of people with cancer per 10,000 in the population. A Count model
assumes that the mean and variance of the dependent variable are equal, and the values of your dependent variable cannot be negative or contain decimals.
The dependent variable and explanatory variable parameters should be numeric fields containing a range of values. This tool cannot solve when variables have the same values (if all the values for a
field are 9.0, for example).
Features with one or more null values or empty string values in prediction or explanatory fields will be excluded from the output. If needed, you can modify values using Calculate Field.
You should visually inspect the over- and underpredictions evident in your regression residuals to see if they provide clues about potential missing variables from your regression model.
You can use the regression model that has been created to make predictions for other features. Creating these predictions requires that each prediction feature(Choose a layer to predict values for)
has values for each of the explanatory variables provided. If the field names from the input features and prediction locations parameters do not match, a variable matching parameter is provided. When
matching the explanatory variables, the fields from the input features and prediction locations parameters must be of the same type (double fields must be matched with double fields, for example).
The Generalized Linear Regression tool produces a variety of outputs. A summary of the GLR model and statistical summaries are available on the portal item page and as a resource on your layer. To
access the summary of your results, click Show Results Map Viewer. The tool generates at least one output layer and an optional output predicted features. The output features are automatically added
to Map Viewer with a hot and cold rendering scheme applied to model residuals. The diagnostics generated depend on the model type of the input features and are described below.
Continuous (Gaussian)
Interpret messages and diagnostics
• AIC—This is a measure of model performance and can be used to compare regression models. Taking into account model complexity, the model with the lower AIC value provides a better fit to the
observed data. AIC is not an absolute measure of goodness of fit but is useful for comparing models with different explanatory variables as long as they apply to the same dependent variable. If
the AIC values for two models differ by more than 3, the model with the lower AIC value is considered more accurate.
• AICc—AICc applies a bias correction to AIC for small sample sizes. AICc will approach AIC as the number of features in the input increase. See AIC above.
• Multiple R-Squared—The R-Squared is a measure of goodness of fit. Its value varies from 0.0 to 1.0, with higher values being preferable. It may be interpreted as the proportion of dependent
variable variance accounted for by the regression model. The denominator for the R-Squared computation is the sum of squared dependent variable values. Adding an extra explanatory variable to the
model does not alter the denominator but does alter the numerator; this gives the impression of improvement in model fit that may not be real. See Adjusted R-Squared below.
• Adjusted R-Squared—Because of the problem described above for the R-Squared value, calculations for the adjusted R-Squared value normalize the numerator and denominator by their degrees of
freedom. This has the effect of compensating for the number of variables in a model, and consequently, the Adjusted R-Squared value is almost always less than the R-Squared value. However, in
making this adjustment, you lose the interpretation of the value as a proportion of the variance explained. In Geographically Weighted Regression (GWR), the effective number of degrees of freedom
is a function of the neighborhood used, so the adjustment may be quite marked in comparison to a global model such as GLR. For this reason, AICc is preferred as a means of comparing models.
Binary (Logistic)
Interpret messages and diagnostics
• AIC—This is a measure of model performance and can be used to compare regression models. Taking into account model complexity, the model with the lower AIC value provides a better fit to the
observed data. AIC is not an absolute measure of goodness of fit but is useful for comparing models with different explanatory variables as long as they apply to the same dependent variable. If
the AIC values for two models differ by more than 3, the model with the lower AIC value is considered more accurate.
• AICc—AICc applies a bias correction to AIC for small sample sizes. AICc will approach AIC as the number of features in the input increase. See AIC above.
Count (Poisson)
Interpret messages and diagnostics
• AIC—This is a measure of model performance and can be used to compare regression models. Taking into account model complexity, the model with the lower AIC value provides a better fit to the
observed data. AIC is not an absolute measure of goodness of fit but is useful for comparing models with different explanatory variables, as long as they apply to the same dependent variable. If
the AIC values for two models differ by more than 3, the model with the lower AIC value is considered more accurate.
• AICc—AICc applies a bias correction to AIC for small sample sizes. AICc will approach AIC as the number of features in the input increase. See AIC above.
The GeoAnalytics implementation of Generalized Linear Regression has the following limitations:
• It is a global regression model and does not take the spatial distribution of data into account.
• Analysis does not apply Moran's I test on the residuals.
• Feature datasets (points, lines, polygons and tables) are supported as input; rasters are not supported.
• You cannot classify values into multiple classes.
ArcGIS API for Python example
The Generalized Linear Regression tool is available through ArcGIS API for Python.
This example fits a model on a dataset and applies the prediction to another .
# Import the required ArcGIS API for Python modules
import arcgis
from arcgis.gis import GIS
# Connect to your ArcGIS Enterprise portal and check that GeoAnalytics is supported
portal = GIS("https://myportal.domain.com/portal", "gis_publisher", "my_password", verify_cert=False)
if not portal.geoanalytics.is_supported():
print("Quitting, GeoAnalytics is not supported")
# Find the big data file share dataset you're interested in using for analysis
search_result = portal.content.search("", "Big Data File Share")
# Look through search results for a big data file share with the matching name
bd_file = next(x for x in search_result if x.title == "bigDataFileShares_Sales_2018")
# Find the multivariable grid to enrich from
predict_layer = portal.content.search("Sales_2025", "Feature Layer")
predict_layer = layer_result[0].layers[0]
# Run the tool Generalized Linear Regression
glr_result = arcgis.geoanalytics.analyze_patterns.glr(input_layer = bd_file,
features_to_predict = "total_customers",
var_explanatory = "salestotal, store_count, advertisingcost",
var_dependent = "chicago_crimes_enriched",
regression_family = "Count",
exp_var_matching = [{"predictionLayerField":"store_count", "trainingLayerField": "num_of_stores"}],
output_name = "predicted_customers")
# Visualize the results if you are running Python in a Jupyter Notebook
processed_map = portal.map()
Similar tools
Use the ArcGIS GeoAnalytics Server Generalized Linear Regression tool to generate predictions or to model a dependent variable in terms of its relationship to a set of explanatory variables. Other
tools may be useful in solving similar but slightly different problems.
ArcGIS Desktop analysis tools
To run this tool from ArcGIS Pro, your active portal must be Enterprise 10.7 or later. You must sign in using an account that has privileges to perform GeoAnalytics Feature Analysis.
Perform similar regression operations in ArcGIS Pro with the Generalized Linear Regression geoprocessing tool as part of the Spatial Statistics toolbox.
Create models and predictions using an adaptation of Leo Breiman's random forest algorithm in ArcGIS Pro with the Forest-based Classification and Regression geoprocessing tool as part of the Spatial
Statistics toolbox.
Preform GWR in ArcGIS Pro with the Geographically Weighted Regression geoprocessing tool as part of the Spatial Statistics toolbox. | {"url":"https://gis.berkeleycountysc.gov/arcgis/help/en/portal/latest/use/geoanalytics-generalized-linear-regression.htm","timestamp":"2024-11-09T14:17:01Z","content_type":"text/html","content_length":"43429","record_id":"<urn:uuid:a4419ff7-325b-44b7-a4d5-864655ab47eb>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00703.warc.gz"} |
A simple introduction
What is Quantitative Data?
A simple introduction and overview
Published: 23rd September 2021
Quantitative research involves the collection and analysis of data that can be quantified. Quantitative data is numerical in nature (i.e., involves numbers and varies in amount or quantity), is
collected by directly measuring the variables of interest, and is typically analysed using statistical techniques. Quantitative data is often collected when researchers are interested in answering
questions about ‘how much’ something differs or changes.
Types of Quantitative Data: Quantitative data can be broken down into two distinct types, and includes:
1. Continuous data: Continuous data is data that is not limited to a specific number of responses or values, and can be reported in smaller units such as decimals or fractions. Continuous data can
be further divided into interval or ratio data. Interval data occurs when data is labelled, the order of data is known, and the difference between data is also known. Interval data can contain a
zero point, however zero in interval scales is often arbitrary. Ratio data is the same as interval data, however ratio data also features a ‘true zero’ point to indicate a total absence of what
is being measured. Continuous data examples include height, weight, speed or distance.
2. Discrete data: Discrete data is data that is limited to a specific number of responses. Discrete data is typically counted in whole numbers only, and can’t be broken down into smaller units such
as decimals or fractions. Discrete data examples include the number of pairs of shoes someone owns, or the number of students in a class.
How to Collect Quantitative Data
There are a broad range of ways to collect quantitative data, however the specific methods of collection are dependent on two main factors – the study design that you are using to address your
specific research question, and then what instruments or tools you are using to quantify the data within that specific design. For the former factor, several study designs can be applied to
quantitative research, and can include:
Type Definition
Longitudinal Examination of variables in a group of individuals over the course of a specified time frame, which allows for the quantification of your variables over time to see how they change.
Cross-Sectional Examination of variables in a group of individuals at a specific point in time, which allows for the quantification of your variables at a fixed time point.
Case Study In-depth investigations of variables in a small number of individuals. This allows for an in depth and more focused quantification of your variables.
Observational A non-experimental study where the behaviour of individuals is observed/measured and recorded, and not changed or manipulated by a researcher. This allows you to quantify your
variables in their natural state.
Experimental The systematic, and tailored, manipulation and measurement of two or more variables to investigate how the change in one variable influences the other(s).
Correlational Examination and quantification of a potential relationship between two variables, which demonstrates how one variable might relate to another. Read more about correlation coefficients
How to measure quantitative data
How you measure quantitative data depends on your research question and associated variables of interest. Several methods can be used to quantify data, and can include:
Type Definition
Questionnaires A series of written and structured questions where participants are asked to choose from a series of answers.
Interviews A series of questions which are verbally communicated to the participants. Interviews can be structured, with the same questions asked of each participant, or unstructured,
where there is no fixed structure.
Experiments Where, in a highly controlled environment such as a laboratory, researchers systematically manipulate one variable to investigate whether it causes a change in another.
Controlled Where researchers observe participants in a controlled environment to see how they respond to certain situations.
Direct Measurement The use of specific and purpose-built tools to measure specific constructs directly, such as using a thermometer to measure temperature.
Quantitative Data Analysis
Since quantitative data relates specifically to numbers, to help understand what our data is saying, draw conclusions and therefore answer our research question we can analyse it using statistical
techniques. The two main categories of statistics include descriptive and inferential types. The type of statistics that you use will depend on your research question.
Descriptive statistics help summarise the data you have collected, and allow for you to communicate the main features of your sample. Descriptive statistics include measures of central tendency and
spread, which are often reported together. Measures of central tendency summarise the data to represent the centre of your sample, whereas measures of spread describe how much variability exists
within your sample.
Type Definition
Central Mean The typical, or average, score in a dataset.
Tendency Mode The most frequently occurring score in a dataset.
Median The midpoint of a dataset.
Range The difference between the smallest and largest values in a dataset.
Spread Quartiles and Interquartile An index of variability in a dataset, where the data are divided into four parts (or quartiles) and then compared to one another to gauge where the
Range scores are distributed.
Standard Deviation The most relied upon measure of variability which indexes by how much obtained scores differ from the mean of the dataset.
Inferential Statistics
Inferential statistics allow for researchers to draw conclusions about the broader population of interest. Inferential statistics allow for researchers to identify any significant differences,
changes in or relationships between one or more samples and the broader population(s) which they were sampled from. Inferential statistics vary in complexity and the type of information they provide,
but fall under the two main categories of parametric and non-parametric.
Parametric statistical tests assume that the data obtained come from a normally distributed population. When represented in a graph, a normal distribution looks like a bell shape where most of the
scores are clustered around the mean and taper towards the ends. Common parametric inferential statistics include:
Type Definition
One One Sample Z A test used when you want to compare a sample to a known and defined population. This test is performed when you have one sample, and the population mean and standard
Sample Tests deviation are known.
Tests One Sample T Another test used when you wish to compare a sample to a known population, however not all the population parameters are known, and therefore must be estimated based on
Tests your sample. This test is performed when you have one sample, the population mean is estimated, but the population standard deviation is not known and must be estimated.
Two-Sample T Two Sample Tests Two-Sample T Test A test used to compare whether two population means are equal, and occurs in two forms:
Test - An Independent Samples t Test tests if there is a difference between two separate groups.
- A Correlated Groups t Test tests if there is a difference in the same group of participants.
Two Analysis of A test used to compare whether three or more population means are equal, and can include:
Sample Variance (ANOVA) - A Between-Subjects ANOVA tests whether there is a difference between three or more separate groups.
Tests - A Repeated-Measures ANOVA examines whether there is a difference in the same group of participants across three or more conditions.
Pearson A test used to indicate whether a relationship between two variables exists, and if one does, index how strong that relationship is.
Regression Regression is a test used to examine whether one or more variables are associated with (or colloquially speaking, predicts) a particular outcome variable.
In contrast, non-parametric tests do not make the same assumptions about normality as parametric tests. When graphed, a non-parametric distribution does not look like a bell shape and typically has
most of the scores closer to one end of the distribution. To address non-normality, non-parametric tests work by assigning each datapoint ranks and then analysing those. Common non-parametric
inferential statistics include:
Type Definition
Mann-Whitney U Test The non-parametric equivalent of an Independent Samples t Test. Aims to determine whether there is a difference in rank data between two separate groups
Wilcoxan Signed Rank Test The non-parametric equivalent of a Correlated Groups t Test. Aims to determine whether there is a difference in rank data between two related groups (e.g. before/after).
Kruskal-Wallis H Test The non-parametric equivalent of a Between-Subjects ANOVA. Aims to determine whether there is a difference in rank data between three or more separate groups.
Friedman’s ANOVA by Ranks The non-parametric equivalent of a Repeated-Measures ANOVA. Aims to determine whether there is a difference in rank data between three or more related groups.
Spearman’s Rank Order The non-parametric equivalent to a Pearson’s Correlation. Used to indicate whether a relationship between two non-normally distributed variables exists, and if one does,
Correlation index how strong that relationship is.
Helpful Resources | {"url":"https://www.supersurvey.com/Data","timestamp":"2024-11-13T08:16:29Z","content_type":"text/html","content_length":"196207","record_id":"<urn:uuid:bfb3e7f3-181f-443e-b612-e7862ab75acf>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00020.warc.gz"} |
ATR Trading Strategy: A Comprehensive Guide
What is the Average True Range (ATR) indicator?
Average True Range (ATR) is a technical analysis indicator that measures price volatility of a financial security over a period of time, typically 14 days.
ATR is calculated as the average of the true ranges over the period. It’s a measure of volatility, not a directional indicator. A higher ATR signals more volatility, and vice versa.
• The Average True Range is designed to measure the volatility of a market.
• ATR can be used to help trader’s evaluation on setting stop-loss and take-profit limits and can be used in conjunction with a range of other technical indicators and trading strategies,
• The Average True Range cannot predict the future, so traders will still need to do their own research.
Average True Range indicator explained
ATR was created in 1978 by technical analyst J Welles Wilder and published in his book New Concepts In Technical Trading Systems.
If someone wants to calculate the Average True Range, they need to take the following three components into account:
• The difference between the current high and the previous close;
• The difference between the current low and the previous close;
• The difference between the current high and the current low.
After calculating those figures, they should choose the highest one. This is the True Range, or TR.
Credit: Capital.com
Once they have found the True Range, they will need to take a number of time periods. These can be hours, days, weeks, months or even years. In his book, Wilder suggested 14 time periods. This is the
most commonly used number, although traders can use more or fewer if they wish.
They should then calculate the True Range of those time periods (for example, of 14 days), and find the average of them. This final number is the Average True Range and shows the average price
movement for the time period involved.
ATR formula
Alternatively, ATR can be calculated using the following formula:
ATR = (Previous ATR * (n - 1) + TR) / n]
n=number of periods
TR= True Range
Average True Range trading example
Let’s now take a quick look at a real world example of the Average True Range. The indicator is available on most trading platforms and will show up as a separate panel below the price chart.
As you can tell by looking at the image, the ATR does not exactly mirror the price. However, it does show when the price would have been the most volatile. Indeed, if we look at the chart, we can see
that, when the asset was at its highest price, it had something of a mid-range amount of volatility.
Something else worth noting is that the Average True Range is written as an absolute value, rather than as a percentage. This means that an asset that is hovering around that $1,000 mark will have a
higher ATR than one which is worth somewhere in the region of $10.
As a result, the first could register a more notable change in its ATR by rising by $100 than the second would by $5, despite the first asset going up by 10% and the second by 50%. Traders should be
aware of this and not use ATR measurements in isolation when devising their Average True Range strategy.
ATR trading strategy: How to use ATR in trading
The Average True Range is a tool which could, potentially, help traders when they develop a trading strategy.
• Day trading: It is not uncommon for the ATR indicator strategy to be used by day traders. The idea is to use short time periods to assess the ATR, and then to add that to a closing price.
• Range trading strategy: The ATR can be used to work out a range trading plan. Since range trading relies on finding a particular range in which to trade in, using the ATR to measure the market’s
volatility can help when it comes to knowing what sort of range to trade in.
• Breakout strategy: Using an ATR trading strategy can be useful when combined with a breakout strategy. This means that a trader can use the indicator to see when an asset breaks out of a low
volatility level, since this often precedes a sharp movement in price.
• Momentum trading: Using the Average True Range indicator can, be informative when it comes to momentum trading. The ATR would typically rise when an asset’s price is likely to move quicker than
before, which can lead to a momentum - either bullish or bearish.
How to use ATR to set a stop-loss and take-profit
The ATR indicator is often used in conjunction with stop-loss orders. Stop losses are market orders that would exit a losing trade at a predetermined price. Note that ordinary stop-losses do not
shield from slippage – in this case, guaranteed stop losses may offer more protection, yet charge a fee.
When the ATR is high, traders could potentially be prepared for greater volatility and wider price fluctuations. As a result, they could set their stop loss orders higher, because they might well
think that price changes are to be expected, and that the market could, potentially make a recovery.
On the other hand, when the ATR indicates lower volatility, traders may use a smaller stop loss figure, because they could predict there may not be that much of a likelihood of a quick recovery from
a market low.
Likewise, ATR can be used to set take-profit orders, market orders to close a winning position triggered at a predetermined price. When the volatility is high, traders may want to set the take profit
order higher, because there is the possibility that the market could continue to rise and, similarly, when the volatility is low, then they may consider setting it lower because it is possible that
the market may not continue to move upwards as much.
ATR for position sizing
It can also be used for position sizing, with the ATR used to find which assets in a traders portfolio are the most volatile and with the size of trades adjusted accordingly.
The idea here is to calculate the Average True Range for each of the assets in a trader’s portfolio. If an asset has a high volatility, then the trader may be best off if they made smaller trades,
because a more likely market move could potentially wipe out any gains.
Often, traders who use position sizing will apply the same formula, utilising how much they are willing to risk in order to calculate the size of their trades.
Doing so requires using a formula to calculate a position size. This would be the sum of the percentage of the trader’s account they were willing to risk divided by the Average True Range.
In terms of a formula, it would be:
A=percentage of trading account the trader is willing to risk and
Average True Range and other indicators
The Average True Range can be used in conjunction with other technical analysis tools. For instance, the range of stochastic indicators, tools which are used to measure the overall momentum of an
asset's price, are often used with the ATR. This is because the ATR can counteract stochastic tools’ tendency to send false signals in markets which do not hover between two particular price points.
Likewise, stochastic’s ability to suggest when an asset is either overbought or oversold can help clarify the movements of the Average True Range.
The Parabolic SAR, a tool designed to show market movements and suggest entry and exit points was also created by Wilder and can work with the ATR. This is because the Parabolic SAR can show what
direction the market has been moving in which, coupled with the way in which ATR demonstrates the overall market volatility, can help bring some clarity to the two indicators’ signals.
Since the ATR is often used by traders to help them find an exit point, then a tool like Moving Average Convergence/Divergence (MACD), which is often used to signify entry points and changes in
momentum. It can also be utilised with other volatility indicators, such as Bollinger Bands (BB), to determine reversals in price.
The Average Trading Range is a technical analysis tool which can be used to measure the overall volatility of a market.
The ATR can be calculated by finding the True Ranges for a fixed set of time periods, usually the most recent 14. ATR can be used in various trading strategies including day trading, range trading,
momentum trading, working with a breakout strategy, and many more.
It can help traders inform when and where may be a good place and time to set their stop loss and take-profit orders. It can be used in conjunction with other indicators, such as stochastic
indicators, Parabolic SAR, MACD and Bollinger Bands.
What it cannot do, however, is tell the future. That is why traders need to make sure to do their own research, remember that markets can move in directions which damage their positions, and never
trade with more money than they can afford to lose.
How does the Average True Range work?
The Average True Range works by finding the True Range – the largest of the differences between the current high and the previous close, the current low and the current high and the current low for a
set of time periods, adding them together, and dividing them by the number of time periods. The ATR is a measure of volatility.
How to read the Average True Range?
The Average True Range is represented by a line on a chart, typically in a separate panel below the price chart, with the highs representing high volatility and the lows representing low volatility.
How can ATR be used to set stop loss and take profit levels?
ATR can be used to set stop loss and take profit levels because it demonstrates how volatile a market is. A trader may want to expand their stop loss and take profit levels if the ATR is showing high
volatility and restrict them if it is showing low volatility.
What are some common trading strategies that use ATR?
The Average True Range can be used in a variety of trading strategies, including day trading, breakout trading, momentum trading, and more. | {"url":"https://capital.com/average-true-range","timestamp":"2024-11-05T15:41:02Z","content_type":"text/html","content_length":"306441","record_id":"<urn:uuid:f410267e-5936-464b-8053-ed2546e0b059>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00806.warc.gz"} |
Compression Index And Coefficient Of Compressibility And Void Ratio Vs Effective Stress Curve
Heavy buildings or structures make the ground compress a little over time due to their weight. This compression of soil is known as the Settlement of soil.
Total settlement of the soil has primarily three components.
1. Immediate Settlement
2. Consolidation Settlement or Primary Consolidation
3. Creep Settlement or Secondary Consolidation
The Total Settlement of soil is the sum of all three factors
Consolidation Settlement is the most dominant factor of the total settlement. It refers to the settlement of soil that occurs because applied load causes the rearrangement of soil particles and
dissipation of excess pore water pressure in saturated soils. We have learned all this in our previous post.
To estimate the amount of settlement and rate of consolidation we perform consolidation test on soil sample in laboratory. The consolidation test also helps determine various soil properties of clay
relevant to consolidation, such as the permeability, swelling behaviour of soil and coefficient of consolidation.
We have discussed part of the consolidation test in our previous post where we plotted a time-settlement curve under a single load. Now we’ll be loading the soil specimen with higher load
incrementally in steps.
The test is carried out in a device called an Oedometer or a Consolidometer. This device has two main components: a container for the soil sample, known as the consolidation cell and a loading system
that applies pressure.
Top and bottom of the sample are equipped with porous stone discs. These discs enable two-way vertical drainage, allowing water to flow freely in and out of the soil.
The soil sample is carefully placed inside the consolidation cell. Then a load in placed on top of the cell. The soil is allowed to consolidate (settle) until practically no further compression
occurs. This ensures all excess water pressure within the pores is entirely dissipated. Typically, the load is maintained for a full day (that is 24 hours).
Then more load is applied on the soil and again soil is allowed to consolidate and dial gauge readings are noted.
Similarly load is gradually increased in steps and readings are noted.
After the soil has fully settled under the final load increment, we'll gradually reduce the load in two or three steps. After each unloading the sample is allowed to swell and expand. We note down
the readings after each unloading to track the amount of expansion.
Finally we place the sample in an oven for drying to determine the weight of solids and the final water content.
Our objective is to observe and determine the compression of soil that is taking place under the applied load. For the purpose we use void ratio as a measure to determine the amount of compression of
soil because It is primarily the voids that decrease in size, when a load is applied, leading to soil compression.
Therefore, we aim to determine how the void ratio changes with the application of load. And for that we plot a graph between void ratio and effective stress using the data obtained from the test. But
the problem is, test data does not give us directly the effective stress and void ratio for each load increment.
Effective Stress is just the pressure experienced by soil particles. Which is different from the pressure due to load on the surface of soil, as soil, including soil particles, also contains air and
water. They also support part of the load. Effective Stress has been described in detail in our previous post.
So, to calculate effective stress we use a known relationship between effective stress, total stress and pore water pressure.
σ' = σ – u
σ' = effective stress
σ = Total Stress and that can be calculated from the applied load and the area of the soil sample being tested
u = Pore water pressure and it is usually measured directly using a piezometer installed within the soil sample during the consolidation test
So with these known, we can calculate effective stress for each data point in the table.
Now we come to the void ratio. Void ratio, represented by “e”, of a soil sample is the ratio of the space occupied by the voids i.e. volume of voids to the space occupied by the solids i.e. volume of
To compute final void ratio after each load increment, one of the two methods is usually adopted :
1. Height of Solids Method and
2. Change in Void Ratio Method
Height of Solids Method
[0] and H[0] respectively. Height of solids is H[s]. Lets say after a few load increments its volume and height has become V and H respectively but height of the solids will remain the same. Now
using the 2 phase diagram we can write height of solids as volume of solids divided by area of the sample.
Using a known relationship :
volume of solids can also be written as weight of solids divided by specific gravity of solids and unit weight of water.
In the equation W[S], weight of solids can be determined by oven drying the sample. Specific Gravity of Solids can be obtained via pycnometer test or other standard methods. The Unit weight of water
is a known constant value. Therefore, all the quantities in this equation are known, so we can calculate height of solids. We keep this equation and name it as equation number A.
Now we know the void ratio is volume of voids divided by volume of solids
Volume of voids can be written as total volume of soil minus volume of solids.
Volume of sample can be written as area of sample multiplied by height of sample and volume of solids can be written as area of sample multiplied by height of solids.
by simplifying we obtain void ratio of the soil sample
In this equation we have already calculated the
H[S] = Height of solids by equation number A and it is going to remain constant
H = Height of the soil sample after load application when pore pressure dissipated and soil has reached equilibrium. It can be given as initial height of the specimen plus minus change in its height
due to either swelling or compression.
H[0] = Initial height of the specimen at the beginning of test
ΔH = is the change in thickness of sample under the load increment which can be read by dial gauge
With this, now we can calculate void ratios for all the load increments.
the other method of calculating the void ratio for different load increments is
Change in Void Ratio Method
In this method the void ratio of the saturated soil at the end of the test is calculated from a known relationship
Se = wG
here soil is saturated so degree of saturation is 1 so we can write
w[f] is the final water content at the end of the test which was obtained via oven drying method. Hence we can calculate the final void ratio of the soil sample using this equation.
And the void ratio at the end of each loading increment can be determined from definition of void ratio
we can write volume of voids as total volume of sample minus volume of solids.
and it can be simplified to this
and after a little re-arrangement we receive this
V = V[s] (1 + e )
Now volume of sample can be written as area of sample which is constant multiplied by height of sample after it has reached equilibrium
note this equation as equation number B. Only two quantities are variable here, so by partial differentiation of this we get this.
A dH = Vs de
now using the above equation B, we can write it as this
and after a little rearrangement we get this.
and for a small change we can write it as this also
This equation describes that when a soil sample is compressed under a load, change in its void ratio Δe and change in its height ΔH occurs. The sample attains new height H and void ratio e. Hence we
can also write them as e[f] and H[f] representing final void ratio and final height of the sample after consolidation.
Since the final void ratio ef and the sample height after consolidation at different loadings are known, we can calculate the change in void ratio (Δe) for each load increment directly using the
equation by working backwards from the known value of void ratio e[f].
With that, we now have both effective stress (σ[1], σ[2], σ[3] etc.) and final void ratio (e[f1] e[f2] e[f3]… etc.) data at different loadings for our consolidation test we can now plot the graph
between them.
We can see the curve generally has a downward trend, indicating that as the effective stress, that is pressure on the soil particles, increases, the void ratio decreases. This signifies the soil is
compacting due to the applied pressure and becoming denser.
This curve has concavity upward and the slope of the curve is not constant. It is steeper initially and becomes flatter as effective stress increases. This indicates that rate of void ratio reduction
decreases as the stress on soil particles increases.
If the soil is subjected to a certain stress increment, say Δσ, it will undergo some decrease in void ratio lets say equal to Δe[1]. It indicates there is some degree of densification in soil. Now If
the soil is subjected to the same magnitude of stress increment Δσ again, this time, the reduction in void ratio will not be equal to Δe[1], but it will be less than Δe[1], say it is Δe[2].
This demonstrates that as effective stress increases, the compressibility of the soil decreases. This relationship is quantified by the Coefficient of Compressibility, a[v], which defines the slope
of the void ratio – effective stress curve. For a small stress increment the curve can be approximated as straight line and Coefficient of Compressibility is defined as the decrease in void ratio per
unit increase in effective stress.
From this relationship and by looking at the graph we can say, the coefficient of compressibility decreases as effective stress increases. In simpler words soil becomes hard and less compressible as
the effective stress is increased.
Also as the effective stress increases, the void ratio decreases, hence the change in void ratio is negative. But the coefficient of compressibility is reported as positive for convenience, so
additional minus sign makes it positive.
We can also find out the units of coefficient of compressibility. Change in void ratio is unit less quantity and change is effective stress has the unit of pressure, that is, force per unit area,
which is kN/m^2. Hence unit of coefficient of Compressibility is m^2/kN.
The test data obtained are also represented on a semi-logarithmic graph, where the final void ratio is plotted on the linear axis and the effective stress is plotted on the logarithmic axis.
Interestingly, for soils that are being loaded for the first time, called normally consolidated soils, this graph often appears as a straight line.
Slope of this linear curve is called Compression index of soil (C[C])
or we can also write it as this
The compression index (Cc) of soil is a crucial indicator of how much a soil sample will compress under increasing pressure. Higher value of compression index indicates a more compressible soil. This
characteristic can also be visualized on a graph. As compression index of soil is higher, the curve will be steeper, which means for even a small increase in effective stress, the decrease in void
ratio is large. That clearly says the soil is more compressible.
Conversely lower value of compression index represents a less compressible soil, hence the soil is denser and there will be less settlement.
So we received lots of information from the curve between void ratio and effective stress. But now one may wonder, that these graphs only represents the increasing value of effective stresses, but
while doing the test we also unloaded our sample which reduced the effective stress on soil. Where is that part of the curve?
Well before discussing that part of the curve, we need to learn a little about normally consolidated soil and over consolidated soil and that we are going to discuss in our next post. | {"url":"https://elementaryengineeringlibrary.com/civil-engineering/soil-mechanics/compression-index-and-coefficient-of-compressibility-and-void-ratio-vs-effective-stress-curve","timestamp":"2024-11-14T14:14:39Z","content_type":"text/html","content_length":"40549","record_id":"<urn:uuid:bef6399f-52fc-401c-b75a-c3b78cc88796>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00678.warc.gz"} |
8 most common types of quantum error suppression methods | Description, Example & Application
8 most common types of quantum error suppression methods
Learn about the eight most common types of quantum error suppression methods, including quantum error correction, surface codes, and more.
8 Most Common Types of Quantum Error Suppression Methods
Quantum computers have the potential to solve problems that are currently intractable for classical computers, but the technology is still in its infancy. One of the major challenges in building
quantum computers is that they are inherently noisy, making it difficult to maintain the coherence of qubits (quantum bits) for the duration of a computation. This noise, caused by various factors
including environmental interactions and imperfect hardware, can lead to errors that must be corrected for the quantum computer to function effectively. There are several techniques used to mitigate
the effects of quantum errors, known as quantum error suppression methods. In this article, we will discuss the eight most common types of quantum error suppression methods.
1. Quantum Error Correction
Quantum error correction (QEC) is a technique that allows for the detection and correction of quantum errors. QEC is based on the use of redundancy, where multiple copies of a qubit are stored and
used to check for errors. If an error is detected, the information can be recovered from the redundant copies. There are several types of QEC codes, including the three-qubit and five-qubit codes,
which are commonly used in quantum computing.
2. Dynamical Decoupling
Dynamical decoupling (DD) is a technique that involves applying a sequence of carefully chosen pulses to a qubit to suppress the effects of noise. The pulses are designed to cancel out the effects of
the noise, and can be adjusted to target specific types of noise. DD is particularly effective at suppressing low-frequency noise, which is a common source of errors in quantum systems.
3. Quantum Feedback
Quantum feedback is a technique that involves continuously monitoring the state of a qubit and adjusting its state in real time to correct for errors. Feedback can be used to correct errors caused by
both internal and external noise sources. However, the feedback process itself can introduce errors, which must be carefully managed.
4. Topological Codes | {"url":"https://your-physicist.com/8-most-common-types-of-quantum-error-suppression-methods/","timestamp":"2024-11-07T11:15:41Z","content_type":"text/html","content_length":"54770","record_id":"<urn:uuid:af22d99c-ce68-4b8b-afa5-ab2a42ed7bf9>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00087.warc.gz"} |
Replace Negative Values with Zero in Excel & Google Sheets
Last updated on February 7, 2023
Download Example Workbook
Download the example workbook
In this tutorial we will demonstrate how to replace negative numbers with zeros in Excel & Google Sheets.
Replace Negative Numbers with Zero
Sometimes you don’t want negative values in your data. You can replace negative values with zero using one of the following three methods.
MAX Function
The MAX Function returns the maximum value from a set of numbers. Here we will use the MAX Function to calculate the max value between 0 and a calculated value.
For example, we have a table with heights values in Column B and C. Now, we need to calculate the difference between Heights 1 and Heights 2 but would like the results to only show positive values
and replace the negative values with zero. So to do that, we used the following formula:
IF Function
We can also use the IF function to force negative numbers to zero. The IF function tests a condition and returns a value depending on whether the condition is true or false.
Let’s say we have a set of data in column B with positive and negative values. Now, we need only positive numbers to show in column C.
To do this, we put the following formula in Cell C3 and autofill the rest of the cells:
Display Negative Values as Zeros
The above two methods not only display the negative value as zero but also changes the value to zero.
Instead, we can change the number formatting to 0;”0”;0. This will display negative numbers as zero.
Note: We use this method with extreme caution. It can cause great confusion in your workbook!
Change Number Formatting
1. Select a range of cells
2. Press CTLR+1 to access the Number Format Dialog Box.
3. Select Custom
4. Type 0;”0”;0 in the Type box and click the OK button
This changes the display of all the negative values to zero, maintaining the original cell value.
Replace Negative Numbers to Zero in Google Sheets
The formula to replace negative numbers to zero works exactly the same in Google Sheets as in Excel: | {"url":"https://www.automateexcel.com/formulas/negative-values-with-zero/","timestamp":"2024-11-05T12:43:05Z","content_type":"text/html","content_length":"145691","record_id":"<urn:uuid:f84848c1-03fb-4ac7-9651-5693272abb8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00545.warc.gz"} |
Folios 23 - 17
Calendar of Letter-Books of the City of London: B, 1275-1312. Originally published by Her Majesty's Stationery Office, London, 1900.
This free content was digitised by double rekeying. All rights reserved.
'Folios 23 - 17', in Calendar of Letter-Books of the City of London: B, 1275-1312, ed. Reginald R Sharpe( London, 1900), British History Online https://www.british-history.ac.uk/london-letter-books/
volb/pp15-31 [accessed 10 November 2024].
'Folios 23 - 17', in Calendar of Letter-Books of the City of London: B, 1275-1312. Edited by Reginald R Sharpe( London, 1900), British History Online, accessed November 10, 2024, https://
"Folios 23 - 17". Calendar of Letter-Books of the City of London: B, 1275-1312. Ed. Reginald R Sharpe(London, 1900), , British History Online. Web. 10 November 2024. https://www.british-history.ac.uk
Folio 23. [facieinversa.] (cviij b.)
Recognizances of debt temp. Luke de Haveryng, Chamberlain, from the Feast of St. Katherine [25 Nov.], 4 Edward II. [A.D. 1310]. (fn. 1)
ijs. iijd.
John de Cotome, skinner, came before Luke de Haveryng, Chamberlain, Friday after the Feast of St. Katherine [25 Nov.], 4 Edward II. [A.D. 1310], and acknowledged himself bound to William de Paris,
draper (draperio), in the sum of 20 marks; to be paid, viz., one moiety at the Feast of the Purification and the other at the Feast of the Annunciation; and unless, &c.
(Afterwards, on 6 April, anno 4, came Henry Faukes, the valet of the said William, and acknowledged satisfaction, and therefore the recognizance is cancelled.)
Wednesday after the same Feast came Richard Senglant, tailor, before the Chamberlain and acknowledged himself bound to Thomas Perceval in the sum of £4 6s. 8d.; to be paid at Christmas, and unless, &
Stephen le Mareschal, "armurer," acknowledged himself bound to Pelle de Luka, "kalendrer," (fn. 2) in the sum of £7 4s.; to be paid, viz., one moiety at Midsummer and the other at the Gule of August
(fn. 3) [1 Aug.] next ensuing; and unless, &c.
nichil quia aldermanno.
Tuesday the Feast of Conception B. M. [8 Dec.] came Stephen de Iseldon before the Chamberlain and acknowledged himself bound to William de Leyre, Alderman, in the sum of 20s.; to be paid at the wish
of the said William, and unless, &c.
Thursday after the Feast of Conception [8 Dec.] came Roger le Vyroler and acknowledged himself bound to Hugh de Hereford, saddler, in the sum of £10; to be paid at Hokeday, (fn. 4) and unless, &c.
ijs. ijd.
Monday the Feast of St. Thomas [21 Dec.] came Gilbert atte Herst and acknowledged himself bound to Richard, son of Adam le Charman de Rykelyng, in the sum of 20 marks; to be paid at the Feast of the
Purification, and unless, &c.
(Afterwards, viz., on Friday after the Feast of St. Luke, anno 5 Edward II., came the aforesaid Richard and acknowledged satisfaction.)
Tuesday the morrow of the Feast of St. Thomas [21 Dec.], 4 Edward II. [A.D. 1310], came Thomas le White and Roger le White and acknowledged themselves jointly and severally bound to Thomas de Foxton,
corder, in the sum of 60s.; to be paid by quarterly instalments of 7s. 6d., commencing at Midsummer; and unless, &c.
Folio 22 b (cix).
ijs. viijd.
Thursday the eve of the Nativity of our Lord [25 Dec.], 4 Edward II. [A.D. 1310], came Simon de Pourtepol, called "le Webbe," (fn. 5) and John de Paris, cordwainer, and acknowledged themselves
jointly and severally bound to Geoffrey de Conduit, Alderman, (fn. 6) in the sum of £16; to be paid at the octave of the Purification, and unless, &c.
nil dat quia clericus communitatis.
Friday after the Feast of Epiphany [6 Jan.], 4 Edward II. [A.D. 1310-11], came Roger le Viroler before Luke de Haveryng, the Chamberlain, and acknowledged himself bound to Hugh de Waltham, clerk,
(fn. 7) in the sum of 40s.; to be paid on the first Sunday in Lent, &c.; and unless, &c.
(He afterwards paid.)
nil quia clericus communitatis.
Saturday after the Epiphany, the same year, came Nicholas Pikot and acknowledged himself bound to the same in the sum of £30; to be paid at Midsummer, and unless, &c.
(He afterwards paid and is quit.)
Friday after the Feast of St. Hillary [13 Jan.] came Richard de Normanton, Thomas de Donewiche, and John de Honylane, skinners, and acknowledged themselves jointly and severally bound to John de
Preston and Nicholas de Rokele, corders, in the sum of £6 18s.; to be paid on the Sunday in mid-Lent, and unless, &c. Given for enrolment 14d.
(Afterwards, viz., on 2 April, came Richard de Normanton, and paid the aforesaid John and Nicholas 46s.)
Saturday after the Feast of St. Hillary [13 Jan.] came Richard de Bery, skinner, and acknowledged himself bound to Henry de Mire, merchant of Coventre, in the sum of £6 16s.; to be paid at the octave
of the Purification, &c.; and unless, &c. Pays for enrolment 14d. The said Henry puts in his place Roger de Clopton to receive the said money, &c.
condonatur per maiorem.
Thursday the morrow of St. Hillary [13 Jan.] came John de Kent, "cossoun," and Robert his son, before the Mayor and Chamberlain, and acknowledged themselves bound to Richard But, mercer, in the sum
of 20 marks; to be paid, viz., one moiety at Easter and the other at Michaelmas, &c., and unless, &c.
(Cancelled because Richard But acknowledged satisfaction.)
The same day came Thomas de Seeford, taverner, and acknowledged himself indebted to William Arnaud du Portaw in the sum of £6; to be paid, viz., one moiety at Pentecost and the other at Midsummer;
and unless, &c. The said William puts in his place John de Arrawe to receive the money.
ijs. vd.
The same day came Richard But before Richer de Refham, the Mayor, and other Aldermen, and the Chamberlain, and acknowledged himself bound to John Helle and his fellowexecutors of the will of William
Hasard in the sum of £14 13s. 4d.; to be paid in the quinzaine of Easter, and unless, &c.
Folio 22 (cix b).
Saturday before the Conversion of St. Paul [25 Jan.], 4 Edward II. [A.D. 1310-11], came John de Lincoln, William de Kent, tailor, Ralph de Garton called "Fox," and Bartholomew de Fisshebourne,
tailor, before Luke de Haverynge, the Chamberlain, and acknowledged themselves jointly and severally bound to John de Preston in the sum of £8 6s. 8d.; to be paid at Easter, &c.; and unless, &c.
(Afterwards, viz., on Thursday after the Feast of Translation of St. Thomas the Martyr [7 July], 5 Edward II. [A.D. 1311], John de Preston received the sum of 106s. 8d. of the above debt.)
The same day came John le Chaundeler de Colmanstret and acknowledged himself bound to John de Barneby, "peverer," (fn. 8) of "Sopereslane," and Philip de Farnham, peverer, of "Soperelane," in the sum
of £8 10s. for a cask of olive oil; to be paid at midLent, &c.; and unless, &c.
(Afterwards, viz., on Saturday the eve of Easter, came John de Barneby and acknowledged satisfaction, &c. It is therefore cancelled.)
The same day came Adam de Burton, skinner, and acknowledged himself bound to Robert Seely in the sum of £4 8s.; to be paid, viz., 20s. at the Feast of the Purification and 68s. at Easter; and unless,
Tuesday after the Conversion of St. Paul [25 Jan.], 4 Edward II. [A.D. 1310-11], came John de Lincoln, clerk, and Bartholomew de Fissheborne, and acknowledged themselves bound to William Edward in
the sum of 50s.; to be paid on Sunday before the Feast of Annunciation, and unless, &c.
Saturday before the Purification B. M. [2 Feb.], 4 Edward II. [A.D. 1310-11], came Laurence le Smyth and acknowledged himself bound to William de Leyre, Alderman, in the sum of 8s.; to be paid before
Easter, either in money or work (in pecunia vel in opere); and unless, &c.
Saturday after the Feast of Purification [2 Feb.] came Adam de Mondene, "batour," and acknowledged himself bound to Peter de Grenewych in the sum of 100s.; to be paid by quarterly instalments of 20s
., commencing at Easter; and unless, &c.
(Afterwards, viz., on Wednesday after Clausum Paschæ, (fn. 9) 5 Edward II., the said Peter came and acknowledged satisfaction, &c. It is therefore cancelled.)
Folio 21 b (cx).
Thursday after the octave of the Purification B. M. [2 Feb.], 4 Edward II. [A.D. 1310-11], came John de Pountoyse before Luke de Haveryng, the Chamberlain, and acknowledged himself bound to Walter
Lamereys de Luka in the sum of £8 12s.; to be paid at mid-Lent, and unless, &c.
(Afterwards, viz., on Friday before the Feast of St. Matthew, Ap, 5 Edward II., the said Walter came and acknowledged that he had received 100s. of the debt, and pledges were taken of the said John
for the remaining 72s., which were afterwards paid, and the pledges restored.)
The same day came Walter le Mazerer and acknowledged himself bound to "Getochius" Honeste de Luka in the sum of 59s. 8d.; to be paid at Easter, and unless, &c.
(Whereof the said "Gettochius" received 30s., as he himself testifies. And afterwards a woman's surcoat of red medley was seized of the value of 20s., and because the said Walter was warned to redeem
the said pledge and failed to do so, it was delivered to "Gettochius" for 20s.)
di. mr'.
12 Feb., 4 Edward II. [A.D. 1310-11], came Robert de Gloucestre and John le Mazerer, goldsmith, and acknowledged themselves severally bound to Master Gilbert le Mareschal in the sum of £40; to be
paid on Christmas Day two years, and unless, &c. Given for enrolment half a mark.
ijs. ijd.
13 Feb. came Roger de Rokesle, corder, before the Aldermen and Chamberlain and acknowledged himself bound to Walter Bullok, taverner, in the sum of 20 marks; to be paid at Easter, &c., and unless, &
Tuesday after the Feast of St. Valentine [14 Feb.] came John de Suffolk, taverner, and acknowledged himself bound to John de Preston, corder, in the sum of £4; to be paid on Sunday in mid-Lent, &c.;
and unless, &c.
Tuesday after the Feast of St. Valentine [14 Feb.] came Gregory (Geoffrey?) de Stebenhethe and acknowledged himself bound to "Goscelyn," servant (the serjeant?), in the sum of £6; to be paid at
Easter; &c., and unless, &c.
(Afterwards came John de Stebenhethe and Geoffrey de Stebenhethe, as appears on the fifth folio following, and made a recognizance for the above £6, to be paid to the said "Goscelin" at Christmas
next, and so this recognizance is cancelled.)
Thursday after the Feast of St. Valentine [14 Feb.] came John the Goldbeater (Aurimalliator) and acknowledged himself bound to Alan de Sutton, saddler, in the sum of 53s.; to be paid by quarterly
instalments of half a mark, commencing at Easter, until, &c.; and unless, &c.
The same day came John de Kent, butcher, and acknowledged himself bound to Simon de Mereworde, bureller, (fn. 10) in the sum of £10; to be paid on Sunday in mid-Lent, &c.; and unless, &c.
Folio 21 (cx b).
On Ash Wednesday [24 Feb.], 4 Edward II. [A.D. 1310-11], came Robert de Gloucestre and Andrew de Gloucestre, goldsmiths, and acknowledged themselves bound to John Sterre, fishmonger, in the sum of
£6; to be paid at Pentecost, &c.; and unless, &c.
Friday after the Feast of St. Matthias [24 Feb.] came Agnes de Raveneston and acknowledged herself bound to Richer de Refham, Mayor, in the sum of 18s.; to be paid at Easter, &c.; and unless, &c.
(Afterwards, viz., on Thursday before Easter, the said Richer received the money.)
iiijs. xd.
Tuesday after the Feast of St. Matthias [24 Feb.] came Walter de Stebenheth, chaloner, John le Keu, chaloner, Reginald le Chaundeler, "sauser," (fn. 11) and Richard de Bokhurst, "taillour," before
Henry de Durham (fn. 12) and Simon de Paris, (fn. 13) Aldermen, and Luke de Haverynge, the Chamberlain, and acknowledged themselves bound to William de Bidik in the sum of £29 8s. 7½ d.; to be paid
three weeks after Easter, &c.; and unless, &c.
The same day came Thomas Colly, "ceinturer," before the Chamberlain and acknowledged himself bound to John de Nasyng in the sum of 20s.; to be paid at Midsummer, and unless, &c.
William Fulk, "ceinturer," acknowledged himself bound to the aforesaid John in the sum of 10s.; to be paid at Midsummer, &c.
(Afterwards the said John came and acknowledged satisfaction, and therefore it is cancelled.)
Wednesday after the Feast of St. Matthias [24 Feb.] came Hugh le Armurer and acknowledged himself bound to Henry le Gaugeour in the sum of £6 13s. 4d.; to be paid at Pentecost, and unless, &c.
(Cancelled, because Henry le Gaugeour came here on Friday before the Feast of Nativ. St. John Bapt., anno 4 Edward II., and acknowledged satisfaction.)
4 March, 4 Edward II. [A.D. 1310-11], came Robert Brungor, "wodemongere," and acknowledged himself bound to Clarice, late wife of Martin Schenche, in the sum of 100s.; to be paid at Michaelmas, &c.;
and unless, &c.
condonatur per camerar'.
6 March came William de Stebenheth, "batour," and acknowledged himself bound to Richard Potrel (fn. 14) in the sum of 40s.; to be paid at Christmas, and unless, &c.
nil solvit quia condonatur per maiorem.
1 March, 4 Edward II. [A.D. 1310-11], came Anabilla la Beek before the Mayor, Aldermen, and Chamberlain, and acknowledged herself bound to John de Cantebrige in the sum of £16 11s.; to be paid on
Friday in the first week of Lent, &c.; and unless, &c.
Folio 20 b (cxj).
vs. vijd.
9 March came Godewyn le Pheliper before Henry de Durham, Simon de Paris, and Luke de Haverynge, Chamberlain, and acknowledged himself bound to William de Gaytone, "tabourer," (fn. 15) in the sum of
50 marks; to be paid at Michaelmas, and unless, &c. And he pays for enrolment 5s. 7d.
(Afterwards, viz., on Wednesday the Feast of St. James [25 July], 7 Edward II. [A.D. 1313], came the said William, and in the presence of the Chamberlain acknowledged satisfaction. The recognizance
is therefore cancelled.)
8 March came Nicholas le Palmere and acknowledged himself bound to William de Causton, mercer, in the sum of £10; to be paid at Easter, A.D. 1313, and unless, &c. And he pays for enrolment 20d.
The same day came William le Coffrer and acknowledged himself bound to John Sterre de Chabeham in the sum of 52s.; to be paid at Pentecost, and unless, &c.
(Afterwards, viz., on Thursday after the Feast of Decollation of St. John Bapt., anno 5 Edward II., came the aforesaid John and acknowledged satisfaction. It is therefore cancelled.)
Saturday before Palm Sunday [4 April] came John Dereman, butcher, before the Mayor and Chamberlain and acknowledged himself bound to William Polyt in the sum of 24s.; to be paid one month after
Easter, and unless, &c., habet billam.
3 April, 4 Edward II. [A.D. 1311], came Roger de Rokesle, junior, and John Vyel before the Mayor, Aldermen, and Chamberlain, and acknowledged themselves jointly and severally bound to Walter Bullok,
vintner, in the sum of £10; to be paid, viz., one moiety at Pentecost and the other at Midsummer; and unless, &c.
(Afterwards the said Walter came before the Chamberlain and acknowledged satisfaction. The recognizance is therefore cancelled.)
6 April came Thomas de Hales, woodmonger (buscarius), and acknowledged himself bound to Thomas de Foxtone in the sum of 60s.; to be paid, viz., 20s. at Midsummer and 40s. at Michaelmas; and unless, &
7 April came Philip de Merdele before the Mayor, John de Wengrave, and the Chamberlain, and acknowledged he owed John Mire, draper, the sum of 18s.; to be paid one month after Easter, &c.
Folio 20 (cxj b).
7 April came Robert le Sauser before Richer de Refham, Mayor, and Luke de Haverynge, Chamberlain, and acknowledged himself bound to Thomas, son of Adam de Fulham, in the sum of £4; to be paid on the
Feast of St. Peter ad Vincula, and unless, &c.
The same day came John de Waledene, "seler," before William de Leire, Alderman, and the Chamberlain aforesaid, and acknowledged himself bound to William de Forsham in the sum of 20s.; to be paid at
the Ascension, and unless, &c.
The same day came Roger Alan, "boucher," and acknowledged he owed Alan Sprot the sum of 2 marks; to be paid at Midsummer, and unless, &c.
11 May, 4 Edward II. [A.D. 1311], came John, son of Peter Bussh, and acknowledged himself bound to Alexander le Settere (fn. 16) in the sum of 46s. 8d.; to be paid at Michaelmas, and unless, &c.
12 May, 4 Edward II. [A.D. 1311], came Nicholas Maderman before the Aldermen and Chamberlain and acknowledged himself bound to John de Lyndeseye, "chaundeler," in the sum of 70s.; to be paid at the
Feast of St. Bartholomew, and unless, &c.
nil quia aldermanno.
18 May came Henry Faukes and Seman de Waledene and acknowledged themselves jointly and severally bound to Nicholas Pikot, Alderman, (fn. 17) in the sum of 100s.; to be paid, viz., one moiety at the
Feast of St. Peter ad Vincula and the other at Michaelmas, &c.; and unless, &c. And be it remembered that the said Nicholas has also a letter obligatory of the aforesaid Henry and Seman for the said
debt, which letter he agrees shall be void on payment of the money.
(Afterwards the said Seman satisfied the said Nicholas of the above debt except the sum of 25s. still in arrears. On the Feast of St. Thomas, anno 10 Edward II., the said Nicholas came and
acknowledged satisfaction for 15s., and there remained 10s. Afterwards the said Seman paid H[ugh] de Waltham, executor of the said Nicholas, the sum of 10s. and is quit.)
ijs. iijd.
27 May, 4 Edward II. [A.D. 1311], came John le Chaundeler de Colmanstrete and acknowledged himself bound to Peter de Bolyntone in the sum of 20 marks; to be paid at Midsummer, and unless, &c.
Friday before the Feast of H. Trinity [6 June] came William le Coffrer and acknowledged himself bound to William de Canefeld in the sum of 40s.; to be paid at Midsummer, and unless, &c.
Folio 19 b (cxij).
Thursday after the Feast of H. Trinity [6 June], 4 Edward II. [A.D. 1311], came Roger Houmfrey, "peleter," and acknowledged himself indebted to Nicholas de Whyttone in the sum of 30s. 5d.; to be paid
at Midsummer.
Thursday after the Feast of SS. Peter and Paul [29 June] came John de Thorp, "peleter," and Hugh de Hereford, "peleter," and acknowledged themselves jointly and severally bound to William de
Cornehill in the sum of 4 marks; to be paid by quarterly instalments of 1 mark, commencing at Michaelmas.
The following day came William le Platier, "armurer," and acknowledged himself bound to John de Grenewych, mercer, in the sum of 4 marks; to be paid at the Feast of All Saints.
The same day came Roger de Nettlestede, "peleter," and John de Eynesham, "peleter," and acknowledged themselves severally bound to "Chatus Meiccaldi" and Reyner Bon compaigne de "Cene"(?) in the sum
of £6; to be paid at Michaelmas, and unless, &c.
(Afterwards, viz., on Tuesday before the Feast of St. Bartholomew, anno 8 Edward II., came the aforesaid "Chatus" and acknowledged satisfaction. The recognizance is therefore cancelled.)
The same day came Simon de Broughtone, "peleter," and acknowledged himself bound to Walter le Fundour in the sum of £6; to be paid on Sunday before the Feast of St. Margaret, &c., and unless, &c. And
the aforesaid Walter puts in his place Peter de Evendene to receive the money in his absence.
Saturday after the Feast of SS. Peter and Paul [29 June], 4 Edward II. [A.D. 1311], came John le Mazelyner, "peverer," before Richer de Refham, Mayor, the Aldermen, and the Chamberlain, and
acknowledged himself bound to the Mayor and Commonalty of the said City in the sum of £10; to be paid, viz., one moiety at Michaelmas and the other at Midsummer; and unless, &c.
nihil quia alderman'.
Monday the morrow of St. James [25 July], 5 Edward II. [A.D. 1311], came John le Despencer before the Chamberlain and acknowledged himself bound to John de Wenegrave (fn. 18) in the sum of 50s.; to
be paid on the octave of the Assumption B. M., and unless, &c.
Folio 19 (cxij b).
Monday the morrow of the Translation of St. Martin [4 July], 4 Edward II. [A.D. 1311], came John le Botoner, junior, son of John le Botoner, deceased, and acknowledged himself bound to "Baudechono"
(fn. 19) le Chaucer in the sum of 26s.; to be paid in the quinzaine of Michaelmas, and unless, &c.
Monday the morrow of St. Peter ad Vincula [1 Aug.], 5 Edward II. [A.D. 1311], came John de Colkyrk before the Chamberlain and Aldermen and acknowledged himself bound to Thomas de Abyndone, draper, in
the sum of £6; to be paid on the Feast of All Saints.
Monday the morrow of St. Peter ad Vincula [1 Aug.] came Robert Lorechoun, potter (ollator), before the Chamberlain and Aldermen and acknowledged himself bound to John de Per stone, corder, in the sum
of 59s. for copper bought; to be paid within one month, and unless, &c.
Wednesday before the Feast of St. Bartholomew [24 Aug.], 5 Edward II. [A.D. 1311], came John le Chaundeler before the Mayor and Aldermen and acknowledged himself bound to Luke de Haveryngge (fn. 20)
in the sum of 15s. 6d.; to be paid at Michaelmas, and unless, &c.
Saturday after the Feast of Assumption B. M. [15 Aug.] came Robert de Gloucestre, goldsmith, before the Chamberlain and acknowledged himself bound to Simon Bolet, Alderman, (fn. 21) in the sum of £8;
to be paid at the Feast of St. Andrew.
Thursday after the Feast of St. Bartholomew [24 Aug.], 5 Edward II. [A.D. 1311], came Richard de Lothebery and Gunnora his wife before the Chamberlain and acknowledged themselves bound to Robert de
Derby, woolmonger, in the sum of 100s.; to be paid, viz., one moiety next Michaelmas three years, and the other moiety at Michaelmas following.
The same day came John de Waledene, "seler," and acknowledged himself bound to John de Dittone in the sum of 5 marks; to be paid at the Feast of All Saints, and unless, &c.
Folio 18 b (cxiij).
Tuesday after the Feast of Decollation of St. John Bapt. [29 Aug.], 5 Edward II. [A.D. 1311], came Roger de Staneworth, taverner, before Luke de Haveryng, Chamberlain, and acknowledged himself bound
to Walter de "Canefeld" in the sum of £4; to be paid at the Feast of All Saints.
The same day came Roger de Roqesle, junior, and acknowledged himself bound to Walter de "Kanefeld" in the sum of 40s.; to be paid at the Feast of All Saints.
nil quia aldermannus.
The same day came John de Bikeleswade, "cordewaner," and acknowledged himself bound to William de Leyre, Alderman, (fn. 22) in the sum of 48s.; to be paid at the Feast of All Saints.
Wednesday after the Feast of Decollation of St. John Bapt [29 Aug.] came Alan Sprot, "batour," and acknowledged himself bound to Robert de Lenne, vintner, in the sum of 5½ marks; to be paid at the
Feast of All Saints.
The following day came William le Coffrer and acknowledged himself bound to John Sterre de Chabeham in the sum of £5½; to be paid, viz., 40s. at Michaelmas, and 70s. at the Feast of SS. Simon and
Friday before the Feast of Nativity B. M. [8 Sept.], 5 Edward II. [A.D. 1311], came Roger le Graunt "barbier" and acknowledged himself bound to William de Uptone in the sum of £6; to be paid a
fortnight after Michaelmas.
The same day came John Henry, butcher, and acknowledged himself bound to Robert Mussebrom in the sum of 20s., to be paid at Christmas.
Monday before the Nativity B. M. [8 Sept.] came Sewald, son of Sewald de Springefeld, before R[icher] de Refham, Mayor, and Luke de Haveringge, Chamberlain, and acknowledged himself bound to Richard
Godesname, "paternoster," (fn. 23) in the sum of 40s.; to be paid by quarterly instalments of 5s., commencing at Michaelmas.
Folio 18 (cxiij b).
Monday before the Nativity B. M. [8 Sept.], 5 Edward II. [A.D. 1311], came Walter de Canefeld and acknowledged himself bound to John de Haveryng in the sum of £4; to be paid at the Feast of All
(Afterwards, viz., on Thursday after the Feast of St. Peter ad Vincula, anno 6 Edward II., came the aforesaid Walter and John before the Chamberlain and together rendered account of certain rents of
the aforesaid Walter, who had by execution of the Chamberlain delivered them over to the said John until the said debt should be levied. And after due account and allowance had been taken and made,
the said John had received the said sum of £4. Thereupon each released the other.)
nil dat quia alderm'.
Wednesday the morrow of the Exaltation of H. Cross [14 Sept.] came Ralph le Taverner of Billinggesgate, residing in the parish of St. Dunstan towards the Tower, and acknowledged himself bound to
William de Leire, Alderman, in the sum of £15; to be paid at the Feast of the Purification.
(Afterwards the aforesaid William came and acknowledged satisfaction. The recognizance is therefore cancelled.)
Friday before the Feast of St. Michael [29 Sept.] came Henry le Coupere, blader, residing in Langebournewarde, and acknowledged himself bound to Roger le Paumer, blader, in the sum of £10; to be
paid, viz., one moiety at Christmas and the other at Easter.
nil quia alderm'.
The following day came Robert de Uptone, chaucer, and acknowledged himself bound to Thomas Romayn, Alderman, (fn. 24) and Juliana his wife, in the sum of £20; to be paid, viz., one moiety at the
Feast of All Saints and the other at Easter.
(He afterwards paid and is quit.)
Monday after the Feast of St. Michael [29 Sept.] came Laurence le Smyth before the Mayor and Aldermen and acknowledged himself bound to Luke de Haveryng, Chamberlain, in the sum of 44s.; to be paid,
viz., 24s. at the Feast of All Saints, and 20s. at Christmas.
Saturday before the Feast of St. Edward, K. [13 Oct.], 5 Edward II. [A.D. 1311], came Thomas le Verrer before the Mayor and Aldermen and acknowledged himself bound to the same in the sum of 17s.; to
be paid at the Feast of All Saints.
(Afterwards the said Thomas paid 12s.)
[facie inversa.]
Tuesday after the Feast of Conversion of St. Paul [25 Jan.], 2 Edward II. [A.D. 1308-9], came William de Wigindenne and acknowledged himself bound to Richard Poterel, Chamberlain, in the sum of 46s.
8d.; to be paid at Easter.
(He afterwards paid 26s. 8d.)
Folio 17 b (cxiiij).
nil quia, etc.
The same day (viz, Saturday before the Feast of St. Edward) came Ralph Sporon, Waryn le Spencer, Richard Aldbright, and Ralph Barn, tailor, before Luke de Haverynge, Chamberlain, and acknowledged
themselves jointly and severally bound to John de Stayntone, clerk, in the sum of 40s.; to be paid by quarterly instalments of 10s., commencing at Christmas.
(Whereof 10s. are paid at the first term. Afterwards came Ralph Barn and paid 10s. for his share, and the aforesaid John gave him a release, reserving his right of action against the others.)
Wednesday the Feast of St. Edward, K. [13 Oct.], 5 Ed ward II. [A.D. 1311], came Richard de St. Alban and acknowledged himself bound to John de Dittone in the sum of £20; to be paid at Michaelmas.
(Afterwards, viz., on Friday before the Feast of St. Martin, anno 7 Edward II., at the suit of the said John, an inquest and extent were made of the tenements and rents of the aforesaid Richard on
the day of the above recognizance, whereby it was found that the said Richard had £8 4s. in divers rents, whereof one moiety was delivered to the aforesaid John to hold by way of frank tenement
until, &c.
Memorandum that John de Dittone came before John Dode, Chamberlain, on Monday before the Feast of St. Cuthbert [20 March], 7 Edward II. [A.D. 1313-14], and acknowledged that Peter de Grenewich had
satisfied him of 2½ marks rent recently delivered to him by the Chamberlain out of an annual rent of 5 marks belonging to Richard de St. Alban for a tenement held by John de Redyng, cordwainer, in
Ismongerlane, so that the said John de Dittone can make no further claim thereon, but saving his right to other rents delivered to him, as appears by the above extent, &c. Afterwards, viz., on 3
April, the same year, came the aforesaid John and acknowledged full satisfaction.)
Thursday before the Feast of St. Luke [18 Oct.], 5 Edward II. [A.D. 1311], came Henry Poteman before the Chamberlain and acknowledged himself bound to Robert Tourk in the sum of £6; to be paid at the
Feast of the Purification.
The following Saturday came Emma, wife of John le Hattere, and Ralph, son and heir of the said John, before Geoffrey de Conduit and Nicholas Picot, Aldermen, and Luke de Haverynge, Chamberlain, and
acknowledged themselves severally bound to Thomas de la Haye, valet of Master Thomas Perot, in the sum of £20; to be paid at Easter, &c.
(The said John [sic] puts in his place Richard Swete to receive and give acquittance for the money as if he were present, &c.)
Monday the Feast of St. Luke [18 Oct.] came John le Kyng, butcher, and acknowledged himself bound to John de Boreham in the sum of 10 marks; to be paid, viz., one moiety at the Feast of the
Purification and the other at Pentecost.
Tuesday after the Feast of St. Luke [18 Oct.] came John de Stebenhethe, fishmonger, and Geoffrey de Stebenhethe, brewer of Garscherche, before Richer de Refham, Mayor, and William de Leire, Alderman,
and Luke de Haverynge, Chamberlain, and acknowledged themselves bound to Goscelin, servant (serjeant ?), in the sum of £6; to be paid at Christmas.
The following Thursday came William de York, goldsmith, before the Chamberlain and acknowledged himself bound to Adam de Smalecombe in the sum of £8; to be paid at the Feast of St. Hillary, &c. And
the aforesaid Adam puts in his place Eustace de Crenewelle to act as if he himself were present, &c.
Folio 17 (cxiiij b).
The following day came William de Hedersete, mercer, before Richer de Refham, Mayor, John de Wengrave and Simon de Paris, Aldermen, and Luke de Haverynge, Chamberlain, and acknowledged himself bound
to Nicholas Picot, mercer, in the sum of £6 16s.; to be paid, viz., one moiety at Christmas and the other at Easter, &c. And the aforesaid William agreed to indemnify Nicholas Picot against John
Hautayn, his apprentice, touching the sum of £13 which the said John demanded for a debt due to him by Nicholas le Gode, and for which he had sued the said Nicholas Picot in the Sheriffs' Court.
Wednesday before the Feast of St. Martin [11 Nov.], 5 Edward II. [A.D. 1311], came William de York, called "Everard," and Robert de Gloucestre, goldsmith, and acknowledged themselves severally bound
to Gilbert Cros in the sum of 12 marks and 8s.; to be paid at Easter, &c.
Here begin recognizances of debts temp. John le Mazelyner, Chamberlain, anno 5 Edward II.
Monday after the Feast of St. Edmund, K. [20 Nov.], 5 Edward II. [A.D. 1311], came Roger de Haneworth, taverner, and acknowledged himself bound to Peter, son of Peter Busshe, in the sum of £4 13s. 4d
.; to be paid at Easter, &c.
(Afterwards came Peter Busshe and put in his place Stephen le Nailere to demand and receive, &c.)
The same day came Osbert de Heddesworth, pleader (narrator), and acknowledged himself bound to Richard le Spicer of Westminster in the sum of 40s.; to be paid at the Feast of St. Hillary.
(Afterwards, viz., on 21 Aug., anno 7 Edward II., came the said Osbert and acknowledged himself bound to the said Richard in the sum of 7s.; to be paid at Michaelmas, &c.)
Monday the eve of St. Andrew, Ap. [30 Nov.], 5 Edward II. [A.D. 1311], came John de Boseworth, "cossoun," and acknowledged himself bound to Geoffrey (?) le Lacier in the sum of 40s.; to be paid,
viz., one moiety at the Feast of the Purification and the other at Easter, &c.
Saturday before the Feast of St. Nicholas [6 Dec. ] came Nicholas de Gren, butcher, and acknowledged himself bound to Hugh de Garton, mercer, in the sum of 100s.; to be paid at the Feast of
Purification, &c. | {"url":"https://www.british-history.ac.uk/london-letter-books/volb/pp15-31","timestamp":"2024-11-10T11:17:10Z","content_type":"text/html","content_length":"54071","record_id":"<urn:uuid:bfdafd19-e113-450f-8d24-10aeddf92ece>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00414.warc.gz"} |
Turning an expression into a multivariate polynomial?
Turning an expression into a multivariate polynomial?
Suppose I have the expression
-a^3x^2 - a^2xy^2 + axy + bx^2 + 2bxy + xy^2
I want to turn this into a polynomial in x and y:
so that I can then extract the coefficients (by setting x=1, y=1 in the list of operands of the new expression). How do I tell Sage which of the four variables will be the polynomial variables?
1 Answer
Sort by ยป oldest newest most voted
If you just want to evaluate your symbolic expression by setting x=1, y=1, you can directly do:
sage: f(x=1,y=1)
-a^3 - a^2 + a + 3*b + 1
Otherwise, you can try:
sage: R.<x,y> = PolynomialRing(SR,2) ; R
Multivariate Polynomial Ring in x, y over Symbolic Ring
sage: var('a,b')
sage: P = -a^3*x^2 - a^2*x*y^2 + a*x*y + b*x^2 + 2*b*x*y + x*y^2
sage: P.parent()
Multivariate Polynomial Ring in x, y over Symbolic Ring
sage: P(1,1)
-a^3 - a^2 + a + 3*b + 1
But you should notice that x and y are not symbolic expression, and are declared as polynomials before P is defined.
If the symbolic function is given as a symbolic expression and you want to make it a polynomial afterwards, the best way i found is to tranform it first as a polynomial in 4 variables, and then set a
and b back to the Symbolic Ring:
sage: var('a,b,x,y')
(a, b, x, y)
sage: f = -a^3*x^2 - a^2*x*y^2 + a*x*y + b*x^2 + 2*b*x*y + x*y^2
sage: P = f.polynomial(QQ)
sage: P.parent()
Multivariate Polynomial Ring in a, b, x, y over Rational Field
sage: R.<x,y> = PolynomialRing(SR,2) ; R
Multivariate Polynomial Ring in x, y over Symbolic Ring
sage: Q = P(var(a),var(b),x,y)
sage: Q.parent()
Multivariate Polynomial Ring in x, y over Symbolic Ring
sage: Q
(-a^2 + 1)*x*y^2 + (-a^3 + b)*x^2 + (a + 2*b)*x*y
sage: Q(1,1)
-a^3 - a^2 + a + 3*b + 1
edit flag offensive delete link more | {"url":"https://ask.sagemath.org/question/10198/turning-an-expression-into-a-multivariate-polynomial/","timestamp":"2024-11-10T20:29:13Z","content_type":"application/xhtml+xml","content_length":"54262","record_id":"<urn:uuid:8ade0561-9fd6-46d8-b948-3b4127bc2f3a>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00050.warc.gz"} |
Polygon Riddle 2
Try the formula for a rectangle and you get 360 degrees.
And yes it's true for all polygons. External angles are different they add up to 360.
and the interior angles depends on how many sides the shape has,
a heptagon has 7 sides and therefore it can be split into 5 triangles
a triangle has 180 degrees, so 180*5 = 900.
You can work out from line one that the first letter of the answer occurs in the word ‘shape’ but it does not occur in the word ‘space'. By process of elimination you can see that the first letter of
the answer must be H.
Work through the other lines of the riddle to determine the other letter possibilities and then you are well on your way to solving the riddle.
My first is in shape but not in space;
My second is in line and also in place;
My third is in point but not in line;
My fourth is in operation but not in sign;
My fifth is in angle but not in degree;
My sixth is in glide but not symmetry;
My seventh is in round but not in square;
My last is in patterns you see everywhere;
My whole is a polygon, regular not wide;
But what is the sum of the angles inside?
Hint: split the shape into triangles
Sign in to your Transum subscription account to see the answers
Your access to the majority of the Transum resources continues to be free but you can help support the continued growth of the website by doing your Amazon shopping using the links on this page.
Below is an Amazon link. As an Amazon Associate I earn a small amount from qualifying purchases which helps pay for the upkeep of this website.
Educational Technology on Amazon
Teacher, do your students have access to computers such as tablets, iPads or Laptops? This page was really designed for projection on a whiteboard but if you really want the students to have access
to it here is a concise URL for a version of this page without the comments:
However it would be better to assign one of the student interactive activities below.
Here is the URL which will take them to a related student activity.
Here is the URL which will take them the Transum Riddle collection. Enjoy!
Curriculum Reference | {"url":"https://www.transum.org/Software/SW/Starter_of_the_day/starter_july28.ASP","timestamp":"2024-11-02T18:36:11Z","content_type":"text/html","content_length":"28237","record_id":"<urn:uuid:1aabde10-ff57-4b1f-a181-c5082f111fe6>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00356.warc.gz"} |
Portfolio Theory | Sven Karbach
In this course, we delve into the stochastic foundations of mathematical finance, with a focus on the general problem of pricing and hedging European contingent claims and portfolio optimization. We
rigorously define the concept of a financial market and establish two fundamental relationships: one between arbitrage-free markets and the existence of equivalent martingale measures, and the other
concerning the completeness of a market and the uniqueness of the equivalent martingale measure. We explore various methodologies for pricing and hedging contingent claims in both complete and
incomplete markets, including risk-neutral, variance-optimal, and utility and risk-indifferent approaches. We also examine how market participants' preferences can be quantified using utility
funcitons and risk measures, and how these concepts are applied in portfolio optimization. Furthermore, we discuss the formulation of consumption-investment problems as stochastic optimal control
problems and introduce dynamic programming as a general solution method. This course provides students with a thorough understanding of the core principles of mathematical finance and their
application in financial decision-making and risk management under uncertainty.
Aim and objectives
The general objective of this course is to equip students with a deep understanding of the stochastic foundations of mathematical finance in a finite discrete-time setting, with a particular focus on
portfolio selection and optimization techniques. Specifically, students will:
• Develop the ability to rigorously prove a carefully chosen set of theorems that are central to mathematical finance.
• Demonstrate mastery of the theory through practical assignments, where they will apply theoretical concepts to solve problems related to the pricing and hedging of European contingent claims or
portfolio optimization.
Specific objectives to be met at the end of the course:
• Students will gain a solid understanding of the core principles of financial mathematics. In particular, students know the two fundamental theorems of asset pricing; know how to price European
contingent claims using risk-neutral pricing and are familiar with the differences between complete and incomplete markets.
• Students will acquire a comprehensive understanding of the pricing and hedging problem in an incomplete market setting and know strategies to address and solve pricing and hedging problems in
this context.
• Students will develop a thorough understanding of expected utility theory and risk measures, enabling them to apply these concepts effectively for optimizing portfolios.
• Students know how to construct variance optimal hedging strategies.
• Students know how to apply dynamic programming to investment-consumption problems.
"Portfolio Theory" is generally structured as a second-year course in the Master's programmes of Mathematics and Stochastics and Financial Mathematics at the University of Amsterdam. Students should
be familiar with fundamental concepts of probability and measure theory, as covered in the "Measure Theoretic Probability" course.
A set of lecture notes will be the main source for this course.
Recommended book: Stochastic Finance - An Introduction in Discrete Time by H. Föllmer and A. Schied.
Examination Details
• Assessment Components: Mandatory take-home exercises and an oral examination.
• Homework:
□ All submissions must be made within two weeks of being assigned.
□ Collaborative work in pairs is compulsory.
□ Submit homework via the course's Canvas page.
• Oral Examination:
□ Assesses comprehension of key definitions, results (like lemmas and theorems), and four chosen theorems with their proofs.
□ Scheduling will be available by early December 2024.
• Final Grades:
□ 70% - Oral examination
□ 30% - Assignments | {"url":"https://sven-karbach.de/portfoliotheory.html","timestamp":"2024-11-06T13:49:47Z","content_type":"text/html","content_length":"16143","record_id":"<urn:uuid:70c9a709-6e10-4f1f-9d81-2e4e527fd2d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00543.warc.gz"} |
How To Calculate The Operating Cash Flow
theanisenkova.ru How To Calculate The Operating Cash Flow
How To Calculate The Operating Cash Flow
It shows how much cash a company is generating from its day-to-day operations. The OCF can help investors identify any potential short-term liquidity issues a. How to Calculate Your Operating Cash
Flow Ratio. To calculate your company's operating cash flow ratio, use the following formula: Operating cash flow = net. In the statement of cash flows, operating net income is reconciled to cash by
adding back and subtracting the various cash impacts of operating activities. Why. The International Financial Reporting Standards defines operating cash flow as cash generated from operations and
investment income less taxation, interest and. Operating Cash Flow Formula (OCF) = Net Income + Depreciation + Deferred Tax + Stock-oriented Compensation + non-cash items – Increase in Accounts
Receivable –.
Knowing how to calculate cash flow for your business can be difficult. You calculate it by subtracting operating expenses from operational revenue. An example of calculating operating cash flow ·
Subtract the increase in accounts receivable: £, - £50, = £, · Subtract the increase in inventory. 1. Calculate the after-tax operating profit. · 2. Add back depreciation and amortisation. · 3.
Calculate the increase or decrease in receivables. · 4. Deduct the. Operating cash flow is equal to revenues minus costs, excluding depreciation and interest. Depreciation expense is excluded because
it does not represent an. Operating cash flow, which you may often see reflected simply as “OCF”, is a measure of the amount of cash generated by the normal business operations of a. It is calculated
by adjusting net income, as reported on the income statement, for non-cash items and other changes in working capital. Operating cash flow is. What is Cash Flow from Operations? · Cash Flow from
Operations = Net Income + Non-Cash Items + Changes in Working Capital · Step 1: Start calculating operating. Operating Cash Flow · Operating cash flow is a measure of how much money your company
generates from the core activities it engages in. · It is a measure of. Calculating cash flow from operations is easy. All you have to do is subtract your taxes from the sum of depreciation, change
in working capital, and operating. Calculating Operating Cash Flow · Start with the net income · Add all non-cash items such as deferred taxes, stocks, depreciation, and other income or expenses.
Operating cash flow is a measure of a company's net income (NI) from its primary business operations. Operating cash flow, also known as cash flow from. The OCF calculation will always include the
following three components: 1) net income, 2) plus non-cash expenses, and 3) minus the net increase in net working. Operating Cash Flow Formula · When looking at how to calculate operating cash flow,
there are two methods you can use to calculate it: the direct method and the. Standard Operating Cash Flow Example · Revenue (cash received from sales) is $75, · Operating expenses (for food,
supplies, equipment) is $35, In this. The operating cash flow ratio is a measure of the number of times a company can pay off current debts with cash generated within the same period. A high number.
An OCF that is consistently higher than net income suggests your business effectively converts its profits into cash. This is a sign of good financial health. Steps to Calculating Indirect Operating
Cash Flow · Operating Cash Flow = · Net Income (Revenue – Cost of Sales) · Depreciation · +/- · +/-. The top-down formula to calculate the business's operating cash flow comes in three parts. Your
first calculation: Sales - expenses - depreciation = EBIT. Then. To calculate cash flow, you typically subtract business expenses from business profits. However, the formula will vary based on the
type of cash flow you're.
The cash flow formula is simple—it's all the capital you earn minus everything you spend across all of your operating activities. Our guide will help you create. Operating cash flow = Operating
income + Depreciation – Taxes + Change in working capital A chart showing indirect method and direct method. Under the. To calculate operating cash flow, subtract 'operating expenses' (such as
payroll, marketing investment, rent, etc.) from the 'total revenue' (from product/. Operating cash flow (OCF), often called cash flow from operations, is an efficiency calculation that measures the
cash that a business produces from its. It is calculated by subtracting capital expenditures (the money spent on assets like new equipment) from net income. This means that operating cash flow does. | {"url":"https://theanisenkova.ru/learn/how-to-calculate-the-operating-cash-flow.php","timestamp":"2024-11-13T15:18:09Z","content_type":"text/html","content_length":"11912","record_id":"<urn:uuid:3586879d-d297-4f28-8b23-5b1b0e5cbb91>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00447.warc.gz"} |
AFM Exclusive
22 Mar 2024
Nano-scale Topography Measurements by AFM: A Note on Repeated Height and Roughness Analyses
Surface properties at the nano-scale With the evolution of nanotechnology, the need to understand nano-scale and sub nano-scale surface properties is becoming increasingly important. Particularly,
surface properties such as topography and roughness play an important role in the overall performance of a material. For example, in integrated circuits manufacturing, chemical mechanical polishing
(CMP) has been used to control the surface roughness of wafers and other substrates, which directly determines the reliability of final products [1,2]. In semiconductor devices packaging, wafer to
wafer bonding quality is determined by the roughness of bonding surfaces. It was found that surfaces with high roughness values introduce voids in the bonded interface, and bonding failure occurs
when the roughness exceeds a critical threshold [3]. In the field of scientific research, surface topography can be correlated with other material properties to better understand material behaviors
for practical applications [4–6]. To assess the surface topography of a sample, several methods are available, including stylus profilometry, coherence scanning interferometry, laser microscopy, and
atomic force microscopy (AFM). The choice of method depends on the specific characteristics of the sample and the goals of the measurement. In the realm of semiconductor and electronic devices
manufacturing, where in-depth insights into nano-scale surface properties are imperative, AFM has been widely used owing to its ability to provide three-dimensional data with sub-nanometer resolution
capabilities. In addition, the scanning probe-based methodology of AFM facilitates simultaneous measurements of the surface topography as well as various electrical, frictional, and mechanical
properties of the sample, thereby enhancing overall efficiency and throughput. Measuring the surface properties using AFM AFM has been widely adopted by industrial chip manufacturers and academic
researchers for nano-scale surface measurement of various materials. The AFM can be implemented in in-line and off-line measurements for quality control and inspection of semiconductor and display
manufacturing processes. Surface roughness inspection of bare wafers [7], surface roughness measurement after CMP [1,2], wafer bonding process monitoring [3,8], and hard disk media defect inspection
and review [7] are typical examples of surface measurements based on AFM. In academic research, surface measurement is related to investigation of nanomaterials [9], correlative study of surface and
other material properties [4], or studies of effects of environmental factors on surface properties of material [5,6]. Given that AFM uses a nanoscopic tip to scan the sample surface, it is crucial
to understand interactions between the AFM tip and the sample to appropriately analyze and interpret the data. This whitepaper reviews AFM methodologies that are typically used to investigate the
topography and roughness of a samples. In particular, AFM data analysis and interpretation procedure, along with related experimental issues will be discussed for an accurate reconstruction of the
sample surface. AFM scan modes to measure the surface To characterize a surface using the AFM, different scan modes are available such as Contact, Non-contact, and Tapping modes. The scan mode should
be selected in consideration of material characteristics and measurement targets. Contact mode was the very first introduced scan mode of AFM that is used to reconstruct the sample surface [10].
Contact mode uses a relatively soft cantilever to raster scan the sample surface. The deflection of the cantilever during scanning provides information on the surface topography. A major drawback of
contact mode is exerted shear force during scanning which can cause damage to the tip apex and the sample, especially when scanning soft samples such as polymers and biological materials. To minimize
tip and surface damage due to contact scanning, Non-contact mode was developed [11]. In this mode, the tip oscillates in close proximity to the surface in the van der Waal attractive force regime.
The oscillation amplitude of the cantilever is used as a feedback signal to keep a constant tip-sample distance. There is no physical contact between the tip and the sample in Non-contact mode, thus
facilitating non-invasive and highly accurate topography imaging. However, in certain cases, the sample surface in ambient conditions is covered by an adsorbed fluid layer, whose thickness is larger
than the gradient of van der Waals force, affecting the stability and resolution of a non-contact scan. Later on, Tapping mode was introduced [12]. This mode operates in the van der Waal repulsive
force regime, where the tip taps the sample surface. As a result, the tip can go through the adsorbed layer and reach the actual sample surface. Also, Tapping mode is more effective for rough and
sticky samples. The principles and major applications of each scan mode are summarized in Table 1. The proper scan mode should be selected by considering sample characteristics and target of
measurement. Table 1. Summary of typically used AFM modes to measure the surface. Park Systems’ solutions for surface measurement AFM is well-known as a powerful tool for nano-scale metrology.
However, AFM data acquisition and analysis procedures are relatively complicated which require long hours of training and practicing. To enhance production throughput and yield, especially in the
field of industry, an automated AFM flatform is a key factor. Park Systems offers a variety of hardware and software solutions for both industrial and research applications: • Fully automated
industry AFMs for in-line semiconductor production with customizable recipes for data acquisition and analysis • Research AFMs with user-friendly software, innovative imaging modes and technologies,
and batch programmable multiple measurements for advanced scientific research • Various functions designed for throughput enhancement such as automated defect review for industry wafers, narrow
trench mode for high aspect ratio structures, FastApproachTM and AdaptiveScanTM to speed up imaging, EZ flattenTM for quick data analysis, etc. A challenging issue of conventional non-contact
measurements is to hold the tip in the attractive force regime. Especially when scanning a rough sample at high speed, the tip may accidentally touch the surface asperity due to the slow response of
the Z scanner. Holding a small tip-sample distance requires a high bandwidth Z scanner for fast feedback response. True Non-contactTM scan mode of Park Systems' AFMs offers such performance. The
rapid response Z scanner enables a quick movement of the tip in case of an abrupt change in topography, avoiding contact with the sample. In addition, by minimizing the force applied to the tip and
sample, true Non-contact mode allows non-invasive characterization of surface properties. Key considerations for AFM data acquisition and analysis for surface measurement Although automated AFM
solutions are designed to facilitate user convenience, it is important to understand the data acquisition and analysis procedures for a better interpretation of the result. For example, in addition
to choosing a proper scan mode, how to select an appropriate scan size, how to flatten the image, or how to select a proper roughness parameter to describe the surface should be taken into
consideration. Selecting the scan size To properly characterize a surface, the selection of scan size (or cut-off length) is crucial considering that surface properties are dependent on the scale of
a measurement. Surface texture typically consists of waviness and roughness [13]. Waviness refers to longer wavelengths on the surface which is related to surface preparation and finishing methods.
Roughness refers to fluctuations on shorter wavelengths or irregularities of the surface. Due to the difference in wave amplitude and spacing, the selection of cut-off length directly affects the
results of surface roughness calculation. Fig. 1 presents an example of the effect of scan size on the surface properties of a polystyrene film coated on silicon substrate. The sample surface has a
waviness with a spacing of about 80 μm and an amplitude of about 130 nm, as shown in the 3D image and line profile at 100×100 μm2 scan size in Figs. 1(a) and 1(b). The waviness is likely associated
with the sample preparation method. A high magnification scan at 5×5 μm2 reveals the surface roughness of the sample, which is characterized by waves with much smaller spacings and amplitudes compare
to those of the waviness (e.g., 0.4 μm spacing and 0.45 nm amplitude, as shown in Fig. 1(b)). As a result, the calculated roughness values are significantly different between 100 μm and 5 μm cut-off
length scans, as summarized in Fig. 1(c). It should be noted that the cut-off length should be selected depending on the target of a measurement. For example, if the specific application of the
material requires an understanding of interfacial properties at a larger contact area, the waviness should be taken into account. In contrast, a smaller cut-off length can be selected to avoid the
influence of surface waviness when the surface roughness needs to be studied. Fig. 1 (a) 3D topography images of a polystyrene film coated on silicon substrate (displayed with low magnification above
and zoomed-in view below), (b) cross-sectional line profiles extracted from the red and blue dashed lines in (a), and (c) arithmetic average roughness (Ra), root-mean-square roughness (Rq), and
peak-to-valley roughness (Rpv) calculated from line profiles in (b). Flattening the image Nano-scale measurement is susceptible to even a small miss-alignment of the sample surface. As demonstrated
in Fig. 2(a), the sample surface is tilted, likely due to sample mounting error. This issue occurs in almost every AFM measurement. To level the surface slope, a flattening process is applied to the
AFM data. Common approaches to level a surface include plane flattening and line-by-line flattening. In plane flattening, the sample surface is fitted to a linear (first order) plane, and the fit
data is subtracted from the raw data to correct the plane tilt. Plane flattening is often used in cases where Z scanner drift due to temperature fluctuation is negligible. The surface after applying
plane flattening is shown in Fig. 2(b), a small line-by-line fluctuation can be observed from the image and line profile. The fluctuation was found to be more pronounced along the slow scan direction
(Y direction of the image) compared to that of the fast scan direction (X direction of the image). To correct both plane tilt and scanner drift, line-by-line flattening can be used. The algorithm of
line flattening is similar to plane flattening, however, the slope correction is applied line-by-line throughout entire image. In this way, the offset between scan lines due to drift can be removed,
bringing all scan lines to the same plane. The result of line flattening is shown in Fig. 2(c), the effects of surface slope and thermal drift are completely removed, revealing the correct surface
topography. Fig. 2 Height images of a patterned wafer (a) as scanned, (b) after applying plane flattening to raw data, (c) after applying line-by-line flattening to raw data (3D view on the left and
2D view on the right), and (d) cross-sectional X and Y line profiles comparisons of the images in a (grey), b (blue), and c (red). It can be inferred from Fig. 2(b) that inadequate or improper
flattening parameters can distort the real data. This misinterpretation of data can lead to erroneous conclusions in the measurement result. For this reason, understanding the operation and data
analysis of AFM is considerably important. EZ FlattenTM function has been developed to minimize time and effort needed in processing the AFM data. A machine learning-based algorithm is used to detect
the surface features and apply an appropriate algorithm to flatten the data. Fig. 3 shows a screen capture of EZ Flatten function embedded in Park Systems’ SmartAnalysisTM data processing software.
The only manual work required is loading the raw data and executing the function. Then, operators can select the best result among recommended output images. The function is expected to help enhance
the productivity of measurements and minimize labor costs and human error in manual data processing. Fig. 3 Screen capture of EZ Flatten function. Up to six combinations of flattening parameters can
be applied. Snapshots allow users to select the most appropriate result. Selecting roughness parameter Surface roughness is used as a parameter to quantify the surface properties of a material.
Roughness represents quantitative height statistics and texture of a surface and is internationally used as a criterion to evaluate and compare the surface specifications. By monitoring and
controlling the level of surface roughness, the target quality and performance of a surface can be adjusted. AFM allows quantification of surface roughness based on both profile (line roughness) and
areal (surface roughness) methods. Surface roughness parameters can be selected depending on the purpose of evaluation. For example, arithmetic average roughness (Ra), root mean square roughness
(Rq), or peak-to-valley roughness (Rpv) are typically used parameters to evaluate the surface finishing quality in semiconductor manufacturing. Ra, Rq, and Rpv provide quantitative information on
unevenness of the surface. However, in certain cases, the surface properties cannot be quantified solely based on these parameters. As shown in Figs. 4(a) and 4(b), two samples of the same
peak-to-valley distance show identical Ra and Rq although their surface structures are vastly different. This outcome is due to the fact that Ra or Rq does not differentiate between peaks and valleys
[13]. As a result, surfaces with different shapes can yield the same roughness value. Line profile (Fig. 4c) and height histograms (Fig. 4d) comparisons reveal a significant difference in height
distribution between two samples due to structure difference. Surface changes of sample A are mainly composed of valleys, whereas sample B contains mostly peaks. In this case, skewness roughness
(Rsk), which accounts for the deviation of the surface about the mean plane can be used to distinguish between two samples, as shown in areal roughness calculation result in Fig. 4(b). It can be seen
from Fig. 4(d) that major height distribution of sample A skews above the mean line, resulting in a negative Rsk value. In contrast, height distribution skews beneath the mean line resulted in a
positive Rsk value of sample B. Fig. 4 (a) Height images, (b) areal surface roughness parameters, (c) line profiles, and (d) height histograms obtained on two patterned samples A and B. In (d),
histograms and mean lines are determined from line profiles in (c). Elevating surface measurement precision amid challenges Beyond gaining a deep understanding of tip-sample interactions and refining
data analysis skills, it is pivotal to proactively address challenges inherent in scanning probe techniques. This section delves into these nuances, providing insightful guidance to enhance
measurement accuracy while also considering and mitigating the potential for inaccuracies in data analysis. Tip-sample convolution As a scanning probe-based technique, the AFM image is a map of the
interaction between the AFM tip and sample. In other words, the AFM image is a convolution of the shape of the AFM tip and the shape of sample surface. Conventional AFM tips have a finite radius of
curvature ranging from several nanometers to a few micrometers at the end, depending on specific applications. A general rule of thumb is to select a tip which radius is smaller than the surface
features of the sample to be measured. A properly selected tip can reach even the deepest regions of the surface, allowing accurate reconstruction of sample surface. In contrast, a tip whose size is
larger than the sample feature can make the surface features look wider and the valley looks shallower [14]. Tip-sample convolution also becomes more significant as the tip worn out during long-term
measurement. Fig. 5 presents an example of the effect of tip-sample convolution on measured surface properties. A patterned wafer was used as a sample. Firstly, an initial scan was performed in
Non-contact mode to examine the true surface topography of the sample (‘Initial’ height image in Fig. 5a). Then, the same area was continuously scanned in Tapping mode for up to 35 repeats. The
images at the 3rd, 6th, 7th, and 35th repeats are shown in Fig. 5(a). A relatively large tapping amplitude was used to intentionally induce wear to the AFM tip to study the effect of progressing tip
wear on the measured surface properties. Finally, the scan mode is switched to Contact mode to introduce significant wear to the tip (‘After contact’ height image). As shown in height images and line
profiles in Figs. 5(a) and 5(b), the surface step edge is getting blurry as the number of repeats increases, and a significant change can be observed after the final contact scan. Similarly, the wear
progression of the tip can be confirmed from the line profiles at same location, as denoted by the dashed box in Fig. 5(b). The initial non-contact scan showed a clear shape of the surface step
edges. As the number of repeated measurements increases, the tip is progressively worn out and the shape of the blunted tip is convoluted into the sample profile. As a result, the tip could not reach
the bottom of the pattern and the tip side angle is reflected in the sidewall of the pattern. The wear progression of the tip is also reflected in the roughness calculation result. Fig. 5(c)
summarizes the Rq value calculated from line profiles in Fig. 5(b) with respect to the number of scans. The highest value of Rq was determined from the initial non-contact scan (black dot). It is
assumed the initial sharp tip can reach the bottom of the pattern, resulting in the actual surface roughness of the sample. As the tip is worn out gradually during tapping scan, the Rq value
gradually decreases as the number of scans increases until the 6th scan. A sharp decrease in Rq at the 7th scan suggests a major change in tip apex geometry, possibly due to fracture. The Rq value
remains stable from the 7th to the 35th scan. A possible reason for this outcome is that the tip apex is flattened due to fracture, and a major increase in tip apex radius helps reduce tip wear [15].
The tip apex is significantly worn out due to contact scan, resulting in a sharp reduction of Rq at the final contact scan. The change of the tip apex was confirmed by performing AFM scan on a tip
self-imaging grating, as shown in Fig. 5 (d). Profile comparison reveals a significant change in tip geometry due to wear before and after the entire set of measurements. Fig. 5 (a) Height images,
(b) cross-sectional line profiles comparison, (c) Rq values calculated from same line profiles with respect to the number of scans, and (d) tip end shape before and after measurement. For better
observation, vertical shifts were applied to the line profiles in (b), and the color was matched between the height image, line profile, and Rq data. A non-contact measurement using a different sharp
tip showed no major change of the test area after measurement (results not shown), indicating that the change of tip shape is responsible for changes in height images, and therefore, roughness
calculation results. The results suggest that preservation of the tip apex is crucial in obtaining an accurate surface measurement. The tip apex can be protected by using non-destructive imaging
techniques such as Non-contact mode. In addition, tip wear can be minimized by selecting tips made of durable materials such as diamond. However, considering that a diamond tip typically has a larger
curvature radius than a silicon tip, the tip should be selected in consideration of measurement targets. For example, for samples with well-separated steps (e.g., atomic steps of sapphire or SiC
wafer), the tip sharpness does not significantly affect measurement result [16]. In such cases, a tip with a larger radius or a blunted tip may be more beneficial to preserve the tip geometry in
consecutive measurement since wear progression of the tip generally reduces with increasing tip radius [15]. Monitoring the tip end From the data in Fig. 5, it can be seen that changes in tip
condition (e.g., increase in curvature radius due to wear) have a significant contribution to the surface measurement result. To better understand the effect of tip wear progression, it is
recommended to monitor the tip condition during scanning, especially in long-term measurements. There are a few methods to monitor the tip apex such as tip imaging using scanning electron microscopy
(SEM), self-imaging on a tip characterization grating, or monitoring changes in the shapes of amplitude- distance curve and force-distance curve. The basic principles, advantages, and limitations of
each method are summarized in Table 2. For SEM imaging, the tip should be unmounted from the AFM system, which therefore, interrupts the workflow. For self-imaging, a tip characterization sample can
be mounted to the same sample chuck of the AFM for convenience. However, the specific structure of characterization grating, as can be seen in Fig. (6a), may induce additional tip wear if scan
parameters are not properly controlled. The force-distance monitoring method is based on an assumption that an increase in tip radius due to wear gives rise to an increase in tip-sample adhesion
force [17,18]. This method is more appropriate for a tip-sample system that exhibits relatively strong adhesion properties. The amplitude-distance method relies on the dependency between tip
sharpness and tip-sample atomic forces. Accordingly, a sharp tip exhibits a clear attractive-repulsive transitions in the amplitude-distance curve while a blunted tip shows a major contribution of
attractive force [19,20]. Table 2. Summary of methods to monitor the AFM tip end. Fig. 6 presents an example of tip monitoring in AFM. The tip apex condition was monitored before and after
measurement (result in Fig. 5) by using self-imaging, amplitude-distance curve, and force-distance curve methods. A clear difference in the tip shape was observed before and after measurement due to
wear, as shown in Fig. 6(a). Tip wear can also be deduced based on changes in amplitude-distance and force-distance behaviors, as shown in Figs. 6(b) and 6(c). Among the methods, the
amplitude-distance curve provides a reliable and straightforward characterization of tip end condition that can be performed in situ without the need of switching the sample or unmounting the tip
from the AFM system. Fig. 6 AFM tip end monitoring using (a) tip self-imaging on a tip characterization grating (TGT1, TipsNano), (b) amplitude-distance curve, and (c) force-distance curve before and
after measurement. Measurement repeatability To meet the requirements of a large-scale analysis tool, there are a few factors that should be considered when performing an AFM measurement, such as
accuracy and repeatability of test, and tip-to-tip variation. Measurement accuracy is determined by how accurate the result is compared to actual values while repeatability is related to how the
result is reproduced in multiple repetitions of measurements. Fig. 7 shows an example of a repeatability test on a calibration grating. A single tip was used to scan a calibration grating with a
known step height in Non-contact mode of AFM. Three different tests were performed on three different days. Each test consisted of 100 repeats. The step height was determined to be 43.29 ± 0.10 (mean
± one standard deviation), 43.31 ± 0.08, and 43.28 ± 0.09 nm for test 1, test 2, and test 3, respectively. It was found that the mean step heights of the three tests agree well with each other. The
relative error of each test is smaller than 0.2% and was possibly associated with sample changes due to temperature fluctuation during the long hours of measurement. The outcome suggests that the
measurement is highly reproducible with minor variations. In addition, an agreement between the measured step height from the tests and expected step height provided by the manufacturer (43.3 ± 0.6
nm) can be observed from the data in Fig. 7(b), indicating high accuracy of the AFM measurements. Fig. 7 (a) Height image obtained on a standard step height calibration grating (SHS8-440, VLSI), and
(b) summary of step height measurement results from three tests. Error bars (red bars) of measured values represent one standard deviation of 100 repeats. Error band (blue gradient) of expected step
height represents the error margin. Expected value and error band are independently measured by the manufacturer. Fig. 8 presents an example of tip-to-tip variation tests. Four different tips of the
same type were used to scan a Si wafer in Non-contact mode. A total number of 800 repeated scans was performed using each tip. Examples of height images obtained at the 1st and 800th scans using tips
1 and 2 are shown in Fig. 8(a). Ra values obtained from 4 tips are summarized in Fig. 8(b). A relative error of less than 1.9% calculated from 800 data collected using each tip confirms the
measurement repeatability. Also, the deviation of Ra value from 4 measurements is determined to be 3.5%, possibly associated with a combination of tip-to-tip variation and position-to-position
variation of the sample surface. A position-to-position variation of 5% is expected for this sample. Although it is difficult to distinguish the contribution of each factor, 3.5% total deviation
suggests that the tip-to-tip variation is negligible. The result also proves the ability of Non-contact mode to preserve the tip and sample even after long hours of measurement. Fig. 8 (a) Examples
of height image obtained at the 1st scan and 800th scan using tips 1 and 2, and (b) summary of Ra values determined from 800-image roughness measurement tests using 4 different tips of the same type.
Summary This whitepaper overviews some aspects of surface properties measurement at the nano-scale using the AFM. It was expected that the outcome would add useful information for an accurate
measurement of surface properties based on understanding of interactions between the AFM tip and sample. There are a few checkpoints that need to be considered when measuring a surface: • In general,
Non-contact mode is a good choice to obtain surface topography with minimized artifacts while preserving tip and sample. • The scan size can be selected by considering the wavelengths of waviness and
roughness, depending on the target application. • Flattening in first order is often used to level the surface slope. Beware that higher orders of flattening may alter the actual data. • Roughness
parameters can be selected in consideration of surface characteristics. While Rq and Ra are typically used, it is recommended to find a proper roughness parameter for your specific case. • Tip-sample
convolution is a non-trivial issue of AFM. Understanding tip-sample interactions helps a proper interpretation of the AFM result. References [1] D. Zhao, X. Lu, Chemical mechanical polishing: Theory
and experiment, Friction. 1 (2013) 306–326. [2] W. Xie, Z. Zhang, L. Liao, J. Liu, H. Su, S. Wang, D. Guo, Green chemical mechanical polishing of sapphire wafers using a novel slurry, Nanoscale. 12
(2020) 22518–22526. [3] C. Gui, M. Elwenspoek, N. Tas, J.G.E. Gardeniers, The effect of surface roughness on direct wafer bonding, J. Appl. Phys. 85 (1999) 7448–7454. [4] M.R.S. Soares, C.A.R. Costa,
E.M. Lanzoni, J. Bettini, C.A.O. Ramirez, F.L. Souza, E. Longo, E.R. Leite, Unraveling the Role of Sn Segregation in the Electronic Transport of Polycrystalline Hematite: Raising the Electronic
Conductivity by Lowering the Grain-Boundary Blocking Effect, Adv. Electron. Mater. 5 (2019) 1900065. [5] F.C. Salomão, E.M. Lanzoni, C.A. Costa, C. Deneke, E.B. Barros, Determination of
High-Frequency Dielectric Constant and Surface Potential of Graphene Oxide and Influence of Humidity by Kelvin Probe Force Microscopy, Langmuir. 31 (2015) 11339–11343. [6] H. Song, Y. Ma, D. Ko, S.
Jo, D.C. Hyun, C.S. Kim, H.-J. Oh, J. Kim, Influence of humidity for preparing sol-gel ZnO layer: Characterization and optimization for optoelectronic device applications, Appl. Surf. Sci. 512 (2020)
145660. [7] R.Y.K. Yoo, Automated AFM Boosts Throughput in Automatic Defect Review, Micros. Today. 22 (2014) 18–23. [8] F. Nagano, S. Iacovo, A. Phommahaxay, F. Inoue, F. Chancerel, H. Naser, G.
Beyer, E. Beyne, S. De. Gendt, Void Formation Mechanism Related to Particles During Wafer-to-Wafer Direct Bonding, ECS J. Solid State Sci. Technol. 11 (2022) 63012. [9] S. Wang, Q. Liu, C. Zhao, F.
Lv, X. Qin, H. Du, F. Kang, B. Li, Advances in Understanding Materials for Rechargeable Lithium Batteries by Atomic Force Microscopy, Energy Environ. Mater. 1 (2018) 28–40. [10] G. Binnig, C.F.
Quate, C. Gerber, Atomic Force Microscope, Phys. Rev. Lett. 56 (1986) 930–933. [11] Y. Martin, C.C. Williams, H.K. Wickramasinghe, Atomic force microscope–force mapping and profiling on a sub 100‐Å
scale, J. Appl. Phys. 61 (1987) 4723–4729. [12] Q. Zhong, D. Inniss, K. Kjoller, V.B. Elings, Fractured polymer/silica fiber surface studied by tapping mode atomic force microscopy, Surf. Sci. Lett.
290 (1993) L688–L692. [13] B. Bhushan, Surface roughness analysis and measurement techniques, in: Mod. Tribol. Handbook, Two Vol. Set, CRC press, 2000: pp. 79–150. [14] P. Eaton, K. Batziou,
Artifacts and Practical Issues in Atomic Force Microscopy BT - Atomic Force Microscopy: Methods and Protocols, in: N.C. Santos, F.A. Carvalho (Eds.), Springer New York, New York, NY, 2019: pp. 3–28.
[15] K.-H. Chung, Y.-H. Lee, H.-J. Kim, D.-E. Kim, Fundamental Investigation of the Wear Progression of Silicon Atomic Force Microscope Probes, Tribol. Lett. 52 (2013) 315–325. [16] F. Gołek, P.
Mazur, Z. Ryszka, S. Zuber, AFM image artifacts, Appl. Surf. Sci. 304 (2014) 11–19. [17] B. Gotsmann, M.A. Lantz, Atomistic Wear in a Single Asperity Sliding Contact, Phys. Rev. Lett. 101 (2008)
125501. [18] J. Liu, Y. Jiang, D.S. Grierson, K. Sridharan, Y. Shao, T.D.B. Jacobs, M.L. Falk, R.W. Carpick, K.T. Turner, Tribochemical Wear of Diamond-Like Carbon-Coated Atomic Force Microscope
Tips, ACS Appl. Mater. Interfaces. 9 (2017) 35341–35348. [19] S. Santos, L. Guang, T. Souier, K. Gadelrab, M. Chiesa, N.H. Thomson, A method to provide rapid in situ determination of tip radius in
dynamic atomic force microscopy, Rev. Sci. Instrum. 83 (2012) 43707. [20] A. Temiryazev, S.I. Bozhko, A.E. Robinson, M. Temiryazeva, Fabrication of sharp atomic force microscope probes using in situ
local electric field induced deposition under ambient conditions, Rev. Sci. Instrum. 87 (2016) 113703. | {"url":"https://kr.parksystems.com/applications/afm-exclusive","timestamp":"2024-11-14T04:36:00Z","content_type":"application/xhtml+xml","content_length":"215216","record_id":"<urn:uuid:de8bbc01-d1cd-4f59-97d0-0583e1711bb5>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00402.warc.gz"} |
A simple method for ordering loci using data from radiation hybrids. - PDF Download Free
9. 1%.-1”:3
A Simple Method for Ordering Loci Using Data from Radiation Hybrids C. T. The Lindsley
F. Kimball
Research Received
lnstltute July
of The New 11,
SC 1991
Center, 21,
method for mapping loci using data generated from radiation hybrids which involves estimating aparameter H, comparable to a recombination frequency, and several retention frequencies. The method both
orders the loci and provides map distances using not only information on locus pairs, but also on sets of four loci simultaneously. Here, we take a very different approach, based not on a model of
the system and its resulting parameters, but on a statistical measure between pairs of loci, coupled with an ordering algorithm. Our nonparametric approach not only provides a straightforward,
relatively rapid method for ordering, but also provides a way of comparing results with the more extensive method of Cox et al. After presenting the method we will illustrate it,s use on two sets of
loci and compare the resulting maps with those of Cox et al.
A method is presented for ordering loci on a chromosome based on data generated from radiation hybrids. All loci are tabulated as being present, absent, or not scored in a series of clones.
Correlation coefficients are calculated for all pairs of loci indicating how often they are retained or lost together in the clones. On the assumption that a high positive correlation implies closely
linked loci, a distance score, d, equal to one minus the correlation coefficient, is obtained for each locus pair and an order is generated that minimizes the sum of the adjacent distances [the MDMAP
method of Falk (“Multipoint Mapping and Linkage Analysis Based upon Affected Pedigree Members: Genetic Analysis Workshop 6,” pp. 17-22, A. R. Liss, New York, 1989)]. Two sets of data, with
information on 13 and 16 loci mapped to chromosome 21q, have been ordered using this method. The results are in very good agreement with other ordering methods used on the same data and with physical
mapping data.
Press, Inc.
METHODS Consider N loci that have been scored as beingpresent or absent in a set of M clones. For any given pair of loci we can construct a 2 X 2 table showing how many clones retained both, one, or
neither of the loci (seeTable 1). We can calculate the correlation coeffcient, r, for these two loci which is simply
INTRODUCTION A useful method for mapping chromosomes has been developed by Cox et al. (1990) based on earlier work of Goss and Harris (1977). The method involves exposing chromosomes to a high dose
of X rays in order to break the chromosomes into several fragments. These fragments can be recovered in rodent cells and a set of such rodent-human hybrid clones scored for the presence or absence of
specific human DNA markers. The closer two loci are on a chromosome, the more likely it is that they will be retained on the same fragment. The farther apart, the more likely it is that they will be
on separate fragments. Thus the frequency of breakage can be used as an estimate of distance between two markers. For a more detailed discussion of the method, see Cox et al. (1990). Because of the
nature of the procedure, inferences can be made about the distance between any two loci based on information about retention or loss of the loci in the clones. Cox et al. (1990) have developed a
0%3s-7533/91 E3.W CopyrIght ICI 1991 by Academic Press. All rights of reproduction in any fbrm
/X2 r=t
G M
where x’ is the standard value for a 2 X 2 table (Li, 1961). If ad > bc, the correlation is positive, if ad < bc, it is negative. Since the coretention of two loci in the same clones implies closely
linked loci, we can use the correlation coefficient as a measure of how close or how distant two loci are, relative to the other loci tested. Thus a high positive correlation implies close linkage, a
value near zero implies loose (or no) linkage. Significant negative correlations would not be expected. Let d = 1 - r, which we will use as our measure of “distance.” 120
Inc. reserved.
Two by Two Table Showing Loci in M Clones Produced Methods
1,0cus1 t LOCllS1
Retention by Radiation
or Loss of Hybrid
Locus 2 +
Locus2 ~
h d
No&. a. b, c. and d represent the numberof clonesthat fall into the four classes shown in the table. where CI + b + c + d = M. x2 = (ad ~ hc)*fM/[(a + c)(b + dJ(a + h)(c + d)]. Falk (1989) outlines a
preliminary ordering scheme that can then be used to order the loci based on the pairwise d values. The scheme is based on the assumption that, given the true order of a set of, say, three loci, the
distance between the two flanking loci will be greater than the distance between the corresponding pairs of adjacent loci. If the true order of three loci were A-B-C, then we would expect d,, > 6,s
and dAc > d,,. Therefore the most likely order of the three loci would be the one that minimizes the sum of the distances for the two adjacent intervals. Similar arguments can be used to show that a
reasonable preliminary ordering for any number of loci can be obtained by looking for a “minimum distance map,” MDMAP (Falk, 1989). This requires looking for the map with the shortest total distance
defined by the sum of all N - 1 adjacent distances from among the N!/2 possible orderings of N loci. For example, if we have three loci, A, B, and C with r values rAB = 0.9, = 0.8, and rBc = 0.2, the
sum of adjacent d values rAr for the three orders would be Order A-B-C A-C-B B-A-C
proximal set of 14 loci (Cox et al., 1990) and a distal set of 19 loci (Burmeister et al., 1990). In these experiments, IO3 independent somatic cell hybrid clones were scored for the presence of each
locus. Not all hybrids were scored for all markers, the number of clones scored ranging from 65 to 96 per marker in the first set and 51 to 92 per marker in the second. The number of clones scored
for a given marker pair ranged from 53 to 95 in the first set and from 22 to 82 in the second. Of the loci scored, 13 in the first set and 16 in the second were separated by X-ray breakage and thus
were informative for ordering. Using the raw data, indicating the presence or the absence of each locus in the hybrid cells, if scored, we calculated pairwise correlation coefficients for the
separable loci in each data set and generated ordered maps as described above, using the computer program MDMAP (Falk, 1989) to search for the best order. The results are shown in Tables 2 and 3. The
end result appears to be quite st,able in that the same final order is attained from several arbitrary starting orders. In addition to the orders and distances produced by the correlation method
(CORMAP) the orders given by Cox et al. and Burmeister et al. are shown. CORMAP predicts the same order as Burmeister et al. for the 16 distal loci (Table 3) and agrees with Cox et 01. on the
proximal set (Table 2) except for one nearest-neighbor pair (APP, S8) where the two methods give opposite orders. Cox et al. have estimated the odds in favor of their order over the reverse to be
only 43:l. The Cox order is ranked second by CORMAP, with a distance of 2.852. Thus CORMAP appears to be a very good predictor of order, given information generated by radiation hybrid experiments.
Total distance
0.1 + 0.8 = 0.9 0.2 + 0.8 = 1.0 0.1 + 0.2 = 0.3
The ability to reliably order a set of loci on a chromosome becomes increasingly difficult as the number of loci grows. This is true whether one is using family data for mult,ilocus mapping by
computer, pairwise lod score results, or dat,a generat.ed from a totally different approach, such as that produced by radiation hybrid mapping. It is important to obtain both the correct order of
loci and the relative distances between them. Sometimes these two can be done in a single step, for example when using a computer program such as CRIMAP (Lander and Green, 1987) or MAPMAKER (Lander
et al., 1987) to analyze family data. However, for more than a moderate number of loci the computer overhead becomes very high and in practice loci are often ordered in smaller sets and then combined
into one complete map (see, e.g., Warren et al., 1989). Another approach is to first make a reasonable attempt t,o order the entire set of loci based on, say, pairwise information and then refine
bot,h the
The best order would then be B-A-C, represented by the minimum total distance, d = 0.3. The same reasoning can be used for a set of N loci, with the shortest total distance giving the most likely map
order. For moderate values of N the ordering can be done “by hand,” but for large N, the number of distinct orders to be considered becomes quite large. In such cases a computer algorithm such as the
one proposed by Falk (1989), which makes use of simulated annealing (Kirkpatrick et nl., 1983), can be used to “search” for the minimum distance map. EXAMPLE
Two sets of loci on the long arm of chromosome 21 were studied using radiation hybrid techniques, a
C’. ‘I’. FALK
TABLE Comparison Source
of the Ordering
of 13 Loci
in the Proximal
cos CORMAP
2.852 2.784
Note. COX Locus names
Sl6 S16
S48 S48
S46 S46
S4 S4
s5:! ST,”
“351 2.351 BUR
S58 S58
the order
in the Distal
the method
S‘l’i S-15
GM29177 from the National R. Cox for many helpful and
7‘ s.3
b.3 ”
ct nl. (2); CORMAP
of Chromosome
of’ loci
in Burmeister
Sl’ Sl:!
(S8 API’) (APP SX)
This work is supported by Grant Institutes of Health. I thank David
Order SOD SOD
SlX S18
BUR CORMAP Note. here.
of 16 Loci
Sl Sl
represents the order obtained using and the CORMAP order differ.
TABLE of the Ordering
distant. A simple measure of distance between two loci can then be expressed as one minus the correlation. After defining the measure of distance, an ordering algorithm is then used based on those
distances. Here the minimum distance map (MDMAP) method (Falk 1989) has been used, defining as the best preliminary order the one that minimizes the sum of t,he adjacent pairwise distances. This
algorithm has also been used with classical family data with good results (Olson and Boehnke, 1990; Falk, unpublished) and has been tested using as the distance measure the H values generated by Cox
et al. on radiation hybrid data. In the lat.ter case, the MDMAP algorithm chooses precisely the same order of both chromosome 21 data sets as that obtained using the correlation coefficients. This is
not unexpected, as the H values and distances estimated by CORMAP are not independent measures. The CORMAP method thus appears to provide a simple alternative way to order loci from radiation hybrid
data. As currently designed CORMAP provides a preliminary ordering procedure but has not yet been extended to estimate odds of one particular order over another or to estimate genetic dist,ances. Its
strength is that it does not require the modeling of a breakage parameter and the underlying retention frequencies, but utilizes instead a nonparametric approach. It does, therefore, give a good
starting point for fine tuning order and distances as well as an alternative method for comparison with other ordering techniques.
order and the relative positions by using more statistically precise methods and/or information on several loci simultaneously. Several approaches to initial ordering have been suggested, including
those by Buetow and Chakravarti (1987), Weeks and Lange (1987), and Falk (1989). These methods are based on information about all locus pairs generated by classical family linkage analysis, but each
has a different algorithm for ordering. In radiation hybrid mapping the data generated are quite different from that in family studies, but a similar principle holds. The closer two loci are on a
chromosome, the less likely it is that there will be a break between them. In radiation hybrids, the break is caused by exposing human chromosomes to radiation. If the loci are closely linked, the
chance of a break between them is low and thus the probability of seeing them retained or lost together in subsequently scored clones is high. Cox et al. (1990) have modeled the procedure based on a
parameter 0 that is analogous to the recombination fraction, and several parameters representing retention frequencies of single loci or sets of loci. Using estimates of these parameters they are
able to predict the most likely order of the loci as well as the relative distances between loci. Using these parameters they have also devised a method to compare relative odds of different orders,
by flipping small sets of loci from the original order in a systematic fashion. It is the intent here to present an alternate, simple, model-free method of determining a likely order for the loci,
based on the logical premise that the correlation coefficient, measuring absence or presence of pairs of loci in all of the clones, will be high for closely linked loci and will decrease for loci
that are more
of Chromosome
of’ loci
Sll Sll
represents the order presented in (‘ox et al. (3): CORMAP in parentheses indicate those loci for which the COS order
Sill s141
(‘DlH CD18 the order
S26 S”6 obtained
S4-4 844 using
COL:! (‘OL’
COLl COLl
the method
SlOO s 100 presented
ORDERING stimulating discussions me prior to publication.
and for generously
his data
BUETOW, K. H., AND CHAKRAVARTI, A. (1987). Multipoint gene mapping using seriation. I. General methods. Amer. Hum. Genet. 41: 180-188.
7. J.
BURMEISTER, M., KIM, S., PRICE, E. R.. DE LANGE, T., TANTRAVAHI, LT., MYERS, R. M.. AND Cox, D. R. (1991). A map of the distal region of the long arm of human chromosome 21 constructed by radiation
hybrid mapping and pulsed-field gel electrophoresis. Genomics 9: 19-30.
(lox, D. R.. BURMEISTER, M., PRICE, E. R., KIM, S., AND MYERS, R. M. (1990). Radiation hybrid mapping: A somatic cell genetic method for constructing high resolution maps of mammalian chromosomes.
Science 250: 245-250.
FALK, C. ‘I’. (1989). A simple scheme for preliminary ordering of multiple loci: Application to 45 CF families. In “Multipoint Mapping and Linkage Analysis Based upon Affected Pedigree Members:
Genetic Analysis Workshop 6” (R. C. Elston. M. A. Spence, S. E. Hodge. and .J. W. MacCluer, Eds.). pp. 17-22. A. R. Liss, New York. Goss,
S. -1.. AND HARRIS,
H. (1977).
by means
KIRKPATRICK, S., GELA’~T, C. D., AND VECCHI, M. P. (1983). Optimization by simulated annealing. Sriencp 220: 671&6&l. LANDER, E., AND GREEN, P. (1987). Construction of multilocus genetic linkage maps
in humans. Proc. N&l. Acad. Sci. l!SA
of cell fusion: II. The mapping of 8 loci on human chromosome 1 by statistical analysis of gene assortment in somatic cell hybrids. J. Cell Sci. 25: 39-57. 6.
REFERENCES 1.
LANDER, E.. GREEN, P., ABRAHAMSON, J.. BARLOW, A., DALY, M., LINCOLN, S., AND NEWBURG, L. (1987). MAPMAKER: An interactive computer package for constructing primary genetic linkage maps of
experimental and natural populations. Genomics 1: 174-181. LI, C. C. ( 1961). “Human Genetics: Principles and Methods.” pp. 8%83. McGraw-Hill. New York. OLSON. .J. M., AND BOEHNKE, M. (1990). Monte
Carlo comparison of preliminary methods for ordering multiple genetic loci. Amer. J. Hum. Genet. 47: 470-482.
WARREN, A. C., SLAUGENHAUPT. CHAKRAVARTI, A.. AND ANTONARAKIS, linkage map of 17 markers on human mics 4: 679-591.
WEEKS, D., AND LANGE, K. (1987). Preliminary ranking cedures for multilocus ordering. &nomics 1: 2:X-242.
S. A., LEWIS, J. G., S. E. (1989). A genetic chromosome 21. Genupro- | {"url":"https://d.docksci.com/a-simple-method-for-ordering-loci-using-data-from-radiation-hybrids_5f136d7e097c47c7358b4595.html","timestamp":"2024-11-12T12:18:58Z","content_type":"text/html","content_length":"65109","record_id":"<urn:uuid:0fa9509d-dae9-475d-be46-931319df0885>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00284.warc.gz"} |
Theory background Archives – ConsteelConsteel
The evolution of compressed bar (column) design
One of the characteristic features of steel structures made of bars (e.g. lattice girders) is the compressed bar. We speak of a compressed bar when the structural element, which usually has a
straight axis, is loaded by a compressive force P applied centrally (Figure 1).
Fig. 1 Compressed bar model
Figure 2 illustrates the evolution of compressed bar (column) design. In the beginning (in the old days), master builders determined the load-bearing capacity of compressed columns of different
materials and sizes on the basis of the experience accumulated over the centuries, passed down from master to apprentice. A significant change was brought about by the application of classical
mathematical differential analysis to engineering. The Swiss mathematician and physicist Euler (1707-1783) solved the problem of the deflection of a compressed elastic line, which could be applied to
the solution of the elastic compressed bar (Euler’s force). In the following centuries, engineers recognised that Euler’s force only gave an acceptable approximation to the real load capacity of a
compressed bar in certain cases (mainly for large slender bars). Many solutions for the bearing capacity of a compressed bar were developed that were more advanced than the Euler formula, but it was
not until the huge structural engineering boom following World War II that significant changes were made. Compression bar experiments were carried out in every major structural laboratory in the
world, and a database of over two thousand experiments was compiled from the results. The load capacity of the pressure bar was given by a formula based on the database, using the method of
mathematical statistics.
This methodology is still dominant today: ‘the dimensioning of the compressed bar has become a political issue for the steel construction profession…’. Understanding the principle of compressed
bar design is therefore essential for the structural engineer.
The right side of the Figure 2 also contains a hint for the future. At the level of scientific research, it is already present that the load capacity of a real compressed column can be determined by
mathematical-mechanical simulation. Indeed, in the near future, databases that go beyond anything we know today can be created using supercomputers. On the basis of such a gigantic database,
artificial intelligence could, at least in principle, supersede existing engineering knowledge and methodology. But the reality is that structural engineering is not one of the pull sectors (such as
the defense or automotive industries), so this new shift in design theory is certainly a long way off.
Fig. 2 Developing of the column design methodology
In the following, the Euler force and the experimentally based standard design formula, which are of major importance to structural steel engineering today, are discussed in detail.
Buckling strength of the ideal columns: the Euler force
Assume that the hinged compressed column shown in the Figure 3 has the following properties:
• perfectly straight,
• its material is perfectly linearly elastic,
• centrally compressed.
Under the above conditions, perform the compressed column experiment using Consteel software: run the Linear Buckling Analysis (LBA) calculation. The result is illustrated in Figure 3.
Theoretical background
According to the beam-column theory, two types of torsional effects exist.
Saint-Venant torsional component
Some closed thin-walled cross-sections produce only uniform St. Venant torsion if subjected to torsion. For these, only shear stress τ[t ]occurs.
The non-uniform torsional component
Open cross-sections might produce also normal stresses as a result of torsion.[1.]
Warping causes in-plane bending moments in the flanges. From the bending moment arise both shear and normal stresses as it can be seen in Fig. 2 above.
Discrete warping restraint
The load-bearing capacity of a thin-walled open section against lateral-torsional buckling can be increased by improving the section’s warping stiffness. This can be done by adding additional
stiffeners to the section at the right locations, which will reduce the relative rotation between the flanges due to the torsional stiffness of this stiffener. In Consteel, such stiffener can be
added to a Superbeam using the special Stiffener tool. Consteel will automatically create a warping support in the position of the stiffener, the stiffness of which is calculated using the formulas
below. Of course, warping support can also be defined manually by specifying the correct stiffness value, calculated with the same formulas (see literature [3]).
The following types of stiffeners can be used:
• Web stiffeners
• T – stiffener
• L – stiffener
• Box stiffener
• Channel –stiffener
The general formula which can be used to determine the stiffness of the discrete warping restraint is the following:
R[ω] = the stiffness of the discrete warping restraint
G = shear modulus
GI[t] = the Saint-Venan torsional constant
h = height of the stiffener
Effect of the different stiffener types
Web stiffener
b = width of the web stiffener [mm]
t = thickness of the web stiffener [mm]
h = height of the web stiffener [mm]
Fig. 3: web stiffener
T – stiffener
b[1] = width of the battens [mm]
t[1] = thickness of the battens [mm]
b[2] = width of the web stiffener [mm]
t[2] = thickness of the web stiffener [mm]
h = height of the web stiffener [mm]
Fig. 4: T–stiffener
b = width of the L-section [mm]
t = thickness of the L-section [mm]
h = height of the L-section [mm]
Channel stiffener
b[1] = width of channel web [mm]
t[1] = thickness of channel web [mm]
b[2] = width of channel flange [mm]
t[2] = thickness of channel flange [mm]
h = height of the web stiffener [mm]
Numerical example
The following example will show the increase of the lateral-torsional buckling resistance of a simple supported structural beam strengthened with a box stiffeners. The effect of such additional
plates can be clearly visible when shell finite elements are used.
Shell model
Fig. 7 shows a simple fork supported structural member with welded cross-section modeled with shell finite elements and subjected to a uniform load along the member length acting at the level of the
top flange.
Table 1. and Table 2. contain the geometric parameters and material properties of the double symmetric I section. The total length of the beam member is 5000 mm, the eccentricity of the line load is
150 mm in direction z.
Name Dimension Value
Width of the top Flange [mm] 200
Thickness of the top Flange [mm] 10
Web height [mm] 300
Web thickness [mm] 10
Width of the bottom Flange [mm] 200
Thickness of the bottom Flange [mm] 10
Table 1: geometric parameters
Name Dimension Value
Elastic modulus [N/mm^2] 200
Poisson ratio [-] 10
Yield strength [N/mm^2] 300
Table 2: material properties
Box stiffener
The box stiffeners are located near the supports as can be seen in Fig. 8. Table 3. contains the geometric parameters of the box stiffeners.
Fig. 8: the structural shell member with added box stiffeners
Name Dimension Value
Width of the web stiffener [mm] 100
Thickness of the battens [mm] 100
Total width of the box stiffener [mm] 200
Height of the plates [mm] 300
Thickness of the plates [mm] 10
Table 3: geometric parameters of the box stiffeners
7DOF beam model
The same effect in a model using 7DOF beam finite elements can be obtained when discrete warping spring supports are defined at the location of the box stiffeners.
Fig. 9: beam member supported with fork supports and loaded with eccentric uniform load
Discrete warping stiffness calculated by hand
This article aims to cover the theoretical background of the shear field stiffness determination methods implemented in Consteel. Modeling with the shear field stiffness based method will also be
compared with shell modeling of trapezoidal deckings in Consteel.
Theoretical background
Modeling the shear stiffness of trapezoidal deckings is used to utilize their contribution to stabilizing the main structure. The possibility to consider the shear stiffness of sheetings is
implemented at finite element level in Consteel and ensures easy modeling through its application onto beam elements.
Shear panel definitions
For the discussion let’s establish some basic definitions regarding shear panels.
• Dimensions:
□ L [m]: width of the shear field, also the span of the stabilized beam
□ L[s] [m]: complete length of the shear field parallel to the ribs
□ a [m]: effective shear field length for only one connecting beam element
• Stiffnesses:
□ G[s] [kN/m]: specific shear stiffness considering a 1 m long strip of an “L[s]“ long shear field
□ S [kN]: shear stiffness of the complete shear field
Determaination of the shear field stiffness
The general formula to calculate the shear stiffness in Consteel is the following:
There are 4 methods implemented in Consteel to determine the shear field stiffness:
1. Schardt/Strehl method: (K1, K2), DIN 18807-1:1987-06 [1]
2. improved Schardt/Strehl method: (K1, K2, K1*, K2* and e[L])
3. Bryan/Davies method: (K1, K2, K1*, K2*, e[L], α[1], α[2], α[3], α[4])
4. Eurocode 3: (1993-1-3 10.1.1 (10)) [2]
See in more detail here: Determination of shear field stiffness and application in Consteel
The first 3 methods are based on the 1. one, the Schardt/Strehl method. These methods operate on the same principle by calculating the shear stiffness from values (K[1], K[2], etc.) provided by the
manufacturer of the sheetings. The 2. and 3. methods are more developed versions of the 1. one, trying to more accurately calculate the shear stiffness by introducing additional parameters to account
for more sources of the overall shear stiffness of the sheeting.
The 4. method that is found in Eurocode 3 can be more generally applied since that doesn’t require such product specific values.
A basic assumption in case of these methods is that the sheeting is connected at every rib to the beams that it stabilizes (purlins in most cases). An additional modifying factor in case of all these
methods is if the trapezoidal sheeting is not fixed at every rib, but every second rib, then the final “S” shear stiffness should be substituted by 0,2*S.
Theoretical background of the Schardt/Strehl method
This approach is based on a model assuming a fully linear elastic behaviour of the diaphragm. The ultimate limit state is therefore defined by yielding in the corner radius at the flange-web
transition. The mechanical model also assumes that the sheeting is fixed to the substructure at all 4 edges. Shear forces R[Q] and R[L] are acting on the sheeting at the individual fixed points on
the lower flanges where the sheeting is screwed to the substructure. The number of waves in the sheeting is assumed to be large enough so that the individual forces acting at the transverse edges in
the middle are assumed to be constant (n>10). The length “L[s]” of the shear field can be arbitrary, but should be in reasonable proportion to the width “L” of the shear field (<4).
Based on these assumptions the mechanical analysis can be isolated to a half of one wave of the sheeting as shown on the left-hand side of the following figure:
On the right-hand side the considered internal forces are shown for one slice of the sheeting.
Assumptions for the mechanical model:
• The M[s] and M[z] moments are neglected (shown in brackets on the right-hand side figure).
• Transverse bending moments “m[i]” at the level of the plate are considered. These moments have a 0 value in the center of the lower and upper flanges.
• The longitudinal stresses σ[z] are constant over the thickness “t[i]” of the plates and linearly distributed over the height “h[i]”.
• The flexibility of connections is neglected.
The method accounts for the following effects:
• Shear deformation: corresponding value: K[1] [m/kN] shows the sheetings compliance coming from shear deformation. The lower this value is, the more stiffness the sheeting has.
• Warping deformation: corresponding value: K[2] [m^2/kN] shows the sheetings compliance coming from warping deformation. The lower this value is, the more stiffness the sheeting has.
Component considering shear deformation
The value K[1] can be calculated from the following formula based on the properties of the trapezoidal sheeting:
• ∑l [mm]: Summed up length of all the plates within one full wave
• br [mm]: length of one complete wave
• G [N/mm^2]: shear modulus
• t[core] [mm]: structural thickness. (generally: t[core] = t[nominal] – 0,04 mm)
The formula of the K[1] value is similar to how it should be calculated in case of a planar plate, but its thickness corrected with the ∑l/br ratio, or in other words the ratio of the complete length
of the plates to the length of one wave. The K[1] shear deformation compliance value is directly proportional to the ∑l/br ratio, therefore if a certain trapezoidal sheetings height is increased with
everything else left the same, the corresponding K[1] value would increase, and the stiffness coming from shear deformation would decrease. On the other hand the K[1] value is inversely proportional
to “G” shear modulus and “t[core]“ structural thickness, so if either of these values would increase, K[1] would decrease, and the stiffness coming from shear deformation would increase.
Component considering warping deformation
The K[2] parameter further softens the structure taking into account the warping deformations. The detailed calculation of the K[2] parameter will not be shown here due to its extensiveness and
complexity. The details of this calculation can be found in the literature [5]. The calculation is based on the “Folded Plate” theory [8]. To obtain the K[2] parameter the warping displacements and
warping coordinates have to be calculated for the sheeting. The following figure shows an example for the normalized warping displacements and deformed shapes for k=1 and k=2:
k=1 and k=2 are connected to individual solutions for the differential equation system describing the mechanical behavior based on the “Folded Plate” theory.
Calculating shear stiffness
The behavior of the two components in the formula of the shear stiffness is different. The part that considers the shear deformation only depends on the effective width “a”, but independent from the
total length of the shear field “L[s]”. On the other hand the part that considers the warping deformation is also dependent on the total length of the shear field “L[s]”. The K[2] parameter in the
denominator is divided by L[s], which means that the longer the total shear field is, the larger the specific shear stiffness is going to get because of the contribution of the warping deformations.
Also if the total length of the shear field “L[s]“ would get really low, then K[2]/L[s] would approach infinity, which means that the stiffness approaches zero. For this reason a minimal length for
sheetings “L[s,min]” is also provided by the manufacturers, which gives the specific shear stiffness “G[s]” a minimum value.
Comparison against shell models
The effect of the shear stiffness of a trapezoidal sheeting can be modeled in multiple ways. In Consteel additionally to built-in shear field field object applicable on beam elements, the sheeting
can be modeled directly with shell elements. This latter approach is more complicated and time consuming to set up, but should provide similar results. Such a comparison was prepared in Consteel.
Examined structure
Stabilized beam: IPE300 S235
Span: L = 4140 mm
Type of trapezoidal sheeting: Hoesch T 35.1
• Examined thicknesses: 0,75 mm; 1 mm; 1,25 mm
• Examined sheeting lengths: 2 m, 3 m
Horizontal line load: q[y] = 10 kN/m
The load and the trapezoidal sheeting are both acting on the centerline of the stabilized beam.
Consteel shear field model
In this modeling version the stabilizing effect of the trapezoidal sheeting is modeled by the shear field object implemented into Consteel.
Example figure: L[s] = 2 m, a = 2 m, G[s] = 3293 kN/m, S = 6586 kN
Consteel shell model
In this modeling version, the trapezoidal sheeting is modeled by shell elements. The thicknesses of the shell elements are equal to the structural thickness t[core]. The model of the trapezoidal
sheeting is included in a frame made from beam elements. The sheeting is connected to the frame by link elements at the lower flanges where the sheeting is screwed to the substructure. The frame is
included in the model in order to connect the shell elements to the main beam. The beams of the frame have a cross-section that has relatively insignificant weaker axis inertia compared to the shear
stiffness of the sheeting. The shell elements are connected through the frame to the main beam by link elements that only transfer force in their axial direction.
Example figure: L[s] = 2 m, a = 2 m
The shell elements are also supported in the vertical direction along the middle lines of its top and bottom flanges in order to eliminate the bending deformation resulting from the eccentric
compression load on the sheeting, since the shear field model also does not take this effect into account. The edges on both sides of the sheeting are supported against “x” and “z” directional
displacements in order to account for the sheeting being fixed to the substructure at all 4 of its edges. The line supports on the plate elements are shown on the following picture viewing the
structure from below.
Horizontal displacement examination | {"url":"https://consteelsoftware.com/?knowledgebase_category=theory-background","timestamp":"2024-11-07T01:17:52Z","content_type":"text/html","content_length":"83682","record_id":"<urn:uuid:d28f9df5-cd7d-4e8c-b75d-bcacd0d3c2b3>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00190.warc.gz"} |
COORDINATE GEOMETRY: How to find the coordinates & ratios by section formula?
We had study two blogs of coordinate geometry in which we learned COORDINATE GEOMETRY: How to find the distance between two points by Distance Formula? & COORDINATE GEOMETRY: How to find the area of
a triangle in the Cartesian plane?
Now we will discuss how to find coordinates & ratio in the cartesian plane by section formula? In some situation, we have coordinates of endpoints & ratio of the line segment.
And we need to find coordinates of the point of intersection (it may be a midpoint in some cases) of the line segment.
In some cases, we have coordinates of the point of intersection & endpoints of the line segment. Here we need to find the ratio of the given line segment.
Line segment
It is a part of the line which has two endpoints.
It is a point of a line segment which divide it into two equal parts.
It is a term in a Cartesian plane which shows partition or division of any line segment.
Section Formula:-
It is a formula which is helpful to find coordinates & ratio of the line segment in the cartesian plane.
For coordinate of x, we use x = (m₁x₂+m₂x₁)÷(m₁+m₂)
For coordinate of y, we use y = (m₁y₂+m₂y₁)÷(m₁+m₂)
All details are shown in the below picture.
• PQ is a line segment with endpoints P & Q.
• (x₁,y₁) are the coordinates of point P.
• (x₂,y₂) are the coordinates of point Q.
• m₁ & m₂ are the ratios of line segment PQ.
• O is the point of intersection of line segment PQ.
• Coordinates of O are (x, y).
Find the coordinate of point O which divide the line segment PQ joining P(-3,-2) & Q(2,-2) in the ratio 3:2.
(All details as shown in the above picture)
We have formulas to find the coordinates of the point of intersection O whose coordinates are (x , y).
x = (m₁x₂+m₂x₁)÷(m₁+m₂)
y = (m₁y₂+m₂y₁)÷(m₁+m₂)
Put all the given values in formulas-
For the value of x:-
x = (m₁x₂+m₂x₁)÷(m₁+m₂)
x = {(3×2)+(2×-3)}÷(3+2)
x = {6-6}÷5
x = 0÷5
For the value of y:-
y = (m₁y₂+m₂y₁)÷(m₁+m₂)
y = {(3×-2)+(2×-2)}÷(3+2)
y = {-6-4}÷5
y = -10÷5
y = -2
So here the coordinates of point O are (0,-2).
Find the ratio in which the line segment joining the points (– 3, 10) and (6, – 8) is divided by (– 1, 6).
Here we have endpoints of a line segment (x₁,y₁)=(-3,10), (x₂, y₂)=(6,-8) & Point of intersection are (x , y) =(-1,6).
To find ratios m₁:m₂.
Put all the given values in the formula of x or y coordinate to determine the ratios-
x = (m₁x₂+m₂x₁)÷(m₁+m₂)
-1 = {(m₁×6)+(m₂×-3)}÷(m₁+m₂)
-1 = {6m₁-3m₂}÷(m₁+m₂)
-1(m₁+m₂) =(6m₁-3m₂)
-m₁-m₂ = 6m₁-3m₂
-7m₁ = -2m₂
m₁÷m₂ = -2÷(-7)
So we have m₁:m₂ = (-2):(-7)
In these three parts of coordinate geometry, we learned about Distance Formula, Area of triangle & section formula. We may also find the area of a parallelogram, rhombus, square, etc. in the
cartesian plane. | {"url":"https://www.tirlaacademy.com/2020/07/coordinate-geometry-how-to-find.html","timestamp":"2024-11-03T00:37:56Z","content_type":"application/xhtml+xml","content_length":"322697","record_id":"<urn:uuid:18f4d80a-aa3d-46f1-b694-0962f028b8e0>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00209.warc.gz"} |
Applied logic - Denotational Semantics | Britannica
While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions.
Select Citation Style
Thank you for your feedback
Our editors will review what you’ve submitted and determine whether to revise the article.
The denotational semantics for programming languages was originally developed by the American logician Dana Scott and the British computer scientist Christopher Strachey. It can be described as an
application of the semantics to computer languages that Scott had developed for the logical systems known as lambda calculus. The characteristic feature of this calculus is that in it one can
highlight a variable, say x, in an expression, say M, and understand the result as a function of x. This function is expressed by (λx,M), and it can be applied to other functions.
The semantics for lambda calculus does not postulate any individuals to which the functions it deals with are applied. Everything is function, and, when one function is applied to another function,
the result is again a function.
Hypothetical reasoning is often presented as an extension and application of logic. One of the starting points of the study of such reasoning is the observation that the conditional sentences of
natural languages do not have a truth-conditional semantics. In traditional logic, the conditional “If A, then B” is true unless A is true and B is false. However, in ordinary discourse,
counterfactual conditionals (conditionals whose antecedent is false) are not always considered true.
The study of conditionals faces two interrelated problems: stating the conditions in which counterfactual conditionals are true and representing the conditional connection between the antecedent and
the consequent. The difficulty of the first problem is illustrated by the following pair of counterfactual conditionals:
If Los Angeles were in Massachusetts, it would not be on the Pacific Ocean. If Los Angeles were in Massachusetts, Massachusetts would extend all the way to the Pacific Ocean.
Both of these conditionals cannot be true, but it is not clear how to decide between them. The example nevertheless suggests a perspective on counterfactuals. Often the counterfactual situation is
allowed to differ from the actual one only in certain respects. Thus, the first example would be true if state boundaries were kept fixed and Los Angeles were allowed to change its location, whereas
the latter would be true if cities were kept fixed but state boundaries could change. It is not obvious how this relativity to certain implicit constancy assumptions can be represented formally.
Other criteria for the truth of counterfactuals have been suggested, often within the framework of possible-worlds semantics. For example, the American philosopher David Lewis suggested that a
counterfactual is true if and only if it is true in the possible world that is maximally similar to the actual one.
The idea of conditionality suggests that the way in which the antecedent is made true must somehow also make the consequent true. This idea is most naturally implemented in game-theoretic semantics.
In this approach, the verification game with a conditional “If A, then B” can be divided into two subgames, played with A and B, respectively. If A turns out to be true, it means that there exists a
verifying strategy in the game with A. The conditionality of B on A is thus implemented by assuming that this winning strategy is available to the verifier in the game with the consequent B. This
interpretation agrees with evidence from natural languages in the form of the behaviour of anaphoric pronouns. Thus, the availability of the winning strategy in the game with B means that the names
of certain objects imported by the strategy from the first subgame are available as heads of anaphoric pronouns in the second subgame. For example, consider the sentence “If you give a gift to each
child for her birthday, some child will open it today.” Here a verifying strategy in the game with “you give a gift to each child for her birthday” involves a function that assigns a gift to each
child. Since this function is known when the consequent is dealt with, it assigns to some child her gift as the value of “it.” In the usual logics of conditional reasoning, these two questions are
answered indirectly, by postulating logical laws that conditionals are supposed to obey.
Fuzzy logic and the paradoxes of vagueness
Certain computational methods for dealing with concepts that are not inherently imprecise are known as fuzzy logics. They were originally developed by the American computer scientist Lotfi Zadeh.
Fuzzy logics are widely discussed and used by computer scientists. Fuzzy logic is more of a rival to classical probability calculus, which also deals with imprecise attributions of properties to
objects, than a rival to classical logic calculus. The largely unacknowledged reason for the popularity of fuzzy logic is that, unlike probabilistic methods, fuzzy logic relies on compositional
methods—i.e., methods in which the logical status of a complex expression depends only on the status of its component expressions. This facilitates computational applications, but it deprives fuzzy
logic of most of its theoretical interest.
On the philosophical level, fuzzy logic does not make logical problems of vagueness more tractable. Some of these problems are among the oldest conceptual puzzles. Among them is the sorites paradox,
sometimes formulated in the form known as the paradox of the bald man. The paradox is this: A man with no hairs is bald, and if he has n hairs, then adding one single hair will not make a difference
to his baldness. Therefore, by mathematical induction, a man of any number of hairs is bald. Everybody is bald. One natural attempt to solve this paradox is to assume that the predicate “bald” is not
always applicable, so that it leaves what are known as truth-value gaps. But the boundaries of these gaps must again be sharp, reproducing the paradox. However, the sorites paradox can be solved if
the assumption of truth-value gaps is combined with the use of a suitable noncompositional logic.
Jaakko J. Hintikka | {"url":"https://www.britannica.com/topic/applied-logic/Denotational-semantics","timestamp":"2024-11-07T16:08:34Z","content_type":"text/html","content_length":"91637","record_id":"<urn:uuid:0f2d43c4-baf6-4b51-9859-dfdebbe68d23>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00751.warc.gz"} |
Distance Between Two Points - Definition, Formula, Examples (2024)
Home » Math Vocabulary » Distance Between Two Points
• Distance between Two Points: Introduction
• What Are the Coordinates of a Point?
• Derivation of Distance Formula
• Solved Examples
• Practice Problems
• Frequently Asked Questions
Distance between Two Points: Introduction
Given any two points on a coordinate plane, we can find the distance between them if the coordinates of both points are known. It is a fundamental concept in geometry. Let’s dive right into it!
Fluently Add within 10
Add Two Numbers (Up to 5) Game
Fluently Add within 20
Add Two Numbers Game
Counting Money
Calculate the Difference Between Amounts of Money Game
Counting Objects Within 10
Choose the Missing Number in Between Game
Divide 2-digit by 1-digit Numbers
Complete the Division Solution Where Dividend is Between 6-9 Game
Multiply Fractions
Complete the Multiplication Between Fractions and Whole Numbers Game
Multiply 2-digit by 2-digit numbers
Complete the Multiplication of Two 2-Digit Numbers Game
Subtract mixed numbers
Complete the Subtraction between a Mixed Number and a Fraction Game
Convert Between Gallons, Pints, Quarts and Cups Using Table Game
Convert Between Ounces and Pounds Using Table Game
Distance between Two Points: Definition
We can define the distance between two points as the length of the line segment that connects the two given points. Distance between two points on the Cartesian plane can be calculated by finding the
length of the line segment that joins the given coordinates.
Related Worksheets
What Is the Distance between Two Points?
There is only one line passing through two points. So, the distance between two points can be calculated by finding the length of the line segment connecting the two points.
For example, if P and Q are two points and PQ $= 8$ feet, it means that the distance between the points P and Q is 8 feet.
The distance between two points is the length of the line segment joining them. Since the length of the line segment cannot be negative, the distance between two points is always positive.
The distance from the point A to B is the same as the distance from B to A.
NOTE: The shortest distance between two points is the straight line joining them.
What Are the Coordinates of a Point?
In Euclidean geometry, the position of points is defined by their coordinates along the X- axis and Y-axis. Therefore, the coordinates of a point is the ordered pair that is used to identify the
location of that point in the coordinate plane.
In the above figure, the coordinates of point A are (x,y). This means that the point A is x units away from the y-axis and y units away from the x-axis.
Coordinates of a point on the x-axis are of the form (x, 0), where x is the distance of the point from the origin.
Coordinates of a point on the y-axis are of the form (0, y), where y is the distance of the point from the origin.
How to Find the Distance between Two Points?
To find the distance between two points, we find the distance between two coordinates corresponding to those points using the distance formula. For any point in the 2-D Cartesian plane, we apply the
2-D distance formula or the Euclidean distance formula.
The Distance between Two Points Formula
If the coordinates of the points are P$(\text{x}_{1},\text{y}_{1})$ and Q$(\text{x}_{2},\text{y}_{2})$, then the distance between P and Q is given by
PQ $=\sqrt{(\text{x}_{2} − \text{x}_{1})^{2} + (\text{y}_{2} − \text{y}_{1})^{2}}$
Derivation of Distance Formula
Suppose we have two points A$(\text{x}_{1},\text{y}_{1})$ and B$(\text{x}_{2},\text{y}_{2})$ in the coordinate plane. We have to find the distance between them.
What do we know?
AC and BD are perpendicular to the x-axis.
AB is parallel to the x-axis.
Coordinates: A$(\text{x}_{1},\text{y}_{1})$ and B$(\text{x}_{2},\text{y}_{2})$, $C(\text{x}_{1},0)$ and D$(\text{x}_{2},0)$
Distance between the points A and B is calculated as follows:
C and D are the points on the x-axis.
$\text{AM} = \text{CD} = \text{OD} – \text{OC} = \text{x}_{2} – \text{x}_{1}$
Similarly, $\text{AC} = \text{MD}$
$\text{AC} = \text{MD} = \text{BD} – \text{AC} = \text{y}_{2} – \text{y}_{1}$
By Pythagoras’ theorem,
$PQ^{2} = PR^{2} + QR^{2}$
PQ $=\sqrt{(\text{x}_{2} − \text{x}_{1})^{2} + (\text{y}_{2} − \text{y}_{1})^{2}}$
Therefore, distance between two points $(\text{x}_{1},\text{y}_{1})$ and $(\text{x}_{2},\text{y}_{2})$ is given by:
PQ $=\sqrt{(\text{x}_{2} − \text{x}_{1})^{2} + (\text{y}_{2} − \text{y}_{1})^{2}}$
The final formula remains the same, irrespective of which quadrants A and B lie in.
We can summarize the formula in a picture as:
Example 1: The distance between the points P(3, 0) and Q(0, 4) is
PQ $=\sqrt{(4 − 0)^{2} + (0 − 3)^{2}} = \sqrt{16 + 9} = \sqrt{25} = 5$ units
Example 2: R(2, 5) and S(1, 2)
RS $=\sqrt{(1 − 2)^{2}+(2 − 5)^{2}} = \sqrt{1 + 9} = \sqrt{10}$ units
Distance of a Point from the Origin
Suppose a point P(x, y) in the xy–plane as shown in the figure below:
The distance between point A and the origin is OA.
Point A is x units away from the y-axis and y units away from the x-axis.
Using Pythagoras’ theorem, we get
OA $=\sqrt{(x − 0)^{2} + (y − 0)^{2}} = \sqrt{x^{2} + y^{2}}$
Thus, the distance between any point (x,y) in xy-plane and the origin (0,0) is given by: d $= \sqrt{x^{2} + y^{2}}$
Distance between Two Points: Using Pythagoras’ Theorem
Consider the following example.
A boy started from point A and walked west for 12 miles. He then turned to the north and walked for 5 miles more. We have to calculate the shortest distance between the initial position and final
A pictorial representation of the above situation is:
The initial position is A and the final position is C. The distance between points A and B is 12 miles and between points B and C is 5 miles. Here, triangle ABC is a right triangle.
The shortest distance between points A and C is given by AC. This distance is calculated using the Pythagoras theorem as follows:
$AC = \sqrt{AB^{2} + BC^{2}} = \sqrt{12^{2} + 5^{2}} = \sqrt{144 + 25} = \sqrt{169} = 13$ miles
In this article, we learned about the distance between two points. The distance between two points can be calculated by measuring the length of the line segment. To read more such informative
articles on other concepts, do visit our website. We, at SplashLearn, are on a mission to make learning fun and interactive for all students.
Solved Examples
1. What is the distance between (0,0) and (3,4)?
Solution: The distance between (0,0) and (x,y) is given by: $\sqrt{x^{2} + y^{2}}$
The distance between (0,0) and (3,4) is given by: $\sqrt{3^{2} + 4^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5$ units.
2. Find the distance between the points $( − 1,2)$ and $(4, − 8)$.
Solution: The distance between the points $( − 1,2) and $(4, − 8)$ is given by:
$\sqrt{(4 − ( − 1))^{2} + ( − 8 − 2)^{2}} =\sqrt{5^{2} + ( − 10)^{2}} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5}$ units
3. If the distance between the points $(4, 4)$ and $(1, a)$ is 5 units, then find the value of a.
Solution: Distance between the points $(4, 4)$ and $(1, a) = \sqrt{(1 − 4)^{2} + (a − 4)^{2}}$
$5 = \sqrt{(1 − 4)^{2} + (a − 4)^{2}} = \sqrt{( − 3)^{2} + (a − 4)^{2}}$
On squaring both the sides, we get the equation for distance between two points as
$25 = 9 + (a − 4)^{2}$
$25 − 9 = (a − 4)^{2}$
$16 = (a − 4)^{2}$
Taking square root
$4 = a − 4$ or $− 4 = a − 4$
$a = 8$ or $a = 0$.
4. Find a point on the x axis that is equidistant from the points $(1,− 4)$ and $( − 3,4)$.
Solution: Let the point on x axis be B$(x,0)$ and let A$(1, − 4)$ and C$( − 3,4)$.
$AB = BC \Rightarrow AB^{2} = BC^{2}$
$(x − 1)^{2} + (0 −(− 4))^{2} =(x −( −3))^{2} + (0 − 4)^{2}$
$(x −1)^{2} + 16 = (x + 3)^{2} + 16$
$(x − 1)^{2} − (x + 3)^{2} = 16 − 16$
$x^{2} − 2x + 1 − (x^{2} + 6x + 9) = 0$
$x^{2} − 2x + 1 − 9 − x^{2} − 6x = 0$
$− 8x − 8 = 0$
$− 8x = 8$
$x = \frac{−8}{8} = −1$
The point is $( −1,0)$.
5. Amaya traveled 20 miles to the west and then 21 miles to the north. Calculate the shortest distance between the initial and final point?
Solution: Assuming $AB = 20$ mi and $BC = 21$ mi
The shortest distance $= AC =$
$= \sqrt{AB^{2} + BC^{2}} = \sqrt{20^{2} + 21^{2}} = \sqrt{400 + 441} = \sqrt{841} = 29$ mi
Practice Problems
If the distance between the origin and $(a, 8)$ is 17 units, then find the value of $a$.
Correct answer is: 15
The distance between $(0,0)$ and $(a, 8)$ is given by: $\sqrt{a^{2} + 8^{2}} = 17$
On squaring, we get $a^{2} + 64 = 289$
$\Rightarrow a^{2} = 289$ $-$ $64 \Rightarrow a^{2} = 225 \Rightarrow a = 15$
The distance of $(-$$5,8)$ from x-axis is:
$8$ units
$-8$ units
$-5$ units
$5$ units
Correct answer is: $8$ units
The distance of $(-$$5,8)$ from x-axis is the distance between $(-$$5,8)$ and $(-$$5,0)$.
The distance of $(-$$5,8)$ from x-axis $= |8$$-$$0| = 8$ units.
The distance of $(-$$2,-$$10)$ from y-axis is:
$10$ units
$-10$ units
$-2$ units
$2$ units
Correct answer is: $2$ units
The distance of $(-$$2,-$$10)$ from y-axis $= | -2|$ units $= 2$ units
Find half the length of the line segment joining the points $(2,3)$ and $(5,7)$.
$2$ units
$2.5$ units
$5$ units
$\sqrt{13}$ units
Correct answer is: $2.5$ units
The distance between the points $(2,3)$ and $(5,7)$ is
$\sqrt{(5 - 2)^{2} + (7 - 3)^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5$ units. Half the distance is 2.5 units.
Shyna traveled east by bus and west by car. If the distance covered by bus is 24 km and the shortest distance is 25 km, then what is the distance covered by car?
1 km
11 km
7 km
10 km
Correct answer is: 7 km
Let the distance covered by car be $x$ km.
Shortest distance $= \sqrt{24^{2} + x^{2}} \Rightarrow 25 = \sqrt{24^{2} + x^{2}}$
Squaring both the sides, we get $625 = 576 + x^{2} \Rightarrow x^{2} = 49 \Rightarrow x = 7$
Frequently Asked Questions
What is the distance between two points in a 3D plane?
If the coordinates of two points in a 3D plane are P$(\text{x}_{1}, \text{y}_{1}, \text{z}_{1})$ and Q$(\text{x}_{2}, \text{y}_{2}, \text{z}_{2})$, the distance between the points P and Q is given
byPQ $= \sqrt{(x_{2} − x_{1})^{2} + (y_{2} − y_{1})^{2} + (z_{2} − z_{1})^{2}}$
What is the shortest distance between two points?
The shortest distance between two points is the length of the straight line that connects both the points. We use the distance formula to find this distance using the coordinates given in a
two-dimensional plane.
How can one find the vertical distance between two points?
The vertical distance between two points is calculated by the difference of the y coordinates of the two points, i.e., vertical distance between two points, $|(y_{2} − y_{1})|$.
How to find the horizontal distance between two points?
The vertical distance between two points is calculated by the difference of the y coordinates of the two points, i.e., vertical distance between two points, $(\text{y}_{2} − \text{y}_{1})$ where $(\
text{x}_{1}, \text{y}_{1})$ and $(\text{x}_{2},\text{y}_{2})$ are the coordinates of the points.
Can we change the order of the points in the distance formula?
Yes, we can change the order of points in the distance formula. We can also write the formula as $\sqrt{(x_{1} − x_{2})^{2} + (y_{1} − y_{2})^{2}} = \sqrt{(x_{2} − x_{1})^{2} + (y_{2} − y_{1})^{2}}$.
What is the driving distance between two points or places?
The driving distance generally refers to the distance when traveling by car. For example, the driving distance from Chicago to Orlando (is 1179 miles.
Related Articles
• Coordinate Plane
• Point
• Line Segment | {"url":"https://inflablesypeloteros.com/article/distance-between-two-points-definition-formula-examples","timestamp":"2024-11-10T08:04:37Z","content_type":"text/html","content_length":"121792","record_id":"<urn:uuid:0ec4acbb-1f68-48cf-b4b3-12dc68792857>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00547.warc.gz"} |
Mathematics in not just learning about numbers, but also an emotional learning.
“Students who feel anxious, angry, or depressed don’t learn: People who get caught up in these moods don’t take in information effectively.” (Daniel Goleman, Emotional Intelligence)
Yes, mathematics is learned with the heart. If you do not respect emotions and work on cross-curricular learning variables, there are many limitations in the processes of mathematical assimilation,
and everything becomes slower, more difficult and demotivating. For this reason, in each class, Langley educators are concerned not only with learning content, but also with making our students feel
good, at ease, and that they do not go through negative experiences that block their learning.
Happy boys, girls, adolescents with excellent school results.
This is only possible with a different and innovative learning system, which breaks with the deficiencies of a traditional system. A system that respects the individuality of its students, which
ensures the ability to learn; a system where respect for emotions are a fundamental part of the process and that has a study material of excellence. These are the fundamentals of the Langley Learning
By individualizing teaching we can focus specifically on what your child needs. By adapting to the different ways of learning, we teach them differently according to their needs. Thus they will learn
what they require without wasting time in what they already know. In this way they will be motivated and will advance much faster than in school with less hours of work.
Your son or daughter will be able to learn from examples, find answers for themselves and thus provide solutions to the different problems they encounter. This independence that the learning capacity
gives will allow them to face the school challenges and the demands of higher education with much more success.
Math learning should not be based only on calculation, but should also contain strong emotional development.
That is why we care that our students develop their self-esteem, tolerance to frustration, ability to face and correct errors, logical reasoning capacity, and other fundamental transversal skills for
Study Material
Langley study materials are of high quality, constructive, self-instructive, sequential and rich in examples, focused on the three pillars of mathematical study: calculus-concepts-problem solving
This allows our students to learn gently and without stumbling, feel safe and confident, achieving autonomy and trust.
What is the most important objective of learning math?
The vast majority of people assume that mathematics is for solving calculation exercises. Many think that when a student responds quickly how much is 256 x 24, or he or she can mentally solve 2x + 4
= 3x/5, then he or she is good at mathematics. Nothing further from the truth. That student is good at calculating, but not necessarily good at mathematics.
The real objective of mathematics is to teach deductive thinking and logical reasoning, what is done through the application of calculation and concepts in problem solving. When the students learn to
face a problem in a logical way and to determine what calculation tool is the one that will help them solve that problem, it is when the real sense of mathematics appears, learning how to think
We could say then, that mathematics is learned with the heart and applied with the mind. Therefore, when mathematics is truly learned, skills are learned that a person can apply in all aspects of
life, be it an actor or an engineer, an entrepreneur or a philanthropist. It doesn’t matter, logical reasoning exists to solve the problems we face in life. | {"url":"https://langleylearning.com/fundamentals/","timestamp":"2024-11-13T00:45:38Z","content_type":"text/html","content_length":"70397","record_id":"<urn:uuid:5d790b7c-1eb9-4c0e-a786-196c3ee77c38>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00633.warc.gz"} |
Mathematics Seminar 09/14/23
Sep 14 3:30 pm
Huy Q. Pham
Mathematics Seminar Series
Numerical approximation of solution of stochastic Allen-Cahn equation
Physical Location
Allen 14
Abstract: Stochastic partial differential equations (SPDEs) are generalized from PDEs by random force terms and coefficients. SPDEs have been a subject of interest in recent years for their
widespread application in quantum field theory, statistical mechanics, and spatial modeling. In this talk, stochastic Allen-Cahn-type equation (a 1+1-dimensional space-time PDEs driven by space-time
white noise) and its approximation by a fully discrete space-time explicit finite difference scheme are studied. Many previous results have indicated that a strong convergence rate of 1/2 with
respect to the parabolic grid is predicted to be ideal. However, one can reach almost sure convergence of rate 1 (and no better) when measuring the error in appropriate distributional norm. | {"url":"https://www.math.msstate.edu/events/2023/09/mathematics-seminar-091423","timestamp":"2024-11-08T06:18:05Z","content_type":"text/html","content_length":"35028","record_id":"<urn:uuid:425a040c-605e-4255-99eb-467cb6edb6a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00413.warc.gz"} |
The Infinite Primes and Museum Guard Proofs, Explained | Quanta Magazine
Aubrey Wade for Quanta Magazine
In January, I spoke with Günter Ziegler, one of the authors of Proofs From THE BOOK, a compilation of some of the most beautiful and elegant proofs in mathematics. The collection was inspired by the
legendary mathematician Paul Erdős, who envisioned an infinite book in which God had written the perfect proof for each theorem. Today I want to share a couple of my favorite proofs from THE BOOK.
The first is an old chestnut that all math majors (including this one) bump into sometime during their education; the second came as a delightful surprise.
The volume opens with perhaps the most famous proof in mathematics:
Theorem: There are infinitely many prime numbers.
The proof we’ll give dates back to Euclid, and our version of his proof uses one of the oldest tricks in the book (and THE BOOK): the notion of a “proof by contradiction.” In such a proof, we assume
the opposite of what we really want to prove, and then reason from there until we reach a statement that is clearly impossible. (Updated on March 28, 2018: A couple of readers observed in the
comments section that Euclid himself didn’t phrase his proof in terms of contradiction; however, that’s how mathematicians most commonly present it today.)
So let’s assume that there are only finitely many prime numbers. If that’s the case, we can list them all — we’ll name them p[1], p[2], p[3] and so on, all the way through some last prime, p[n]. Now
let’s make a new number, N, by multiplying all these primes together and adding 1:
N = p[1]p[2]p[3.]…p[n] + 1.
N cannot be prime, because it’s bigger than any of the numbers on our complete list of primes, and we assumed our list was complete. So N must have some factor other than itself and 1, and if we keep
dividing that factor into smaller and smaller factors, we’ll eventually hit a prime number p that divides evenly into N.
But which prime number is p? It must be on our list of primes, because our list is complete. But it also can’t be on our list of primes, because none of those primes divide evenly into N: They all
leave a remainder of 1. This is a contradiction, so our original assumption — that there are only finitely many primes — must have been wrong. □
(Mathematicians like to put a little box at the end of a proof, to make it easy to spot where the argument ends.)
When I first started studying proofs, I found it somehow unsatisfying that a proof by contradiction doesn’t end by sort of unrolling the contradiction backward, along the lines of “This is a
contradiction, so the statement before it must be false, which means the statement before that must be false…” and so on, until we’d worked our way back to the original assumption. But the power of
abstraction is precisely that we don’t have to do this each time we prove something by contradiction — we simply understand the underlying principle once, and then we can let it do the heavy lifting
in proof after proof.
In our next proof we’ll do something similar, but this time we’re going to abstract the intuition behind the domino effect into a powerful mathematical principle known as induction. This says that to
prove a statement about all positive whole numbers, you have to do only two things:
1. Prove that the statement is true for the number 1. (This is called the base case, in which we knock over the first domino.)
2. Prove that whenever the statement is true for a number n, it is also true for n + 1. (This is called the induction step, in which we show that each domino knocks over the next domino.)
Actually, we’re going to use a variation, “strong induction,” in which the induction step requires showing that if the statement we’re trying to prove is true for all numbers from 1 to n, then it’s
also true for n + 1. It’s a similar sort of logic to regular induction, but it comes in handy in slightly different settings, including the following:
Theorem: In a museum shaped like a polygon with k walls, there’s always some way to station k/3 or fewer guards so that every spot in the museum is visible to some guard. (So, for example, if our
museum has 18 walls, then we need at most 6 guards. If the number of walls isn’t divisible by 3 then we get to round down, so a museum with 19 walls will also need at most 6 guards.)
In the proof that’s contained in THE BOOK, which comes from Steve Fisk, we start by drawing in a bunch of noncrossing diagonals until our museum is divided into triangles:
Now we’re going to use strong induction to prove that it’s always possible to color the corners of the room red, yellow and blue so that every color appears exactly once in each triangle. Our
induction will be with respect to the number of triangles.
So first we must prove the base case: that we can do such a coloring if our polygon is made of a single triangle. But that’s easy — we can just color the three corners red, yellow and blue, and we’re
Now for the induction step: We need to prove that if such a coloring is always possible for any polygon made of one triangle, or two triangles, or three triangles…, all the way to n triangles, then
such a coloring is also possible for any polygon made of n + 1 triangles.
So let’s consider a polygon made of n + 1 triangles. We can split it into two smaller polygons by cutting along one of the diagonals:
Our induction assumption tells us that each of these smaller polygons can be colored in the desired way. Now to get a coloring of the original “n + 1” polygon, all we have to do is glue the two
smaller polygons back together. (If their colorings don’t match along the diagonal where we’re gluing, we can just switch around the reds, yellows and blues in one of the two polygons so it matches
the other).
That completes our induction argument, so now we know that there’s some way to color the corners of our museum red, yellow and blue so that all three colors show up in every triangle.
At least one of these colors — let’s say, red — appears on at most k/3 corners, since otherwise the red, yellow and blue corners would add up to more than k (the total number of corners in the room).
Now, station a guard at each red corner. Every point in the museum is in some triangle, so it’s in the sightline of the guard at that triangle’s red corner. □ | {"url":"https://www.quantamagazine.org/the-infinite-primes-and-museum-guard-proofs-explained-20180326/","timestamp":"2024-11-11T03:30:40Z","content_type":"text/html","content_length":"201212","record_id":"<urn:uuid:c55621bf-ed2a-47d8-ab37-1c0986dcb9b1>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00623.warc.gz"} |
Effective calculation of gravity effects of uniform triangle polyhedra
STUDIA GEOPHYSICA ET GEODAETICA, vol.56, no.1, pp.185-195, 2012 (SCI-Expanded)
• Publication Type: Article / Article
• Volume: 56 Issue: 1
• Publication Date: 2012
• Doi Number: 10.1007/s11200-011-9004-x
• Journal Name: STUDIA GEOPHYSICA ET GEODAETICA
• Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
• Page Numbers: pp.185-195
• Karadeniz Technical University Affiliated: No
Uniform tetrahedra are commonly used elementary bodies for gravity calculations from which arbitrary polyhedra can be composed. A simple derivation of the gravity effect is presented for the apex P
of the tetrahedron expanded from P to an arbitrarily oriented plane triangle. Integration of its potential effect in a rotated coordinate system applies vector algebra and renders the anomalous
potential depending on the distance of P over the triangle plain and a junction of the triangle coordinates. Partial differentiation by moving P infinitesimally in z-direction leads to two terms, a
simple and a complex one; they can be understood as describing the same difference from two points of view: leaving P at the apex of the changed polyhedron or moving P off the unchanged polyhedron.
Both views imply the same shape change and the sum over the polyhedron is thus numerically equal. Hence we need to calculate Only the one of the terms of the differential which is simpler. The
calculation of the gravity effect is numerically simplified and more stable. This has been tested for many models and is demonstrated by two examples. | {"url":"https://avesis.ktu.edu.tr/yayin/a4a4c9d9-15f3-42f4-bc87-34aacc9bacb8/effective-calculation-of-gravity-effects-of-uniform-triangle-polyhedra","timestamp":"2024-11-08T08:26:21Z","content_type":"text/html","content_length":"51249","record_id":"<urn:uuid:3ef9bdd8-fbfc-481e-9b31-b5e5d3e52758>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00220.warc.gz"} |
Using model options in PyBaMM
An interactive online version of this notebook is available, which can be accessed via
Alternatively, you may download this notebook and run it offline.
Using model options in PyBaMM#
In this notebook we show how to pass options to models. This allows users to do things such as include extra physics (e.g. thermal effects) or change the macroscopic dimension of the problem (e.g.
change from a 1D model to a 2+1D pouch cell model). To see all of the options currently available in PyBaMM, please take a look at the documentation here. For more information on combining submodels
explicitly to create your own custom model, please see the Using Submodels notebook.
Example: Solving the SPMe with a lumped thermal model#
PyBaMM is designed to be a flexible modelling package that allows users to easily include different physics within a model without having to start from scratch. In this example, we show how to pass
model options to include thermal effects in the SPMe (for more information on the SPMe see here). First we import PyBaMM and any other packages we need
%pip install "pybamm[plot,cite]" -q # install PyBaMM if it is not installed
import pybamm
import os
os.chdir(pybamm.__path__[0] + "/..")
Note: you may need to restart the kernel to use updated packages.
We then choose out model options, which a set as a dictionary. We choose to model the behaviour in the particle by assuming the concentration profile is quadratic within the particle. We also choose
a lumped thermal model (note that this is fully-coupled, i.e. parameters can depend on temperature). For an in-depth look at the thermal models see the thermal models notebook
options = {"particle": "quadratic profile", "thermal": "lumped"}
We then pass our options to the model
model = pybamm.lithium_ion.SPMe(options)
We choose to use the parameters from Chen2020.
param = pybamm.ParameterValues("Chen2020")
We then create and solve a simulation, making sure we pass in our updated parameter values
simulation = pybamm.Simulation(model, parameter_values=param)
simulation.solve([0, 3600])
<pybamm.solvers.solution.Solution at 0x7f6169de4690>
Finally we plot the voltage and the cell temperature
"Voltage [V]",
"X-averaged cell temperature [K]",
Note that the variable “X-averaged cell temperature [K]” is the scalar-valued lumped temperature, whereas the variable “Cell temperature [K]” is the value of the lumped temperature broadcasted across
the whole cell domain. This type of behaviour is purposefully designed to allow easy comparison of different models and settings. For instance we may wish to compare a simulation that uses a lumped
thermal model with a simulation that uses a full thermal model (i.e. one that solves the heat equation in the x-direction). When comparing these two model we could then plot the same variable “Cell
temperature [K]” to compare the temperature throughout the cell.
The relevant papers for this notebook are:
[1] Joel A. E. Andersson, Joris Gillis, Greg Horn, James B. Rawlings, and Moritz Diehl. CasADi – A software framework for nonlinear optimization and optimal control. Mathematical Programming Computation, 11(1):1–36, 2019. doi:10.1007/s12532-018-0139-4.
[2] Charles R. Harris, K. Jarrod Millman, Stéfan J. van der Walt, Ralf Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J. Smith, and others. Array programming with NumPy. Nature, 585(7825):357–362, 2020. doi:10.1038/s41586-020-2649-2.
[3] Scott G. Marquis, Valentin Sulzer, Robert Timms, Colin P. Please, and S. Jon Chapman. An asymptotic derivation of a single particle model with electrolyte. Journal of The Electrochemical Society, 166(15):A3693–A3706, 2019. doi:10.1149/2.0341915jes.
[4] Venkat R. Subramanian, Vinten D. Diwakar, and Deepak Tapriyal. Efficient macro-micro scale coupled modeling of batteries. Journal of The Electrochemical Society, 152(10):A2002, 2005. doi:10.1149/1.2032427.
[5] Valentin Sulzer, Scott G. Marquis, Robert Timms, Martin Robinson, and S. Jon Chapman. Python Battery Mathematical Modelling (PyBaMM). ECSarXiv. February, 2020. doi:10.1149/osf.io/67ckj. | {"url":"https://docs.pybamm.org/en/stable/source/examples/notebooks/models/using-model-options_thermal-example.html","timestamp":"2024-11-04T05:57:08Z","content_type":"text/html","content_length":"44995","record_id":"<urn:uuid:e6f0fa08-fdc5-45e3-9d77-c470523c17e7>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00147.warc.gz"} |
Learners - eLearn
The platform empowers learners to engage with a wide range of curriculum-aligned audio-visual resources. Providing access to materials tailored to different learning styles and facilitating
independent learning.
At the end of this lesson, the students should be able to understand Fraction
At the end of this lesson, learner will know what soil is and types of soil and uses of soil.
At the end of this lesson, learner will know what soil is and types of soil
Geometrical Construction IV
At the end of this lesson, learners should be able to understand Geometrical Construction
Geometrical Construction III
At the end of this lesson, learners should be able to understand Geometrical Construction
Geometrical Construction II
At the end of this lesson, learners should be able to understand Geometrical Construction
Matrices and determinants
The video helps define a matrix stating the order and notation of a matrix, types, operation of addition, subtraction of matrices, multiply matrix …
Matrices and determinants
The video helps define determinant of 2 x 2 and 3 x 3 metrics, also types, operation of addition, subtraction of matrices, multiply matrix …
Matrices and determinants
The video helps define a matrix stating the order and notation of a matrix, types, operation of addition, subtraction of matrices, multiply matrix …
Simple and Compound Interest
At the end of this lesson, learners should be able to understand Simple and Compound Interest
Value Ordering and Rounding
At the end of this lesson, learners should be able to understand Value ordering and Rounding
Geometrical Construction I
At the end of this lesson, learners should be able to understand Geometrical Construction
Partial Variation (Reciprocal Variation)
At the end of this lesson, learners should be able to understand Partial Variation (Reciprocal Variation)
At the end of this lesson, learners should be able to understand Variation (Inverse Variation)
Variation (Inverse Variation)
At the end of this lesson, learners should be able to understand Variation (Inverse Variation)
Fractions, Decimals and Percentage II
At the end of this lesson, learners should be able to understand Fractions, Decimals and Percentage
Fractions, Decimals and Percentage
At the end of this lesson, learners should be able to understand Fractions, Decimals and Percentage
Shapes and Geometric Reasoning
At the end of this lesson, learners should be able to understand Shapes and Geometric Reasoning
At the end of this lesson, learners should be able to understand Ratio And Proportion
Calculation and Mental Strategies II
At the end of this lesson, learners should be able to understand Calculation and Mental Strategy
Calculation and Mental Strategy
At the end of this lesson, learners should be able to understand Calculation and Mental Strategy
At the end of this lesson, learners should be able to understand Ratio and Proportion
At the end of this lesson, learners should be able to understand Binary Number System | {"url":"https://elearn.education.gov.ng/learners/","timestamp":"2024-11-06T07:10:38Z","content_type":"text/html","content_length":"325075","record_id":"<urn:uuid:f62a1f96-92a7-4ff1-8987-37dd840d083f>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00808.warc.gz"} |
GTP 2014
GTP 2014: Fifth Workshop on Game-Theoretic Probability and Related Topics
November 12 - 16, 2014, CIMAT (Centro de Investigación en Matemáticas, or Mathematics Research Center), Guanajuato, Mexico. The arrival date is November 12 and the departure date is November 16; the
actual talks will be spread over three days, November 13 - 15.
General information Objectives Game-theoretic probability Related topics Links
This time the workshop is organized under the auspices of CIMAT. For local information, please click here. For a list of attendees (most of whom will give talks), please click here. Registration is
now closed.
For the program please click here. Some abstracts are here.
This workshop will bring together researchers studying game-theoretic probability with others who have also been studying probability-free frameworks and frameworks depending on weak probabilistic
assumptions. This year's workshop will have "Probability-free finance" as its area of special emphasis. Previous workshops with the same title, Game-Theoretic Probability and Related Topics, were
held in 2006, 2008, and 2012 in Tokyo and in 2010 in the London area.
Traditionally, this series of workshops have included the following topics related to game-theoretic probability: imprecise probabilities, prequential statistics, on-line prediction, and algorithmic
randomness. Another related topic is conformal prediction. However, this year, due to the emphasis on probability-free finance, not all of them will be represented.
The objective of the workshop will be to make researchers using different approaches to probability-free finance (both mathematical and philosophical) aware of each others' work and to explore
commonalities between their frameworks and points of view.
What is game-theoretic probability? Like the better known measure-theoretic framework, the game-theoretic framework for probability can be traced back to the 1654 correspondence between Blaise Pascal
and Pierre Fermat, often said to be the origin of mathematical probability. In their correspondence, Pascal and Fermat explained their different methods for solving probability problems. Fermat's
combinatorial method is a precursor of the measure-theoretic framework, now almost universally accepted by mathematicians as the result of work by Borel, Kolmogorov, Doob, Martin-Lof, and others.
Pascal's method of backward recursion, using prices at each step in a game to derive global prices, can be seen as the precursor of the game-theoretic framework, to which von Mises, Ville,
Kolmogorov, Schnorr, and Dawid contributed further ingredients.
The game-theoretic framework was presented in a comprehensive way, as an alternative to the measure-theoretic framework, by Shafer and Vovk in 2001 and Takeuchi in 2004 (see
www.probabilityandfinance.com). As these authors explain, a classical probability theorem tells us that some event is very likely or even certain to happen. In the measure-theoretic framework, such a
theorem becomes a statement about the measure of a set: the set has measure near or equal to one. In the game-theoretic framework, it is instead a statement about a game: a player has a strategy that
multiplies the capital it risks by a large or infinite factor if the event fails to happen. The mere fact that we can translate theorems from measure theory into game theory in this way is of limited
interest, but the translation opens up new ways of using probability theory and provides new insights into the meaning of probability and into many existing applications and related fields.
Some of the new insights come from the greater generality of the game-theoretic picture. Many classical theorems hold in generalized form even in games where relatively few payoffs are priced. In
classical probability, all payoffs are priced (we call them random variables, and we call their prices expected values). This is not necessary in the game-theoretic picture. An investor in a
financial market, for example, plays a game in which he can buy some payoffs (corresponding to various securities traded in the market) but not others. Because probabilities (or upper and lower
probabilities) for global events are defined even in these situations, we see these probabilities as features that emerge from the structure of the game, not as features of objective reality or
subjective belief external to the game.
One of the most active recent topics in game-theoretic probability, not adequately treated in the monographs by Shafer, Vovk, and Takeuchi, is continuous-time processes. The mathematical idea
underlying recent work has been described by Takeuchi as high-frequency limit order trading. A player divides his capital among many different strategies, all of which rebalance a portfolio of bets
when a continuous function reaches various discrete levels, but some of which operate at a much higher frequency than others. Various classical properties of stochastic processes emerge merely from
the basic assumption that a strategy will not multiply the capital it risks by a very large factor.
These related topics some of which might be represented in this workshop: (1) imprecise probabilities, (2) prequential statistics, (3) on-line prediction, (4) algorithmic randomness, and (5)
conformal prediction.
1. Imprecise probabilities (see www.sipta.org) is now a fairly broad field, which includes Walley's upper and lower probabilities, Dempster-Shafer theory, and other approaches to loosening the
classical axioms of probability. The imprecise-probabilities community has accumulated expertise in studying various classes of set functions, including several important classes of Choquet
capacities. Inasmuch as game-theoretic probability leads to upper and lower probabilities for events, it can be considered a topic within imprecise probabilities, and the extent to which other
work on imprecise probabilities can be understood game-theoretically is an interesting and sometimes open question.
2. Philip Dawid introduced prequential ("predictive sequential") statistics in the 1980s. It helped inspire the development of game-theoretic probability in the 1990s, because it re-conceptualized
the notion of a probability distribution for a sequence of events as a strategy for a forecaster in a sequential forecasting game, which the forecaster can also play without thinking through a
complete strategy. It has recently attracted renewed attention, because it brings statistics closer to the spirit of machine learning, shifting emphasis from parameter estimation and model
selection to predictive performance.
3. On-line prediction is a computer-science counterpart of prequential statistics. Performance of prediction strategies is measured either by evaluating a loss function or by measuring calibration
and resolution. There are three important recent threads in on-line prediction: (i) prediction with expert advice, (ii) well-calibrated prediction, and (iii) defensive forecasting. All three are
related to game-theoretic probability, and an elucidation of the relation may help the three learn from each other.
4. The algorithmic theory of randomness, which continues to develop, shares with game-theoretic probability roots in the pioneering work by Ville in the 1930s and Schnorr in the 1970s. It still
provides a theoretical underpinning and a convenient testbed for game-theoretic probability and its applications. It also contributes to prequential statistics, on-line prediction, and conformal
prediction. This workshop will help the number of connections, already significant, to grow further.
5. Conformal prediction is a method of producing prediction sets that can be applied on top of a wide range of prediction algorithms. The method has a guaranteed coverage probability under the
standard IID assumption regardless of whether the assumptions (often considerably more restrictive) of the underlying algorithm are satisfied. The method shares the origins and techniques with
game-theoretic probability and algorithmic randomness. There are annual workshops on conformal prediction (third planned for October 2014) for discussing applied aspects of conformal prediction.
Recent theoretical work has included the analysis of the method of "conformalizing" for Bayesian algorithms. For the method to be really useful it is desirable that in the case where the
assumptions of the underlying algorithm are satisfied, the conformal predictor loses little in efficiency as compared with the underlying algorithm (whereas being a conformal predictor, it has
the stronger guarantee of validity). Asymptotic results have been obtained for Bayesian ridge regression, and work is underway for other classes of prediction algorithms. This workshop might
serve as a forum for discussing new results and approaches in theoretical conformal prediction.
The two previous workshops:
Some further links:
This page is maintained by Vladimir Vovk. Last modified on 20 December 2014 | {"url":"https://gtfp.net/GTP2014/index.html","timestamp":"2024-11-05T07:27:07Z","content_type":"text/html","content_length":"14055","record_id":"<urn:uuid:18393ab6-57bd-46e7-bc42-b986d06f4a85>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00031.warc.gz"} |
Math Colloquia - <정년퇴임 기념강연> Some remarks on PDEs and Numerical Analysis: some results developed at SNU
줌 회의실: 889 8813 5947 (https://snu-ac-kr.zoom.us/j/88988135947)
초록: As this is my last colloquium at SNU, I will explain the meanings and backgrounds of selected theories that I have achieved in this department during the last 30 years. First I will describe a
generalized Green's Theorem from which the exact Sobolev function spaces to which the traces of H(curl,D) belong. Here, D is a bounded Lipschitz domain.
Then several nonconforming Finite Element Spaces will be discussed.
I will also explain numerical inversion of Laplace transforms, which motivated the exponential convergent algorithms. If time permits, I will mention some results on inverse problems and other | {"url":"http://my.math.snu.ac.kr/board/index.php?mid=colloquia&page=9&sort_index=date&order_type=asc&l=en&document_srl=871001","timestamp":"2024-11-08T06:04:47Z","content_type":"text/html","content_length":"45591","record_id":"<urn:uuid:4602533a-d144-4657-99a9-47effdbb0eb8>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00628.warc.gz"} |
I. Introduction
In the era of digital transformation, the application of smart algorithms and advanced technologies has become ubiquitous across various sectors worldwide. The field of education is no exception to
this trend, with numerous research works demonstrating the potential of these technologies in optimizing educational processes and outcomes. One such example is the work of Tajbakhsh et al. (2022),
who proposed an accelerator-aware in-network load balancing system, P4Mite, to improve application performance [
]. Another inspiring example of the application of smart algorithms and advanced technologies is the work of Jamali et al. (2024). They introduced a new global online platform for sharing ideas and
collaborating. This platform transcends geographical boundaries and disciplinary barriers, creating an engaging space where individuals worldwide can exchange ideas, receive valuable feedback, and
collaborate on exciting projects [
Inspired by these advancements, this paper aims to contribute to the literature by proposing a novel approach to optimize classroom resource. Classroom resource allocation is a complex and
challenging problem in educational institutions. The task of allocating resources to classrooms for lectures, exams, meetings, and events requires a careful consideration of various factors,
including classroom capacity, availability, location, equipment, scheduling preferences, and cost. These factors are typically subject to frequent changes, leading to a dynamic and unpredictable
environment that requires quick and efficient decision-making. In recent years, cloud computing has emerged as a promising solution to optimize the classroom resource allocation process.
Cloud computing is a model of delivering computing services over the internet, where resources such as servers, storage, and applications are provided on-demand to users. This model offers numerous
benefits, including cost-effectiveness, flexibility, scalability, and reliability. In the context of classroom resource allocation, cloud computing can enable resource providers to allocate resources
to users in a timely and efficient manner, based on the users’ needs and preferences. This can result in improved resource utilization, reduced costs, and enhanced user satisfaction [
The classroom resource allocation problem is a longstanding challenge for educational institutions. In a traditional classroom environment, resource allocation was done manually, which was a
time-consuming and inefficient process. Resource optimization algorithms have been proposed to help solve this problem. Salp Swarm Algorithm (SSA) has shown promising results in optimization problems
in various areas such as engineering, clustering, feature selection, and machine learning. In this paper, we used a modified SSA provided by Jamali et al. [
] to solve the classroom resource allocation problem. The proposed algorithm generates a population of potential solutions and updates them iteratively to find the optimal solution. In this section,
we provide background on the classroom resource allocation problem and overview existing optimization algorithms. We also include a comprehensive review of salp swarm optimization algorithms.
Finally, we present an overview of the using modified SSA algorithm.
A. Salp Swarm Optimization
One of the key challenges in the optimal allocation of classroom resources is the task scheduling problem. This problem involves assigning tasks to resources in a way that minimizes the overall
completion time while satisfying various constraints, such as resource availability, capacity, and compatibility. Traditional approaches to the task scheduling problem involve heuristics,
mathematical programming, and meta-heuristic optimization algorithms. However, these approaches may not always be effective in solving the problem, especially in dynamic and complex environments.
Recently, meta-heuristic optimization algorithms have gained popularity in solving the task scheduling problem due to their ability to find near-optimal solutions quickly. One such algorithm is the
Modified Salp Swarm Algorithm (MSSA), which is inspired by the swarming behavior of salps. The MSSA algorithm involves generating a population of salps representing potential solutions to the
problem, and updating the positions of the salps iteratively based on a fitness function and mathematical equations. This algorithm has been shown to be effective in solving various optimization
problems, including resource allocation problems [
Figure 1. Structure of a Salp - An illustration of the unique swarming and propulsion behavior of sea salps, which inspired the development of the Salp Swarm Algorithm for solving complex
optimization problems.
B. Classroom Resource Allocation
Classroom resource allocation optimization involves distributing available resources of classrooms in a way that they are adequately configured, students are assigned to appropriate classrooms, and
the scheduling of lectures and practicals is done in an efficient manner. In modern education environments, there is a critical need for effective resource allocation due to the increasing number of
students, courses, and classes. An efficient allocation of resources impacts not only the classroom experience of the students but also the institutional performance as a whole. Fair and efficient
resource allocation can help an institution to maximize the use of its resources, minimize idle time, and reduce overall operational costs.
In previous studies, various optimization techniques have been proposed to solve classroom allocation problems. Linear programming has been used to optimize the allocation process of the finite
available classrooms effectively. For instance, in [
], linear programming is applied to the allocation of the classroom in tertiary institutions in Nigeria. In another study [
], linear programming is used to allocate classroom space in Premier Nurses Training College in Kumasi. Other optimization techniques that have been explored include genetic algorithms and simulated
annealing. However, these methods require a good understanding of programming languages, mathematical equations, as well as experience in coding, which might limit their applicability for
Salp swarm optimization (SSO) was first introduced by Mirjalili et al. in 2017 [
] and has since been applied in a variety of problems, including engineering optimization, clustering, and feature selection [
]. SSA is a relatively new nature-inspired meta-heuristic optimization technique. It’s inspired by the swarming behavior of sea salps, which are barrel-shaped planktic tunicates that belong to the
family of Salpidae. Salps move by contracting, thereby pumping water through their gelatinous body [
]. They often form a stringy colony by aligning next to each other, referred to as a salp chain. SSO is a population-based optimization algorithm that tries to find the optimum solution by imitating
the salp colony behavior [
The key to SSA is to mimic this behavior in solving complex optimization problems. Swarm Intelligence is the collective behavior of decentralized, self-organized systems. These systems can be natural
or artificial but typically consist of simple agents interacting locally with one another and their environment. While the agents follow simple rules, their interaction leads to the emergence of
intelligent global behavior [
The optimization problems in real life are often very complex in nature. The search space they generate is often quite tricky for mathematical programming to find a globally optimal solution in
limited time and resources. This is where metaheuristics like SSA can promise to find suitable solutions in less computational effort. Although, they do not guarantee to find the globally optimal
SSA has been successfully utilized in a wide range of optimization problems in different fields, such as machine learning, engineering design, wireless networking, image processing, and power energy.
It’s an effective single-objective optimization algorithm that was inspired by the navigating and foraging behaviors of salps in their natural habitats [
The basic version of SSA involves generating a population of salps in a random search space and then updating the position of each salp based on a fitness function and the mathematical equations
inspired by the salp movement. The positions of the salps represent candidate solutions. In each iteration, the algorithm implements global and local search strategies to explore the search space and
improve the optimization process. SSO has shown promising results in comparison with other well-known optimization algorithms, such as Particle Swarm Optimization (PSO), Ant Colony Optimization
(ACO), and Genetic Algorithms (GA).
In this paper, we used a modified version of SSA [
] to suit the classroom resource allocation problem. Different from the basic version of SSA, our proposed algorithm involves adding a teaching-learning-based optimization (TLBO) strategy to increase
the speed of convergence and improve the performance of the algorithm. TLBO is an operator that involves grouping the salps into pairs and emphasizing the sharing and learning of information between
them. Following the TLBO operator, we perform an adaptive strategy that changes the radius of movement of salps according to the problem characteristics. From hereon, the algorithm is called the
Modified Salp Swarm Algorithm with adaptive and teaching-learning-based optimization (MSSTA) × TLBO.
The proposed algorithm will be evaluated on real-world datasets to measure its efficiency in solving the classroom resource allocation problem. Specifically, the proposed algorithm’s performance in
terms of completion time of tasks, use of resources, and overall operational cost will be compared to that of existing optimization algorithms. The results will offer a comparison of different
optimization algorithms’ efficiency in solving classroom resource allocation problems [
The remainder of this paper is organized as follows. In the next section, we will provide a brief review of the related literature on classroom resource allocation and optimization algorithms.
Section 3
will describe our proposed approach in detail, including the mathematical formulas, fitness function, and simulation settings.
Section 4
will present the results of the simulations and analyze the performance of the proposed algorithm. Finally, in
Section 5
, we will provide the conclusions of our work and discuss the potential for future research in this area.
II. Related Literature
Classroom resource allocation is an essential aspect of educational institutions, as it can impact the academic success and satisfaction of both students and teachers. Several studies have proposed
optimization algorithms and strategies to improve the classroom resource allocation process and achieve optimal allocation outcomes.
One approach that has gained popularity in recent years is the use of meta-heuristic optimization algorithms to solve the classroom resource allocation problem. A meta-heuristic algorithm is a
mathematical procedure that enhances the efficiency of traditional optimization algorithms by optimizing their parameters based on randomization. One such algorithm is the Modified Salp Swarm
Algorithm (MSSA), which is inspired by the behavior of salps, a type of oceanic invertebrate.
Several studies have used the optimize resource algorithms to task scheduling problem in cloud computing, which shares similarities with the classroom resource allocation problem. For instance,
Mapetu et al. [
] proposed a binary variant of the Particle Swarm Optimization algorithm to solve the task scheduling problem in cloud computing and achieve optimal load balancing.
In addition to meta-heuristic optimization algorithms, other studies have approached the classroom resource allocation problem using mathematical programming models and heuristics. For instance, Chen
et al. [
] proposed a user-priority guided Min-Min scheduling algorithm for load balancing in cloud computing. The algorithm considers user preferences and priorities to ensure fair resource allocation and
optimal performance. Similarly, Lavanya et al. [
] proposed a multi-objective task scheduling algorithm based on Service Level Agreements and processing time suitable for cloud environments.
Saeedi et al. [
] proposed an improved Many-Objective Particle Swarm Optimization algorithm to solve the task scheduling problem in cloud computing. The algorithm considers multiple objectives, including cost,
makespan, and completion time, simultaneously, and optimizes them using a multi-objective approach. A survey conducted by Sharma and Tyagi [
] reviewed existing heuristic approaches for task scheduling in cloud computing, including Genetic Algorithm, Simulated Annealing, Ant Colony Optimization, and Particle Swarm Optimization. The survey
analyzed the strengths and weaknesses of each approach and explored potential directions for future research.
Overall, the existing literature shows that the classroom resource allocation problem is a complex and challenging problem that can be optimized using various optimization algorithms and techniques.
The MSSA algorithm has shown promise in solving resource allocation problems in cloud computing and can be extended to optimize the classroom resource allocation process.
III. Proposed Approach
In this section, we describe the proposed approach in detail, including the mathematical formulas, fitness function, and simulation settings.
A. Mathematical Formulas
The Modified Salp Swarm Algorithm (MSSA) is inspired by the coordinated movement of salps, oceanic invertebrates that swarm and move in a coordinated manner [?]. The MSSA algorithm involves
generating a population of salps representing potential solutions to the optimization problem and updating their positions iteratively based on a fitness function and mathematical equations. We
propose using the MSSA algorithm to optimize the classroom resource allocation problem.
Leader Update
: The positions of leader salps are updated as follows:
$X leader _ next = X leader + velocity _ leader × cos ( 2 π × random _ number ( ) ) × ( F food − X leader )$
Follower Update
: The positions of follower Salps are updated as follows:
$X follower _ next = X follower + velocity _ follower × cos ( 2 π × random _ number ( ) ) × ( F food − X follower ) + velocity _ follower × sin ( 2 π × random _ number ( ) ) × ( X leader − X follower
Velocity Update
: The velocity of salps is updated using the following equation:
$velocity _ next = c 1 × velocity + c 2 × random _ number ( ) × ( X best − X ) + c 3 × random _ number ( ) × ( X worst − X )$
B. Fitness Function
The fitness function evaluates the quality of a potential solution based on constraints and objectives. In the classroom resource allocation problem, the fitness function should consider classroom
capacity, availability, location, equipment, scheduling preferences, and cost.
We define the fitness function as:
$fitness = ( 1 − α ) × makespan ( X ) + α × cos t ( X )$
• X represents a candidate solution,
• $makespan ( X )$ is the total time required for completing all tasks with all constraints satisfied,
• $cos t ( X )$ is the total cost of using the allocated resources,
• $α$ is the weight parameter representing the trade-off between the two objectives.
C. Simulation Settings
We conducted simulations to evaluate the proposed algorithm’s performance in optimizing classroom resource allocation. The simulations were conducted on various scenarios with 150-300 tasks and 2-15
virtual machines. We compared the MSSA algorithm with three other algorithms: Ant Colony Optimization with Reservation (ACOr), Particle Swarm Optimization (PSO), and Genetic Algorithm (GA).
The performance was evaluated based on fitness values and average computation time per iteration. The MSSA algorithm was run for 100 iterations, and the results were averaged over 20 independent
runs. Python with NumPy and Matplotlib libraries was used as the simulation environment.
The parameters used in the MSSA algorithm were set as follows: $c 1 = 0.5$, $c 2 = 0.5$, $c 3 = 1$, $salp _ size = 0.1$, and $max _ iter = 500$.
IV. Results and Analysis
In this section, we present the results of the simulations conducted to evaluate the performance of the Modified Salp Swarm Algorithm (MSSA) in optimizing classroom resource allocation. We compare
the MSSA algorithm with three other algorithms: Ant Colony Optimization with Reservation (ACOr), Particle Swarm Optimization (PSO), and Genetic Algorithm (GA). The simulations were conducted on
various scenarios with 150-300 tasks and 2-15 virtual machines.
A. Simulation Results
Table 1
summarizes the fitness values, average computation times, and improvements achieved by each algorithm in the different scenarios.
Figure 2
illustrates the comparison of fitness values across the scenarios, while
Figure 3
shows the average computation times.
B. Analysis
The results demonstrate that MSSA consistently outperforms ACOr, PSO, and GA in terms of fitness value across all scenarios. MSSA achieves fitness values of 2387, 2409, 2442, and 2533 for scenarios
150-2, 200-5, 250-10, and 300-15, respectively. In comparison, ACOr, PSO, and GA achieve fitness values of 2780, 2750, 2760, 2810; 2684, 2600, 2650, 2668; and 2800, 2850, 2870, 2950, respectively,
for the same scenarios. The improvement percentages for MSSA over ACOr, PSO, and GA are 13.5%, 7.5%, and 16.6%, respectively.
Regarding average computation time, MSSA also performs well, with average times of 12.0, 12.2, 13.4, and 14.8 ms for scenarios 150-2, 200-5, 250-10, and 300-15, respectively. ACOr, PSO, and GA have
average times of 16.0, 16.5, 16.9, 19.1; 15.0, 15.8, 16.3, 17.0; and 18.0, 19.0, 19.3, 21.0 ms, respectively, for the same scenarios. Although MSSA’s average computation time is slightly higher than
that of PSO, it achieves significantly better fitness values.
These results indicate that MSSA is a promising algorithm for optimizing classroom resource allocation, offering a good balance between solution quality and computational efficiency. Further
fine-tuning of the algorithm’s parameters and exploration of different problem instances could potentially lead to even better performance.
V. Conclusion and Future Work
In this paper, we proposed a Modified Salp Swarm Algorithm (MSSA) for optimizing classroom resource allocation. Through simulations and comparisons with other algorithms, we demonstrated that MSSA
outperforms existing approaches in terms of fitness values and average computation times. The results indicate that MSSA is a highly effective and efficient method for optimizing classroom resource
Future research in this area could explore several directions to further improve the optimization of classroom resource allocation. One direction is to enhance the MSSA algorithm by incorporating
more advanced strategies for updating the positions of salps or by introducing hybrid approaches that combine MSSA with other metaheuristic algorithms. Additionally, research could focus on extending
the application of MSSA to other optimization problems in educational institutions, such as course scheduling or student assignment.
Furthermore, the integration of machine learning techniques could enhance the performance of the algorithm by providing more accurate predictions of resource demands and constraints. Another avenue
for future research is to consider the dynamic nature of classroom resource allocation, where resource demands and constraints may change over time. Developing adaptive algorithms that can adjust to
these changes in real-time could further improve the efficiency and effectiveness of classroom resource allocation.
In conclusion, the MSSA algorithm shows great potential for optimizing classroom resource allocation, and future research could further enhance its capabilities and applicability in educational
1. Tajbakhsh, H.; Parizotto, R.; Neves, M.; Schaeffer-Filho, A.; Haque, I. Accelerator-Aware In-Network Load Balancing for Improved Application Performance. In Proceedings of the 2022 IFIP
Networking Conference (IFIP Networking); 2022; pp. 1–9. [Google Scholar] [CrossRef]
2. Jamali, H.; Dascalu, S.M.; Harris, F.C. Fostering Joint Innovation: A Global Online Platform for Ideas Sharing and Collaboration. arXiv arXiv:2402.12718, 2024. [CrossRef]
3. Jamali, H.; Karimi, A.; Haghighizadeh, M. A new method of Cloud-based Computation Model for Mobile Devices: Energy Consumption Optimization in Mobile-to-Mobile Computation Offloading. In
Proceedings of the 6th International Conference on Communications and Broadband Networking, Singapore, 24–26 February 2018; pp. 32–37. [Google Scholar] [CrossRef]
4. Jamali, H.; Shill, P.C.; Feil-Seifer, D.; Harris, F.C.; Dascalu, S.M. A Schedule of Duties in the Cloud Space Using a Modified Salp Swarm Algorithm. In Internet of Things. Advances in Information
and Communication Technology; IFIPIoT 2023. IFIP Advances in Information and Communication Technology; Puthal, D., Mohanty, S., Choi, B.Y., Eds.; Springer: Cham, 2024. [Google Scholar] [CrossRef]
5. Oladejo, N.K.; Abolarinwa, A.; Salawu, S.O.; Bamiro, O.M.; Lukman, A.F.; Bukari, H.I. Application of optimization principles in classroom allocation using linear programming. International
Journal of Mechanical Engineering and Technology (IJMET) 2019, 10, 874–885. [Google Scholar]
6. Mtonga, K.; Twahirwa, E.; Kumaran, S.; Jayavel, K. Modelling Classroom Space Allocation at University of Rwanda—A Linear Programming Approach. Applications and applied mathematics: an
international journal (AAM) 2021, 16, 40. [Google Scholar]
7. Salgotra, R.; Singh, U.; Singh, S.; Singh, G.; Mittal, N. Self-adaptive salp swarm algorithm for engineering optimization problems. Applied Mathematical Modelling 2021, 89, 188–207. [Google
Scholar] [CrossRef]
8. Abualigah, L.; Shehab, M.; Alshinwan, M.; et al. Salp swarm algorithm: a comprehensive survey. Neural Comput & Applic 2020, 32, 11195–11215. [Google Scholar] [CrossRef]
9. Ponnusamy, M.; Bedi, P.; Suresh, T.; et al. Design and analysis of text document clustering using salp swarm algorithm. J Supercomput 2022, 78, 16197–16213. [Google Scholar] [CrossRef]
10. Dagal, I.; Akın, B.; Akboy, E. MPPT mechanism based on novel hybrid particle swarm optimization and salp swarm optimization algorithm for battery charging through simulink. Scientific Reports
2022, 12, 2664. [Google Scholar] [CrossRef] [PubMed]
11. Mirjalili, S.; Gandomi, A.H.; Mirjalili, S.Z.; Saremi, S.; Faris, H.; Mirjalili, S.M. Salp Swarm Algorithm: A bio-inspired optimizer for engineering design problems. Advances in Engineering
Software 2017, 114, 163–191. [Google Scholar] [CrossRef]
12. Abed-alguni, B.H.; Paul, D.; Hammad, R. Improved Salp swarm algorithm for solving single-objective continuous optimization problems. Applied Intelligence 2022, 52, 17217–17236. [Google Scholar] [
13. Mapetu, J.P.; Chen, Z.; Kong, L. Low-time complexity and low-cost binary particle swarm optimization algorithm for task scheduling and load balancing in cloud computing. Applied Intelligence 2019
, 49, 3308–3330. [Google Scholar] [CrossRef]
14. Chen, H.; Wang, F.Z.; Helian, N.; Akanmu, G. User-priority guided Min-Min scheduling algorithm for load balancing in cloud computing. In Proceedings of the 2013 National Conference on Parallel
Computing Technologies (PARCOMPTECH); 2013; pp. 1–8. [Google Scholar] [CrossRef]
15. Lavanya, M.; Shanthi, B.; Saravanan, S. Multi-objective task scheduling algorithm based on SLA and processing time suitable for cloud environment. Computer Communications 2020, 151, 183–195. [
Google Scholar] [CrossRef]
16. Saeedi, S.; Khorsand, R.; Ghandi Bidgoli, S.; Ramezanpour, M. Improved many-objective particle swarm optimization algorithm for scientific workflow scheduling in cloud computing. Computers &
Industrial Engineering 2020, 147, 159–187. [Google Scholar] [CrossRef]
17. Sharma, S.; Tyagi, S. A Survey on Heuristic Approach for Task Scheduling in Cloud Computing. International Journal of Advanced Research in Computer Science 2017, 8, 1089–1092. [Google Scholar]
Algorithm Fitness Value Avg. Comp. Time (ms) Improvement (%)
MSSA 2387 12.0 -
ACOr 2780 16.0 13.5
PSO 2684 15.0 7.5
GA 2800 18.0 16.6
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s).
MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http:/ | {"url":"https://www.preprints.org/manuscript/202403.1568/v1","timestamp":"2024-11-11T17:57:03Z","content_type":"text/html","content_length":"563285","record_id":"<urn:uuid:72da5342-6bd4-4d57-9f27-5165591e4bb8>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00599.warc.gz"} |
AI Subset
🧠 Machine Learning
Jan 16, 2023 3:57 AM
Last updated
Jan 16, 2023 4:18 AM
In mathematics, a tangent is a straight line that touches a curve at only one point. It is used to understand the slope, or steepness, of a curve at a specific point.
Imagine you're standing at the bottom of a hill and looking up. The slope of the hill at your feet is the steepness of the hill at that point. If you take a step forward, the slope may change, it
could be steeper or less steep. The tangent line is a mathematical representation of the slope of the hill at that specific point.
Tangents are often used in geometry, calculus and other branches of mathematics, and it is also used to understand the behavior of a curve.
Tangent in machine learning
In machine learning, tangents are used in a technique called gradient descent to optimize the parameters of a model. A model's parameters are the values that determine its behavior, such as the
weights and biases in a neural network. To make a model perform well, we need to find the best values for these parameters.
In order to find the best parameters, the algorithm starts with a set of initial values and then repeatedly updates the parameters by taking small steps in the direction of the tangent line of the
model's performance curve.
The tangent line represents the direction of the steepest descent of the curve, and the algorithm aims to reach the lowest point of the curve, which represents the best model parameters.
For each step, the algorithm calculates the gradient of the performance curve at the current parameter values. The gradient is a vector that points in the direction of the steepest ascent. The
algorithm then takes a step in the opposite direction of the gradient, which is the direction of the steepest descent. This is where the term gradient descent comes from.
In summary, tangent is used in machine learning to optimize the parameters of a model by using a technique called gradient descent. The algorithm starts with initial values and repeatedly updates the
parameters by taking small steps in the direction of the tangent line of the model's performance curve until it finds the best parameters. | {"url":"https://internettools.ai/ai/ai-terms/tangent","timestamp":"2024-11-03T11:53:15Z","content_type":"text/html","content_length":"200338","record_id":"<urn:uuid:e79b4441-53e7-4f3f-aebb-f325010b72be>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00256.warc.gz"} |
Class 5 Math
ICSE /Class 5
Maths MCQ Based On Chord of a circle
Our free online Maths test quiz for Class 5, ICSE will assist you to improve your Maths skills on every concept in a fun interactive way.
50K +
Happy students
10K +
Awesome classes
ICSE Class 5 Maths Chord of a circle
At JustTutors, we believe in the power of digital technology to help students get personalized learning and attention from India's best-in-class science, english and math tutors. We are focused on
creating a top-class e-learning platform that brings together the best teachers, technology, media, content for creating a seamless and world-class experience for every student. | {"url":"https://www.justtutors.com/test-your-knowledge/icse-class-5-maths-chord-of-a-circle","timestamp":"2024-11-11T06:39:51Z","content_type":"text/html","content_length":"332663","record_id":"<urn:uuid:cd67baaa-1620-4478-943b-82a35a34bf89>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00181.warc.gz"} |
• 140 ressources ont été trouvées. Voici les résultats 1 à 10
2 3 4 5 6 Page suivante >> >| documents par page
Tri : Date Editeur Auteur Titre
Giovanni Alberti - Introduction to minimal surfaces and finite perimeter sets (Part 1)
/ Fanny Bastien / 15-06-2015 / Canal-u.fr
Alberti Giovanni
Voir le résumé
In these lectures I will first recall the basic notions and results that are needed to study minimal surfaces in the smooth setting (above all the area formula and the first variation of the
area), give a short review of the main (classical) techniques for existence results, and then outline the theory of Finite Perimeter Sets, including the main results of the theory (compactness,
structure of distributional derivative, rectifiability). If time allows, I will conclude with a few applications.
Mot(s) clés libre(s) : Grenoble, école d'été, mathématique, institut fourier, summer school, geometric measure theory, calculus of variation
Accéder à la ressource
Giovanni Alberti - Introduction to minimal surfaces and finite perimeter sets (Part 5)
/ Fanny Bastien / 18-06-2015 / Canal-u.fr
Alberti Giovanni
Voir le résumé
In these lectures I will first recall the basic notions and results that are needed to study minimal surfaces in the smooth setting (above all the area formula and the first variation of the
area), give a short review of the main (classical) techniques for existence results, and then outline the theory of Finite Perimeter Sets, including the main results of the theory (compactness,
structure of distributional derivative, rectifiability). If time allows, I will conclude with a few applications.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation
Accéder à la ressource
Nicholas Alikakos - On the structure of phase transition maps : density estimates and applications
/ Fanny Bastien / 02-07-2015 / Canal-u.fr
Alikakos Nicholas
Voir le résumé
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation
Accéder à la ressource
Lars Andersson - Geometry and analysis in black hole spacetimes (Part 1)
/ Fanny Bastien / 16-06-2014 / Canal-u.fr
Andersson Lars
Voir le résumé
Black holes play a central role in general relativity and astrophysics. The problem of proving the dynamical stability of the Kerr black hole spacetime, which is describes a rotating black hole in
vacuum, is one of the most important open problems in general relativity. Following a brief introduction to the evolution problem for the Einstein equations, I will give some background on geometry
of the Kerr spacetime. The analysis of fields on the exterior of the Kerr black hole serve as important model problems for the black hole stability problem. I will discuss some of the difficulties
one encounters in analyzing waves in the Kerr exterior and how they can be overcome. A fundamentally important as pect of geometry and analysis in the Kerr spacetime is the fact that it is
algebraically special, of Petrov type D, and therefore admits a Killing spinor of valence 2. I will introduce the 2 spinor and related formalisms which can be used to see how this structure leads to
the Carter constant and the Teukolsky system. If there is time, I will discuss in this context some new conservation laws for fields of non zero spin.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis
Accéder à la ressource
Lars Andersson - Geometry and analysis in black hole spacetimes (Part 2)
/ Fanny Bastien / 17-06-2014 / Canal-u.fr
Andersson Lars
Voir le résumé
Black holes play a central role in general relativity and astrophysics. The problem of proving the dynamical stability of the Kerr black hole spacetime, which is describes a rotating black hole in
vacuum, is one of the most important open problems in general relativity. Following a brief introduction to the evolution problem for the Einstein equations, I will give some background on geometry
of the Kerr spacetime. The analysis of fields on the exterior of the Kerr black hole serve as important model problems for the black hole stability problem. I will discuss some of the difficulties
one encounters in analyzing waves in the Kerr exterior and how they can be overcome. A fundamentally important as pect of geometry and analysis in the Kerr spacetime is the fact that it is
algebraically special, of Petrov type D, and therefore admits a Killing spinor of valence 2. I will introduce the 2 spinor and related formalisms which can be used to see how this structure leads to
the Carter constant and the Teukolsky system. If there is time, I will discuss in this context some new conservation laws for fields of non zero spin.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis
Accéder à la ressource
Lars Andersson - Geometry and analysis in black hole spacetimes (Part 3)
/ 18-06-2014 / Canal-u.fr
Andersson Lars
Voir le résumé
Black holes play a central role in general relativity and astrophysics. The problem of proving the dynamical stability of the Kerr black hole spacetime, which is describes a rotating black hole in
vacuum, is one of the most important open problems in general relativity. Following a brief introduction to the evolution problem for the Einstein equations, I will give some background on geometry
of the Kerr spacetime. The analysis of fields on the exterior of the Kerr black hole serve as important model problems for the black hole stability problem. I will discuss some of the difficulties
one encounters in analyzing waves in the Kerr exterior and how they can be overcome. A fundamentally important as pect of geometry and analysis in the Kerr spacetime is the fact that it is
algebraically special, of Petrov type D, and therefore admits a Killing spinor of valence 2. I will introduce the 2 spinor and related formalisms which can be used to see how this structure leads to
the Carter constant and the Teukolsky system. If there is time, I will discuss in this context some new conservation laws for fields of non zero spin.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis
Accéder à la ressource
Lars Andersson - Symmetry operators and energies
/ 02-07-2014 / Canal-u.fr
Andersson Lars
Voir le résumé
Black holes play a central role in general relativity and astrophysics. The problem of proving the dynamical stability of the Kerr black hole spacetime, which is describes a rotating black hole in
vacuum, is one of the most important open problems in general relativity. Following a brief introduction to the evolution problem for the Einstein equations, I will give some background on geometry
of the Kerr spacetime. The analysis of fields on the exterior of the Kerr black hole serve as important model problems for the black hole stability problem. I will discuss some of the difficulties
one encounters in analyzing waves in the Kerr exterior and how they can be overcome. A fundamentally important as pect of geometry and analysis in the Kerr spacetime is the fact that it is
algebraically special, of Petrov type D, and therefore admits a Killing spinor of valence 2. I will introduce the 2 spinor and related formalisms which can be used to see how this structure leads to
the Carter constant and the Teukolsky system. If there is time, I will discuss in this context some new conservation laws for fields of non zero spin.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis
Accéder à la ressource
Alain Bachelot - Waves in the Anti-de Sitter space-time Ads
/ Fanny Bastien / 04-07-2014 / Canal-u.fr
Bachelot Alain
Voir le résumé
In this talk we address some issues concerning the wave propagation in the 4D+1 anti de Sitter space time : the role of the conformal boundary, the representation of the fields in term of Kaluza
Klein tower, the existence of new dynamics associated with a family of novel boundary conditions, the linear stability of a De Sitter brane.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis
Accéder à la ressource
Thomas Backdahl - Symmetry operators, conserved currents and energy momentum tensors
/ Fanny Bastien / 04-07-2014 / Canal-u.fr
Backdahl Thomas
Voir le résumé
Conserved quantities, for example energy and momentum, play a fundamental role in the analysis of dynamics of particles and fields. For field equations, one manifestation of conserved quantities in
a broad sense is the existence of symmetry operators, i.e. linear differential operators which take solutions to solutions. A well known example of a symmetry operator for the scalar wave equation
is provided by the Lie derivative along a Killing vector field. It is important to note that other kinds of objects can generate symmetry operators. For waves in the Kerr spacetime there is a
symmetry operator associated with Carter's constant. This symmetry, which is "hidden" in the sense that it arises from a Killing spinor, satisfying a generalization of the Killing vector equation,
rather than a Killing vector, was an essential ingredient in a proof of decay of scalar waves on the Kerr background by Andersson and Blue. In this talk we will consider what conditions on a
spacetime are necessary for existence of symmetry operators for the conformal wave equation, the Dirac Weyl equation, and the Maxwell equation, i.e. for massless test fields of spins 0, 1/2 and 1.
We will investigate how the conditions for the symmetry operators for the different field equations are related, and how they are related to existence of conserved currents. Furthermore, these tools
lead to the construction of a new energy momentum tensor for a Maxwell field on a Kerr background. This will provide a powerful tool for the study of decay of Maxwell fields on the Kerr spacetime.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis
Accéder à la ressource
Giovanni Alberti - Introduction to minimal surfaces and finite perimeter sets (Part 4)
/ Giovanni Alberti / 18-06-2015 / Canal-u.fr
Bastien Fanny
Voir le résumé
In these lectures I will first recall the basic notions and results that are needed to study minimal surfaces in the smooth setting (above all the area formula and the first variation of the
area), give a short review of the main (classical) techniques for existence results, and then outline the theory of Finite Perimeter Sets, including the main results of the theory (compactness,
structure of distributional derivative, rectifiability). If time allows, I will conclude with a few applications.
Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, institut fourier, summer school, geometric measure theory, calculus of variation
Accéder à la ressource
2 3 4 5 6 Page suivante >> >| documents par page | {"url":"http://indexation.univ-fcomte.fr/ori-oai-search/thematic-search.html?menuKey=lom&search=true&id=summer_school&submenuKey=keywords&sort_field=editor&sort_field_ascending=true","timestamp":"2024-11-12T16:46:40Z","content_type":"application/xhtml+xml","content_length":"31754","record_id":"<urn:uuid:34f02f76-48d2-4438-968b-9ae0b97c118f>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00531.warc.gz"} |
Data Compression/Coding - Wikibooks, open books for an open world
Most people think that compression is mostly about coding. Well at one time it was. Before information theory, people spent years developing the perfect code to store data efficiently. To understand
the limits of coding as a compression mechanism, we have to understand what coding is.
The ASCII binary translation of the word Wikipedia morphs into the English translation.
The computer really understands only one type of data, strings of 0s and 1s. You know 0 is off, 1 is on, and all that rot. These actually have to do with the state of a memory cell, where 0 is a low
voltage output of the cell, and 1 is a slightly higher voltage. The actual voltage spread involved is getting smaller as the size of the circuits gets smaller and the voltage needed to jump the gap
between parallel circuits gets smaller.
Hardware configuration used to define the size of a code, by limiting the number of bits as the two state memory element is called, that could be moved at a single pass. An example is the 4bit 8bit,
16bit 32bit, 64bit, and now 128bit architectures found during the development of microcomputers.
Now essentially everything in a computer is stored as sequences of bits. The storage capacity of the computer is limited by how many of those bits, are wasted. Ideally we don't want to waste bits, so
we tend to use codes that minimize the number of bits that are wasted for any particular architecture of computer they will run on. The ultimate code is the code that covers the exact amount of
variations that are possible in a data type, with the least amount of wasted bits.
It is interesting to find that this is not a consideration in the human brain, in fact we have had to define a term to explain the redundant information carried in the codes for the brain, and that
term is degenerate coding, a pejorative term that really just means that there is more than one code for the same information.
Consider a string of numbers, that mean something. Are they a string of digits, in which case we want to code them to minimize the storage for each digit, An integer value, in which case we want to
code them to minimize the storage for an arbitrarily sized integer, A floating point value, in which case we want to code them to minimize the storage for an arbitrarily sized mantissa, and include
information on what power the number is raised to, and how many bits past the point we are going to store. Or an address string in which case we want to code it to minimize the number of bits it
takes for a character in a specific language.
To a human all of these different types of storage look the same on the printed page. But it takes time to convert between the different types of storage, so you might not want to translate them into
the tightest coding possible while you are using them. However later when you want to store them away for posterity, it looks like a good idea to compress them down to the smallest size possible.
That is the essence of compression.
limits to coding density
So the limit to coding density, is determined by the type of data, you are trying to code and also by the amount of information that is available from a separate location.
As an example, the military have the practice to limit radio chatter during a campaign, this is often considered good thing. Even taking it to the point of limiting themselves to a click on their
mike, to trigger the next phase of the attack. To a person who is fully briefed on it, a click on a mike will be all that is needed to send a wealth of information about the state of the operation.
The event will have no meaning to the enemy waiting in the trenches, so that the first evidence they have of attack is when guns start going off.
Ideally all storage could be achieved in one bit files that contained whole encyclopedias. In actuality that is not likely to happen. Coding is still important, but it is very limited in what it can
now do for compression. New algorithms for compression are increasingly hard to come by, hence the path of specialization on the source data new approaches take, true advances have recently been
reduced to the hardware where the code will run.
bits, bytes, symbols, pixels, streams, files
“ Suffice it to say that real compression algorithms are a complex technology, bordering on an art form. ”
—Leo A. Notenboom, [1]
In the field of data compression, it is convenient to measure the "size" of data in terms of the fundamental unit of data: bits.
3 basic common primitive types char,short int,long int.
Many data compression algorithms produce a compressed data stream that is a stream of bits -- with no particular alignment to any other size.
It is sometimes convenient to consider the input data in terms of "symbols". Some algorithms compress English text in terms of the symbols from an input and process them converting them in a
reversible representation on the compressed output. Symbols are usually sets of bits but they in turn can represent any type of character representation/encoding, even pixels or waveforms.
When compressing CD-quality audio, the symbols are the 2^16 possible amplitude levels -- the 2^16 possible physical positions of the speaker cone.
Some file compression utilities consider the uncompressed file in terms of 257 different symbols. At any point in the file, the "next symbol" could be any one of 257 symbols: one of the 256 possible
bytes -- or we could have already reached the end of the file, the 257th symbol indicating "end-of-file".
When comparing image compression routines, sometimes the term "bpp" (bits per pixel) is used. An uncompressed full-color bitmap image is 24 bpp. An uncompressed 2 color bitmap image (containing only
black pixels and white pixels) is 1 bpp. Some image compression algorithms can compress some images to much less than 0.1 bpp. The same image compression algorithm may be doing pretty good to
compress some other image to 7.5 bpp.
variable-length coding
With variable-length coding, we can make some symbols very short -- shorter than any fixed-length encoding of those symbols. This is great for compressing data.
However, variable-length coding almost always end up make other symbols slightly longer than best fixed-length encoding of those symbols.
Using more bits store a symbol sounds ridiculous at first -- Why don't we simply store a shorter fixed-length version of such symbols?
OK, yeah, *other* symbols could be shorter. So ... why can't we use the variable-length version when that would be shorter, and use the fixed-length version when that would be shorter? The best of
both worlds? Perhaps use a "1" prefix to indicate "the fixed-length representation follows" ?
We discuss variable-length coding in more detail in the entropy coding section of this book.
Representing integers
Digital information can be encoded as a series of integers. With modern digital electronics, we typically first (1) encode the data (etc.) in some relatively simple (but verbose) encoding as a series
of integers, then (2 & 3) data compression algorithms take that series of integers, and find some other way to represent the entire series in fewer bits.
For example, when we store music in a MP3 file, the machine often first (1) digitizes the analog sound with a microphone and an ADC as a (relatively simple, but verbose) 16-bit PCM encoding at 44.1
kHz sampling rate per channel (Compact Disc digital audio, uncompressed WAV audio, etc.). Then the software processes that raw data first to (2) transform it into a representation that requires fewer
integers, then (3) represents each of those integers as bits in a way that the entire song can be stored in relatively few bits.
(In the early days of computers, people spent a lot of time figuring out how to encode music (etc.) into a reasonable number of keystrokes, both compressing and encoding the data "manually" with
human brainpower before entering it into the computer).
People have invented a surprisingly large number of ways of representing integers as bits.
... (say a few words here about 32-bit unsigned integers) ...
... (say a few words here about 2's complement signed integers) ...
Floating-point numbers are often stored as pairs of signed integers -- the significand (the significant digits) and the exponent.
A fixed-length code is the simplest way to represent a series of integers.
A comma code is one way to represent a series of arbitrarily long integers.
Each integer is represented by a series of digits.
A special symbol -- called the comma -- is used between each code word, to mark the end of one integer and the beginning of the next.
SCDC is one way to represent a series of arbitrarily long integers.
The (s,c) codes, also called (s,c)-dense codes (SCDC), have lengths that are multiples of 8 bits. ^[1]
SCDC has a single parameter s chosen as 0 < s < 256. Each codeword consists of a sequence of 8-bit bytes, where the last byte in the codeword has a value less than s, and the other bytes (if any)
have a value greater than or equal to s.
The end-tagged dense code (ETDC), also called variable-length quantity (VLQ), is the special case of SCDC for which s=128.
Because every SCDC codeword is aligned on byte boundaries, SCDC decompression is simpler and faster than Huffman decompression.^[1]
Both SCDC and Fibonacci codes support direct search in the compressed file.^[2]^[1]
ZigZag encoding is one way to represent signed integers. It maps "small" signed integers (close to zero) to "small" unsigned integers (the most-significant bits are all zero).^[3]^[4] It's designed
to have a one-to-one mapping between 32-bit ZigZag-encoded signed integers and 32-bit 2's complement signed integers. ZigZag encoding (like delta encoding) doesn't save any space by itself, but it
often transforms data in such a way that other downstream compression algorithms work better. (Delta encoding followed by ZigZag encoding often produces better results -- with the same downstream
compression algorithm -- than either one alone).^[5]
encoded integer
0000 0005 : -3
0000 0003 : -2
0000 0001 : -1
0000 0000 : 0
0000 0002 : +1
0000 0004 : +2
0000 0006 : +3 | {"url":"https://en.m.wikibooks.org/wiki/Data_Compression/Coding","timestamp":"2024-11-06T12:24:22Z","content_type":"text/html","content_length":"52537","record_id":"<urn:uuid:0b63f4fd-9182-47d1-9cc8-5a267a1341c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00070.warc.gz"} |
What is an example of prisoner dilemma?
What is an example of prisoner dilemma?
What is an example of prisoner dilemma?
The U.S. debt deadlock between the Democrats and Republicans that springs up from time to time is a classic example of a prisoner’s dilemma. Let’s say the utility or benefit of resolving the U.S.
debt issue would be electoral gains for the parties in the next election.
What is the general idea behind the Prisoners dilemma in game theory?
A prisoner’s dilemma is a decision-making and game theory paradox illustrating that two rational individuals making decisions in their own self-interest cannot result in an optimal solution.
What is the repeated prisoner’s dilemma?
The iterated prisoner’s dilemma is an extension of the general form except the game is repeatedly played by the same participants. An iterated prisoner’s dilemma differs from the original concept of
a prisoner’s dilemma because participants can learn about the behavioral tendencies of their counterparty.
What is chicken game theory?
A chicken game is a game theory set up that typically decribes two players heading toward each other. If the players continue on the same path, they bump into each other; if one swerves out of the
way and other doesn’t, the swerver “loses” and is labeled the chicken, while the second, implicitly braver player, wins.
What is the conclusion in the prisoners dilemma?
Outcome: The players stay at the noncooperative outcome. When play starts cooperatively, neither player will defect, because if he does, the other player will also defect, and they both will end up
worse off. Thinking ahead, therefore, neither player will defect. Outcome: The players stay at the cooperative outcome.
What do killer balls do in Golden Balls?
7 Then the first round starts: 12 golden balls are drawn from a lottery and four “killer” balls are added, i.e., inside the balls there is written either a cash amount (in £) or the word “killer”.
Killer balls are the worst for the players, because these may damage the jackpot in the final round. | {"url":"https://drinksavvyinc.com/essay-samples/what-is-an-example-of-prisoner-dilemma/","timestamp":"2024-11-13T03:08:33Z","content_type":"text/html","content_length":"35770","record_id":"<urn:uuid:5cafe818-65d3-4197-837f-bc29a319592a>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00253.warc.gz"} |
Graph Isomorphism
Biologically-related RNA molecules tend to have similar secondary structures. Thus, comparing RNA graphs can help identify RNA molecules that are structurally, functionally and evolutionarily
related. The problem of deciding algorithmically whether two graphs are isomorphic or structurally equivalent is known as the graph isomorpism problem. Many heuristic methods exist to determine
structurally equivalent graphs, but the general solution to the problem has, so far, eluded mathematicians.
We use Laplacian eigenvalue spectra to compare and find structurally similar graphs (see RNA Matrix program). Two graphs are deemed to be isomorphic when they have the same eigenvalue spectrum. This
method is imperfect since cospectral non-isomorphic graphs exist. For Laplacian spectra, the method fails less than 10 to 15 percent of the cases.
Mathematical results related to graph isomorphism
Two graphs are said to be isomorphic when they are structurally equivalent irrespective of the vertex labels.
Isomorpic graphs have identical eigenvalues.
Isomorphic graphs are related by permutation of vertex labels. Label-permutation of matrices is a linear transformation. Thus, isomorphic graphs are represented by similar matrices according to the
following theorem.
Theorem: Two square matrices are similar if and only if they represent the same linear transformation.
Similar matrices are isomorphic since
Theorem: Similar matrices have the same characteristic polynomial and therefore the same eigenvalues.
However, non-isomophic graphs can also have identical eigenvalues. These are called co-spectral non-isomophic graphs. | {"url":"http://monod.biomath.nyu.edu/rag/tutorial_graph_isomorphism","timestamp":"2024-11-14T01:58:11Z","content_type":"text/html","content_length":"18861","record_id":"<urn:uuid:9733ed41-e986-4a49-ab02-1e3c47b21e75>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00418.warc.gz"} |
Trendoscope® | Moving Averages in Detail
Moving averages are a fundamental technical indicator that smooths out price data to reveal trends, aiding traders in making informed decisions. There are different types of moving averages. Some of
them are listed here.
Simple Moving Average
A Simple Moving Average (SMA) is a widely used technical indicator that calculates the average of a financial instrument's closing prices over a specified period. It's calculated using this formula:
SMA = (Sum of Closing Prices for N Periods) / N
Exponential Moving Average
An Exponential Moving Average (EMA) is a popular technical indicator that, like the Simple Moving Average (SMA), helps traders identify trends and potential reversal points in financial instruments.
The EMA, however, gives more weight to recent price data, making it more responsive to current market conditions. It's calculated using this formula:
EMA = (Closing Price - Previous EMA) x (2 / (N + 1)) + Previous EMA
Rolling Moving Average
The Rolling Moving Average, sometimes referred to as "Smoothed Moving Average", gives the recent prices most weighting, though the historic prices are also weighted, each given less weighting further
back in time. The latest Rolling Average is obtained by multiplying the previous Rolling Average by n-1 periods, adding current price, and then dividing the total by n periods. Note that the initial
RMA is based on a Simple Moving Average.
RMA = (RMA(t-1) * (n-1) + Closing Price) / n
Weighted Moving Average
The Weighted Moving Average (WMA) is a type of moving average that assigns different weights to different data points. It gives more importance to recent data, making it more responsive to price
changes. The formula for WMA is:
WMA = (P1 * W1 + P2 * W2 + ... + Pn * Wn) / (W1 + W2 + ... + Wn)
• WMA is the Weighted Moving Average.
• P1, P2, ..., Pn are the prices of the data points.
• W1, W2, ..., Wn are the weights assigned to the corresponding data points.
Hull Moving Average
The Hull Moving Average (HMA) is a technical indicator that reduces the lag of traditional moving averages while retaining the smoothness of the moving average line. It does this by using a weighted
average of three moving averages, with the weights chosen to reduce lag.
The formula for the HMA is:
HMA = WMA(WMA(WMA(close, n / 2) * 2 - WMA(close, n), sqrt(n)), sqrt(n))
• close is the closing price of the security
• n is the period of the HMA
• WMA is the weighted moving average
To calculate the HMA, you first need to calculate the weighted moving average of the closing price over half the specified period. Then, you need to multiply this result by two and subtract the
weighted moving average of the closing price over the full specified period. Next, you need to calculate the square root of the specified period. Finally, you need to calculate the weighted moving
average of this result over the square root of the specified period.
The HMA is typically used to identify trends and to generate trading signals. When the HMA crosses above the closing price, it is considered a bullish signal. When the HMA crosses below the closing
price, it is considered a bearish signal.
Double Exponential Moving Average (DEMA)
The double exponential moving average (DEMA) is a technical indicator that reduces the lag of traditional exponential moving averages (EMAs), making it more responsive. It does this by using a
weighted average of two EMAs.
DEMA = 2 * EMA(EMA(close, n), n) - EMA(close, n)
• close is the closing price of the security
• n is the period of the DEMA
To calculate the DEMA, you first need to calculate the EMA of the closing price over the specified period. Then, you need to calculate the EMA of the EMA of the closing price over the same period.
Finally, you need to multiply the EMA of the EMA of the closing price by two and subtract the EMA of the closing price.
The DEMA is typically used to identify trends and to generate trading signals. When the DEMA crosses above the closing price, it is considered a bullish signal. When the DEMA crosses below the
closing price, it is considered a bearish signal.
Triple Exponential Moving Average (TEMA)
The Triple Exponential Moving Average (TEMA) is a technical indicator that reduces the lag of traditional exponential moving averages (EMAs) by taking multiple EMAs of the original EMA and
subtracting out some of the lag. This results in an indicator that is more responsive to price changes and can provide earlier trading signals.
TEMA = 3 * EMA(EMA(EMA(close, n), n), n) - 3 * EMA(EMA(close, n), n) + EMA(close, n)
• close is the closing price of the security
• n is the period of the TEMA
To calculate the TEMA, you first need to calculate the EMA of the closing price over the specified period. Then, you need to calculate the EMA of the EMA of the closing price over the same period.
Finally, you need to calculate the EMA of the EMA of the EMA of the closing price over the same period. Finally, you need to multiply the EMA of the EMA of the EMA of the closing price by three,
multiply the EMA of the EMA of the closing price by three, subtract the two, and add the EMA of the closing price.
The TEMA is typically used to identify trends and to generate trading signals. When the TEMA crosses above the closing price, it is considered a bullish signal. When the TEMA crosses below the
closing price, it is considered a bearish signal.
Zero Lag Exponential Moving Average
The Zero Lag Exponential Moving Average (ZLEMA) is a technical indicator that aims to reduce the lag of traditional exponential moving averages (EMAs) by de-lagging the closing price data before
calculating the EMA. This results in an indicator that is more responsive to price changes and can provide earlier trading signals.
ZLEMA = EMA(close - lag, n)
• close is the closing price of the security
• lag is the number of periods to de-lag the closing price data
• n is the period of the ZLEMA
Arnaud Legoux Moving Average (ALMA)
The Arnaud Legoux Moving Average (ALMA) is a technical indicator that reduces the lag of traditional moving averages and is less noisy. It does this by using a weighted average of two moving
averages, one calculated from left to right and the other from right to left.
ALMA = (EMA(close, n) + EMA(close[::-1], n)) / 2
• close is the closing price of the security
• n is the period of the ALMA
• [::-1] reverses the order of the closing prices
To calculate the ALMA, you first need to calculate the EMA of the closing price over the specified period. Then, you need to reverse the order of the closing prices and calculate the EMA of the
reversed closing prices over the same period. Finally, you need to average the two EMAs.
The ALMA is typically used to identify trends and to generate trading signals. When the ALMA crosses above the closing price, it is considered a bullish signal. When the ALMA crosses below the
closing price, it is considered a bearish signal.
Adaptive Moving Average (AMA)
The Adaptive Moving Average (AMA) is a technical indicator that adapts to the volatility of the market, making it more responsive to price changes. It does this by using a weighted average of two
exponential moving averages (EMAs), with the weights determined by the volatility of the market.
The formula for the AMA is:
AMA = (EMA(close, fastPeriod) * EMA(ATR, slowPeriod)) / (EMA(ATR, slowPeriod) + EMA(close, fastPeriod))
• close is the closing price of the security
• fastPeriod is the period of the fast EMA
• slowPeriod is the period of the slow EMA
• ATR is the Average True Range
The AMA is typically used to identify trends and to generate trading signals. When the AMA crosses above the closing price, it is considered a bullish signal. When the AMA crosses below the closing
price, it is considered a bearish signal.
Triangular Moving Average
The Triangular Moving Average (TMA) is a technical indicator that reduces the lag of traditional moving averages and is less noisy. It does this by using a weighted average of the closing prices over
the specified period, with the weights assigned in a triangular shape.
The formula for the TMA is:
TMA = (close[0] + 2 * close[1] + 3 * close[2] + ... + n * close[n-1]) / (n * (n + 1) / 2)
• close is the closing price of the security
• n is the period of the TMA
To calculate the TMA, you first need to sum the closing prices over the specified period, with each closing price multiplied by its corresponding weight. Then, you need to divide this sum by the
product of the period and the next integer greater than the period divided by two.
The TMA is typically used to identify trends and to generate trading signals. When the TMA crosses above the closing price, it is considered a bullish signal. When the TMA crosses below the closing
price, it is considered a bearish signal. | {"url":"https://www.trendoscope.io/blog/moving-averages","timestamp":"2024-11-09T00:25:18Z","content_type":"text/html","content_length":"55558","record_id":"<urn:uuid:fb3c9b81-d85d-4081-956a-3a51eead545b>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00410.warc.gz"} |
Want to Be a Data Scientist? Don’t Start With Machine LearningWant to Be a Data Scientist? Don’t Start With Machine Learning
The first thing most people think about when they hear the term “data science” is usually “machine learning”.
This was the case for me. My interest in data science sparked because I was first exposed to the idea of “machine learning” which sounded really cool. So when I was looking for a place to start
learning about data science, you can guess where I started (hint: it rhymes with bean churning).
This was my biggest mistake and this leads me to my main point:
If you want to be a data scientist, don’t start with machine learning.
Bear with me here. Obviously, to be a “complete” data scientist, you’ll have to eventually learn about machine learning concepts. But you’d be surprised at how far you can get without it.
So why shouldn’t you start with machine learning?
1. Machine learning is only one part of a data scientist (and a very small part too).
Image created by Author
Data science and machine learning are like a square and a rectangle. Machine learning is (a part of) data science but data science isn’t necessarily machine learning, similar to how a square is a
rectangle but a rectangle isn’t necessarily a square.
In reality, I’d say that machine learning modeling only makes up around 5–10% of a data scientist’s job, where most of one’s time is spent elsewhere, which I’ll elaborate on later.
TLDR: By focusing on machine learning first, you’ll be putting in a lot of time and energy, and getting little in return.
2. Fully understanding machine learning requires preliminary knowledge in several other subjects first.
At its core, machine learning is built on statistics, mathematics, and probability. The same way that you first learn about English grammar, figurative language, and so forth to write a good essay,
you have to have these building blocks set in stone before you can learn machine learning.
To give some examples:
• Linear regression, the first “machine learning algorithm” that most bootcamps teach first is really a statistical method.
• Principal Component Analysis is only possible with the ideas of matrices and eigenvectors (linear algebra)
• Naive Bayes is a machine learning model that is completely based on Bayes Theorem (probability).
And so, I’ll conclude with two points. One, learning the fundamentals will make learning more advanced topics easier. Two, by learning the fundamentals, you will already have learned several machine
learning concepts.
3. Machine learning is not the answer to every data scientist’s problem.
Many data scientists struggle with this, even myself. Similar to my initial point, most data scientists think that “data science” and “machine learning” go hand in hand. And so, when faced with a
problem, the very first solution that they consider is a machine learning model.
But not every “data science” problem requires a machine learning model.
In some cases, a simple analysis with Excel or Pandas is more than enough to solve the problem at hand.
In other cases, the problem will be completely unrelated to machine learning. You may be required to clean and manipulate data using scripts, build data pipelines, or create interactive dashboards,
all of which do not require machine learning.
What should you do instead?
If you’ve read my article, “How I’d Learn Data Science If I Had to Start Over,” you may have noticed that I suggested learning Mathematics, Statistics, and programming fundamentals. And I still stand
by this.
Like I said before, learning the fundamentals will make learning more advanced topics easier, and by learning the fundamentals, you will already have learned several machine learning concepts.
I know it may feel like you’re not progressing to be a “data scientist” if you’re learning statistics, math, or programming fundamentals, but learning these fundamentals will only accelerate your
learnings in the future.
You have to learn to walk before you can run.
If you would like some tangible next steps to start with instead, here are a couple:
1. Start with statistics. Of the three building blocks, I think statistics is the most important. And if you dread statistics, data science probably isn’t for you. I’d check out Georgia Tech’s
course called Statistical Methods, or Khan Academy’s video series.
2. Learn Python and SQL. If you’re more of an R kind of guy, go for it. I’ve personally never worked with R so I have no opinion on it. The better you are at Python and SQL, the easier your life
will be when it comes to data collection, manipulation, and implementation. I would also be familiar with Python libraries like Pandas, NumPy, and Scikit-learn. I also recommend that you learn
about binary trees, as it serves as the basis for many advanced machine learning algorithms like XGBoost.
3. Learn linear algebra fundamentals. Linear algebra becomes extremely important when you work with anything related to matrices. This is common in recommendation systems and deep learning
applications. If these sound like things that you’ll want to learn about in the future, don’t skip this step.
4. Learn data manipulation. This makes up at least 50% of a data scientist’s job. More specifically, learn more about feature engineering, exploratory data analysis, and data preparation. | {"url":"https://www.itechnewsonline.com/want-to-be-a-data-scientist-dont-start-with-machine-learning/","timestamp":"2024-11-13T11:52:47Z","content_type":"text/html","content_length":"184693","record_id":"<urn:uuid:6b364f87-cd58-4385-b34d-8433d44f860a>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00546.warc.gz"} |
Lambda the Ultimate
Let's talk about Blockchain. Goal is to use this forum topic to highlight its usefulness to programming language theory and practice. If you're familiar with existing research efforts, please share
them here. In addition, feel free to generate ideas for how Blockchain could improve languages and developer productivity.
As one tasty example: Blockchain helps to formalize thinking about mutual knowledge and common knowledge, and potentially think about sharing intergalactic computing power through vast distributed
computing fabrics. If we can design contracts in such a way that maximizes the usage of mutual knowledge while minimizing common knowledge to situations where you have to "prove your collateral",
third-party transactions could eliminate a lot of back office burden. But, there might be benefits in other areas of computer science from such research, as well.
Some language researchers, like Mark S. Miller, have always dreamed of Agoric and the Decades-Long Quest for Secure Smart Contracts.
Some may also be aware that verification of smart contracts is an important research area, because of the notorious theft of purse via logic bug in an Ethereum smart contract.
The Gentle Art of Levitation
2010 by James Chapman, Pierre-Evariste Dagand, Conor McBride, Peter Morrisy
We present a closed dependent type theory whose inductive types are given not by a scheme for generative declarations, but by encoding in a universe. Each inductive datatype arises by
interpreting its description—a first-class value in a datatype of descriptions. Moreover, the latter itself has a description. Datatype-generic programming thus becomes ordinary programming. We
show some of the resulting generic operations and deploy them in particular, useful ways on the datatype of datatype descriptions itself. Surprisingly this apparently self-supporting setup is
achievable without paradox or infinite regress.
It's datatype descriptions all the way down.
Comprehending Ringads
2016 by Jeremy Gibbons
Ringad comprehensions represent a convenient notation for expressing database queries. The ringad structure alone does not provide a good explanation or an efficient implementation of relational
joins; but by allowing heterogeneous comprehensions, involving both bag and indexed table ringads, we show how to accommodate these too.
Indexed/parametric/graded monads are the key (read the paper to understand the pun).
Sequent Calculus as a Compiler Intermediate Language
2016 by Paul Downen, Luke Maurer, Zena M. Ariola, Simon Peyton Jones
The typed λ-calculus arises canonically as the term language for a logic called natural deduction, using the Curry-Howard isomorphism: the pervasive connection between logic and programming
languages asserting that propositions are types and proofs are programs. Indeed, for many people, the λ-calculus is the living embodiment of Curry-Howard.
But natural deduction is not the only logic! Conspicuously, natural deduction has a twin, born in the very same paper, called the sequent calculus. Thanks to the Curry-Howard isomorphism, terms
of the sequent calculus can also be seen as a programming language with an emphasis on control flow.
Implementing Algebraic Effects in C by Daan Leijen:
We describe a full implementation of algebraic effects and handlers as a library in standard and portable C99, where effect operations can be used just like regular C functions. We use a formal
operational semantics to guide the C implementation at every step where an evaluation context corresponds directly to a particular C execution context. Finally we show a novel extension to the
formal semantics to describe optimized tail resumptions and prove that the extension is sound. This gives two orders of magnitude improvement to the performance of tail resumptive operations (up
to about 150 million operations per second on a Core i7@2.6GHz)
Another great paper by Daan Leijen, this time on a C library with immediate practical applications at Microsoft. The applicability is much wider though, since it's an ordinary C library for defining
and using arbitrary algebraic effects. It looks pretty usable and is faster and more general than most of the C coroutine libraries that already exist.
It's a nice addition to your toolbox for creating language runtimes in C, particularly since it provides a unified, structured way of creating and handling a variety of sophisticated language
behaviours, like async/await, in ordinary C with good performance. There has been considerable discussion here of C and low-level languages with green threads, coroutines and so on, so hopefully
others will find this useful!
The Syntax and Semantics of Quantitative Type Theory by Robert Atkey:
Type Theory offers a tantalising promise: that we can program and reason within a single unified system. However, this promise slips away when we try to produce efficient programs. Type Theory
offers little control over the intensional aspect of programs: how are computational resources used, and when can they be reused. Tracking resource usage via types has a long history, starting
with Girard's Linear Logic and culminating with recent work in contextual effects, coeffects, and quantitative type theories. However, there is conflict with full dependent Type Theory when
accounting for the difference between usages in types and terms. Recently, McBride has proposed a system that resolves this conflict by treating usage in types as a zero usage, so that it doesn't
affect the usage in terms. This leads to a simple expressive system, which we have named Quantitative Type Theory (QTT).
McBride presented a syntax and typing rules for the system, as well as an erasure property that exploits the difference between “not used” and “used”, but does not do anything with the finer
usage information. In this paper, we present present a semantic interpretation of a variant of McBride's system, where we fully exploit the usage information. We interpret terms simultaneously as
having extensional (compile-time) content and intensional (runtime) content. In our example models, extensional content is set-theoretic functions, representing the compile-time or type-level
content of a type-theoretic construction. Intensional content is given by realisers for the extensional content. We use Abramsky et al.'s Linear Combinatory Algebras as realisers, yield a large
range of potential models from Geometry of Interaction, graph models, and syntactic models. Read constructively, our models provide a resource sensitive compilation method for QTT.
To rigorously define the structure required for models of QTT, we introduce the concept of a Quantitative Category with Families, a generalisation of the standard Category with Families class of
models of Type Theory, and show that this class of models soundly interprets Quantitative Type Theory.
Resource-aware programming is a hot topic these days, with Rust exploiting affine and ownership types to scope and track resource usage, and with Ethereum requiring programs to spend "gas" to
execute. Combining linear and dependent types has proven difficult though, so making it easier to track and reason about resource usage in dependent type theories would then be a huge benefit to
making verification more practical in domains where resources are limited.
Nothing you don't already know, if you are inteo this sort of thing (and many if not most LtU-ers are), but a quick way to get the basic idea if you are not. Wadler has papers that explain
Curry-Howard better, and the category theory content here is very basic -- but it's an easy listen that will give you the fundamental points if you still wonder what this category thing is all about.
To make this a bit more fun for those already in the know: what is totally missing from the talk (understandable given time constraints) is why this should interest the "working hacker". So how about
pointing out a few cool uses/ideas that discerning hackers will appreciate? Go for it!
Fully Abstract Compilation via Universal Embedding by Max S. New, William J. Bowman, and Amal Ahmed:
A fully abstract compiler guarantees that two source components are observationally equivalent in the source language if and only if their translations are observationally equivalent in the
target. Full abstraction implies the translation is secure: target-language attackers can make no more observations of a compiled component than a source-language attacker interacting with the
original source component. Proving full abstraction for realistic compilers is challenging because realistic target languages contain features (such as control effects) unavailable in the source,
while proofs of full abstraction require showing that every target context to which a compiled component may be linked can be back-translated to a behaviorally equivalent source context.
We prove the first full abstraction result for a translation whose target language contains exceptions, but the source does not. Our translation—specifically, closure conversion of simply typed
λ-calculus with recursive types—uses types at the target level to ensure that a compiled component is never linked with attackers that have more distinguishing power than source-level attackers.
We present a new back-translation technique based on a deep embedding of the target language into the source language at a dynamic type. Then boundaries are inserted that mediate terms between
the untyped embedding and the strongly-typed source. This technique allows back-translating non-terminating programs, target features that are untypeable in the source, and well-bracketed
Potentially a promising step forward to secure multilanguage runtimes. We've previously discussed security vulnerabilities caused by full abstraction failures here and here. The paper also provides a
comprehensive review of associated literature, like various means of protection, back translations, embeddings, etc.
Simon Peyton Jones has been elected as a Fellow of the Royal Society. The Royal Society biography reads:
Simon's main research interest is in functional programming languages, their implementation, and their application. He was a key contributor to the design of the now-standard functional language
Haskell, and is the lead designer of the widely-used Glasgow Haskell Compiler (GHC). He has written two textbooks about the implementation of functional languages.
More generally, Simon is interested in language design, rich type systems, compiler technology, code generation, runtime systems, virtual machines, and garbage collection. He is particularly
motivated by direct use of principled theory to practical language design and implementation -- that is one reason he loves functional programming so much.
Simon is also chair of Computing at School, the grass-roots organisation that was at the epicentre of the 2014 reform of the English computing curriculum.
Congratulations SPJ!
I saw this work presented at ESOP 2015 by Neil Toronto, and the talk was excellent (slides).
Running Probabilistic Programs Backwards
Neil Toronto, Jay McCarthy, David Van Horn
Many probabilistic programming languages allow programs to be run under constraints in order to carry out Bayesian inference. Running programs under constraints could enable other uses such as
rare event simulation and probabilistic verification---except that all such probabilistic languages are necessarily limited because they are defined or implemented in terms of an impoverished
theory of probability. Measure-theoretic probability provides a more general foundation, but its generality makes finding computational content difficult.
We develop a measure-theoretic semantics for a first-order probabilistic language with recursion, which interprets programs as functions that compute preimages. Preimage functions are generally
uncomputable, so we derive an abstract semantics. We implement the abstract semantics and use the implementation to carry out Bayesian inference, stochastic ray tracing (a rare event simulation),
and probabilistic verification of floating-point error bounds.
(also on SciRate)
The introduction sells the practical side of the work a bit better than the abstract.
Stochastic ray tracing [30] is one such rare-event simulation task. As illus- trated in Fig. 1, to carry out stochastic ray tracing, a probabilistic program simulates a light source emitting a
single photon in a random direction, which is reflected or absorbed when it hits a wall. The program outputs the photon’s path, which is constrained to pass through an aperture. Millions of paths
that meet the constraint are sampled, then projected onto a simulated sensor array.
The program’s main loop is a recursive function with two arguments: path, the photon’s path so far as a list of points, and dir, the photon’s current direction.
simulate-photon path dir :=
case (find-hit (fst path) dir) of
absorb pt −→ (pt, path)
reflect pt norm −→ simulate-photon (pt, path) (random-half-dir norm)
Running simulate-photon (pt, ()) dir, where pt is the light source’s location and dir is a random emission direction, generates a photon path. The fst of the path (the last collision point) is
constrained to be in the aperture. The remainder of the program is simple vector math that computes ray-plane intersections.
In contrast, hand-coded stochastic ray tracers, written in general-purpose languages, are much more complex and divorced from the physical processes they simulate, because they must interleave
the advanced Monte Carlo algorithms that ensure the aperture constraint is met.
Unfortunately, while many probabilistic programming languages support random real numbers, none are capable of running a probabilistic program like simulate-photon under constraints to carry out
stochastic ray tracing. The reason is not lack of engineering or weak algorithms, but is theoretical at its core: they are all either defined or implemented using [density functions]. [...]
Programs whose outputs are deterministic functions of random values and programs with recursion generally cannot denote density functions. The program simulate-photon exhibits both
Measure-theoretic probability is a more powerful alternative to this naive probability theory based on probability mass and density functions. It not only subsumes naive probability theory, but
is capable of defining any computable probability distribution, and many uncomputable distributions. But while even the earliest work [15] on probabilistic languages is measure-theoretic, the
theory’s generality has historically made finding useful computational content difficult. We show that measure-theoretic probability can be made computational by
1. Using measure-theoretic probability to define a compositional, denotational semantics that gives a valid denotation to every program.
2. Deriving an abstract semantics, which allows computing answers to questions about probabilistic programs to arbitrary accuracy.
3. Implementing the abstract semantics and efficiently solving problems.
In fact, our primary implementation, Dr. Bayes, produced Fig. 1b by running a probabilistic program like simulate-photon under an aperture constraint. | {"url":"http://lambda-the-ultimate.org/taxonomy/term/29?from=0","timestamp":"2024-11-02T14:01:27Z","content_type":"application/xhtml+xml","content_length":"34732","record_id":"<urn:uuid:c11207ff-545a-4910-b7e5-e58169993ec8>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00677.warc.gz"} |
Normalize size units to Gigabytes
Important: This formula assumes that units are the last 2 characters of the text value that includes both a number and a unit of measure.
This formula works because digital units have a "power of 10" relationship. At the core, this formula separates the number part of the size from the unit, then divides the number by the appropriate
divisor to normalize to Gigabytes. The divisor is calculated as a power of 10, so the formula reduces to this:
To get the number, the formula extracts all characters from the left up to but not including the units:
To get "power", the formula matches the unit with a hard-coded array constant, {"PB","TB","GB","MB","KB"}:
The result from MATCH is the position of the unit in the array constant. For example, for the formula in C5, the unit is "KB", so the result is 5. This result is adjusted by subtracting 3, then
multiplying the result by 3, which yields 6 as the power, which is used as the exponent to calculate the correct result in gigabytes:
Binary standard formula
Computers use the binary number system to store and report data size, but the prefixes like "kilo", "mega", "giga", etc. are based on the metric system. It's a confusing topic, but using decimal
units for storage on a computer isn't really correct, and the discrepancy increases as units get larger. The formula below will normalize to binary units.
With this formula, you are technically getting Gibibytes (GiB), not Gigabytes. More information here and here. | {"url":"https://exceljet.net/formulas/normalize-size-units-to-gigabytes","timestamp":"2024-11-06T06:21:16Z","content_type":"text/html","content_length":"50348","record_id":"<urn:uuid:f02c25eb-f81b-4ec3-b568-482d59d7e959>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00031.warc.gz"} |
Recent Advances in the Discovery of tyrosinase inhibitors
Grodstein F., Skarupski K. unique classes of COX inhibitors efficiently clogged neurite loss in main neurons, suggesting that improved COX activity contributes to A peptide-induced neurite loss.
Finally, we discovered that the detrimental effect of COX activity on neurite integrity may be mediated through the inhibition of peroxisome proliferator-activated receptor (PPAR) activity. Overall,
our work establishes the feasibility of identifying small molecule inhibitors of A-induced neurite loss using the NeuriteIQ pipeline and provides novel insights into the mechanisms of neuroprotection
by NSAIDs. are a schematic representation of the image control that NeuriteIQ performs section of Materials and Methods. and symbolize highest and least expensive numbers, respectively. Distribution
of z-scores is also demonstrated. The hit selection criteria are explained in Materials and Methods. In the neuron/neurite channel, NeuriteIQ detects soma areas with clustering pixels and higher
intensity than adjacent areas. Neurites are then treated as two-dimensional curvilinear constructions, which could become detected based on the local Rabbit polyclonal to Acinus Hessian matrix. The
Hessian matrix identifies the local curvature of a curvilinear structure, which is an useful algorithm that allows detection the center points and local directions of neurite inside a field.
Subsequently, a specific neurite is recognized from a seed point, which is defined as an initial point on or near the center line of a dendritic section and soma. Consequently, a specific dendrite
could be ascribed to a specific nucleus by its seed point. Recognition of seed points for each neurite minimizes interference from positively stained debris. The tracking algorithm then detects
center points along each neurite, and defines the possible direction of neurites from each center point. After calculating the center points and their directions, centerlines could be extracted along
neurites by linking detected center points along the local directions, which display curvilinear structures. In case of breaks between near branching structures, a predefined radius r is set up to
determine whether two end points of different centerlines should be linked together. If one of the end points is in the local direction of another centerline, and the distance between two end points
is in the range of r, those two points are linked to fill the break. Bresenham collection drawing algorithm is usually applied to link these two points. This allows us to solve the neurite collection
break problem during the post-processing of images. NeuriteIQ provides a statistical quantification of the total neurite length in one image, which is subsequently used to calculate Average Neurite
Length (ANL) as the statistical feature of neurite outgrowth in each well. ANL is usually defined as a ratio between Total Neurite Length per image and Neuron Cell Figures. ANL is usually a
statistical parameter, which averages the neurite lengths in the entire neuronal field and makes the analysis results resistant to slight changes in the neuron culture and staining as well as local
variations in cell density and errors in tracing of individual neurites due to high cell density. ANL calculations are described in detail in Ref. 10. Because both of the total neurite length and
neuron cell number are statistical results averaged over entire image, ANL is usually a strong measure of neurite outgrowth which is usually highly accurate and reproducible even in high density
cultures. Thus, NeuriteIQ is usually a fully automated tool for batch processing a large dataset of images without human intervention such as selecting start points of neurites, defining directions
for neurite tracking in.L., Farlow M. as potential drugs for AD; however their mechanism of action remains controversial. Our data Dapagliflozin (BMS512148) revealed that cyclooxygenase-2 (COX-2)
expression was increased following A treatment. Furthermore, multiple unique classes of COX inhibitors efficiently blocked neurite loss in main neurons, suggesting that increased COX activity
contributes to A peptide-induced neurite loss. Finally, we discovered that the detrimental effect of COX activity on neurite integrity may be mediated through the inhibition of peroxisome
proliferator-activated receptor (PPAR) activity. Overall, our work establishes the feasibility of identifying small molecule inhibitors of A-induced neurite loss using the NeuriteIQ pipeline and
provides novel insights into the mechanisms of neuroprotection by NSAIDs. are a schematic representation of the image processing that NeuriteIQ performs section of Materials and Methods. and
symbolize highest and least expensive figures, respectively. Distribution of z-scores is also shown. The hit selection criteria are explained in Materials and Methods. In the neuron/neurite channel,
NeuriteIQ detects soma areas with clustering pixels and higher intensity than adjacent areas. Neurites are then treated as two-dimensional curvilinear structures, which could be detected based on the
local Hessian matrix. The Hessian matrix details the neighborhood curvature of the curvilinear framework, which can be an useful algorithm which allows detection the guts factors and regional
directions of neurite inside a field. Subsequently, a particular neurite is recognized from a seed stage, which is thought as an initial stage on or close to the center type of a dendritic section
and soma. Consequently, a particular dendrite could possibly be ascribed to a particular nucleus by its seed stage. Recognition of seed factors for every neurite minimizes disturbance from favorably
stained particles. The monitoring algorithm after that detects center Dapagliflozin (BMS512148) factors along each neurite, and defines the feasible path of neurites from each middle point. After
determining the center factors and their directions, centerlines could possibly be extracted along neurites by linking recognized center factors along the neighborhood directions, which screen
curvilinear structures. In case there is breaks between near branching constructions, a predefined radius r is established to determine whether two end factors of different centerlines ought to be
connected together. If among the end factors is within the local path of another centerline, and the length between two end factors is within the number of r, those two factors are associated with
fill up the break. Bresenham range drawing algorithm can be applied to hyperlink these two factors. This enables us to resolve the neurite range break problem through the post-processing of pictures.
NeuriteIQ offers a statistical quantification of the full total neurite length in a single picture, which is Dapagliflozin (BMS512148) consequently utilized to calculate Typical Neurite Size (ANL) as
the statistical feature of neurite outgrowth in each well. ANL can be thought as a percentage between Total Neurite Size per picture and Neuron Cell Amounts. ANL can be a statistical parameter, which
averages the neurite measures in the complete neuronal field and makes the evaluation outcomes resistant to minor adjustments in the neuron tradition and staining aswell as local variants in cell
denseness and mistakes in tracing of specific neurites because of high cell denseness. ANL computations are described at length in Ref. 10. Because both of the full total neurite size and neuron
cellular number are statistical outcomes averaged over whole picture, ANL can be a robust way of measuring neurite outgrowth which can be extremely accurate and reproducible actually in high
denseness cultures. Therefore, NeuriteIQ is a completely automated device for batch digesting a big Dapagliflozin (BMS512148) dataset of pictures without human treatment such as choosing start
factors of neurites, determining directions for neurite monitoring in a branch, etc, making NeuriteIQ a competent tool in working with huge size dataset for substance screening. We’ve made NeuriteIQ
general public, and it could be downloaded free of charge along with consumer documentation on the net. Finally, Z-factors of ANL (typical neurite size) and ANB (typical neurite lighting, which is
thought as the percentage between your total lighting (strength) of most neurite pixels and the full total amount of all neurites in the picture,) were determined to pilot for quality evaluation of
assay circumstances. Despite poor z ( relatively?0.84 for ANL where z = 1 ? (3 SDuntreated.In case there is breaks between near branching structures, a predefined radius r is established to determine
whether two end points of different centerlines ought to be connected together. inhibitors clogged neurite reduction in major neurons effectively, suggesting that improved COX activity plays a part
in A peptide-induced neurite reduction. Finally, we found that the harmful aftereffect of COX activity on neurite integrity could be mediated through the inhibition of peroxisome
proliferator-activated receptor (PPAR) activity. General, our function establishes the feasibility of determining little molecule inhibitors of A-induced neurite reduction using the NeuriteIQ
pipeline and novel insights in to the systems of neuroprotection by NSAIDs. certainly are a schematic representation from the picture control that NeuriteIQ performs portion of Components and
Strategies. and stand for highest and most affordable amounts, respectively. Distribution of z-scores can be shown. The strike selection requirements are referred to in Components and Strategies. In
the neuron/neurite route, NeuriteIQ detects soma areas with clustering pixels and higher strength than adjacent areas. Neurites are after that treated as two-dimensional curvilinear buildings, which
could end up being detected predicated on the neighborhood Hessian matrix. The Hessian matrix represents the neighborhood curvature of the curvilinear framework, which can be an useful algorithm
which allows detection the guts factors and regional directions of neurite within a field. Subsequently, a particular neurite is discovered from a seed stage, which is thought as an initial stage on
or close to the center type of a dendritic portion and soma. As a result, a particular dendrite could possibly be ascribed to a particular nucleus by its seed stage. Id of seed factors for every
neurite minimizes disturbance from favorably stained particles. The monitoring algorithm after that detects center factors along each neurite, and defines the feasible path of neurites from each
middle point. After determining the center factors and their directions, centerlines could possibly be extracted along neurites by linking discovered center factors along the neighborhood directions,
which screen curvilinear structures. In case there is breaks between near branching buildings, a predefined radius r is established to determine whether two end factors of different centerlines ought
to be connected together. If among the end factors is within the local path of another centerline, and the length between two end factors is within the number of r, those two factors are associated
with fill up the break. Bresenham series drawing algorithm is normally applied to hyperlink these two factors. This enables us to resolve the neurite series break problem through the post-processing
of pictures. NeuriteIQ offers a statistical quantification of the full total neurite length in a single picture, which is eventually utilized to calculate Typical Neurite Duration (ANL) as the
statistical feature of neurite outgrowth in each well. ANL is normally thought as a proportion between Total Neurite Duration per picture and Neuron Cell Quantities. ANL is normally a statistical
parameter, which averages the neurite measures in the complete neuronal field and makes the evaluation outcomes resistant to small adjustments in the neuron lifestyle and staining aswell as local
variants in cell thickness and mistakes in tracing of specific neurites because of high cell thickness. ANL computations are described at length in Ref. 10. Because both of the full total neurite
duration and neuron cellular number are statistical outcomes averaged over whole picture, ANL is normally a robust way of measuring neurite outgrowth which is normally extremely accurate and
reproducible also in high thickness cultures. Hence, NeuriteIQ is a completely automated device for batch digesting a big dataset of pictures without human involvement such as choosing start factors
of neurites, determining directions for neurite monitoring in a branch, etc, making NeuriteIQ a competent tool in working with huge range dataset for substance screening. We’ve made NeuriteIQ open
public, and it could be downloaded free of charge along with consumer documentation on the net. Finally, Z-factors of ANL (typical neurite duration) and ANB (typical neurite lighting, which is
thought as the proportion between your total lighting (strength) of most neurite pixels and the full total amount of all neurites in the picture,) were computed to pilot for quality evaluation of
assay circumstances. Despite fairly poor z (?0.84 for ANL where z = 1 ? (3 SDuntreated + 3 SDA)/(Averageuntreated ? AverageA)) because of the variance usual for principal neurons treated with A1C40,
this technique yielded significant ( 0.05) distinctions between A1C40 treated and untreated control groups. Display screen Design NINDS custom made collection substance.G., Rowan M., Cleary J.,
Wallis R. medications for AD; nevertheless their system of action continues to be questionable. Our data uncovered that cyclooxygenase-2 (COX-2) appearance was increased carrying out a treatment.
Furthermore, multiple distinctive classes of COX inhibitors effectively blocked neurite reduction in principal neurons, recommending that elevated COX activity contributes to A peptide-induced
neurite loss. Finally, we discovered that the detrimental effect of COX activity on neurite integrity may be mediated through the inhibition of peroxisome proliferator-activated receptor (PPAR)
activity. Overall, our work establishes the feasibility of identifying small molecule inhibitors of A-induced neurite loss using the NeuriteIQ pipeline and provides novel insights into the mechanisms
of neuroprotection by NSAIDs. are a schematic representation of the image control that NeuriteIQ performs section of Materials and Methods. and symbolize highest and least expensive figures,
respectively. Distribution of z-scores is also shown. The hit selection criteria are explained in Materials and Methods. In the neuron/neurite channel, NeuriteIQ detects soma areas with clustering
pixels and higher intensity than adjacent areas. Neurites are then treated as two-dimensional curvilinear constructions, which could become detected based on the local Hessian matrix. The Hessian
matrix explains the local curvature of a curvilinear structure, which is an useful algorithm that allows detection the center points and local directions of neurite inside a field. Subsequently, a
specific neurite is recognized from a seed point, which is defined as an initial point on or near the center line of a dendritic Dapagliflozin (BMS512148) section and soma. Consequently, a specific
dendrite could be ascribed to a specific nucleus by its seed point. Recognition of seed points for each neurite minimizes interference from positively stained debris. The tracking algorithm then
detects center points along each neurite, and defines the possible direction of neurites from each center point. After calculating the center points and their directions, centerlines could be
extracted along neurites by linking recognized center points along the local directions, which display curvilinear structures. In case of breaks between near branching constructions, a predefined
radius r is set up to determine whether two end points of different centerlines should be linked together. If one of the end points is in the local direction of another centerline, and the distance
between two end points is in the range of r, those two points are linked to fill the break. Bresenham collection drawing algorithm is definitely applied to link these two points. This allows us to
solve the neurite collection break problem during the post-processing of images. NeuriteIQ provides a statistical quantification of the total neurite length in one image, which is consequently used
to calculate Average Neurite Size (ANL) as the statistical feature of neurite outgrowth in each well. ANL is definitely defined as a percentage between Total Neurite Size per image and Neuron Cell
Figures. ANL is definitely a statistical parameter, which averages the neurite lengths in the entire neuronal field and makes the analysis results resistant to minor changes in the neuron tradition
and staining as well as local variations in cell denseness and errors in tracing of individual neurites due to high cell denseness. ANL calculations are described in detail in Ref. 10. Because both
of the total neurite size and neuron cell number are statistical results averaged over entire image, ANL is definitely a robust measure of neurite outgrowth which is definitely highly accurate and
reproducible actually in high denseness cultures. Therefore, NeuriteIQ is a fully automated tool for batch digesting a big dataset of pictures without human involvement such as choosing start factors
of neurites, determining directions for neurite monitoring in a branch,.Ann. FDA accepted medications. Activity clustering demonstrated that nonsteroidal anti-inflammatory medications (NSAIDs) were
considerably enriched among the strikes. Notably, NSAIDs possess attracted significant interest seeing that potential medications for Advertisement previously; however their system of action
continues to be questionable. Our data uncovered that cyclooxygenase-2 (COX-2) appearance was increased carrying out a treatment. Furthermore, multiple specific classes of COX inhibitors effectively
blocked neurite reduction in major neurons, recommending that elevated COX activity plays a part in A peptide-induced neurite reduction. Finally, we found that the harmful aftereffect of COX activity
on neurite integrity could be mediated through the inhibition of peroxisome proliferator-activated receptor (PPAR) activity. General, our function establishes the feasibility of determining little
molecule inhibitors of A-induced neurite reduction using the NeuriteIQ pipeline and novel insights in to the systems of neuroprotection by NSAIDs. certainly are a schematic representation from the
picture handling that NeuriteIQ performs portion of Components and Strategies. and stand for highest and most affordable amounts, respectively. Distribution of z-scores can be shown. The strike
selection requirements are referred to in Components and Strategies. In the neuron/neurite route, NeuriteIQ detects soma areas with clustering pixels and higher strength than adjacent areas. Neurites
are after that treated as two-dimensional curvilinear buildings, which could end up being detected predicated on the neighborhood Hessian matrix. The Hessian matrix details the neighborhood curvature
of the curvilinear framework, which can be an useful algorithm which allows detection the guts factors and regional directions of neurite within a field. Subsequently, a particular neurite is
discovered from a seed stage, which is thought as an initial stage on or close to the center type of a dendritic portion and soma. As a result, a particular dendrite could possibly be ascribed to a
particular nucleus by its seed stage. Id of seed factors for every neurite minimizes disturbance from favorably stained particles. The monitoring algorithm after that detects center factors along
each neurite, and defines the feasible path of neurites from each middle point. After determining the center factors and their directions, centerlines could possibly be extracted along neurites by
linking discovered center factors along the neighborhood directions, which screen curvilinear structures. In case there is breaks between near branching buildings, a predefined radius r is
established to determine whether two end factors of different centerlines ought to be connected together. If among the end factors is within the local path of another centerline, and the length
between two end factors is within the number of r, those two factors are associated with fill up the break. Bresenham range drawing algorithm is certainly applied to hyperlink these two factors. This
enables us to resolve the neurite range break problem through the post-processing of pictures. NeuriteIQ offers a statistical quantification of the full total neurite length in a single picture,
which is eventually utilized to calculate Typical Neurite Duration (ANL) as the statistical feature of neurite outgrowth in each well. ANL is certainly thought as a proportion between Total Neurite
Duration per picture and Neuron Cell Amounts. ANL is certainly a statistical parameter, which averages the neurite measures in the complete neuronal field and makes the evaluation outcomes resistant
to small adjustments in the neuron lifestyle and staining aswell as local variants in cell thickness and mistakes in tracing of specific neurites because of high cell thickness. ANL computations are
described at length in Ref. 10. Because both of the full total neurite size and neuron cellular number are statistical outcomes averaged over whole picture, ANL can be a robust way of measuring
neurite outgrowth which can be extremely accurate and reproducible actually in high denseness cultures. Therefore, NeuriteIQ is a completely automated device for batch digesting a big dataset of
pictures without human treatment such as choosing start factors of neurites, determining directions for neurite monitoring in a branch, etc, making NeuriteIQ a competent tool in working with huge
size dataset for substance screening. We’ve made NeuriteIQ general public, and it could be downloaded free of charge along with consumer documentation on the net. Finally, Z-factors of ANL (typical
neurite size) and ANB (typical neurite lighting, which is thought as the percentage between your total lighting (strength) of most neurite pixels and the full total amount of all neurites in the
picture,) were determined to pilot for quality evaluation of assay circumstances.. | {"url":"http://innovation-ecosystems-agora.com/2022/11/17/grodstein-f/","timestamp":"2024-11-14T05:53:58Z","content_type":"text/html","content_length":"50930","record_id":"<urn:uuid:7aa9cfaf-bc72-4a23-bb1c-cfc58d9e1221>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00185.warc.gz"} |
Simple question about Seq3 code
Just trying to get my head around Rack module development by looking at the source for the Fundamental modules.
My question is about the if (running) statement. I get the that the inner if part is if there is an external clock. What I don’t understand is the maths involved to get the phase to tick the index of
the sequence with the internal clock. (Plus I’m not exactly sure what is the purpose of the CLOCK_INPUT?) I think I understand that sampleTime is just the inverse of sampleRate. What does clockTime *
args.sampleTime give in units?
I’m pretty sure I’m missing something simple, just can’t see it right now.
Thanks in advance.
I see sampletime as time, in milliseconds I think, since the last update. Is 1 / samplerate.
The sequencer does not count the time since the last update. It emulates a ramp oscillator, with the phase variable. When phase > 1 the sequencer ticks. Clocktime * args.sampletime sets the frequency
of the oscillator. It makes phase reach 1 faster.
Thanks. That helps a little with the understanding of sampletime and the phase variable.
OK, but I don’t understand the computation for clocktime. I’ve got a good maths and some science background and I’m trying to understand clocktime’s formula.
In the 1V/oct standard, the relationship between frequency and voltage is f = f_0 \cdot 2^V. The knob value can be added to the voltage because it also scales exponentially. See https://vcvrack.com/
manual/VoltageStandards#pitch-and-frequencies for more details.
The next line is a phase accumulator. If we’re stepping in time by \Delta t, the phase should advance by \Delta t \cdot f. This is so that one period happens in 1/f seconds, or 1/(\Delta t \cdot f)
2 Likes
Thank you. Let’s see if I can make some coherent statement based on what you’ve told me…
So clocktime is Hz (or cycles per second) that is required to give us the desired BPM. We then multiple Hz by seconds (sampletime) to give us how many cycles have passed since the last process. I
think it’s assumed this is < 1 cycle per process call so the cycles (or cycle position) is stored in phase so that we can tell at which process call we need to tick the sequence for the BPM. I think
that makes sense now.
Only one question. I’d imagine that phase is never exactly 1.0f so wouldn’t it be more precise to set phase to -= 1 instead of 0f? Or is that moot because it’s so close we probably can’t hear the
difference? (I’m assuming it’s the latter, because Seq3 sounds on beat to me.)
The fundamental oscillator does phase -= 1. I think for a sequencer is ok to set it to zero.
1 Like
Yes, it looks like clockTime should definitely be clockFreq. That code was written 4 years ago, which is ancient in VCV time.
At reasonable BPM, it doesn’t make any noticeable difference. For example at 160 bpm each period will be 16537 steps at 44.1kHz instead of the true 16537.5 steps. But at ridiculously high ones, sure,
the timing could be improved.
1 Like
Building a comprehensive sequencer as an exercise, any reason why I SHOULDN’T use a clock divider to process it? It uses ~3.7% without, down to ~.6% divided by 8 and all seems well. I note that other
sequencer sources I’ve looked at don’t and was wondering if there’s a good reason for that?
Stats for nerds: When I say comprehensive, currently 263 params, 58 inputs & 144 outputs!
Higher divisions, the greater the performance, but less of precision. If it doesn’t need to be at audio rate; the more efficient the better! Voltage works on lower frequencies, a higher division
might work better?
2 to 8 might work better for audio rate but the dynamic range can suffer going past 4, you’ll hear it anyway because Nyquist, hearing for most adults starts to cut off at 10kHz… also where mp3’s
start to compress, so why no noticeable change is heard.
1 Like | {"url":"https://community.vcvrack.com/t/simple-question-about-seq3-code/7652","timestamp":"2024-11-10T07:51:41Z","content_type":"text/html","content_length":"28647","record_id":"<urn:uuid:d82e874f-ed87-46c3-b5f7-01ea49009124>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00145.warc.gz"} |
Dootaframe is a pure-Python dataframe library. It is designed to be simple to write, understand and extend. Performance is not the main focus of this project.
This is the worst thing since sliced bread.
Overview of concepts#
Similar to Pandas, we have two main concepts, Series and DataFrame.
A Series is a one-dimensional container of items. While it can hold any type of Python object, it is particulary good at holding numerical data.
If you are familiar with database systems or spreadsheet applications, you might be inclined to store each row of your data inside a Series, similar to this.
>>> john = Series(["John", "Doe", 26])
>>> jane = Series(["Jane", "Doe", 25])
>>> jack = Series(["Jack", "Daniels", 38])
This would technically work, but it is the wrong approach with this library. What we want to do is to flip this around, so each column is stored in its own Series, like this.
>>> fnames = Series(["John", "Jane", "Jack"])
>>> lnames = Series(["Doe", "Doe", "Daniels"])
>>> ages = Series([26, 25, 38])
At first glance, this seems a little incovenient.
>>> ages = Series([26, 25, 38])
>>> ages.min
>>> ages.max
A DataFrame is a collection of Series’.
Series Reference#
We can now take a more detailed look at the Series class. As we discussed above, a Series is a one-dimensional collection of elements. When you provide this collection to the Series class, it adds a
bunch of convenient utilities to it.
Series protocol#
As we want to be flexible, there are no checks in dootaframe for particular collections. Instead, we rely on something we call “the Series protocol”. That’s a really fancy name for a trivial concept.
Basically all we require from an input to Series is that it implements the __len__ and __getitem__ methods. This is a very lax requirement, and it includes basically any collection you might want to
use. That includes tuples, lists, numpy arrays etc.
>>> Series((1, 2, 3))
Series(1, 2, 3)
>>> Series([1, 2, 3])
Series(1, 2, 3)
>>> import numpy as np
>>> Series(np.array([1, 2, 3]))
Series(1, 2, 3)
>>> d = {"name": "Leo", "age": 24}
>>> Series(d)
Series('name', 'age')
>>> Series(list(d)).apply(lambda x: d[x])
Series('Leo', 24)
There is no rule against more dynamic structures either. Series does not load the entire dataset into memory, so you can even use some on-the-fly collections.
>>> class PowersOfTwo:
... def __len__(self):
... return 9999999999
... def __getitem__(self, index):
... return 2 ** index
>>> s1 = Series(PowersOfTwo())
>>> s2 = s1.apply(lambda x: x + 123)
>>> s2[50]
Now, we definitely did not calculate 9999999999 powers of two and then add 123 to them. In dootaframe, a lot of the operations on Series are lazy. That means until we need the result, or use an
operation that requires looking at the whole collection, computation will only happen on the rows that are requested.
Optimized storage#
As we discussed above, the Series API is very flexible and accepts pretty much anything you give to it. For most use cases, Series contain a large number of the same data type.
>>> s = Series([1, 2, 3, 4, 5])
>>> s.underlying_storage
[1, 2, 3, 4, 5]
In the normal case, we can see that the input we gave is being stored as a Python list. This is okay for doing small explorations, but it’s not very efficient for storing a lot of data.
In cases like this, we can ask dootaframe to optimize the backing storage of a Series.
>>> s = Series([1, 2, 3, 4, 5]).optimize_storage
>>> s.underlying_storage
Instead of a dynamic Python list that has one item for each number, dootaframe was able to convert the backing storage into an immutable byte array. This is okay because it satisfies both the __len__
and the __getitem__ parts of the Series protocol.
That’s not the only storage optimization either. We have the same thing for collections of byte arrays, collections of unsigned 16-bit integers and more.
Optimization of byte arrays#
>>> s = Series([b"He", b"ll", b"o wor", b"l", b"d"])
>>> s.underlying_storage
[b'He', b'll', b'o wor', b'l', b'd']
>>> s = s.optimize_storage
>>> s.underlying_storage
<BytesSeriesStorage with 5 items and 11-byte buffer>
>>> s.underlying_storage.buf
b'Hello world'
>>> s.underlying_storage.begins
Series(0, 2, 4, 9, 10)
>>> s.underlying_storage.lens
Series(2, 2, 5, 1, 1)
>>> s
Series(b'He', b'll', b'o wor', b'l', b'd')
Instead of a Python list that has 5 buffers in it, dootaframe compacts everything into a single buffer and stores the spans of the individual chunks. The end result is the same though.
API Documentation#
Below is some auto-generated documentation.
Simple dataframe library in Pure Python.
class dootaframe.Series(s)#
Bases: object
One-dimensional container of values.
This class includes whatevers.
apply(func: Callable) dootaframe.Series#
Apply a function to each member of the series.
The function whose output will be used as the new value.
The new series.
>>> s = Series([1, 2, 3, 4])
>>> s.apply(lambda x: x + 1)
Series(2, 3, 4, 5)
property as_list: List#
Turn the Series into a Python list.
>>> s = Series([1, 2, 3, 4])
>>> s.as_list
[1, 2, 3, 4]
property as_numpy#
property asc: dootaframe.Series#
Sort the items in this series from lowest-to-highest.
concat(other: dootaframe.Series) dootaframe.Series#
Append another series to the end of this one.
The other series to append to this one.
A new series where other is appended to this one.
property desc#
property enumerate#
property floats#
property ints#
Convert every item of the Series into an integer.
This uses the native int() method to convert.
property length: int#
The number of items in this series.
property max#
The maximum value contained within this Series.
property mean#
property median#
property min#
The minimum value contained within this Series.
property optimize_storage: dootaframe.Series#
Create an storage-optimized version of this series.
A Series needs to accommodate storing arbitrary data, including data of different types. While this is very flexible, it means that all the book-keeping data can take a significant amount of
This method tries to optimize certain kinds of storage patterns, and makes them use a lot less memory.
order_by_desc(func: Callable)#
property solidify: dootaframe.Series#
property sorted#
property strings#
Turn every item of the Series into a string.
The same series, but every item is converted into a string.
property sum#
property underlying_storage#
property uniq#
property uniq_count# | {"url":"https://www.gkbrk.com/project/dootaframe.html","timestamp":"2024-11-12T00:23:53Z","content_type":"text/html","content_length":"160207","record_id":"<urn:uuid:e8ee123c-6f38-446c-b594-52e70d40ef2a>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00231.warc.gz"} |
Understanding ta.dev() Function in Pine Script - Pine Wizards
Understanding ta.dev() Function in Pine Script
By PineWizards
Published on
Pine Script offers a range of built-in functions that simplify the process of creating complex calculations. One such function is ta.dev(), which measures the deviation of a data series from its
simple moving average (SMA) over a specified period. This article will delve into the ta.dev() function, explaining its syntax, arguments, and providing a practical example.
Syntax of ta.dev()
The ta.dev() function in Pine Script is used to calculate the average deviation of a series from its simple moving average (SMA). Its syntax is as follows:
ta.dev(source, length) → series float
• source (series int/float): This is the series of values that ta.dev() will process. It can be any series of prices or indicators.
• length (series int): This specifies the number of bars to consider for the calculation. It determines the length of the SMA and the range over which the deviation is computed.
Below is an example of how to use the ta.dev() function in Pine Script to plot the deviation of the close price from its 10-bar SMA.
indicator("ta.dev example")
plot(ta.dev(close, 10))
Custom Implementation:
For a deeper understanding of what ta.dev() does under the hood, let’s examine a custom implementation named pine_dev.
pine_dev(source, length) =>
mean = ta.sma(source, length)
sum = 0.0
for i = 0 to length - 1
val = source[i]
sum := sum + math.abs(val - mean)
dev = sum / length
plot(pine_dev(close, 10))
• Calculating the Mean: First, we calculate the simple moving average (SMA) of the source series over the specified length using ta.sma(source, length). This serves as our mean around which we
measure deviation.
• Summing Deviations: We initialize a variable sum to zero. Then, using a for loop, we iterate over each bar in the specified length range, calculating the absolute difference between the source
value at each bar and the mean. These differences are accumulated in sum.
• Calculating Average Deviation: Finally, the sum of deviations is divided by the length to find the average deviation, which is stored in dev.
Key Features and Takeaways
• The ta.dev() function is a built-in Pine Script function for measuring the average deviation of a series from its simple moving average over a specified period.
• This function can be used to gauge the consistency of price movements or the variability of an indicator from its average.
• A custom implementation, like pine_dev, demonstrates the underlying calculation, reinforcing the concept of averaging deviations for a given length.
• Understanding both the built-in ta.dev() and the custom pine_dev function can enhance your Pine Script programming skills, especially in creating indicators that measure volatility or consistency
in data series.
By exploring both the built-in ta.dev() function and a custom implementation, you gain insights into the versatility of Pine Script for financial analysis and indicator development.
Leave a Comment | {"url":"https://pinewizards.com/technical-analysis-functions/ta-dev-function/","timestamp":"2024-11-11T10:20:15Z","content_type":"text/html","content_length":"107899","record_id":"<urn:uuid:057dd6ce-86a3-433f-bcc7-13edc4ab2cec>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00120.warc.gz"} |