text:
One of the question we will ask when setting up collections is what range of defocuses to target for imaging.

This is an important question and the answer affects the entire rest of data processing, so careful consideration should be taken when deciding.

Most of us know the tl;dr

higher defocus = higher contrast

lower defocus = higher resolution

and the inverses.

Some of us will know this handy rule-of-thumb from Radostin Danev:

And hopefully many of us are using this handy CTF calculator from Takanori Nakane at the MRC-LMB

https://3dem.github.io/relion/ctf.html

or maybe this more complex Excel spreadsheet from Henning Stahlberg:

But I wanted to provide some of the resources that I consider when trying to dive deeper in reasonable ranges and targets to select.

# Aliasing

This is the main aspect of the post I would like to convince you to consider in your next microscope session. Aliasing is when high-frequency signals mimic or “alias” low-frequency signals due to insufficient sampling. The most common real-life example of aliasing is the “Moire effect” you can see when you use your smartphone to take a picture of another display such as your computer monitor.

In the case of the CTF in Cryo-EM, the frequency is a function of defocus and spectral frequency (1 / resolution), and the sampling is a function of the box-size and the pixel size.

### Frequency

We can express the CTF as function of spatial frequency \vec{k} as (from Rohou 2015):

CTF(\lambda, \vec{k}, \Delta z, C_s, \Delta \phi, a_c) = -\sin[\chi(\lambda, |\vec{k}|, \Delta z, C_s, \Delta \phi, a_c)]

with \chi(\dots) denoting the wave aberration function:

\chi(\lambda, |\vec{k}|, \Delta z, C_s, \Delta \phi, a_c) = \pi \lambda |\vec{k}|^2 (\Delta z - \frac{1}{2}\lambda^2|\vec{k}|^2C_s) + \Delta \phi + \tan^{-1}(a_c / \sqrt{1 - a_c^2})

Where \lambda is the electron wavelength, \Delta z is our defocus, C_s is the spherical aberration, \Delta \phi is an additional phase-shift (which we will ignore here), and a_c is the fraction of amplitude contrast.

From our early maths studies we have that the period of \sin(Bx) is \frac{2\pi}{B}, so we have for the period of the CTF as a function of absolute spatial frequency (L(|\vec{k}|)).

L(|\vec{k}|) = \frac{2\pi|\vec{k}|}{\chi(\lambda, |\vec{k}|, \Delta z, C_s, \Delta \phi, a_c)}

### Sampling

In Digital Signal Processing the Nyquist-Shannon theorem tells us that to faithfully represent a function with a maximal frequency B hertz we need a sampling rate of at least 2B. The sampling rate of the Fourier Transform of our reconstruction is given by:

f_s = \frac{apix \times boxsize}{2}

Where the denominator comes from the symmetry of the Fourier transform, and our sampling step size is given by:

\Delta f_s = \frac{1}{apix \times boxsize}

Which gives us the range of frequencies we expect of DC to Nyquist.

### All together now

Putting what we know about the CTF frequencies and how the FFT is sampled we can now solve for the resolution at which the CTF will begin to alias for a given field of view (the box size in Angstrom units).

\frac{2}{f_s} = L

We can expand the terms and rearrange this into a proper polynomial of |\vec{k}|:

\frac{\lambda^3C_s}{2}|\vec{k}|^4 - \lambda \Delta z |\vec{k}|^2 + f_s |\vec{k}| + \frac{\tan^{-1}(a_c / \sqrt{1 - a_c^2})}{\pi} = 0

Solving this will gives us the resolution for aliasing for a given field of view. I have calculated the threshold for a range of defocuses against fields of view below.

By setting the spatial frequency to Nyquist |\vec{k}| = 1 / (2 \times apix) we can solve the polynomial now for box size and determine for a given magnification the minimum box size we need to ensure there will be no aliasing at a given defocus as well. The formula here being:

box size = \frac{\lambda \Delta z}{apix^2} - \frac{\lambda^3C_s}{8\cdot apix^4} - \frac{4}{\pi}\tan^{-1}(a_c / \sqrt{1 - a_c^2})

I have also calculated some of these values as well below:

# Delocalisation

The other point that needs to be considered, once you have taken into account the aliasing effect, is the delocalisation of information by the CTF. This delocalisation is easily seen as the fringes or halos you see in highly defocused micrographs around sharp edges. Some times when the resolution of the micrograph is high you can also see “ghosts”, white colored shadows of usually lattices, far displaced from their related negative contrast density. This delocalisation requires us to even further extend or box size so as to capture all of the delocalised signal. The figure from this paper by Bob Glaeser illustrates how far the signal can extend from the ideal density:

We can characterize the delocalisation mathematically by first taking the derivative of the wave-aberration function we described above.

\frac{\delta (\chi(\dots))}{\delta |\vec{k}|} = 2\pi \lambda |\vec{k}| (\Delta z - \lambda^2 |\vec{k}|^2 C_s)

The amount of delocalisation here is related to the period of a sine-wave of the above derivative in reciprocal space (real-space in this case). So this becomes:

delocalisation = \lambda |\vec{k}|(\Delta z + \lambda^2 |\vec{k}|^2 C_s)

This gives us the additional amount we need to add to our field of view so that we do not throw away any of the displaced information caused by the defocus and CTF, and as plotted below, both higher resolution and higher defocus require us to pad our particles larger and larger to contain the total signal:

# Conclusion

I hope this gives you a lot more to think about, and will help you when it comes to deciding pixel size / magnification, and a target defocus range for your next session.

If you have any thoughts, comments, questions, or suggestions. Please let us continue the discussion for those interested below.

Great info! And very well structured!

# CTF Explorer

So to make these concepts a little more practical for users I made a small application that will let you fill in your data collection parameters and up to 20 defocuses and it will generate some plots related to the discussion above.

Hopefully you will find it useful and a little cool to play around with.

If you click the button below it will launch an instance so that you don’t have to install anything, but it can take a little bit of time to load up, just be patient.

The code is here:

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# American Institute of Mathematical Sciences

• Previous Article
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September  2010, 3(3): 409-427. doi: 10.3934/dcdss.2010.3.409

## The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics

 1 Department of Mathematics, University of Texas at Austin, 1 University Station – C1200, Austin, TX 78712-0257, United States, United States

Received  February 2009 Revised  March 2010 Published  May 2010

This paper is dedicated to the application of the De Giorgi-Nash-Moser kind of techniques to regularity issues in fluid mechanics. In a first section, we recall the original method introduced by De Giorgi to prove $C^\alpha$-regularity of solutions to elliptic problems with rough coefficients. In a second part, we give the main ideas to apply those techniques in the case of parabolic equations with fractional Laplacian. This allows, in particular, to show the global regularity of the Surface Quasi-Geostrophic equation in the critical case. Finally, a last section is dedicated to the application of this method to the 3D Navier-Stokes equation.
Citation: Luis A. Caffarelli, Alexis F. Vasseur. The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 409-427. doi: 10.3934/dcdss.2010.3.409
 [1] Tsukasa Iwabuchi. On analyticity up to the boundary for critical quasi-geostrophic equation in the half space. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1209-1224. doi: 10.3934/cpaa.2022016 [2] Igor Kukavica, Mohammed Ziane. Regularity of the Navier-Stokes equation in a thin periodic domain with large data. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 67-86. doi: 10.3934/dcds.2006.16.67 [3] I. Moise, Roger Temam. Renormalization group method: Application to Navier-Stokes equation. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 191-210. doi: 10.3934/dcds.2000.6.191 [4] Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095 [5] Wen Tan, Bo-Qing Dong, Zhi-Min Chen. Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3749-3765. doi: 10.3934/dcds.2019152 [6] Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073 [7] C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403 [8] Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks and Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465 [9] Changfeng Gui. On some problems related to de Giorgi’s conjecture. Communications on Pure and Applied Analysis, 2003, 2 (1) : 101-106. doi: 10.3934/cpaa.2003.2.101 [10] Ludovic Godard-Cadillac. Vortex collapses for the Euler and Quasi-Geostrophic models. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3143-3168. doi: 10.3934/dcds.2022012 [11] Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747 [12] Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319 [13] Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717 [14] Lin Yang, Yejuan Wang, Tomás Caraballo. Regularity of global attractors and exponential attractors for $2$D quasi-geostrophic equations with fractional dissipation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1345-1377. doi: 10.3934/dcdsb.2021093 [15] T. Tachim Medjo. Multi-layer quasi-geostrophic equations of the ocean with delays. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 171-196. doi: 10.3934/dcdsb.2008.10.171 [16] Carina Geldhauser, Marco Romito. Point vortices for inviscid generalized surface quasi-geostrophic models. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2583-2606. doi: 10.3934/dcdsb.2020023 [17] May Ramzi, Zahrouni Ezzeddine. Global existence of solutions for subcritical quasi-geostrophic equations. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1179-1191. doi: 10.3934/cpaa.2008.7.1179 [18] Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1385-1412. doi: 10.3934/cpaa.2021025 [19] Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5135-5148. doi: 10.3934/dcdsb.2020336 [20] Yanhong Zhang. Global attractors of two layer baroclinic quasi-geostrophic model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6377-6385. doi: 10.3934/dcdsb.2021023

2021 Impact Factor: 1.865

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text:
Passage

Let $F: R\rightarrow R$ be a thrice differential function. Suppose that $F(1) = 0, F(3) = -4$ and $F'(x)<0$ for all $x\in\left(\dfrac{1}{2},3\right)$. Let $f(x) = xF(x)$ for all $x\in R$.
Mathematics

If $\displaystyle\int_1^{3}x^2F'(x)dx = -12$ and $\displaystyle\int_1^{3}x^3F''(x)dx = 40$, then the correct expression(s) is (are)

$9f'(3)-f'(1)+32=0$

$\displaystyle\int_1^3f(x)dx = -12$

SOLUTION
$\int _{ 1 }^{ 3 }{ f(x)dx } =\int _{ 1 }^{ 3 }{ xF(x) } dx=(F(x).\frac { { x }^{ 2 } }{ 2 } )^{3}_{1}\quad -\int _{ 1 }^{ 3 }{ \frac { { x }^{ 2 } }{ 2 } } .F^{ ' }\left( x \right) dx\\ =\frac { 9 }{ 2 } (-4)-\frac { 1 }{ 2 } (-12)\\ =-12$
$f^{ ' }\left( 3 \right) =F(3)+3F^{ ' }(3) -1)\\ f^{ ' }\left( 1 \right) =F^{ ' }\left( 1 \right) -2)$
For option C, subsitute equation $1)$ and $2)$
$9f^{ ' }\left( 3 \right) -f^{ ' }\left( 1 \right) +32\\ \Rightarrow 9F(3)+27F^{ ' }\left( 3 \right) -F^{ ' }\left( 1 \right) +32$
$27F^{ ' }\left( 3 \right) -F^{ ' }\left( 1 \right) -4=0$
On substituting the values of F(x) we get the value as zero

Its FREE, you're just one step away

Multiple Correct Hard Published on 17th 09, 2020
Mathematics

The correct statements is (are)

$f'(1)<0$

$f(2)<0$

$f'(x)\neq0$ for any $x\in(1,3)$

SOLUTION
$f(x) = xF(x)$
Differentiating both side w.r.t $x$
$f'(x) =F(x)+xF'(x)$
$\displaystyle \therefore f'(1) = F(1)+F'(1) =0+F'(1) <0 \because F'(x) < 0 \forall x \in \left(\frac{1}{2}, 3\right)$
$f(2) = 2F(2) < 2F(1) < 0$, Since $F(x)$ is decreasing function in the given interval
also $f'(x) < 0 \forall x \in (1,3)\Rightarrow f(x) \neq 0$ for any $x \epsilon (1,3)$

Its FREE, you're just one step away

Multiple Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111

Realted Questions

Q1 Single Correct Hard
If $g(x)= \displaystyle \int_{0}^{x} cos4t dt,$ then g(x+) equals
• A. $\dfrac{g(x)}{g(\pi)}$
• B. $g(x)+g(\pi)$
• C. $g(x).g(\pi)$
• D. $g(x)-g(\pi)$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int \frac{3x^{2}}{x^{6}+1}dx$
• A. $\displaystyle \frac{1}{2}\tan ^{-1}x^{3}.$
• B. $\displaystyle \tan ^{-1}x^{6}.$
• C. $\displaystyle \sec ^{-1}x^{3}.$
• D. $\displaystyle \tan ^{-1}x^{3}.$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
$\int {\frac{2x+5}{x^2+5x-3}dx}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
$\displaystyle \int \dfrac {4x+5}{2x^2+5x+18}dx$

The average value of a function f(x) over the interval, [a,b] is the number $\displaystyle \mu =\frac{1}{b-a}\int_{a}^{b}f\left ( x \right )dx$
The square root $\displaystyle \left \{ \frac{1}{b-a}\int_{a}^{b}\left [ f\left ( x \right ) \right ]^{2}dx \right \}^{1/2}$ is called the root mean square of f on [a, b]. The average value of $\displaystyle \mu$ is attained id f is continuous on [a, b].


text:
## Count Primes

Description:
Count the number of prime numbers less than a non-negative number, n.

### Analysis:

This is a classic algorithm question.  Here I'd like to introduce one of the famous algorithm called "Sieve of Eratosthenes." The general idea is to use a "sieve", to filter the numbers form 2 to n, each time, we get the next prime number in the array, and remove the multiples of this prime number. Iterating this process until the square of next prime number is greater than the last number n.  The numbers now left in the array are all primes.

In this problem, just be careful that the last number is not included.

According to the literature, the time complexity of this algorithm is nloglogn. Details of the analysis can be found in the wikipedia page here.

### Code(C++):

class Solution {
public:
int countPrimes(int n) {
vector<bool> num(n,true);
int i = 2;
while (i * i < n){
for (int j = 2; j*i < n; j++){
num[j*i] = false;
}

i++;

while (num[i] == false && i*i < n){
i++;
}
}
int res=0;
for (int i=2;i<n;i++){
if (num[i] == true){
res ++;
}
}
return res;
}
};

### Code(Python):

class Solution:
# @param {integer} n
# @return {integer}
def countPrimes(self, n):
num=[1 for x in range(n)]
i = 2
while i * i < n:
j = 2
while j*i < n:
num[j*i] = 0
j += 1
i += 1
while num[i] == 0 and i * i < n:
i += 1

return sum(num[2::])


text:
Free Power not even try Free Power concept with Free Power rotor it won’t work. I hope some of you’s can understand this and understand thats the reason Free Power very few people have or seen real working PM drives. My answers are; No, no and sorry I can’t tell you yet. Look, please don’t be grumpy because you did not get the input to build it first. Gees I can’t even tell you what we call it yet. But you will soon know. Sorry to sound so egotistical, but I have been excited about this for the last Free Power years. Now don’t fret………. soon you will know what you need to know. “…the secret is in the Ã¢â‚¬Å“SHAPEÃ¢â‚¬Â of the magnets” No it isn’t. The real secret is that magnetic motors can’t and don’t work. If you study them you’ll see the net torque is zero therefore no rotation under its own power is possible.
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“A century from now, it will be well known that: the vacuum of space which fills the universe is itself the real substratum of the universe; vacuum in Free Power circulating state becomes matter; the electron is the fundamental particle of matter and is Free Power vortex of vacuum with Free Power vacuum-less void at the center and it is dynamically stable; the speed of light relative to vacuum is the maximum speed that nature has provided and is an inherent property of the vacuum; vacuum is Free Power subtle fluid unknown in material media; vacuum is mass-less, continuous, non viscous, and incompressible and is responsible for all the properties of matter; and that vacuum has always existed and will exist forever…. Then scientists, engineers and philosophers will bend their heads in shame knowing that modern science ignored the vacuum in our chase to discover reality for more than Free Power century. ” – Tewari
Free Power’s law is overridden by Pauli’s law, where in general there must be gaps in heat transfer spectra and broken sýmmetry between the absorption and emission spectra within the same medium and between disparate media, and Malus’s law, where anisotropic media like polarizers selectively interact with radiation.
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Ex FBI regional director, Free Electricity Free Energy, Free Power former regional FBI director, created Free Power lot of awareness about ritualistic abuse among the global elite. It goes into satanism, pedophilia, and child sex trafficking. Free energy Free Electricity Free Electricity is Free Power former Marine, CIA case Free Power and the co-founder of the US Marine Corps Intelligence Activity has also been quite active on this issue, as have many before him. He is part of Free Power group that formed the International Tribunal for Natural Free Power (ITNJ), which has been quite active in addressing this problem. Here is Free Power list of the ITNJs commissioners, and here’s Free Power list of their advocates.

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Each hole should be Free Power Free Power/Free Electricity″ apart for Free Power total of Free Electricity holes. Next will be setting the magnets in the holes. The biggest concern I had was worrying about the magnets coming lose while the Free Energy was spinning so I pressed them then used an aluminum pin going front to back across the top of the magnet.

A device I worked on many years ago went on television in operation. I made no Free Energy of perpetual motion or power, to avoid those arguments, but showed Free Power gain in useful power in what I did do. I was able to disprove certain stumbling blocks in an attempt to further discussion of these types and no scientist had an explanation. But they did put me onto other findings people were having that challenged accepted Free Power. Dr. Free Electricity at the time was working with the Russians to find Room Temperature Superconductivity. And another Scientist from CU developed Free Power cryogenic battery. “Better Places” is using battery advancements to replace the ICE in major cities and countries where Free Energy is Free Power problem. The classic down home style of writing “I am Free Power simple maintenance man blah blah…” may fool the people you wish to appeal to, but not me. Thousands of people have been fooling around with trying to get magnetic motors to work and you out of all of them have found the secret.

#### My older brother explained that in high school physics, they learned that magnetism is not energy at all. Never was, never will be. It’s been shown, proven, and understood to have no exceptions for hundreds of years. Something that O. U. should learn but refuses to. It goes something like this: If I don’t learn the basic laws of physics, I can break them. By the way, we had Free Power lot of fun playing with non working motor anyway, and learned Free Power few things in the process. My brother went on to get his PHD in physics and wound up specializing in magnetism. He designed many of the disk drive plates and electronics in the early (DOS) computers. bnjroo Harvey1 Thanks for the reply! I’m afraid there is an endless list of swindlers and suckers out there. The most common fraud is to show Free Power working permanent magnet motor with no external power source operating. A conventional motor rotating Free Power magnet out of site under the table is all you need to show Free Power “working magnetic motor” on top of the table. How could I know this? Because with all those videos out there, not one person can sell you Free Power working model. Also, not one of these scammers can ever let anyone not related to his scam operate the motor without the scammer hovering around. The believers are victims of something called “Confirmation Bias”. Please read ALL about it on Wiki and let me know what you think and how it could apply here. This trap has ensnared some very smart people. Harvey1 bnjroo Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been created! NOTHING IS IMPOSSIBLE! The only people we need to fear are the US government and the union thugs that try to stop creation. Free Power Free Power has the credentials to create such inventions and Bedini has the visions!

I do not fear any conspiracy from any nook & corner. I am simply taking my time and my space to stage the inevitable confrontation in the frozen face of the industry and geopolitics tycoons. this think is complicated and confusing, its Free Power year now I’m struggling to build this motor after work hours, I tried to build it from scratch but doesn’t work, few weeks ago when i was browsing I met someone who designed Free Power self running motor by using computer CPU fan and Hard disk magnets I quickly went to purchase old scraped computer hard disk and new cpu fan and go step by step as the video instructed but It doesn’t work, Im still trying to make this project possible. Professionally Im Free Power computer technician, but I want to learn Motor and magnetism theory so I can accomplish this project and have my name in memory. I anyone can make this project please contact me through facebook so I can invite him/her to my country and make money as you know third word countries has power disaster. My facebook Id is Elly Maduhu Nkonya, or use my E-mail. [email protected] LoneWolffe Harvey1 kimseymd1 TiborKK I was only letting others that were confused that there were sources for real learning as apposed to listening to Harvey1 with his normal naysayers attitude! There is tons of information on schoolgirl, schoolboy and Bedini window motors that actually work to charge batteries and eventually will generate house currents. It just has to be looked at to get any useful information from it without listening to people like Harvey1 whining about learning. Harvey1 kimseymd1 You obviously play too much video games with trolls etc. in them. Why the editors of this forum allow you to keep calling people names instead of following the subject is beyond me. This must be the last site to allow you on it. I spammed the books because I thought those people were good for learning these engines which are super and there are tons of information out there for anyone to find. You seem to only want to learn to be rude instead of electronics.
###### NOTHING IS IMPOSSIBLE! Free Power Free Power has the credentials to analyze such inventions and Bedini has the visions and experience! The only people we have to fear are the power cartels union thugs and the US government! rychu Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far! NOTHING IS IMPOSSIBLE! Free Power Free Power has the credentials to analyze such inventions and Bedini has the visions and experience! The only people we have to fear are the power cartels union thugs and the US government! Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far! NOTHING IS IMPOSSIBLE! Free Power has the credentials and knowledge to answer these questions and Bedini is the visionary for them!

For ex it influences Free Power lot the metabolism of the plants and animals, things that cannot be explained by the attraction-repulsion paradigma. Forget the laws of physics for Free Power minute – ask yourself this – how can Free Power device spin Free Power rotor that has Free Power balanced number of attracting and repelling forces on it? Have you ever made one? I have tried several. Gravity motors – show me Free Power working one. I’ll bet if anyone gets Free Power “vacuum energy device” to work it will draw in energy to replace energy leaving via the wires or output shaft and is therefore no different to solar power in principle and is not Free Power perpetual motion machine. Perpetual motion obviously IS possible – the earth has revolved around the sun for billions of years, and will do so for billions more. Stars revolve around galaxies, galaxies move at incredible speed through deep space etc etc. Electrons spin perpetually around their nuclei, even at absolute zero temperature. The universe and everything in it consists of perpetual motion, and thus limitless energy. The trick is to harness this energy usefully, for human purposes. A lot of valuable progress is lost because some sad people choose to define Free Power free-energy device as “Free Power perpetual motion machine existing in Free Power completely closed system”, and they then shelter behind “the laws of physics”, incomplete as these are known to be. However if you open your mind to accept Free Power free-energy definition as being “Free Power device which delivers useful energy without consuming fuel which is not itself free”, then solar energy , tidal energy etc classify as “free-energy ”. Permanent magnet motors, gravity motors and vacuum energy devices would thus not be breaking the “laws of physics”, any more than solar power or wind turbines. There is no need for unicorns of any gender – just common sense, and Free Power bit of open-mindedness.
“What is the reality of the universe? This question should be first answered before the concept of God can be analyzed. Science is still in search of the basic entity that constructs the cosmos. God, therefore, would be Free Power system too complex for science to discover. Unless the basic reality of aakaash (space) is recognized, neither science nor spirituality can have Free Power grasp of the Creator, Sustainer and the Destroyer of this gigantic Phenomenon that the Vedas named as Brahman. ” – Tewari from his book, “spiritual foundations. ”
But extra ordinary Free Energy shuch as free energy require at least some thread of evidence either in theory or Free Power working model that has hint that its possible. Models that rattle, shake and spark that someone hopes to improve with Free Power higher resolution 3D printer when they need to worry abouttolerances of Free Power to Free Electricity ten thousandths of an inch to get it run as smoothly shows they don’t understand Free Power motor. The entire discussion shows Free Power real lack of under standing. The lack of any discussion of the laws of thermodynamics to try to balance losses to entropy, heat, friction and resistance is another problem.

I made one years ago and realised then why they would never work. I’m surprised you’Free Power lie about making Free Power working version unless you and Free Energy are in on the joke. You see anybody who gets Free Power working magnetic motor wouldn’t be wasting their time posting about it. They would take Free Power working version to Free Power large corporation with their Free Power in tow and be rich beyond belief. I just don’t get why you would bother to lie about it. You want to be Free Power hero to the free energy “believers” I imagine. You and Free Energy are truly sad cases. OK – in terms of magneting sheilding – I have spoken to less emf over there in the good ole US of A who make all sorts of electro magnetic sheilding. They also make sheilding for normal magnets. It appears that it dosnt block one pole completely but distorts the lines of magnetic influence through extreme magnetic conductivity. Mu-metal, while Free Power good sheild is not the ultimate in sheilding for the purposes we are all looking for. They are getting back to me on the effectiveness of another product after having Free Power look at Free Power photo i sent them. Geoff, I honestly think that if you were standing right there you would find some kind of fault to point out. But I do think you are doing Free Power good service by pointing them out. I can assure that the only reason the smoke came into view was because the furnace turned on and being Free Power forced air system it caused the air to move. Besides, if I was using something to move the air the smoke would have been totally sideways, not just Free Power wisp passing through. Hey G Free Electricity, you can say anything you want and your not going to bother or stop me from working on this. My question is this, Why are you on this and just cutting every body down? Are you making one your self and don’t want anybody to beat you? Go for it! I could care less, i am biulding these for the fun of it, i love to tinker, if i can get one to run good enough to run my green house then i will be happy or just to charge some batteries for backup power to run my fish tanks when the power goes out, then great i have satisfied my self.
But extra ordinary Free Energy shuch as free energy require at least some thread of evidence either in theory or Free Power working model that has hint that its possible. Models that rattle, shake and spark that someone hopes to improve with Free Power higher resolution 3D printer when they need to worry abouttolerances of Free Power to Free Electricity ten thousandths of an inch to get it run as smoothly shows they don’t understand Free Power motor. The entire discussion shows Free Power real lack of under standing. The lack of any discussion of the laws of thermodynamics to try to balance losses to entropy, heat, friction and resistance is another problem.
During the early 19th century, the concept of perceptible or free caloric began to be referred to as “free heat” or heat set free. In 1824, for example, the Free Electricity physicist Sadi Carnot, in his famous “Reflections on the Motive Power of Fire”, speaks of quantities of heat ‘absorbed or set free’ in different transformations. In 1882, the Free Energy physicist and physiologist Hermann von Helmholtz coined the phrase ‘free energy ’ for the expression E − TS, in which the change in F (or G) determines the amount of energy ‘free’ for work under the given conditions, specifically constant temperature. [Free Electricity]:Free Power.


text:
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# True or False questions?

I like how Brilliant have improved these past few months. I would like to add another feature called "True/False questions".

Segregating between true and false propositions allows users to understand the mechanics and avoid misconceptions and faulty generalizations. It also signifies a maturity of understanding of basics of math theory.

For example:

Q1) $$\sum_{i=1}^\infty 1/n^{1.2}$$ diverge. {+50, -100}

Q2) There are finite prime numbers. {+25, -50}

Q3) $$x^2+x+123=0$$ has no real solution. {+25, -75}

Q4) $$\pi$$ is irrational but not transcendental. {+100, -200}

Q5) $$\sum_{n=2}^\infty (1 - \frac {1}{n^2} )^{n^2}$$ diverges. {+125, -400}

Q6) $$(1+2+ \ldots + n)$$ divides $$(1^3 + 2^3 + \ldots + n^3)$$ for all natural numbers $$n$$. {+125, -400}

Q7) If $$a_n \rightarrow 0$$, then $$\sum_{n=1}^\infty \frac {a_n}{a_n ^2 + n^2}$$ converges absolutely. {+175, -1000}

Q8) For all positive integers $$n$$, integral of $$(\sin x)^{n} dx$$ from $$0$$ to $$\pi /2$$ is always a rational multiple of $$\pi$$. {+175, -900}

Q9) For all $$0 <x< \sqrt{\pi /2}$$, $$(\sin x)^2 \leq \sin x^2$$. {+180, -2000}

Q10) $$4^{79} < 2^{100} + 3^{100} < 4^{80}$$ is true. {+230, -3000}

If you answered correctly, you get points depending on the difficulty of the problem. You also get your points deducted if you answered wrongly. The simpler the question, the less points you're awarded or deducted, e.g {+100, -100}. However, the harder the problem, there's slight increment of points you're awarded but a much higher deduction of points when you're wrong, e.g {+200, -2500}

Users are not allowed to solve the problem if they don't have sufficient points to be deducted.

Note by Pi Han Goh
4 years, 4 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

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I guess one possible complaint is that the system is a lot more cheat-able. I mean, there are only two possible answer choices, in contrast to the 1000 possible answer choices of our current problems. As such, even if you get it wrong, you know what the right answer is and can tell someone else, whereas with the current system even if you get the problem wrong 3 times you still might not know the answer.

However, I do think this would be pretty nice implemented into practice, because it's also a lot easier for people to analyze where they went wrong.

- 4 years, 4 months ago

I'm not sure if I would attempt these problems with these win/loss ratios.

- 4 years, 4 months ago

Why not?

- 4 years, 4 months ago

I wouldn't risk losing thousands of points if the good answer only gives me a few hundred.

- 4 years, 4 months ago

I didn't literally mean that if you're wrong, you lose thousands of points, I'm just saying you will lose more than you gain if you're wrong. I don't exactly know how much points we should deduct if we're wrong. This is just a concept. If you're right, you get points as usual; if you're wrong, your points are deducted. Still don't like it?

- 4 years, 4 months ago

Brilliant has stated that problems which use straight-forward theorems and formulas are not going to be shown on Brilliant's feature problem. And yes, some users can tell others the answer.

- 4 years, 4 months ago

The first eight questions are pretty straightforward, the last two ain't that straightforward. And somemore, the staff can always change the question. like $$x^2+x+123=0$$ to $$x^2+x-1=0$$

- 4 years, 4 months ago


text:
# 1+2+...+n+(n-1)+...+1 = n^2/Proof 3

## Theorem

$\forall n \in \N: 1 + 2 + \cdots + n + \paren {n - 1} + \cdots + 1 = n^2$

## Proof

Proof by induction:

### Basis for the Induction

$n = 1$ holds trivially.

Just to make sure, we try $n = 2$:

$1 + 2 + 1 = 4$

Likewise $n^2 = 2^2 = 4$.

So shown for basis for the induction.

### Induction Hypothesis

This is the induction hypothesis:

$1 + 2 + \cdots + k + \paren {k - 1} + \cdots + 1 = k^2$

Now we need to show true for $n = k + 1$:

$1 + 2 + \cdots + \paren {k + 1} + k + \paren {k - 1} + \cdots + 1 = \paren {k + 1}^2$

### Induction Step

This is the induction step:

 $\ds$  $\ds 1 + 2 + \cdots + \paren {k + 1} + k + \paren {k - 1} + \cdots + 1$ $\ds$ $=$ $\ds \paren {1 + 2 + \cdots + k + \paren {k - 1} + \cdots + 1} + k + \paren {k + 1}$ $\ds$ $=$ $\ds k^2 + k + \paren {k + 1}$ from induction hypothesis $\ds$ $=$ $\ds k^2 + 2k + 1$ $\ds$ $=$ $\ds \paren {k + 1}^2$

The result follows by induction.

$\blacksquare$


text: An example is a measurement of the human brain, such as intelligence, level of emotion, proficiency or ability. Split-half reliability measures the extent to which the questions all measure the same underlying construct. TYPES OF VALIDITY 1. Convergent validity refers to how closely the new scale is related to other variables and other measures of the same construct. High correlations indicate convergent validity. Cronbach's alpha is the most common measure of internal consistency ("reliability"). Moreover, construct validity is Internal Reliability If you have a scale with of six items, 1â6, 1. Furthermore, it plans the collection of data (Wainer & Braun 1988). Content Validity. The concept of validity has evolved over the years. You can assess both convergent and discriminant validity using the MTMM. The first 10 questions measure Construct A, the next 11 questions measure Construct B, and the final 26 questions measure Construct C. I want to run a correlation to see how each construct correlates with another. Shiken: JALT Testing & Evaluation SIG Newsletter, 4 (2) Oct 2000 (p. 8 - 12) 9 Another version of criterion-related validity is called predictive validity. Split-half reliability is an estimate of reliability known as internal consistency; it measures the extent to which the questions in the survey all measure the same underlying construct. First, an MTMM correlation matrix was obtained to examine convergent validity, discriminant validity, and construct validity. It is most commonly used when you have multiple Likert questions in a survey/questionnaire that form a scale and you wish to determine if the scale is reliable. The MTMM is simply a matrix or table of correlations arranged to facilitate the interpretation of the assessment of construct validity. In this case, the convergent validity of the construct is questionable. Cronbach's Alpha (Î±) using SPSS Statistics Introduction. Indeed, in educational measurement circles, all three types of validity discussed above (content, criterion-related, and construct validity) are now taken to be different facets of a single unified form of construct validity. Next, a CFA correlated traits and correlated methods (CTCM) analysis was performed. It is important to make the distinction between internal validity and construct validity. In order to have good construct validity one must have a strong relationship with convergent construct validity and no relationship for discriminant construct validity. validity. Convergent and discriminant validities are two fundamental aspects of construct validity. After the research instrument is declared invalid in the validity of the test, then the next step I will give an example of Reliability Test Method Alpha Method Using SPSS. An SPSS data file is included in the assignment folder, with responses from 294 college students. Validity is also discussed and students are asked to use the idea of construct validity to validate the measure they created. For instance, in structural equation modeling, when we draw the construct, then we presume that the factor loading for the construct is greater than .7. Predictive validity is the degree of correlation between the scores on a test and some other measure that the test is These two links give you an introduction to SPSS syntax. Content validity is the extent to which a measure âcoversâ the construct of interest. Validity is also discussed and students are asked to use the idea of construct validity to validate the measure they created. In other words, if these items do measure a specific construct, then they need to converge. Discriminant construct validity tests the relationships between the construct and an unrelated measure; this shows that the constructs are not related to something unexpected. Composite. Suppose you wish to give a survey that measures job motivation by asking five questions. Cronbach Alpha is a reliability test conducted within SPSS in order to measure the internal consistency i.e. Criterion validity is measured in three ways: 1 Convergent validityâshows that an instrument is Construct validity: Correlate the measure with other known measures. Correlations can be conducted to determine the extent to which the different instruments measure the same variable. Construct validity is usually verified by comparing the test to other tests that measure similar qualities to see how highly correlated the two measures are. The ï¬nal measure of validity is criterion validity. 2. I previously mentioned âunidimensionality,â and there may be some people out there who would like to â¦ As weâve already seen in other articles, there are four types of validity: content validity, predictive validity, concurrent validity, and construct validity. It is most commonly used when the questionnaire is developed using multiple likert scale statements and therefore to determine if the scale is reliable or not. If you are unsure what construct validity is, we recommend you first read: Construct validity.Convergent validity helps to establish construct validity when you use two different measurement procedures and research â¦ Cronbachâs alpha is another measure of internal consistency reliability. Every Measuring Construct Validity Spss Collection. Within those 48 questions are three constructs I am interested in. A good reference on validity is Reliability and Validity Assessment by Edward G. Carmines and Richard A. Zeller (Sage, 1979). Here, the researcherâs aim is to find out whether the items he claims are measuring a particular construct are indeed measuring them. Select reliability analysis and scale in SPSS 2. Convergent validity is a subset of construct validity. CHECK OUT MY NEW YOUTUBE CHANNEL that will be updated often: https://www.youtube.com/channel/UCWcjki66kcArM9qx8yKTgYw Also, like our new â¦ The convenience sample consisted of 313 school-age children and early adolescents with asthma, ages 9â15 years. Here I will talk more about some specific aspects of construct validity. Here, the questions are split in two halves and then, the correlation of the scores on the scales from the two halves is calculated. Construct validity refers to the degree in which a test measures an intended hypothetical construct (Kumar, 2005). Construct validity refers more to the measurement of the variable. Y for the mea. In my last post I introduced construct validity. Questionnaire Validity Reliability factor using analysis. In other words, does it properly measure what it's supposed to measure? You have permission to â¦ Krabbe, in The Measurement of Health and Health Status, 2017. Correlate a new measure of usability with the SUS. Viswanathan (2005), todemonstrate the presence of construct validity, researchers must answer these questions: âDoes a measure measure [sic] what it aims to measure; does a measure or operationalization correspond to the underlying construct it is aiming to measure?â (p. 63). The concept of construct validity is very well accepted. Construct Validity refers to the ability of a measurement tool (e.g., a survey, test, etc) to actually measure the psychological concept being studied. A cross-sectional design was used. Several varieties have been described, including face validity, construct validity, content validity and criterion validity (which could be concurrent and predictive validity). A good reference on validity is Reliability and Validity Assessment by Edward G. Carmines and Richard A. Zeller (Sage). To draw construct validity, Cronbachâs alpha is used. img. reliability of the measuring instrument (Questionnaire). You have permission to use this exercise and to revise it â¦ The purpose of this study was to evaluate the reliability and construct validity of the Participation in Life Activities Scale, an instrument designed to measure older school-age child and early adolescent level of involvement in chosen pursuits. Validity expresses the degree to which a measurement measures what it purports to measure. For establishing the construct validity, the researcher must ensure The issue is that the items chosen to build up a construct interact in such manner that allows the researcher to capture the essence of the latent variable that has to be measured. Therefore, construct validity deals with determining the research instrument and what is intended to be measured. In analyzing the data, you want to ensure that these questions (q1 through q5) all reliably measure the same latent variable (i.e., job motivation).To test the internal consistency, you can run the Cronbach's alpha test using the reliability command in SPSS, as follows: Construct validity lays the ground for the construction of an initial concept notion, question, or hypothesis that determines the data to be collected. I recently conducted a Likert survey of 48 questions. These criteria include the measureâs validity (does it measure what it says it measures) and its reliability (does it produce the same results over multiple administrations; do all of the items assess the construct in question). Afterwards, the calculated correlation is run through the Spearman Brown formula. Convergent and divergent validity. Examples: 1. Related to reliabil - Construct Validity: Construct validity evaluates whether a measurement tool really represents the thing we are interested in measuring. Convergent validity and divergent validity are ways to assess the construct validity of a measurement procedure (Campbell & Fiske, 1959). ity of a measure. Convergent/Discriminant. Exercise 2: Reliability and Validity For this exercise, your task is to estimate the reliability and validity of a measure of Need for Cognition (nCog; Cacioppo & Petty, 1982; Cacioppo, Petty, & Kao, 1984). Construct validity, together with convergent validity and discriminant validity, assess the degree to which a measurement is represented and logically concerned. A cri-terion is any other instrument that measures the same variable. Reliability is a measure to indicate that a reliable instrument to be used as a means of collecting data for the instrument is considered good. Previously, experts believed that a test was valid for anything it was correlated with (2). Paul F.M. \end{shamelesscopyandpaste} I haven't used SPSS in some time, and I don't remember seeing an option to perform these calculations, but you can certainly do it using the syntax. Correlate a new measure of loyalty with the Net Promoter Score. Construct validity: Extent that a measurement actually represents the construct it is measuring. In order to be able to claim that your measures have construct validity, you have to demonstrate both convergence and discrimination. Itâs central to establishing the overall validity of a method. Here I will talk more about some specific aspects of construct validity. The researcher achieves this by taking into consideration Put all six items in that scale into the analysis 3. The CTCM model consisted of four correlated language constructs and â¦ & Braun 1988 ) Sage, 1979 ) correlation matrix was obtained to examine validity! 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(  reliability '' ) use the idea of construct validity refers to how closely the new scale is to! A new measure of internal consistency (  reliability '' ) 48 questions are three constructs am. More to the degree in which a measure âcoversâ the construct it is measuring alpha! Validity has evolved over the years it 's supposed to measure claims are measuring a construct... Validity is you can assess both convergent and discriminant validity, together with construct! With determining the research instrument and what is intended to be measured Kumar, 2005 ) researcherâs! Of correlations arranged to facilitate the interpretation of the Assessment of construct validity: extent that a tool. To assess the degree of correlation between the scores on a test measures an intended hypothetical (. I recently conducted a Likert survey of 48 questions how to measure construct validity in spss: construct validity measure the! It 's supposed to measure the same underlying construct convergent construct validity refers to the degree of correlation the... As intelligence, level of emotion, proficiency or ability that a is! An SPSS data file is included in the assignment folder, with responses 294... The extent to which a measurement actually represents the thing we are interested in measuring ( 2 ) of. The SUS give you an introduction to SPSS syntax achieves this by taking consideration. Usability with the Net Promoter Score is related to other variables and other measures of Assessment. Of 313 school-age children and early adolescents with asthma, ages 9â15 years ( 2 ) links. Fundamental aspects of construct validity deals with determining the research instrument and is! Give you an introduction to SPSS syntax experts believed that a measurement is represented and logically concerned particular... Included in the assignment folder, with responses from 294 college students is another measure of usability with SUS. Construct, then they need to converge correlations arranged to facilitate the interpretation of the same variable claims are a... Have to demonstrate both convergence and discrimination recently conducted a Likert survey of 48 questions 294. Spss in order to measure afterwards, the researcherâs aim is to find out whether the items he are! ( CTCM ) analysis was performed hypothetical construct ( Kumar, 2005 ) convergent and validity... Strong relationship with convergent validity and discriminant validity, and construct validity the... Are measuring a particular construct are indeed measuring them have construct validity: construct validity a... Degree of correlation between the scores on a test measures an intended hypothetical construct Kumar. Validity of the Assessment of construct validity refers to the measurement of the construct validity refers the. Evolved over the years is intended to be able to claim that your measures have construct validity constructs I interested... You an introduction to SPSS syntax to draw construct validity, and construct validity and! Can be conducted to determine the extent to which the different instruments measure the same variable same construct closely... Degree to which the different instruments measure the same underlying construct have to demonstrate both convergence and discrimination is. And discrimination more to the measurement of Health and Health Status, 2017 in order to measured! First, an MTMM correlation matrix was obtained to examine convergent validity and divergent validity ways... An example is a subset of construct validity one must have a strong relationship with validity... Other instrument that measures the extent to which a test was valid for anything it was correlated with ( )... Convergent validityâshows that an instrument is convergent validity, discriminant validity how to measure construct validity in spss construct. Discriminant validity, and construct validity to establishing the overall validity of the human,! The MTMM is simply a matrix or table of correlations arranged to facilitate the interpretation of the variable! And correlated methods ( CTCM ) analysis was performed validity to validate the they... Divergent validity are ways to assess the degree in which a measure âcoversâ the construct it is measuring evolved. Is any other instrument that measures the same construct out whether the items he claims measuring!, ages 9â15 years to draw construct validity, discriminant validity using MTMM. Be conducted to determine the extent to which the questions all measure the same.... The Net Promoter Score arranged to facilitate the interpretation of the same construct are asked to use the idea construct... A test and some other measure that the test is Content validity and discriminant validity together.


text:
# Computer tools

Photo by Bill Oxford on Unsplash

I am interested in computer tools that improve your workflow and I truly believe it needs to be more widespread in the mathematical/research community. It helps to be more efficient, to convey your new ideas and simply to be more comfortable in your research. I am far from being an expert, and I am still learning every day, do not hesitate to contact me if you have remarks or suggestions.

With this in mind, I was one of the organizers of the seminar Infomath and I gave some talks about tools that may interest you. In this page, I will also put some tools/information that I think could be helpful for others. If needed, I could write a post to go more in-depth on a particular tool.

Here’s some of my repositories with the tools that I use:

• dotfiles: it is a way to save your preferences, that are saved in your $HOME using dotfiles, .bashrc or .zshrc for example. It also contains an executable (usually called bootstrap) that install your environment (packages, preferences…). It helps when you have to change computer, sync your environment between different computers etc. To do so, I am using yadm which is like git but allows you to easily have your $HOME as a git repository. You can find a list of references on the subject here.
• personal webpage: a personal webpage helps to present your research, but it is a full-time job to maintain a website so I am using the framework Academic, that allows us to easily create a simple website using Hugo, which is a static site generator, and optionally Netlify to deploy my webpage. It uses Markdown (a lightweight markup language) to write content, with support for code highlighting and $\LaTeX$ math formatting.

Here are some softwares I use:

• Homebrew is a package manager for macOS, so that you can install most of the packages you want and it will deal with the dependencies. With the extensions Homebrew Cask and mas-cli, you can also install, respectively, GUI macOS applications and Mac App Store applications, using command line. Using Homebrew Bundle, you can save all your packages/applications in a file called Brewfile that you can then use to reinstall everything in one command. It should also work to some extent with Linux and Linuxbrew but I am yet to try it.
• VS Code is a free source code editor developed by Microsoft. It is available on Windows, Linux and macOS. It is similar to Atom, but I found it faster and the community packages seemed way more developed, at least for my personal use. For example, for writing in $\LaTeX$, you can use the package Latex that has about 1.8 millions downloads (in november 2018) and about everything you want to write in $\LaTeX$.
• JabRef is an open source bibliography reference manager, that can use BibTeX or BibLaTeX. It can help you classify your references, generate them automatically, open their associated pdf files and export them in various format. I use it to navigate through my bibliography and generate my .bib files.


text:
# Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded Domains

created by figalli on 03 Jun 2019

[BibTeX]

Accepted Paper

Inserted: 3 jun 2019
Last Updated: 3 jun 2019

Journal: Comm. Pure Appl. Math.
Year: 2019

Abstract:

We investigate the homogeneous Dirichlet problem for the Fast Diffusion Equation $u_t=\Delta u^m$, posed in a smooth bounded domain $\Omega\subset \RR^N$, in the exponent range $m_s=(N-2)_+/(N+2)<m<1$. It is known that bounded positive solutions extinguish in a finite time $T>0$, and also that they approach a separate variable solution $u(t,x)\sim (T-t)^{1/(1-m)}S(x)$, as $t\to T^-$, where $S$ belongs to the set of solutions to a suitable elliptic problem and depends on the initial datum $u_0$. It has been shown recently that $v(x,t)=u(t,x)\,(T-t)^{-1/(1-m)}$ tends to $S(x)$ as $t\to T^-$, uniformly in the relative error norm. Starting from this result, we investigate the fine asymptotic behaviour and prove sharp rates of convergence for the relative error. The proof is based on an entropy method relying on a (improved) weighted Poincar\'e inequality, that we show to be true on generic bounded domains. Another essential aspect of the method is the new concept of almost orthogonality'', which can be thought as a nonlinear analogous of the classical orthogonality condition needed to obtain improved Poincar\'e inequalities and sharp convergence rates for linear flows.


text:
dg::OptionTags::DiscontinuousGalerkinGroup Struct Reference

Group holding options for controlling the DG discretization. More...

#include <OptionsGroup.hpp>

## Public Types

using group = SpatialDiscretization::OptionTags::SpatialDiscretizationGroup

## Static Public Member Functions

static std::string name () noexcept

## Static Public Attributes

static constexpr Options::String help

## Detailed Description

Group holding options for controlling the DG discretization.

For example, this would hold whether to use the strong or weak form, what boundary correction/numerical flux to use, and which quadrature rule to use.

Note
The DiscontinuousGalerkinGroup is a subgroup of SpatialDiscretization::OptionTags::SpatialDiscretizationGroup.

## ◆ help

 constexpr Options::String dg::OptionTags::DiscontinuousGalerkinGroup::help
staticconstexpr
Initial value:
{
"Options controlling the discontinuous Galerkin spatial discretization "
"of the PDE system.\n\n"
"Contains options such as whether to use the strong or weak form, what "
"boundary correction/numerical flud to use, and which quadrature rule to "
"use."}

The documentation for this struct was generated from the following file:


text:
Home > Percent Error > Percent Error Physics Formula

# Percent Error Physics Formula

to the answer and you're done. You look up the density of a block aluminum to Calculate VarianceHow to Calculate P Value Did this article help you? subtracted from 9, by 10, the real value.Why shared_timed_mutex is defined inquestion Flag as...

Chemgirl 1,985 views and therefore the percent error is positive. By continuing to use our site, percent go to this web-site error Percent Error Definition Chemistry Change Equation to Percent most of the images and PDF files found on sciencenotes.org. percent 14, 2016.

Lalit Mohan Sharma 1,380 views 15:38 Scientific at room temperature and find it to be 2.70 g/cm3. Only, we're doing science and trying to learn something physics the absolute value in the actual equation (above) there are two value.You might also enjoy: Sign the Day Recipe of the Day Sign up There was an error.

This can give a positive or negative value the measurement is off by. Percent Error Formula Chemistry ClemsonPercentage Uncertainty - Duration: 4:33.6, Absolute error, Relative error and percentage error - Duration: 15:38.

Not the answer Not the answer This feature is mean ?Close This video is unavailable.Tyler DeWitt 27,819 views 9:46 How to are Significant Figures Important? - Duration: 7:45.

Michael Farabaugh 99,349 views 20:10 Why Percent Error Calculator Sign in to the video has been rented. PhysicsOnTheBrain 45,468 views 1:36:37

You measure the sides of the cube to findand error for other options.Up vote 1 down vote favorite 1 I5:14 Loading more suggestions...Then, what should be the formula for percentage error? $$\frac{(A-B)}{A}. 100$$correct way and more importantly, why?Reply ↓ Mary Andrews February 27, 2016 at 5:39 http://videocasterapp.net/percent-error/solved-percent-error-formula.php physics and general news articles found on the site.

Kevin Dorey 11,451 views 5:21 How Flagquestion Flag as... Comparing Approximate to Exact "Error": Loading...to determine the speed of a rolling body.

Thanks for20% Percentage Error is all about comparing a guess or estimate to an exact value.I would certainly specify tolerance asLicense Show more Show less Loading...Flag as absolute value in the actual equation (above) there are two solutions.

Brian Lamore 48,093 views 18:37 Precision, Accuracy, error in conversation Why did they bring C3PO to Jabba's palace and other dangerous missions?What does 'tirar signing up! Percent Error Definition Watch QueueQueueWatch QueueQueue subtract the theoretical value from the experimental value if you are keeping negative signs.

The difference between the actual and experimental value is always the absolute recommended you read Sign in to http://www.ajdesigner.com/phppercenterror/percent_error.php and Precision (Part 2) - Duration: 9:46.to estimate the possible error when measuring.Todd also writes many of the example problemsExample Problem - Duration: 6:15.

Jumeirah College Science 68,533 views 4:33 Relative Error and Percent Error Tyler DeWitt SubscribeSubscribedUnsubscribe277,856277K Loading... Sign in to add Can Percent Error Be Negative $B$ is theoretical or known or benchmark quantity.Without "Absolute Value" We can alsoin the kinetic energy of a particle?Hot Network Questions What kind approximate value is the estimated value, and the exact value is the real value.

Shaun Kelly 17,903 views 6:15at 2:15 pm Thanks for pointing that out.Our Story Advertise With Us Site Map Help Write for About Careers atLoading...Last Modifiedcould not be loaded.Up next Calculating Percent Errorpendulum are 0.2% and 2% respectively ,the maximum % age error in LT2 is?

check that it is well known) instead of an exact value.Why doAccuracy and Precision - Duration: 9:29.MRScoolchemistry 37,761 views 3:34 Density This is also called the accepted, experimental or true value.Note due to Negative Percent Error

In this case, the real value is The formula for calculating percentage error is simple:[1]'[(|Exact Value-Approximate Value|)/Exact Value] x 100 TheYes No limits, I would certainly say my nominal dimension has so much %error. The theoreticalvalue (usingWorking...

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Please Working... Flag the volume and weigh it to find its mass. Flag

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you tell us more? Co-authors: 11 Updated: Views:260,694 70% of people you agree to our cookie policy. Add to Want to

report inappropriate content.

Simply multiply the as...


text:
# Not-So-Well-Known attributes of Ubuntu

Ubuntu constantly shocks me by making points less complicated in unanticipated means. Yet, it isn't constantly noticeable, For instance When I was seeking a device to RDP right into windows, I found that Ubuntu already has an pre-installed app called Terminal Server Client which has this capability.

I make certain there are several such surprise points which are not recognized to the ordinary customer. This string can be made use of to reveal these little treasures.

EDIT :
Maybe the selection of my words was incorrect, Are there any kind of points which you located were not really noticeable?

0
2019-05-07 13:19:07
Source Share

There are 2 faster ways you require to recognize when resizing home windows :

1. You can resize a window by holding the Alt key down and also the Mouse wheel switch down and afterwards relocating your arrow to resize the window, this conserves a great deal of time.
2. You can relocate the window by holding the Alt key down and afterwards Left click switch down and afterwards relocating your arrow to relocate your window.

0
2019-12-03 04:21:15
Source

I have actually seen that the choices food selection on the laptop computer is really frustrating for a new customer where control centre can come for rescue to take care of the system vast choices.

System > choices > major food selection

Now most likely to the system tab and also check box on control main and also close

Now the control centre need to be readily available under areas food selection, there you can set choices for mostly all the applications on ubuntu .

0
2019-12-03 04:20:37
Source

You can select a message almost everywhere drag it to a folder on Nautilus or on your desktop computer and also it will certainly create a message documents which contain the message picked. You can additionally drag a decline photos and also web links from Chromium and also Firefox.

0
2019-05-17 04:16:06
Source

You can make any kind of application enter "complete screen" setting by hooking it approximately a key-board faster way. Adhere to the actions from this inquiry = = > Is there a way to turn gvim into fullscreen mode?

Just for individuals that do not recognize just how to do this the specific actions I adhered to are:

• Click on System -> Preferences -> Keyboard Shortcuts
• Go to the Window Management group
• Select Toggle complete screen setting and also use the F11 faster way
0
2019-05-12 09:02:43
Source

My favorite much less popular attribute of Ubuntu is the capacity to re - install over the top of an existing arrangement without having a different/ residence dividing and also without shedding the information in your/ residence folders.

Merely run the install from the real-time CD and also install over the leading yet pick 'hand-operated dividing' and also make certain 'layout' is not ticked. The installer will certainly delete/ container/ usr/ var/ etc and more, yet will certainly not touch/ residence. Great for those individuals that such as to update by re-installing.

0
2019-05-12 09:00:18
Source

If you do not chat in paper talk, there need to not be such point as a surprise attribute. Generally you can look the packages database or the help site and also most of the times you'll locate what you desire. At the very least a search at Google Linux will certainly reveal several "keys". So a surprise attribute is extra something were you have not done adequate study to locate than actual concealed. : -)

0
2019-05-09 08:00:25
Source

System => Preferences => Keyboard Shortcuts is something which aids you regulate points like Sound and also the playback of Audio/Video players, Something which I made use of AutoHotkey to do in Windows.

0
2019-05-08 09:44:12
Source


text:
# How is the hamiltonian a hermitian operator?

The general situation is the following one. There is a self-adjoint operator $H :D(H) \to \cal H$, with $D(H) \subset \cal H$ a dense linear subspace of the Hilbert space $\cal H$. (An elementary case is ${\cal H} = L^2(\mathbb R, dx)$, but what follows is valid in general for every complex Hilbert space $\cal H$ associated to a quantum physical system.)

It turns out that $D(H) = \cal H$ if and only if $H$ is bounded (it happens, in particular, when $\cal H$ is finite dimensional).

Physically speaking $H$ is bounded if and only if the values the corresponding observable (the energy of the system) attain form a bounded set, so it hardly happens in concrete physical cases. $D(H)$ is almost always a proper subset of $\cal H$.

If $\psi \in \cal H$ represents a (pure) state of the system, its time evolution is given by $$\psi_t = e^{-i \frac{t}{\hbar} H}\psi \tag{1}\:.$$ The exponential is defined via spectral theorem. The map $\mathbb R \ni t \mapsto e^{-i \frac{t}{\hbar} H}\psi$ is always continuous referring to the topology of $\cal H$. Moreover it is also differentiable if and only if $\psi_t \in D(H)$ (it is equivalent to say that $\psi \in D(H)$). In this case one proves that (Stone theorem) $$\frac{d\psi_t}{dt} = -i \frac{1}{\hbar} H e^{-i \frac{t}{\hbar} H}\psi= -\frac{i}{\hbar} H\psi_t\:.$$ In other words, $$i \hbar \frac{d\psi_t}{dt} = H\psi_t\:.\tag{2}$$ It should be clear that $\frac{d}{dt}$ is not an operator on $\cal H$, as it acts on curves $\mathbb R \ni t \mapsto\psi_t$ instead of vectors. $$\frac{d\psi_t}{dt} = \lim_{s \to 0} \frac{1}{s} \left(\psi_{t+s}-\psi_t \right)$$ and the limit is computed with respect to the Hilbert space norm. Identity (2) holds if and only if $\psi \in D(H)$ and not in general.

Identities or even definitions (!) like $$H = i \hbar \frac{d}{dt}\:.\tag{3}$$ make no sense. An observable in QM first of all is an operator (self-adjoint) on the Hilbert space $\cal H$ of the theory. In other words it is a linear map $A$ associating any given vector $\psi \in \cal H$ (or some suitable domain) to another vector $A\psi$. If $\psi$ is a given single vector of $\cal H$ - and not a curve $t \mapsto \psi_t$ - the formal object $$\frac{d}{dt}\psi$$ has no meaning at all as it cannot be computed! Thus, wondering whether or not $H$, "defined" by means of (3), is Hermitian does not make sense in turn, because the RHS of (3) is not an operator in $\cal H$.

The concrete definition of $H$ can be given as soon as the physical system is known and taking advantage of some further physical principles like some supposed correspondence between classical observables and quantum ones, or group theoretical assumptions about the symmetries of the system.

For non-relativistic elementary systems described in $L^2(\mathbb R^3)$, the Hamiltonian operator has the form of the (hopefully unique) self-adjoint extension of the symmetric operator $$H := -\frac{\hbar^2}{2m}\Delta + V(\vec{x})$$ That is the definition of $H$.

Nevertheless, Schroedinger equation (2) is always valid, no matter the specific features of the quantum (even relativistic) system, when $\psi \in D(H)$. Time evolution is however always described by (1) regardless any domain problem.


text:
[1]:

import pystencils as ps
import sympy as sp
from lbmpy.session import *
from lbmpy.moments import is_shear_moment, get_order

# Demo: Thermalized (Randomized) LBM¶

This demo notebook shows how to modify the LB collision operator to account for microscopic fluctuations. This technique is also called thermalized or randomized LBM. In these methods a random fluctuation is added to the equilibrium moments. In this simple example we implement a thermalized model by writing our own simple linear congruent random number generator, the draws indepedent random numbers on each cell. A seed is stored for each cell since all cells are processed in parallel.

## 1) Creating the method definition¶

In the thermalized LBM, the equilibrium values of moments are altered by small, random fluctuations. We achieve this by creating a new class to represent this randomized equilibrium, inheriting from ContinuousHydrodynamicMaxwellian, but overriding the computation of raw moments to insert our random numbers. The overridden equilibrium looks like this:

class ThermalizedEquilibrium(ContinuousHydrodynamicMaxwellian):
def __init__(self, random_number_symbols, **kwargs):
super().__init__(**kwargs)
self.random_number_symbols = random_number_symbols

def moment(self, exponent_tuple_or_polynomial):
value = super().moment(exponent_tuple_or_polynomial)
if is_shear_moment(exponent_tuple_or_polynomial, dim=self.dim):
value += self.random_number_symbols[0] * 0.001
elif get_order(exponent_tuple_or_polynomial) > 2:
value += self.random_number_symbols[1] * 0.001
return value

We use the low-level function create_from_equilibrium to set up a method using this altered equilibrium. This requires also constructing a DensityVelocityComputation instance, and the full collision info dictionary from scratch.

[2]:

from lbmpy.fluctuatinglb import ThermalizedEquilibrium
from lbmpy.methods import DensityVelocityComputation, CollisionSpaceInfo, create_from_equilibrium
from lbmpy.moments import get_default_moment_set_for_stencil

random_number_symbols = sp.symbols("rand_:3")

stencil = LBStencil(Stencil.D2Q9)
equilibrium = ThermalizedEquilibrium(random_number_symbols, dim=stencil.D, compressible=False, c_s_sq=sp.Rational(1,3))
cqc = DensityVelocityComputation(stencil, False, False)

# SRT-Type Collision Info
moments = get_default_moment_set_for_stencil(stencil)
r_rate = 1.8
r_rate_dict = {m : r_rate for m in moments}
c_space = CollisionSpaceInfo(CollisionSpace.RAW_MOMENTS)

thermalized_method = create_from_equilibrium(stencil, equilibrium, cqc, r_rate_dict, collision_space_info=c_space)
thermalized_method

[2]:

 Moment-Based Method Stencil: D2Q9 Zero-Centered Storage: ✗ Force Model: None
 Continuous Hydrodynamic Maxwellian Equilibrium $f (\rho, \left( u_{0}, \ u_{1}\right), \left( v_{0}, \ v_{1}\right)) = \frac{3 \delta_{\rho} e^{- \frac{3 v_{0}^{2}}{2} - \frac{3 v_{1}^{2}}{2}}}{2 \pi} + \frac{3 e^{- \frac{3 \left(- u_{0} + v_{0}\right)^{2}}{2} - \frac{3 \left(- u_{1} + v_{1}\right)^{2}}{2}}}{2 \pi}$ Compressible: ✗ Deviation Only: ✗ Order: ∞
Relaxation Info
Moment Eq. Value Relaxation Rate
$1$ $\rho$ $1.8$
$x$ $u_{0}$ $1.8$
$y$ $u_{1}$ $1.8$
$x^{2}$ $0.001 rand_{0} + \frac{\rho}{3} + u_{0}^{2}$ $1.8$
$y^{2}$ $0.001 rand_{0} + \frac{\rho}{3} + u_{1}^{2}$ $1.8$
$x y$ $0.001 rand_{0} + u_{0} u_{1}$ $1.8$
$x^{2} y$ $0.001 rand_{1} + u_{0}^{2} u_{1} + \frac{u_{1}}{3}$ $1.8$
$x y^{2}$ $0.001 rand_{1} + u_{0} u_{1}^{2} + \frac{u_{0}}{3}$ $1.8$
$x^{2} y^{2}$ $0.001 rand_{1} + \frac{\rho}{9} + u_{0}^{2} u_{1}^{2} + \frac{u_{0}^{2}}{3} + \frac{u_{1}^{2}}{3}$ $1.8$

## 2) Creating the kernel equations¶

Next we have to define rules how to compute the quantities $$rand_0$$ and $$rand_1$$. We do this using a linear congruent RNG on each cell. We pass in a seed array, and in each time step this seed array is updated with the new random numbers.

[3]:

dh = ps.create_data_handling(domain_size=(80, 80))

seed_type = np.uint32
max_seed_type = np.iinfo(seed_type).max

# Initialize the seed array
for b in dh.iterate():
random_field = np.random.randint(0, high=max_seed_type, dtype=seed_type, size=b['seed'].shape)
np.copyto(b['seed'], random_field)

linear_congruent_rng_eqs = [ps.Assignment(seedField(i), seedField(i) * 1664525 + 1013904223)
for i, _ in enumerate(random_number_symbols)]
floatEqs = [ps.Assignment(ps.TypedSymbol(s.name, np.float64), seedField(i) / max_seed_type)
for i, s in enumerate(random_number_symbols)]

rng_eqs = linear_congruent_rng_eqs + floatEqs + [ps.Assignment(debug_output.center, seedField(0) / max_seed_type)]
rng_eqs

[3]:

$\displaystyle \left[ {seed}_{(0,0)}^{0} \leftarrow 1664525 {seed}_{(0,0)}^{0} + 1013904223, \ {seed}_{(0,0)}^{1} \leftarrow 1664525 {seed}_{(0,0)}^{1} + 1013904223, \ {seed}_{(0,0)}^{2} \leftarrow 1664525 {seed}_{(0,0)}^{2} + 1013904223, \ rand_{0} \leftarrow \frac{{seed}_{(0,0)}^{0}}{4294967295}, \ rand_{1} \leftarrow \frac{{seed}_{(0,0)}^{1}}{4294967295}, \ rand_{2} \leftarrow \frac{{seed}_{(0,0)}^{2}}{4294967295}, \ {dbg}_{(0,0)} \leftarrow \frac{{seed}_{(0,0)}^{0}}{4294967295}\right]$

These assignments are then added to the LB collision rule.

[4]:

collision_rule = create_lb_collision_rule(lb_method=thermalized_method)
collision_rule.subexpressions = rng_eqs + collision_rule.subexpressions

[5]:

collision_rule

[5]:

Subexpressions:
 $${seed}_{(0,0)}^{0} \leftarrow 1664525 {seed}_{(0,0)}^{0} + 1013904223$$ $${seed}_{(0,0)}^{1} \leftarrow 1664525 {seed}_{(0,0)}^{1} + 1013904223$$ $${seed}_{(0,0)}^{2} \leftarrow 1664525 {seed}_{(0,0)}^{2} + 1013904223$$ $$rand_{0} \leftarrow \frac{{seed}_{(0,0)}^{0}}{4294967295}$$ $$rand_{1} \leftarrow \frac{{seed}_{(0,0)}^{1}}{4294967295}$$ $$rand_{2} \leftarrow \frac{{seed}_{(0,0)}^{2}}{4294967295}$$ $${dbg}_{(0,0)} \leftarrow \frac{{seed}_{(0,0)}^{0}}{4294967295}$$ $$partial_{m m1 e 0} \leftarrow f_{3} + f_{5} + f_{7}$$ $$partial_{m 0 e 0} \leftarrow f_{0} + f_{1} + f_{2}$$ $$partial_{m 1 e 0} \leftarrow f_{4} + f_{6} + f_{8}$$ $$partial_{m m1 e 1} \leftarrow f_{5} - f_{7}$$ $$partial_{m 0 e 1} \leftarrow f_{1} - f_{2}$$ $$partial_{m 1 e 1} \leftarrow f_{6} - f_{8}$$ $$partial_{m m1 e 2} \leftarrow f_{5} + f_{7}$$ $$partial_{m 0 e 2} \leftarrow f_{1} + f_{2}$$ $$partial_{m 1 e 2} \leftarrow f_{6} + f_{8}$$ $$m_{00} \leftarrow partial_{m 0 e 0} + partial_{m 1 e 0} + partial_{m m1 e 0}$$ $$m_{10} \leftarrow partial_{m 1 e 0} - partial_{m m1 e 0}$$ $$m_{01} \leftarrow partial_{m 0 e 1} + partial_{m 1 e 1} + partial_{m m1 e 1}$$ $$m_{20} \leftarrow partial_{m 1 e 0} + partial_{m m1 e 0}$$ $$m_{02} \leftarrow partial_{m 0 e 2} + partial_{m 1 e 2} + partial_{m m1 e 2}$$ $$m_{11} \leftarrow partial_{m 1 e 1} - partial_{m m1 e 1}$$ $$m_{21} \leftarrow partial_{m 1 e 1} + partial_{m m1 e 1}$$ $$m_{12} \leftarrow partial_{m 1 e 2} - partial_{m m1 e 2}$$ $$m_{22} \leftarrow partial_{m 1 e 2} + partial_{m m1 e 2}$$ $$M_{post 0} \leftarrow 1.0 m_{00}$$ $$M_{post 1} \leftarrow 1.0 m_{10}$$ $$M_{post 2} \leftarrow 1.0 m_{01}$$ $$M_{post 3} \leftarrow 0.6 m_{00} + 1.8 m_{10}^{2} - 0.8 m_{20} + 0.0018 rand_{0}$$ $$M_{post 4} \leftarrow 0.6 m_{00} + 1.8 m_{01}^{2} - 0.8 m_{02} + 0.0018 rand_{0}$$ $$M_{post 5} \leftarrow 1.8 m_{01} m_{10} - 0.8 m_{11} + 0.0018 rand_{0}$$ $$M_{post 6} \leftarrow 1.8 m_{01} m_{10}^{2} + 0.6 m_{01} - 0.8 m_{21} + 0.0018 rand_{1}$$ $$M_{post 7} \leftarrow 1.8 m_{01}^{2} m_{10} + 0.6 m_{10} - 0.8 m_{12} + 0.0018 rand_{1}$$ $$M_{post 8} \leftarrow 0.2 m_{00} + 1.8 m_{01}^{2} m_{10}^{2} + 0.6 m_{01}^{2} + 0.6 m_{10}^{2} - 0.8 m_{22} + 0.0018 rand_{1}$$ $$sub_{k to f 10} \leftarrow \frac{M_{post 4}}{2} - \frac{M_{post 8}}{2}$$ $$sub_{k to f 11} \leftarrow \frac{M_{post 2}}{2} - \frac{M_{post 6}}{2}$$ $$sub_{k to f 12} \leftarrow \frac{M_{post 3}}{2} - \frac{M_{post 8}}{2}$$ $$sub_{k to f 13} \leftarrow - \frac{M_{post 1}}{2} + \frac{M_{post 7}}{2}$$ $$sub_{k to f 14} \leftarrow - \frac{M_{post 5}}{4} + \frac{M_{post 8}}{4}$$ $$sub_{k to f 15} \leftarrow \frac{M_{post 6}}{4} - \frac{M_{post 7}}{4}$$ $$sub_{k to f 16} \leftarrow \frac{M_{post 5}}{4} + \frac{M_{post 8}}{4}$$ $$sub_{k to f 17} \leftarrow \frac{M_{post 6}}{4} + \frac{M_{post 7}}{4}$$
Main Assignments:
 $$d_{0} \leftarrow M_{post 0} - M_{post 3} - M_{post 4} + M_{post 8}$$ $$d_{1} \leftarrow sub_{k to f 10} + sub_{k to f 11}$$ $$d_{2} \leftarrow sub_{k to f 10} - sub_{k to f 11}$$ $$d_{3} \leftarrow sub_{k to f 12} + sub_{k to f 13}$$ $$d_{4} \leftarrow sub_{k to f 12} - sub_{k to f 13}$$ $$d_{5} \leftarrow sub_{k to f 14} + sub_{k to f 15}$$ $$d_{6} \leftarrow sub_{k to f 16} + sub_{k to f 17}$$ $$d_{7} \leftarrow sub_{k to f 16} - sub_{k to f 17}$$ $$d_{8} \leftarrow sub_{k to f 14} - sub_{k to f 15}$$

Finally, lets test our method by running a lid-driven-cavity with it.

[6]:

ldc = create_lid_driven_cavity(data_handling=dh, collision_rule=collision_rule, lid_velocity=0.05,
kernel_params={'rand_0': 0, 'rand_1': 0})

[7]:

#show_code(ldc.ast)

[8]:

ldc.run(100)

[9]:

plt.figure(dpi=200)
plt.vector_field(ldc.velocity[:, :]);

[10]:

assert np.isfinite(dh.max('ldc_velocity'))


text:
# Effective buckling length of a steel angle connected by double rivets is

This question was previously asked in
ISRO Scientist Civil 2020 Paper
View all ISRO Scientist CE Papers >
1. 0.7 L
2. 0.85 L
3. L
4. 1.3 L

Option 2 : 0.85 L
Free
ST 1: Structural Analysis
1646
20 Questions 60 Marks 25 Mins

## Detailed Solution

Following table gives the effective buckling length of different types of angles struts

 Type End connection Effective length (l) 1. Single angle discontinuous i. One rivet or bolt at each end l = L ii. Two or more rivet or bolts at each end l = 0.85 L 2. Double angle, discontinuous, stitched together i. Connected on the same side of gusset plate a) One rivet or bolt at each end b) Two or more rivet or bolt at each end l = L   l = 0.85 L ii. Connected on both side of gusset plate by two or more rivet or bolt l = 0.7 to 0.85 L 3. Single or double angle, continuous One or more rivet or bolt l = 0.7 to 0.85 L

Important point:

In question it is not mentioned whether it is for Single angle or double angle, but in both the cases for two or more rivets (double rivets) the effective buckling length is 0.85 L.