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https://www.usgs.gov/media/images/a-closer-look-one-steam-sources-crack-which-st
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# A closer look at one of the steam sources. The crack from which st...
## Detailed Description
A closer look at one of the steam sources. The crack from which steam is issuing is not visible through the thick vegetation.
## Details
Image Dimensions: 5184 x 3456
Date Taken:
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2021-09-25 13:11:03
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http://codac.io/manual/07-graphics/02-figtube.html
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# Display tubes and trajectories w.r.t. time¶
This page presents a tool for displaying tubes and trajectories objects with respect to time.
## The class VIBesFigTube¶
VIBesFigTube creates a figure for displaying temporal objects such as trajectories and tubes. The x-axis of this 2d figure is time.
This class inherits from VIBesFig: the methods provided in VIBesFig can be used here.
To create and show a $$600\times300$$ figure located at $$100,100$$:
fig = VIBesFigTube("Tube")
fig.set_properties(100, 100, 600, 300)
# add graphical items here ...
show()
VIBesFigTube fig("Tube");
fig.set_properties(100, 100, 600, 300);
// add graphical items here ...
show();
To add temporal objects (before the method .show()), use the methods .add_tube() or .add_trajectory() with three arguments:
• the first argument refers to the object to be shown
• the second one is the name of the object in the view
• (optional) the third argument defines the color of the object (edge_color[fill_color])
fig.add_tube(tube_x, "x", "blue[yellow]")
# Where tube_x and traj_x are respectively Tube and Trajectory objects
fig.add_tube(&tube_x, "x", "blue[yellow]");
// Where tube_x and traj_x are respectively Tube and Trajectory objects
// In C++, the first argument is a pointer to the object to display
Draw slices
The .show() method accepts a boolean argument. If set to true, then the slices of the tubes will be displayed (instead of a polygon envelope). For instance:
dt = 0.1
tdomain = Interval(0,10)
traj = TrajectoryVector(tdomain, TFunction("(sin(t) ; cos(t) ; cos(t)+t/10)"))
y = Tube(traj[0], dt)
x = Tube(traj[1], traj[2], dt)
beginDrawing()
fig = VIBesFigTube("Tube")
fig.set_properties(100, 100, 600, 300)
fig.show(True)
endDrawing()
float dt = 0.1;
Interval tdomain(0,10);
TrajectoryVector traj(tdomain, TFunction("(sin(t) ; cos(t) ; cos(t)+t/10)"));
Tube y(traj[0], dt);
Tube x(traj[1], traj[2], dt);
vibes::beginDrawing();
VIBesFigTube fig("Tube");
fig.set_properties(100, 100, 600, 300);
fig.show(true);
vibes::endDrawing();
which produces:
Draw a set of objects on the same figure
Several objects can be drawn on the same figure with successive calls to the .add_...() methods. It is also possible to project all components of a vector object on the same figure with .add_tubes() or .add_trajectories().
The following code:
dt = 0.001
tdomain = Interval(0,10)
f = TFunction("(cos(t) ; cos(t)+t/10 ; sin(t)+t/10 ; sin(t))") # 4d temporal function
traj = TrajectoryVector(tdomain, f) # 4d trajectory defined over [0,10]
# 1d tube [x](·) defined as a union of the 4 trajectories
x = Tube(traj[0], dt) | traj[1] | traj[2] | traj[3]
beginDrawing()
fig = VIBesFigTube("Tube")
fig.set_properties(100, 100, 600, 300)
fig.show()
endDrawing()
float dt = 0.001;
Interval tdomain(0.,10.);
TFunction f("(cos(t) ; cos(t)+t/10 ; sin(t)+t/10 ; sin(t))"); // 4d temporal function
TrajectoryVector traj(tdomain, f); // 4d trajectory defined over [0,10]
// 1d tube [x](·) defined as a union of the 4 trajectories
Tube x = Tube(traj[0], dt) | traj[1] | traj[2] | traj[3];
vibes::beginDrawing();
VIBesFigTube fig("Tube");
fig.set_properties(100, 100, 600, 300);
fig.show();
vibes::endDrawing();
produces:
Technical documentation
See the C++ API documentation of this class:
## The class VIBesFigTubeVector¶
More content coming soon.
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2022-01-18 16:42:59
|
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https://en.algorithmica.org/hpc/cpu-cache/mlp/
|
Memory-Level Parallelism - Algorithmica
# Memory-Level Parallelism
Memory requests can overlap in time: while you wait for a read request to complete, you can send a few others, which will be executed concurrently with it. This is the main reason why linear iteration is so much faster than pointer jumping: the CPU knows which memory locations it needs to fetch next and sends memory requests far ahead of time.
The number of concurrent memory operations is large but limited, and it is different for different types of memory. When designing algorithms and especially data structures, you may want to know this number, as it limits the amount of parallelism your computation can achieve.
To find this limit theoretically for a specific memory type, you can multiply its latency (time to fetch a cache line) by its bandwidth (number of cache lines fetched per second), which gives you the average number of memory operations in progress:
The latency of the L1/L2 caches is small, so there is no need for a long pipeline of pending requests, but larger memory types can sustain up to 25-40 concurrent read operations.
### #Direct Experiment
Let’s try to measure available memory parallelism more directly by modifying our pointer chasing benchmark so that we loop around $D$ separate cycles in parallel instead of just one:
const int M = N / D;
int p[M], q[D][M];
for (int d = 0; d < D; d++) {
iota(p, p + M, 0);
random_shuffle(p, p + M);
k[d] = p[M - 1];
for (int i = 0; i < M; i++)
k[d] = q[d][k[d]] = p[i];
}
for (int i = 0; i < M; i++)
for (int d = 0; d < D; d++)
k[d] = q[d][k[d]];
Fixing the sum of the cycle lengths constant at a few select sizes and trying different $D$, we get slightly different results:
The L2 cache run is limited by ~6 concurrent operations, as predicted, but larger memory types all max out between 13 and 17. You can’t make use of more memory lanes as there is a conflict over logical registers. When the number of lanes is fewer than the number of registers, you can issue just one read instruction per lane:
dec edx
movsx rdi, DWORD PTR q[0+rdi*4]
movsx rsi, DWORD PTR q[1048576+rsi*4]
movsx rcx, DWORD PTR q[2097152+rcx*4]
movsx rax, DWORD PTR q[3145728+rax*4]
jne .L9
But when it is over ~15, you have to use temporary memory storage:
mov edx, DWORD PTR q[0+rdx*4]
mov DWORD PTR [rbp-128+rax*4], edx
You don’t always get to the maximum possible level of memory parallelism, but for most applications, a dozen concurrent requests are more than enough.
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2022-10-05 12:12:43
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http://www.gap-system.org/Manuals/pkg/forms/doc/chap1.html
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Goto Chapter: Top 1 2 3 4 5 Bib Ind
### 1 Introduction
#### 1.1 Philosophy
Forms is a package for computing with sesquilinear and quadratic forms on finite vector spaces. It provides users with the basic algebraic tools to work with classical groups and polar geometries, and enables one to specify a form and its corresponding geometry. The functionality of the package includes:
• the construction of sesquilinear and quadratic forms;
• operations which allow a user to change coordinates, that is, to change form'' and work in an isometric (or similar) formed vector space; and
• a way to determine the form(s) left invariant by a matrix group (up to a scalar).
#### 1.2 Overview over this manual
The next chapter (2) gives some basic examples of the use of this package. In "Background Theory of Forms" (Chapter 3) we revise the basic notions of the theory of sesquilinear and quadratic forms, where we also set the notation and conventions adopted by this package. In "Constructing forms and basic functionality" (Chapter 4), we describe all operations to construct sesquilinear and quadratic forms and basic attributes and properties that do not require morphisms. In "Morphims of forms" (Chapter 5) we revise the basic notions of morphisms of forms, and the classification of sesquilinear and quadratic forms on vector spaces over finite fields. Operations, attributes and properties that are related to the computation of morphisms of forms, are also described in this chapter.
#### 1.3 How to read this manual
We have tried to make this manual pleasant to read for the general reader. So it is inevitable that we will use Greek symbols and simple mathematical formulas. To make these visible in the HTML version of this documentation, you may have to change the default character set of your browser to UTF-8.
#### 1.4 Release notes
Version 1.2.1 of Forms contains some changed and extra functionality with relation to trivial forms. The changed and new functionality is described completely in Section 4.9. We gratefully acknowledge the useful feedback of Alice Niemeyer.
In version 1.2.2 of Forms a minor bug, pointed out by John Bamberg, in the code of IsTotallyIsotropicSubspace is repaired. On the occasion of the release of the first beta versions of GAP4r5, we changed the names of some global functions such that a name clash becomes unlikely. Version 1.2.2 of Forms is compatible with GAP4r4 and GAP4r5.
Version 1.2.3 contains a new operation TypeOfForm. Together with this addition, some parts of the documentation, especially concerning degenerate and singular forms, have been edited. A bug found in the methods for \^ applicable on a pair of vectors and a hermitian form, and a pair of matrices and a hermitian form has been fixed. A series of test files is now included in the tst directory. Alexander Konovalov pointed out the the init.g and read.g files had windows line breaks, this is also fixed. Finally, the documentation has been recompiled with the MathJax option.
Goto Chapter: Top 1 2 3 4 5 Bib Ind
generated by GAPDoc2HTML
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2017-11-25 02:15:20
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https://gamedev.stackexchange.com/questions/29260/transform-matrix-multiplication-order
|
# Transform Matrix multiplication order
I am experiencing difficulties trying to figure out the correct multiplication order for a final transform matrix. I always get either strange movement or distorted geometry. My current model is explained below:
For a single node my multiplication order is:
L = S * R * T
where
L = local transformation matrix
S = local scale matrix
R = local rotation matrix
T = local translate matrix
For a node's world transformation:
W = P.W * L
where
W = world transformation matrix
P.W = parent world transformation matrix
L = the local transformation matrix calculated above
When rendering, for each node I calculate the matrix :
MV = Inv(C) * N.W
where
MV = the model view transformation matrix for a particular node
Inv(C) = the inverse camera transformation matrix
N.W = the node's world transformation matrix calculated above.
Finally, in the shader I have the fallowing transformation:
TVP = PRP * MV * VP
where
TVP = final transformed vertex position
PRP = perspective matrix
MV = the node's world transformation matrix calculated above
VP = untransformed vertex position.
With the current model, child nodes which have local rotation, rotate strangely when transforming the camera. Where did I go wrong with the multiplication order?
• Just to note something here for any newcomers. The OP assumes right-handed conventions. Oct 13 '20 at 11:32
Any combination of the order S*R*T gives a valid transformation matrix. However, it is pretty common to first scale the object, then rotate it, then translate it:
L = T * R * S
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2022-01-29 11:14:59
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|
https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-6-systems-of-equations-and-inequalities-6-5-linear-inequalities-practice-and-problem-solving-exercises-page-393/11
|
## Algebra 1
To find this, insert (0,1) into the inequality $y\gt x - 1$ $(1)\gt(0) - 1$ Then solve $1\gt-1$ This now reads 1 is greater than -1. Since that is a true statement, (0,1) is a solution to the inequality.
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2018-08-20 13:49:27
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https://stats.stackexchange.com/questions/282665/should-optimized-parameters-of-a-maximum-likelihood-estimation-match-the-optimiz
|
# Should optimized parameters of a maximum likelihood estimation match the optimized parameters of a minimized chi square for a given distribution?
Setup: I have data that seems to follow a lognormal distribution. By logging every data point, I can also generate a corresponding normal distribution. I wrote a python script to use maximum likelihood estimation to find the optimized parameters mu and sigma. This script is also used to generate a contour plot of an error metric (z-axis) against the parameters (x- and y- axes), for which the error metric is the negative log likelihood and the reduced chi square value. (I will eventually change this such that the z-axis will be the p-value that corresponds to the reduced chi square value). My full code is a few hundred lines long, but I can provide code upon request; I think my problem is more concept-oriented than code-oriented.
Problem: I noticed that the parameters that optimize the distribution using the maximum likelihood estimation are slightly different than the parameters that optimize the distribution using chi square, as can be seen in the contour plots of lognormal
and normal distribution
(NOTE: I'm aware that I need to fix the probability; typo -- the title of the lognormal contour plot of chi square is the chi square value, NOT the probability). Is this normal or expected behavior? Is this more likely if there is group of outliars in the fitted distribution, or perhaps a mixed model? Most importantly, when plotting a distribution fit, which method is more conventional or preferred - maximum likelihood estimation or minimizing chi square?
EDIT:
SOLUTION: I realized that I had overlooked a requirement of measuring chi square. By excluding the observed and expected counts at indices for which the expected counts are below a threshold, the optimized parameters become closer in agreement.
Using a threshold of 5, these are the updated plots that serve as an example of the above:
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2019-10-20 00:51:36
|
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https://math.stackexchange.com/questions/2630397/how-many-bit-strings-of-length-n-are-there-with-two-or-more-consecutive-zeros
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# How many bit strings of length n are there with two or more consecutive zeros? [closed]
$a_n = 2^n - (a_{n-2} + a_{n-1})$
I have read this formula somewhere but don't know how its used here $a_n$ is the number of bit-strings of length $n$
## closed as off-topic by Did, Shailesh, Leucippus, Namaste, N. F. TaussigFeb 1 '18 at 2:14
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Shailesh, Leucippus, Namaste, N. F. Taussig
If this question can be reworded to fit the rules in the help center, please edit the question.
• Are you sure about the formula: $a_1=0$, $a_2=1$, $a_3=3$ and it doesn't hold? – asdf Jan 31 '18 at 22:13
• @asdf Doesn't $a_3=4?$ Did you not count $000?$ – saulspatz Jan 31 '18 at 22:22
• @saulspatz $000$, $001$, $100$. What is the fourth? – Clement C. Jan 31 '18 at 22:23
• $001, 100, 000$ – asdf Jan 31 '18 at 22:23
• @ClementC. You're right. I overlooked "consecutive". – saulspatz Jan 31 '18 at 22:26
The formula I get is this:
$$a_n=2^{n-2}+a_{n-1}+a_{n-2}$$
Reasoning:
Consider all such strings of length $n\geq2$:
If the first entry of such string is $1$, then the first entry doesn't contribute with anything to the property "have at least $2$ consecutive $0$'s" since and thus we can pretend it doesn't exist. Hence the number of such strings that start with $1$ is $a_{n-1}$.
If a string starts with $01$, then again, we need $2$ consecutive $0$'s and the $01$ has no contribution, so we could just erase the first $2$ entries and get that the number of such sequences is $a_{n-2}$
Finally, if it starts with $00$, then it doesn't matter what the other $n-2$ entries are, since we already have the $2$ consecutive $0$'s.
Since these are all possible cases we are done.
https://www.wolframalpha.com/input/?i=f(n)%3D2%5E(n-2)%2Bf(n-1)%2Bf(n-2),+f(1)%3D0,+f(2)%3D1
This gives a general formula which looks quite ugly to me.
Hopefully this helps
• It says it's the n-th Fibonacci number plus the n-th Lucas number, which is just $F_{n-1}+F_n+F_{n+1},$ which doesn't look so ugly to me, especially for the solution to a recurrence relation, but I guess ugliness is in the eye of the beholder. :-) – saulspatz Jan 31 '18 at 22:41
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2019-05-20 22:22:42
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https://eprint.iacr.org/2021/991
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### Fake it till you make it: Data Augmentation using Generative Adversarial Networks for all the crypto you need on small devices
Naila Mukhtar, Lejla Batina, Stjepan Picek, and Yinan Kong
##### Abstract
Deep learning-based side-channel analysis performance heavily depends on the dataset size and the number of instances in each target class. Both small and imbalanced datasets might lead to unsuccessful side-channel attacks. The attack performance can be improved by generating traces synthetically from the obtained data instances instead of collecting them from the target device. Unfortunately, generating the synthetic traces that have characteristics of the actual traces using random noise is a difficult and cumbersome task. This research proposes a novel data augmentation approach based on conditional generative adversarial networks (cGAN) and Siamese networks, enhancing in this way the attack capability. We present a quantitative comparative machine learning-based side-channel analysis between a real raw signal leakage dataset and an artificially augmented leakage dataset. The analysis is performed on the leakage datasets for both symmetric and public-key cryptographic implementations. We also investigate non-convergent networks' effect on the generation of fake leakage signals using two cGAN based deep learning models. The analysis shows that the proposed data augmentation model results in a well-converged network that generates realistic leakage traces, which can be used to mount deep learning-based side-channel analysis successfully even when the dataset available from the device is not optimal. Our results show potential in breaking datasets enhanced with faked'' leakage traces, which could change the way we perform deep learning-based side-channel analysis.
Available format(s)
Publication info
Preprint. MINOR revision.
Keywords
Machine learning-based Side-channel AttacksASCADElliptic Curves CryptographyData AugmentationSignal Processing
Contact author(s)
naila abbasi06 @ gmail com
lejla @ cs ru nl
picek stjepan @ gmail com
History
Short URL
https://ia.cr/2021/991
CC BY
BibTeX
@misc{cryptoeprint:2021/991,
author = {Naila Mukhtar and Lejla Batina and Stjepan Picek and Yinan Kong},
title = {Fake it till you make it: Data Augmentation using Generative Adversarial Networks for all the crypto you need on small devices},
howpublished = {Cryptology ePrint Archive, Paper 2021/991},
year = {2021},
note = {\url{https://eprint.iacr.org/2021/991}},
url = {https://eprint.iacr.org/2021/991}
}
Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.
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2022-08-09 11:21:04
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http://studywell.com/maths/pure-maths/proof/proof-by-exhaustion/
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## Proof by Exhaustion
In maths, proof by exhaustion means that you can prove that something is true by showing that it is true for each and every case that could possibly be considered. This is different to proof by deduction in that we do not use algebraic symbols to represent any number and showing that it is true for the symbol infers that it is true for all numbers – we must show that it is true for each number in consideration.
Example – Prove that $(n+1)^3\geq 3^n$ for $n\in{\mathbb N}, n\leq 4$.
We must show that $(n+1)^3$ is greater than $3n$ for all $n$ that is a natural number less than or equal to 4. The natural numbers less than or equal to 4 include 1, 2, 3, and 4. Proving the above by exhaustion will involve showing that it is true for 1, 2, 3 and 4:
1. For $n=1$, $(n+1)^3=(1+1)^3=2^3=8$ and $3^n=3^1=3$. Since, $8\geq 3$, the above is true when $n=1$.
2. For $n=2$, $(n+1)^3=(2+1)^3=3^3=27$ and $3^n=3^2=9$. Since, $27\geq 9$, the above is true when $n=2$.
3. For $n=3$, $(n+1)^3=(3+1)^3=4^3=64$ and $3^n=3^3=27$. Since, $64\geq 27$, the above is true when $n=3$.
4. For $n=4$, $(n+1)^3=(4+1)^3=5^3=125$ and $3^n=3^4=81$. Since, $125\geq 81$, the above is true when $n=4$ and thus concluding the proof by exhaustion.
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2017-06-29 05:39:29
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http://jnfa.mathres.org/archives/1100
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#### Mouataz Billah Mesmouli, Abdelouaheb Ardjouni, Ahcene Djoudi, Stability in neutral nonlinear differential equations, Vol. 2017 (2017), Article ID 4, pp. 1-17
Full Text: PDF
DOI: 10.23952/jnfa.2017.4
Received August 3, 2016; Accepted November 20, 2016
Abstract. In this paper, we use a modification of Krasnoselskii’s fixed point theorem introduced by Burton and the Carath\'{e}odory condition to obtain stability results of the zero solution of the neutral nonlinear differential equations with variable delay $x'(t)=-a(t)h(x(t))+c(t)x'(t-\tau(t))+G(t,x(t),x(t-\tau(t))).$ The stability of the zero solution of this eqution provided that $h\left(0\right) =G\left( t,0,0\right)=0$.
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2018-07-16 06:29:28
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https://space.stackexchange.com/questions/21122/how-much-energy-is-needed-to-bring-phobos-closer-to-mars/30183#30183
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How much energy is needed to bring Phobos closer to Mars?
Because Phobos has always the same face turned toward Mars, an electric propulsion system could be placed at the front to slow down the orbital speed.
But what would be the energy needed to supply the force of the propulsion system ?
• If you would allow the engine to be put in a different location on Phobos, consider momentum instead of energy, and don't care about anything ON the Martian surface, then...
– uhoh
Apr 17 '17 at 9:33
It is possible, but too costly!
And the orbital energy reduction is done best by retro-thrust, and that means, the exhaust of the propulsion device must be on the face in the direction of orbital motion (so that thrust is retrograde).
The energy needed would be:
$m \left(\frac{v_1^2}{2}-\frac{v_2^2}{2}-GM\left(\frac{1}{r_1}-\frac{1}{r_2}\right)\right)$
And this is in the best scenario, the propulsive energy that is not the same as solar energy input (considering solar-electric propulsion), because propulsion efficiency is not 100%.
For mining, It is indeed better to keep the Phobos up! Because it is much easier to loft materials off the Phobos' weak gravity and use it for construction in Martian orbit. And also, the debris of Phobos falling on Mars would wreak havoc in Martian atmosphere and climate and might pose danger to Martian colonies. Because when the Phobos falls apart, it's orbital height would be much lower and decays much faster. Also, after falling apart, Kessler syndrome happens and makes low Mars orbit a very dangerous place. And Mars gets its own ring after all! cool ;)
• Could you specify m,v1,v2,G,M,r1,r2 ? I have read that Phobos is 'falling' 2 meters in a hudred years. Would it be too costly to keep Phobos up ? Apr 17 '17 at 16:19
• @Conelisinspace "m" is the Phobos' mass, "v1" is the initial orbital velocity and "v2" is final. "r1" is the initial orbital radius and "r2" is final. "G" is the universal gravitational constant. "M" is the mass of Mars. And no, mv^2/2 is not all the energy terms of the Phobos, it is just kinetic and you need to consider the potential term (GM/r) too... Apr 18 '17 at 17:26
• Would it be enough to, instead of using an engine, to paint half of Phobos black and the other hemisphere white, so that the sunlight heats it when approaching the Sun, braking its speed by light reflection and evaporation from heating? Is it straight off clear that such effects cannot compete with Mars' tidal forces? Sep 15 '17 at 7:13
• @uhoh It's been some time since, but still I would very much like tto know he right answer from you. Indeed, with the formula in this answer one still doesn't know the values of v2 and r2. Jul 7 at 10:35
• @Cornelis I think AliRD explained those in a comment You can get the velocities from the vis-viva equation and you'll see that that equation is related to this expression for energy. It's been a few years, but I think my objection was only on the use of the word "energy" since going to a lower orbit means the spacecraft loses energy. I am not 100% sure if that's what I meant but I think so. In some old answer I remember putting vis-viva into this equation, I'll look for it.
– uhoh
Jul 7 at 11:59
First of all, the best place to place any such rockets would be at the point where the rocket is facing the direction that Phobos is rotating around Mars. That would give you about 4 x more bang for your buck to deorbiting Phobos than pointing directly at Mars, due to orbital mechanics.
Secondly, Phobos is actually being slowly deorbited naturally. It will take about 30 million years, but eventually it will land.
If you really wanted to deorbit Phobos, there is one major issue. Phobos is believed to not really be a cohesive body. If one put too much strain on it via a propulsion maneuver, it would probably break apart. Also, bringing it closer to Mars will almost certainly have the same effect.
Lastly, anything is in fact possible, if you get a large enough engine. To bring it from it's current 6,000 km orbit to a 3,000 km orbit would require a delta-v of around 660 m/s (I can't find an exact value, but that should be pretty close) The mass is about 10$$^1^6$$ kg. To have that much change in velocity over 10 years (315360000 s), it would require an average acceleration of 2.08 μm/s$$^2$$. That would require a thrust of about $$2.1*10^{10} N$$, which is a considerable amount. All that really needs to be done is to hook up an engine that can maintain that thrust for 10 years, and you would accomplish your goal.
• Assume that because Phobos is more like a large asteroid than a "celestial body" that it could ... become enclosed in a material that prevented it from breaking apart. Now assume that I want it at 160 km and I want to have it be stationary over a single position such that I could build a Space Elevator. Would that same engine have the ability to keep it there? Apr 17 '17 at 14:50
• Stationary at low altitude requires an ENORMOUS amount of energy. Far better would be to raise it up. But if you wanted to do that, then use Deimos. Apr 17 '17 at 14:53
• Ok, I assume your saying Deimos because it is what roughly 20% of the mass of Phobos and 67% the size so it becomes easier to package and less mass to manipulate and at 8 Miles in diameter, it would be sufficiently large to work as a habitat for the space end of the Elevator. Besides Deimos is leaving Mars and why let that resource just ... leave. So you convinced me so now it is the same question would that same engine be sufficient to keep Deimos there IF it is needed at all. I did not originally take into consideration the density of the Martian Atmosphere. Apr 17 '17 at 16:30
• The deltaV for a hohmann transfer to a 3000km orbit would be approximately 210m/s + 260m/s so a total dV of ~470m/s, you could add a bit to that for the spiralling trajectory but I doubt it'd be much, 500m/s would probably be a fair estimate. Apr 17 '17 at 17:51
• Applying a low constant magnitude thrust that always points directly toward at Mars would have almost exactly the same effect as would a low constant magnitude thrust that always points directly away from Mars. Both are incredibly inefficient mechanisms for slowly raising Phobos's orbit. Aug 19 '18 at 11:30
Phobos is tidally locked now, but if you started changing Phobos' orbit, you'd be changing its orbital period. Tidal locking is a very slow process, so pretty soon Phobos will lose tidal lock and will start to spin relative to Mars. That means your rocket engine will be in the wrong position to provide retrograde thrust for most of Phobos' rotation period. So you end up either taking a very long time to deorbit Phobos or having to build a rocket engine that can travel across Phobos' surface.
• What about orbital changes ? en.wikipedia.org/wiki/Tidal_locking#Orbital_changes And since huge amounts of energy are involved, the change in orbit will be very slow,slow enough for the tidal locking to compensate in time i think. Aug 19 '18 at 11:31
• Or just make your rocket aimable by a fraction of a fraction of a fraction of a degree. Jul 8 at 0:02
First, one assumptions:
The acceleration is so low that instantaneous impulse solutions are out of the question, and the trajectory can be modelled as a very gentle spiral.
This is quite reasonable, as an absolutely enormous amount of thrust would be necessary to provide high acceleration to a $$1.0659×10^{16} kg$$ rock.
So let's get started then. First, we need the delta-v.
For gentle continuous thrust spirals between circular orbits, the equation is surprisingly simple:
$$\Delta v = v_0 - v_1$$
Yes, just the difference between the initial and final orbital velocities.
To bring Phobos down to say, half the orbital radius, we need to supply 885 m/s (orbital velocity scales with the inverse square root or radius, so the velocity difference for half the radius is $$\sqrt{2} - 1$$ times the orbital velocity of Phobos, which is 2.14 km/s). We can then see what exhaust velocities are needed to spend only 1% of the mass of Phobos.
We take the rocket equation...
$$\Delta v = v_e \cdot ln\left(\frac{m_1}{m_0}\right)$$
... and turn it around!
$$v_e = \frac{\Delta v}{ln\left(\frac{100}{99}\right)} = 88.1 km/s$$
The energy required for this would be:
$$E = \frac{m_{propellant} \cdot v_e^2}{2}$$
For the half orbital radius, 1% Phobos mass example, that would be $$4.14 ×10^{23} J$$, or 4000x the World's yearly energy production.
Note that you do not want to use a higher specific impulse than necessary, as at small propellant mass fractions, the energy requirements go up linearly with the exhaust velocity.
As a general equation:
$$E = \frac{m_{propellant} \cdot \left(\frac{\Delta v}{R_{mass}}\right)^2}{2}$$
Were the mass ration $$R_{mass}$$ is:
$$R_{mass} = ln\left(\frac{1.0659 \cdot 10^{16} kg}{1.0659 \cdot 10^{16} kg - m_{propellant}}\right)$$
And the $$\Delta v$$ is:
$$\Delta v = \sqrt{\frac{\mu}{r_{final}}}- 2138 m/s$$
In the range 0 m/s (no change) to 1400 m/s (gracing the surface of Mars).
• Thank you, I asked for formula, now I got them ! But to learn from this answer could you explain where you got them from, in particular those with the ln expression ? And how did you calculate the 885 m/sec ? Jul 7 at 12:32
• @Cornelis Both added. Jul 7 at 16:13
• I think an exhaust velocity of 88 km/sec is not realistic, is it ? Could a much lower one not bring down the required energy ? Jul 7 at 16:53
• @Cornelis The energy would indeed go down, but the mass would go up! If the exhaust velocity is less than 88 km/s you have to use more than 1% of the mass of Phobos as propellant. That's a lot of mass! Jul 7 at 20:02
• Clear, what a waste of energy ! Jul 11 at 13:13
Goal: lower Phobos's orbit.
Status: ACHIEVED!
By the time you read this line, Phobos' orbital altitude is already lower than when you started reading this question.
Tidal deceleration is dropping Phobos by about 2cm per year, and in less than 50 million years Phobos will impact Mars.
Well, actually it won't. In only about 20-25 million years, Phobos will descend below its Roche's limit, and turn itself into a nice rocky ring for Mars.
You should be worrying about how much energy is needed to keep Phobos UP, and prevent armageddon from raining fiery death down on your newly-terraformed Mars!
(spoiler: you need about 60N of continuous thrust to counteract the Tidal deceleration)
• We don't have the time to wait a million years ! And by the time Phobos is rurned into a "nice rocky ring" will there still be tidal deceleration to "rain it down" ? Jul 7 at 22:13
• When Phobos starts falling apart, probably valuable material could be extracted.from the inside. Jul 8 at 13:03
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2021-10-17 13:49:08
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http://mathhelpforum.com/calculus/209189-integrate.html
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# Math Help - integrate!
1. ## integrate!
Hi I have problems integrating two questions, how do I integrate sqrt(x^2-4) / x? And (4lnx) / (x(1+(lnx)^2)? Thank you very much!!
2. ## Re: integrate!
For the second problem, first use t= ln(x), then use z = 1+ t^2. Hope this helps!
3. ## Re: integrate!
Hello, Tutu!
$\int\frac{\sqrt{x^2-4}}{x}\,dx$
It should be obvious that a Trig Substitution is called for.
Let $x \,=\,2\sec\theta \quad\Rightarrow\quad dx \,=\,2\sec\theta\tan\theta\,d\theta \quad\Rightarrow\quad \sqrt{x^2-4} \,=\,2\tan\theta$
Substitute: . $\int \frac{2\tan\theta}{2\sec\theta}(2\sec\theta\tan \theta\,d\theta) \;=\;2\int\tan^2\theta\,d\theta$
. . . . . . . . $=\;2\int(\sec^2-1)\,d\theta \;=\;2(\tan\theta - \theta) + C$
Now back-substitute.
4. ## Re: integrate!
Originally Posted by coolge
For the second problem, first use t= ln(x), then use z = 1+ t^2. Hope this helps!
Which is, of course, the same as the single substitution z= 1+ (ln(x))^2.
5. ## Re: integrate!
Originally Posted by HallsofIvy
Which is, of course, the same as the single substitution z= 1+ (ln(x))^2.
My two cents. I prefer to use two separate substitutions rather than one more complicated one. I can keep better track that way. Or maybe I'm just a coward.
-Dan
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2014-09-01 20:27:08
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https://www.samabbott.co.uk/tags/reproducible-research/
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# reproducible-research
## prettypublisher
prettypublisher is an R package that aims to improve your workflow by allowing an easier transition from literate code to a paper draft ready for journal submission.
## tbinenglanddataclean
R package containing the scripts required to clean data from the Enhanced Tuberculosis Surveillance system, and the Labour Force Survey, and to then calculate Tuberculosis incidence.
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2019-08-24 14:38:03
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https://academy.vertabelo.com/course/ms-sql-window-functions/partition-by-order-by/partition-by-ranking/rank-partition-by-order-by
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Kickstart 2020 with new opportunities! - hours only!Up to 80% off on all courses and bundles.-Close
Introduction
Quick Refresher
PARTITION BY ORDER BY with Ranking
7. RANK() with PARTITION BY ORDER BY
PARTITION BY ORDER BY with Window Frames
Using PARTITION BY ORDER BY with Analytical Functions
Summary and Review
## Instruction
Excellent! Let's get started with the new stuff.
In Part 4, you learned ranking functions. These are one place where you can apply PARTITION BY and ORDER BY together.
So far, all the rankings we calculated were performed for all the rows from the query result. With that knowledge, we could have calculated the position of each store in the global network based on their ratings:
SELECT
Id,
Country,
City,
Rating,
RANK() OVER(ORDER BY Rating DESC) AS Rank
FROM Store;
Now, we can add PARTITION BY to calculate the positions independently for each country:
SELECT
Id,
Country,
City,
Rating,
RANK() OVER(PARTITION BY Country ORDER BY Rating DESC) AS Rank
FROM Store;
In this way, we create a separate ranking for each country. Paris and Frankfurt can both get Rank = 1 because there are separate rankings for France and Germany.
## Exercise
For all sales between August 10 and August 14, 2016, show the following information: StoreId, Day, number of customers and rank (based on the number of customers in that store). Name the column Ranking.
### Stuck? Here's a hint!
Use PARTITION BY StoreId ORDER BY Customers ASC.
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2020-02-25 13:31:36
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https://puzzling.stackexchange.com/questions/1856/draw-a-line-through-all-doors/1881
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# Draw a line through all doors
I saw the following problem on 4chan and couldn't solve it:
It's very likely to be some kind of troll (no solution).
I'm hoping to see some rigorous proofs that disprove the existence of such a line.
• Well, since a 'trough' is 'a channel used to convey a liquid', I'm a bit confused on how to solve this puzzle. It may be a typo! – Doug.McFarlane Aug 24 '15 at 19:55
It is impossible.
Quite the same problem is "Seven Bridges of Königsberg", it was solved (proven) by Euler.
1. Suppose you have drawn such a line and follow it from one room to another. Since you must use each door you must have a look at each room out of 5. What are these rooms?
2. There will be 3 (at least) rooms you always go through - if you enter it you always exit it later.
Indeed, 1 (at most) room you can start at, and another 1 (at most) room you can finish at, but others you must go through: $5-1-1 = 3$.
3. Since you use each door exactly once, the mentioned 3 rooms must have an even number of doors, since you entry them the same number of times you exit them. But you have only 2 rooms with an even number of doors, the others have 5 doors. So you could not draw such a line.
• I cannot understand which "other" room you're referring to in "You can start at one room, and end at the other." – Gabriel Romon Jul 3 '14 at 21:48
• @G.T.R, what do you mean which? Any. (I changed it to "another", may be "the" was the problem?) – klm123 Jul 3 '14 at 21:49
• OK, but why do you need to go through at least 3 of them ? – Gabriel Romon Jul 3 '14 at 21:54
• @G.T.R, not of them, but of the rest. Because you need to go through all 5 rooms eventually. – klm123 Jul 3 '14 at 21:59
• Everything makes sense now, thanks ! – Gabriel Romon Jul 3 '14 at 22:02
It works when you use a really wide line:
• This one cracked me up! – Shashank Sawant Jul 26 '14 at 18:28
Just for fun:
Actually, there is solution, which formally satisfies all the rules. You just need to walk through a wall!
Hard, but possible!
• +1. I think this is the troll solution envisioned by the poster in 4chan, because it satisfies the rules! Nobody says you can't walk through a wall. Hard, but possible! Haha – justhalf Jul 4 '14 at 2:49
• HA!, draw a ine through one wall. – Jasen Mar 29 '18 at 9:55
Similar to klm's solution, this one requires a little bit of lateral thinking but doesn't actually involve walking through a wall. Instead, you have to fold the corner of the piece of paper that the picture is drawn on, to form a bridge over a wall.
You can draw a graph with six vertices, one for each room and one for the outside. Draw an edge to represent each door. The puzzle asks for an Eulerian path, which can be done if no more than two vertices have an odd number of edges coming in. In this graph the top two rooms, the middle bottom room, and the outside all have an odd number of edges coming in, so in the usual way there is no such path. A sketch is below. ABCDE are the rooms in the same locations as in the picture, F is outside, and the lines are connections through doors. ABD and F all have an odd number of edges coming in.
Alternatively, going "through" a door need not necessarily be interpreted in the same way as one would in a house.
This single, continuous line passes exactly once through each door, which is the constraint in the original question. Of course, it also conveniently side-steps making any other assumptions and takes certain liberties with the walls.
Credit goes to mathgrant for this answer.
It is actually impossible if you consider walking through walls impossible as stated in the other answers. That can be explained by graphs or in another similar, but yet easier to understand way:
Ignoring the connections between the rooms, you have 5 rooms:
3 rooms with 5 doors and 2 rooms with 4 doors.
• if a room has an even amount of doors, there are two ways it can be part of the solution: either the line starts and ends within that room and all other doors are used to both enter and leave the room or the line passes through (amount of doors / 2) times.
• if a room has an uneven amount of doors, there are another two ways it can be part of the solution: either the line starts within that room and all the other doors are used to both enter and leave or the line ends within that room and all the other doors are used to both enter and leave.
This gives us the insight that there are 3 rooms that need to contain either the start or the end of the line and there are 2 rooms that might contain both start and end. Following that logic, we have at least 2 of either start or end of the line and thus we can't use just a single line to solve this problem.
• How is this different from accepted answer? – klm123 Jul 4 '14 at 9:02
• I believe that my answer is easier to understand and that mine is more organised. Additionally, in my case, it does not matter if the rooms are adjacent or placed far away from each other since I only consider the connectors (the doors) and don't bother with the meta concept of doors and paths. This would only be neccessary if there was a room that only connects to other rooms and not to the "outside world". My answer is directed mostly at the people that don't get what the poster of the accepted answer wanted to say. – Fredchen777 Jul 4 '14 at 9:16
• @klm123 This answer is much clearer and easier to understand. – DisgruntledGoat Jul 4 '14 at 10:05
Begin wherever you want and draw one line through all of the doors, but you cannot go through the same door twice. Hard, but possible!
• Welcome to puzzling.se. I downvoted your answer before I understood it. Now I think you have a good point - that this question could have been posted on 4chan as a lateral thinking question. (If you do an edit on your answer, I can then change to an upvote.) – Len Feb 8 '15 at 7:21
As far as I can tell there are no doors in this image, therefore any line will cross every doors and no line will cross a door twice: just draw a line anywhere. Actually you might have already draw some lines of length 0.
EDIT: some people seem to think this is a troll, maybe they simply don't know what universal and existential quantification mean, or they disagree with the fact that there are no door on the image ?
• +1. I think this is valid answer, taking into account formulation of the question. – klm123 Jul 7 '14 at 16:20
• Ara, I think it'd help if you explicitly said "There are no doors, just doorways" rather than just "there are no doors" without replacing them. – Bobson Jul 7 '14 at 19:12
• This is not a pipe – Engineer Toast Feb 6 '15 at 19:00
• Alternatively, one may figure that most of the gaps on exterior walls are more likely to be windows than doors. – supercat Apr 23 '15 at 16:36
Even assuming that it is not allowed to draw the line through a "wall", there is another kind of troll solution. You are forbidden to go through the same door twice, but you can go through that door four times. Then it is easy to do it.
Or you could maintain that there are no door at all in the drawing, just $12$ disconnected pieces of "wall". This makes it a non-problem.
• To go through a door four times you need to go through it twice first, don't you? – klm123 Jul 4 '14 at 15:28
• Yes, but upon going through it a third time, you're no longer going through it "twice" anymore. – Joe Z. Jul 6 '14 at 7:31
If we take this puzzle in the spirit in which it was originally created, and not quibble about whether openings are windows or doors, and if we acknowledge that this house represents a house that we could walk from room to room only through the given doors, then, as presented, there is no answer.
IMAGINE, if you will, that this is a life-sized house.
When you start, every door is standing open.
1. Start anywhere and go through each door, but as you go through, close the door behind you so that it becomes a wall.
2. You cannot walk through a wall.
3. The only way to solve this puzzle is by cheating.
Euler proved that when a vertex (a room in this example) has an odd number of lines (or in this example, a room that has an odd number of open doors leading out of that room) coming from that point, the answer must either start or end on that vertex.
When we have zero odd vertexes, you can start anywhere and you will find an answer that ends back in the room you started in that will have gone through every door.
When we have two odd vertexes (vertices), the path MUST start in one and end in the other.
When we have four odd vertices, there will be two "odd vertices" that have one unused line, one unused door between them, and no way to get to either of those rooms.
HOW TO CHEAT:
1. Fold a corner over so that the corner of the diagram is suddenly covered by the blank other side of the paper.
2. Build your house over a basement. Put a square hole in the floor of the kitchen (one of the five-door rooms, then put a "Wizard of Oz" exit from the basement leading to the outside. (The doors flat in the ground that you lift out of the way to climb up from under the house after a tornado.)
3. Draw the diagram on a skewed (not evenly baked) bagel, with the bagel hole in the middle of the middle room in the bottom (a room with five doors). Topologically, the outside of the diagram and the inside of the room are now one. You can draw one continuous line starting in the top left room, eventually when you enter that "hole" room the third time, continue that line down through the hole and around to the outside of the house, then go back in the house by one of the other unused. Eventually, you will only have one door left, which will lead you into the top right room.
Of these three cheats, the bagel is my favorite, but if you use permanent ink marker, I would recommend you don't eat the bagel after math class.
• If I recall, the donut hole can actually be in any of the rooms and it will still work. – ben-Nabiy Derush May 18 '17 at 0:29
This "puzzle" can be described as an Euler Trail, and was discussed in a video by James Grime in which he proves why the puzzle is impossible:
The gist of it is, every time you enter a room, you have to leave it as well, which means all doors come in pairs (the door you entered the room with, and the door you left the room with). The only time you can have an odd number of doors is in two cases: if you start the path in that room, or end in that room.
If we look at the image, two of the rooms have 4 doors (an even number), while three of the rooms have 5 doors (an odd number). This means if we start in the top left room, make our path, then end in the top right room, the bottom center room will have been walked through an even amount of times, meaning one of the doors will have remained unused and inaccessible. Therefore, it is indeed a "troll puzzle".
• Although it doesn't apply to this example, you can also have a valid room layout where there are an even number of doors in all rooms, in that case you go back to and end in the room you started in. – IQAndreas Jul 6 '14 at 8:30
According to graph theory
Given a graph G, is it possible to find a walk that traverses each line exactly once, goes through ail points, and ends at the starting point? A graph for which this is possible is called eulerian graph.
Thus, an eulerian graph has an eulerian trail, a closed trail containing all points and lines.
Theorem: The following statements are equivalent for a connected graph G:
(1) G is eulerian.
(2) Every point of G has even degree.
In this problem there exists 4 points(nodes) with odd degree. So this is not eulerian graph.
i.e: There no solution existing.
If I remember correctly, this question was originally posed by an MIT professor to his class. I think only 1 student got it by the end of the year.
As I understood it, the question was phrased "With a single line, pass through each plane without breaking each plane twice." There was not any more restrictions on it. I think this version was a simplified version with the doors showing where the planes are to pass through, but it is not limited to a single 2 dimensions.
Take a sewing thread and stitch your way through the planes, going in and out where needed in the "rooms" to cross each plane only once. If you want to do it with pen and paper, draw like normal through each "door" and then poke a hole in the paper and then continue your line on the other side of the paper and then poke back through to cross the last plane...
• this seems similar to the bagel solution from Marc Williamson – Jasen Mar 29 '18 at 19:10
If u Shift this line segment (wall) on either far side u can solve this. However , it was not given anywhere that u can't modify the figure. After all, there are still 5 rooms. In Euler's trail the main cause of failing is this edge's position in the middle. Or else the solution of this puzzle lie in 3d format only.
You guys have to think out-side the box (or in this case the 2D screen). If this was on paper we can fold the paper and solve the problem, as shown below. Unless there is a change to the original configuration of doors and walls, this problem is impossible. Hence the warning of "Hard, but possible!"
• But ... that doesn't actually go through all the doors, does it? How is this "solution" any different from arbitrarily deciding it's ok to take scissors to the paper and cut off the right half entirely, as you've effectively folded it out of the puzzle? – Rubio Mar 23 '18 at 12:28
It took me quite a while to do but I worked it out 😁
• Your floorplan is not the same as the one in the question. – Gareth McCaughan Feb 19 at 13:49
• No I just took the picture the wrong way up. If you turn it upside down it will look right. I could delete this post and then take another picture the right way up 🤷♀️ – PuzzleMaster Feb 19 at 13:52
• Oops, sorry, I was confused by the upside-down-ness. But what definitely is wrong is that you have used one door twice (near the bottom left in your version of the picture), which you're explicitly forbidden to do. – Gareth McCaughan Feb 19 at 13:52
• Oh ok thanks for telling me 😂 I’m going to try again now lol – PuzzleMaster Feb 19 at 13:55
## protected by Community♦Feb 19 at 13:51
Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
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2019-10-19 13:05:18
|
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https://www.gamedev.net/forums/topic/325521-matrices-shaders-and-pointers/
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This topic is 4872 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
Okay, this is one of those problems that shouldnt be a problem, or maybe an anomoly in the earths magnetic field has developed around my house making all kinds of weird things happen. Yes, that must be it. Okay, so I've got this engine working pretty decently so far (still very early on). I've set up my own math related classes for doing 3d math related things in my engine so that I can send those numbers to the renderer which is decoupled from any specific API (although designed around the features of directx, as that is the API I am working with atm). All that, in theory, works well. My 3D math skills are pretty lame, but I think I know how to find the basic formulas in books and other resources and transfer those into code (the only true use for derivitives, imo). Anywho, on to my problems. What I am currently doing is merely rendering a test cube while I set up all the features of my renderer, so I can see when things go wrong. This has evolved throughout developement to utilize the current interfaces of my renderer and other engine components. Currently, I am setting the transform matrices from the main loop in absense of a more proper way just yet (such as a scene graph, etc.), using my own matrix and vector structures. It looks something like this:
Interface->SetTransform( TT_WORLD, FMatrix::Identity());
Interface->SetTransform( TT_VIEW, FMatrix::Identity().CameraLookAtMatrixLH(FVector( 0.0f, 3.0f,-10.0f ),FVector(),FVector(0.0f,1.0f,0.0f)));
Interface->SetTransform( TT_PROJECTION, FMatrix::Identity().ProjectionMatrixPerspectiveFOVLH(PI/4.0f,Viewport->SizeX/Viewport->SizeY,1.0f,1000.0f ));
What this is doing is calling the Identity() method, which is a static function that returns an instance of the struct initialized to an identity matrix. It works as it is supposed to. That value can then be used to set a camera look-at matrix, or a perspective matrix. The FVector struct initializes to 0,0,0 with a default constructor, and in this instance is the look at parameter. These numbers have been used since the early beginnings of my renderer, so I know they are valid and should work, and they do work ;) So, what is the problem, you ask? Well, to transform the cube I'm rendering, I've been using a vertex shader, and sending the matrix to the shader constants as you are supposed to. There are two problems with this procedure, however. First, the matrix I am giving to the hardware is World*View*Projection, which I believe would be correct, except it isnt. Maybe I read wrong (in several different places), but I am quite sure that the matrices are set correctly, and are correctly referenced when multiplying. Like so: WantedState.WorldViewProjection = (WantedState.Matrices[TS_WORLD] * WantedState.Matrices[TS_VIEW]) * WantedState.Matrices[TS_PROJECTION]; The result is, however, incorrect, but is correct if the order projection and view matrices are reversed. I know matrix multiplication isnt commutative, but isnt it supposed to be world*view*projection, or am I just crazy, illiterate, and blind? My projection matrix code, or even the multiplication code could be wonky, but it seems to work (I suck at math, so I cant do it long hand to check). I'll post more code if anyone thinks it would help. Second, I'm setting the result of the multiply/transpose in to a struct in my render state management code which holds the shader constant data needed to make the call (StartRegister, Data, and RegisterCount) until the render state is about to be commited to hardware (thus no redundant values are ever set). The matrix is passed as a float* and the pointer is valid in the debugger. This all happens after the vertices are sent to the vertex buffer. The problem is, calling CreateIndexBuffer causes the pointer to my matrix data to become invalid. It's valid right before the call, but never right after, but the matrix that the pointer is referring to (the pointer is actually pointing to float m[16]; in the matrix, which gets returned in the float* cast). Is this some trick of pointers that I'm missing, or some fluke, or some evil that harasses me for sport? I can reset the pointer after creating the index buffer, and it stays valid through the call to DIP, and works fine, OR allocating the pointer data with new works as well, but that either creates a memory leak, or I have to constantly delete [] the pointer before setting new data to it, thus making my design cumbersome.
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2018-10-15 12:38:06
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https://www.esaral.com/q/let-be-the-binary-operation-on-n-defined-by-a-b-h-c-f-of-a-and-b-87589/
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Let * be the binary operation on N defined by a * b = H.C.F. of a and b.
Question:
Let * be the binary operation on defined by * = H.C.F. of and b. Is * commutative? Is * associative? Does there exist identity for this binary operation on N?
Solution:
The binary operation * on N is defined as:
* b = H.C.F. of a and b
It is known that:
H.C.F. of $a$ and $b=$ H.C.F. of $b$ and $a \& m n F o r E ; a, b \in \mathbf{N}$.
$\therefore a^{*} b=b^{*} a$
Thus, the operation * is commutative.
For $a, b, c \in \mathbf{N}$, we have:
$\left(a^{*} b\right)^{*} c=(\text { H.C.F. of } a \text { and } b)^{*} c=$ H.C.F. of $a, b$, and $c$
$a^{*}\left(b^{*} c\right)=a^{*}($ H.C.F. of $b$ and $c)=$ H.C.F. of $a, b$, and $c$
Thus, the operation * is associative.
Now, an element $e \in \mathbf{N}$ will be the identity for the operation ${ }^{*}$ if $a^{*} e=a=e^{*} a \forall a \in \mathbf{N}$.
But this relation is not true for any $a \in \mathbf{N}$.
Thus, the operation * does not have any identity in $\mathbf{N}$.
|
2022-09-26 15:29:10
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http://algrepair-math.blogspot.com/2017/03/underground-from-portland-to-new-york_10.html
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## Friday, March 10, 2017
### Underground from Portland to New York (Part 2)
Underground from Portland to New York (Part 2) This post is part two of what will be a three part series on the derivation of the equations of an underground tunnel that minimizes travel time when only gravity is used for propulsion. We left our subterranean tunnel problem with the following equation we wished to minimize. \eqalign{ I &= \int_a^b\frac{\sqrt{(dr)^2+(rd\theta)^2}}{\sqrt{\frac{g}{R}(R^2-r^2)}} } Our first issue is figuring out what to do with $$\sqrt{(dr)^2+(rd\theta)^2}$$ so that we can integrate it. We could choose to factor out either the $$dr$$ or the $$d\theta$$ and could solve the problem either way. We'll choose to factor out the $$dr$$ because that leaves us an integrand that is not a function of the path variable (in this case $$\theta$$) which means that $$\pard{F}{\theta} = 0$$ and $$\pard{F}{\theta'} = constant$$. So, we have the Euler-Lagrange equation for this problem (remember that $$r$$ is our variable of integration): \eqalign{ \frac{d}{dr}\pard{F}{\theta'} - \pard{F}{\theta} = 0 \cr \text{but}\enspace \pard{F}{\theta} = 0 \cr \text{therefore}\enspace \frac{d}{dr}\pard{F}{\theta'} = 0 \cr \text{so}\enspace \pard{F}{\theta'} = k } $$\pard{}{\theta'} \left(\frac{\sqrt{1+(r\theta')^2}}{\sqrt{\frac{g}{R}(R^2-r^2)}}\right) = k$$ Performing the differentiation we have $$\frac{1}{\sqrt{\frac{g}{R}(R^2-r^2)}}\cdot\frac{r^2\theta'}{\sqrt{1+(r\theta')^2}} = k$$ Squaring both sides and solving for $$(r\theta')^2$$ first $$\frac{1}{\frac{g}{R}(R^2-r^2)}\cdot\frac{(r^2\theta')^2}{1+(r\theta')^2} = k^2$$ $$$$\frac{\frac{k^2g}{R}(R^2-r^2)}{r^2-\frac{k^2g}{R}(R^2-r^2)} = (r\theta')^2 \label{eq:rt2}$$$$ We solve for $$(r\theta')^2$$ and mark that equation because in the end, our original integral for the minimum time is a function of $$(r\theta')^2$$ and we don't want to have to recompute it. Going the final step to solve for $$\theta'$$ we have $$$$\theta' = \frac{1}{r}\sqrt{\frac{\frac{k^2g}{R}(R^2-r^2)}{r^2-\frac{k^2g}{R}(R^2-r^2)}} \label{eq:tp}$$$$ At this point we pause to consider the variable and limits of integration. If you look at the figure below, we can see that to traverse the curve, $$r$$ goes from $$R$$ to $$R$$. That doesn't help us much because the definite integral would end up being 0.
We do notice, however, that the curve must be symmetric (the path shouldn't matter on which side you start) and so the deepest part of the curve is in the middle at $$r_0$$. At this value of $$r=r_0$$, we can see that $$\frac{dr}{d\theta} = 0$$ since $$r(\theta)$$ is mimimized. Conversely, this means that $$\theta' = \frac{d\theta}{dr} \rightarrow \infty$$. This happens when the denominator of equation \ref{eq:tp} becomes zero. So, solving for the constants in this case when $$r=r_0$$ $$r_0^2 - \frac{k^2g}{R}(R^2-r_0^2) = 0$$ yields $$\frac{k^2g}{R} = \frac{r_0^2}{R^2-r_0^2}$$ Before we find the path, we'll find the time it takes to traverse the curve. Of course, this time will be a function of $$r_0$$ so we will still have some work to do. Rewriting equation \ref{eq:rt2} with the above substitution and then cleaning up some of the fractions yields $$(r\theta')^2 = \frac{\frac{r_0^2}{R^2-r_0^2}(R^2-r^2)}{r^2-\frac{r_0^2}{R^2-r_0^2}(R^2-r^2)}$$ yields $$$$(r\theta')^2 = \frac{r_0^2(R^2-r^2)}{r^2(R^2-r_0^2)-r_0^2(R^2-r^2)} \label{eq:rt2r0}$$$$ Now, going back to our original functional we factor out the $$dr$$ and put in the actual limits of integration, multiplying the whole thing by 2 because we're only going halfway in the integral. $$\Delta t = 2\int_{R}^{r_0}\frac{\sqrt{1+(r\theta')^2}}{\sqrt{\frac{g}{R}(R^2-r^2)}}dr$$ Substituting \ref{eq:rt2r0} into the above and moving some constants over to the left hand side $$\frac{1}{2}\sqrt{\frac{g}{R}}\Delta t = \int_{R}^{r_0}\frac{1}{\sqrt{(R^2-r^2)}}\cdot\frac{r\sqrt{R^2-r_0^2}}{\sqrt{r^2(R^2-r_0^2) - r_0^2(R^2-r^2)}}dr$$ And after a little more algebra, factoring out the $$(R^2-r_0^2)$$ $$\frac{1}{2}\sqrt{\frac{g}{R}}\Delta t = \int_{R}^{r_0}\frac{r}{\sqrt{(R^2-r^2)(r^2 - \frac{r_0^2}{R^2-r_0^2}(R^2-r^2))}}dr$$ Sometimes at points like this it helps to look at the units to make sure we at least have a sanity check. On the left hand side we have $$\sqrt{\frac{m}{s^2}\cdot\frac{1}{m}}\cdot s$$ which is unitless. On the right, we have $$\frac{m\cdot m}{\sqrt{m^4}}$$ which is also unitless. So we at least have that going for us. We're almost home for this part. We now have a relatively straightforward integration. We let $$u=R^2-r^2$$, $$du = -2rdr$$ and $$r^2 = R^2 - u$$. These substitutions result in $$\frac{1}{2}\sqrt{\frac{g}{R}}\Delta t = -\frac{1}{2}\int_{R}^{r_0}\frac{1}{\sqrt{u((R^2-u) - \frac{r_0^2}{R^2-r_0^2}u)}}dr$$ Simplifying the denominator $$\frac{1}{2}\sqrt{\frac{g}{R}}\Delta t = -\frac{1}{2}\int_{R}^{r_0}\frac{1}{\sqrt{u(R^2-(1+\frac{r_0^2}{R^2-r_0^2})u)}}dr$$ A little more $$\frac{1}{2}\sqrt{\frac{g}{R}}\Delta t = -\frac{1}{2}\int_{R}^{r_0}\frac{1}{\sqrt{u(R^2-\frac{R^2}{R^2-r_0^2}u)}}dr$$ A little more algebra by factoring out an $$R^2$$ from the to get the integral into a more recognizable form $$\frac{1}{2}\sqrt{\frac{gR^2}{R}}\Delta t = -\frac{1}{2}\int_{R}^{r_0}\frac{1}{\sqrt{u(1 - \frac{1}{R^2-r_0^2}u)}}dr$$ Simplifying the left a little and with a little help from Wolfram-Alpha we have (putting back the $$u=R^2-r^2$$) \eqalign { \frac{1}{2}\sqrt{gR}\Delta t &= 2\sqrt{(R^2-r_0^2)}\sin^{-1}\left(\sqrt{\frac{R^2-r^2}{R^2-r_0^2}}\right) \Bigg \bracevert_R^{r_0} \cr \frac{1}{2}\sqrt{gR}\Delta t &= 2\sqrt{(R^2-r_0^2)}(\frac{\pi}{2} - 0) \cr \frac{1}{2}\sqrt{gR}\Delta t &= \pi\sqrt{R^2-r_0^2} } And finally solving for $$\Delta t$$ $$\Delta t = 2 \pi \sqrt{\frac{R^2 - r_0^2}{gR}}$$ Here, we can ask the question of how long will it take to go from one side of the earth to the other, straight through the core. At that point, $$r_0 = 0$$ so $$\Delta t = 2\pi\sqrt{\frac{R}{g}}$$. With $$g=9.8\frac{\text{m}}{\text{s}^2}$$ and the radius of the earth $$R=6.4\cdot10^6 \text{m}$$ $$\Delta t = 2\pi\sqrt{\frac{6.4\cdot 10^6}{9.8}} \approx 5078 \text{s} \approx 85 \text{min}$$ Here is a graph of the transit time in minutes versus the depth as a fraction of $$R$$.
Solving for the path is coming in part 3.
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2020-02-27 17:34:26
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https://www.bakpax.com/assignment-library/assignments/applying-ratios-in-right-triangles-hofot-fituz
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# Applying Ratios in Right Triangles
ID: hofot-fituz
Illustrative Mathematics, CC BY 4.0
Subject: Geometry
Standards: HSG-SRT.CHSG-SRT.C.8HSN-Q.A.2
6 questions
# Applying Ratios in Right Triangles
Classroom:
Due:
Student Name:
Date Submitted:
##### Tilted Triangle
Find the indicated parts of the triangle.
1) Find the length of $AC$. Round to the nearest tenth of a unit.
2) Find the length of $BC$. Round to the nearest tenth of a unit.
##### Tallest Tower
3) The tallest building in the world is the Burj Khalifa in Dubai (as of April 2019).
If you’re standing on the bridge 250 meters from the bottom of the building, you have to look up at a 73 degree angle to see the top. How tall is the building? Round your answer to the nearest whole meter.
4) Explain or show your reasoning.
5) The tallest masonry building in the world is City Hall in Philadelphia (as of April 2019). If you’re standing on the street 1,300 feet from the bottom of the building, you have to look up at a 23 degree angle to see the top. How tall is the building? Round your answer to the nearest whole foot.
6) Explain or show your reasoning.
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2020-09-29 21:19:41
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https://www.aimsciences.org/article/doi/10.3934/dcds.1999.5.301
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Article Contents
Article Contents
# Multiple solutions of Neumann elliptic problems with critical nonlinearity
• The paper is concerned with a class of Neumann elliptic problems, in bounded domains, involving the critical Sobolev exponent. Some conditions on the lower order term are given, sufficient to guarantee existence and multiplicity of positive solutions without any geometrical assumption on the boundary of the domain.
Mathematics Subject Classification: 35J65, 35J20, 35J25.
Citation:
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2023-03-23 23:45:37
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https://www.physicsforums.com/threads/inverse-function.66395/
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Inverse function
1. Mar 7, 2005
yoyo
what is the inverse of of y=sqrt(x^3+x^2+x+1)
i know u are suppose to solve for x but having trouble....help please
2. Mar 8, 2005
dextercioby
Why don't u do it?
$$x^{3}+x^{2}+x+1=y^{2}$$
U need to solve this cubic for "x".Use Cardano's formulae.
Daniel.
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2017-11-23 19:09:02
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https://codereview.stackexchange.com/questions/115/subroutine-to-call-other-subroutines
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# Subroutine to call other subroutines
I have a Perl application that allows the user to choose between two different formats for data output (and there will likely be more in the near future). In the heart of the algorithm, the code makes a call to a print subroutine.
my $stats = analyze_model_vectors($reference_vector, $prediction_vector ); print_result($stats, $tolerance ); The print_result subroutine simply calls more specific methods. sub print_result { if($outformat eq "text")
{
print_result_text(@_);
}
elsif($outformat eq "xml") { print_result_xml(@_); } else { # Should not reach this far if input checking is done correctly printf(STDERR "Error: unsupported output format '%s'\n",$outformat);
}
}
Is this good practice? What other alternatives are there and what are their pros/cons? I could think of the following alternatives.
• Test for output format in the heart of the algorithm, and call the appropriate printing subroutine there.
• I've never used subroutine references before, but perhaps when I could store a reference to the correct subroutine in a scalar variable and call the print method with that scalar in the heart of the algorithm.
• Place code for all output formats in a single subroutine, separated by if/elsif/else statements.
Keep in mind there may be more output formats required in the near future.
• Big if/elsif/else chains are ugly. The canonical method of handling this case in Perl is to use a dispatch table: a hash containing subroutine references. See Charles Bailey's response. A tempire offers a more exotic approach, that uses Perl's internal symbol tables for the lookup. – daotoad May 30 '11 at 18:26
• How/where is $outformat specified? – 200_success Oct 13 '14 at 18:33 ## 4 Answers If you can pass in the subroutine it makes the code a lot simpler, you also don't have to deal with an unknown string format as the subroutine itself has been passed in. sub print_result { my$subroutine = shift;
&$subroutine( @_ ); } print_result( \&print_result_text,$arg1, $arg2 ) Otherwise I think I'd go with with a hash of subroutine references. It's easily readable and simple to update. sub print_result { my %print_hash = ( text => \&print_result_text, xml => \&print_result_xml ); if( exists($print_hash{ $outformat } ) ) { &{$print_hash{ $outformat }}( @_ ); } else { printf(STDERR "Error: unsupported output format '%s'\n",$outformat);
}
}
• I think I'll try using the subroutine references. – Daniel Standage Jan 22 '11 at 17:15
I find that Perl's flexibility can help you eliminate many IF/ELSIF/* code constructs, making code much easier to read.
sub print_result {
my ($stats,$tolerance, $outformat) = @_; my$name = "print_result_$outformat"; print "$outformat is not a valid format" and return
if !main->can($name); no strict 'refs'; &$name(@_);
}
sub print_result_xml { ... }
sub print_result_text { ... }
sub print_result_whatever { ... }
Walkthrough
print "$outformat is not a valid format" and return if !main->can($name);
This checks the main namespace (I presume you're not using classes, given your code sample) for the $name subroutine. If it doesn't exist, print an error message and get out. The sooner you exit from a subroutine, the easier your code will be to maintain. no strict 'refs'; no strict 'refs' turns off warnings & errors that would be generated for creating a subroutine reference on the fly (You're using 'use strict', aren't you? If not, for your own sanity, and for the children, start). In this case, since you've already checked for it's existence with main->can, you're safe. &$name(@_);
Now you don't need any central registry of valid formatting subroutines - just add a subroutine with the appropriate name, and your program will work as expected.
If you want to be super hip (some might say awesome), you can replace the last 5 lines of the subroutine with:
no strict 'refs';
main->can($name) and &$name(@_) or print "$outformat is not a valid format"; Whether you find that more readable or not is a simple personal preference; just keep in mind the sort of folk that will be maintaining your code in the future, and make sure to code in accordance with what makes the most sense to them. Perl is the ultimate in flexibility, making it inherently the hippest language in existence. Make sure to follow http://blogs.perl.org and ironman.enlightenedperl.org to keep up with the latest in Modern::Perl. On a separate note, it's Perl, not PERL. The distinction is important in determining reliable & up-to-date sources of Perl ninja-foo. • Why not use eval? eval { &$name(@_); 1 } or print "$outformat is not a valid format\n"; can() may be fooled by autoloaded functions and other magics, but simply trying the call is as authoritative as it gets. – daotoad May 30 '11 at 18:30 • eval is brute force. Calling the function outright assumes it's idempotent and has no unintended side effects. Also, it seems to me errors should always be avoided. Using eval/try/catch is for the !@#$ moments that should never happen. – tempire Jun 3 '11 at 3:39
I'd use anonymous subroutines to make the code cleaner:
my %output_formats = (
'text' => sub {
# print_result_text code goes here
},
'xml' => sub {
# print_result_xml code goes here
},
# And so on
);
sub print_result {
my ($type,$argument1, $argument2) = @_; if(exists$output_formats{$type}) { return$output_formats{$type}->($argument1, $argument2); } else { die "Type '$type' is not a valid output format.";
}
}
I try to avoid Perl if I can, so this is more of a general answer: I've coded like this in an old VB6 app. Each output function had a wrapper function that then called the required implementation using a series of IFs. Sometimes a particular output method wouldn't need anything - eg. "Start New Line" is relevant for text file output, but not Excel worksheet output.
I'm currently in the process of porting/re-writing that particular app in C#/.NET 4, where I've been able to take a much more object oriented approach. I have defined an "output base class" with a standard interface. This is then inherited by the various implementations. As I start the output, I can create the required implementation using a factory method/class, and pass data to it using the standard interface.
This particular implementation is actually multi-threaded, so the bulk of the output base class is actually the thread & queue management. Data is then passed in using a small "data chunk" class, and queued for output using a thread-safe queue.
• The OO approach is definitely making things easier to handle, conceptually and practically. Thanks! – Daniel Standage Jan 22 '11 at 17:16
• Can't downvote this yet, but the OOP approach is completely overkill and unnecessary here. Languages like VB6 and Java (not too sure about C#) don't have first-class functions so that you are supposed to do some OOP magic. Perl has first-class functions and thus you can use them as any other variable. The answer provided by Charles Bailey is thus more correct and also more appropriate for the provided code. – Nikolai Prokoschenko Feb 9 '11 at 13:06
• Seriously, I would down-vote this twice if I could. ( I don't have enough rep to down-vote yet ) – Brad Gilbert Dec 16 '11 at 23:57
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2020-09-22 08:15:10
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http://clay6.com/qa/30584/which-of-the-following-is-a-natural-fibre-
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Browse Questions
# Which of the following is a natural fibre?
$\begin {array} {1 1} (a)\;Starch & \quad (b)\;Cellulose \\ (c)\;Rubber & \quad (d)\;Nylon-6 \end {array}$
Natural polymeric materials such as shellac, amber, wool, silk and natural rubber have been used for centuries. A variety of other natural polymers exist, such as . Cellulose is the most abundant organic compound on Earth, and its purest natural form is cotton.
The woody parts of trees, the paper we make from them, and the supporting material in plants and leaves are also mainly cellulose. Like amylose, it is a polymer made from glucose monomers
Ans : (b)
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2017-04-29 15:35:25
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https://socratic.org/questions/how-do-you-write-the-mixed-expression-r-1-3r-as-a-rational-expression
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# How do you write the mixed expression r+1/(3r) as a rational expression?
Nov 15, 2017
$\frac{3 {r}^{2} + 1}{3 r}$
#### Explanation:
$\text{before we can add the 2 fractions we require them to}$
$\text{have a "color(blue)"common denominator}$
$\text{to obtain this multiply the numerator/denominator of}$
$r \text{ by } 3 r$
$\Rightarrow \frac{r}{1} \times \frac{3 r}{3 r} = \frac{3 {r}^{2}}{3 r}$
$\Rightarrow r + \frac{1}{3 r}$
$= \frac{3 {r}^{2}}{3 r} + \frac{1}{3 r} \leftarrow \textcolor{b l u e}{\text{common denominator of 3r}}$
$\text{add the numerators leaving the denominator as it is}$
$= \frac{3 {r}^{2} + 1}{3 r}$
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2019-01-21 17:42:03
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https://www.nature.com/articles/s41598-019-43619-3?fbclid=IwAR0xSjlCwH6Gh-5Bvy9F0p_-75ofXTQFes4v5ENtDrJt2LlW2CGxXowTh5M&error=cookies_not_supported&code=3fc19a0e-de09-48c7-a7cf-2788e67d3ea9
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Article | Open | Published:
# Statistical Properties and Predictability of Extreme Epileptic Events
## Abstract
The use of extreme events theory for the analysis of spontaneous epileptic brain activity is a relevant multidisciplinary problem. It allows deeper understanding of pathological brain functioning and unraveling mechanisms underlying the epileptic seizure emergence along with its predictability. The latter is a desired goal in epileptology which might open the way for new therapies to control and prevent epileptic attacks. With this goal in mind, we applied the extreme event theory for studying statistical properties of electroencephalographic (EEG) recordings of WAG/Rij rats with genetic predisposition to absence epilepsy. Our approach allowed us to reveal extreme events inherent in this pathological spiking activity, highly pronounced in a particular frequency range. The return interval analysis showed that the epileptic seizures exhibit a highly-structural behavior during the active phase of the spiking activity. Obtained results evidenced a possibility for early (up to 7 s) prediction of epileptic seizures based on consideration of EEG statistical properties.
## Introduction
Extreme events are rare significant deviations of a system variable from its mean value. This fundamental phenomenon is inherent in many real-life systems and manifests itself as rogue waves in the ocean, extreme rainfalls, financial crisis, traffic jams, monster blackouts in power grids, etc.1,2,3,4. From the physical point of view, a study of extreme events is useful for revealing hidden underlying mechanisms responsible for abnormally large fluctuations. The knowledge of these mechanisms can help in the development of efficient methods for predicting and controlling the system’s extreme behavior.
Extreme events were observed and extensively studied in many deterministic and stochastic systems. Different scenarios of the emergence of extreme events have been discovered in model equations, including coupled oscillators and complex networks5,6,7,8, and evidenced in several physical experiments with fluids, nanophotonics and optical systems9,10,11,12,13,14,15. Sudden climatic changes, epidemics and epilepsy16,17,18,19,20,21 have recently received significant attention from the viewpoint of extreme event theory.
In this work, we focus on epilepsy as a clinical manifestation of extreme events characterized by a recurrent and sudden malfunction of the brain caused by excessive and hyper-synchronous neuron activity in the brain. Almost 50 million people are currently suffering from this disease, which can put the individual’s life at risk due to recurrent and sudden incidence of seizures, loss of consciousness and motor control22. Modern medicine is only able to improve the state of about two thirds of the patients, and surgery can help very few of them. However, no therapy can help one quarter of epileptic patients. Therefore, the prediction of epileptic seizures can greatly improve the life quality of these patients and open new therapeutic possibilities23,24. Furthermore, the solution of this challenging and still open problem would provide benefits ranging from pure fundamental ones, related to the understanding of epileptic seizure origin, to the application of methods for seizure forecasting and control.
In this paper, we consider a special form of epilepsy known as absence epilepsy characterized by the occurrence of spontaneous seizures in the form of spike-wave discharges (SWDs) in cortical and thalamic EEGs25, which is extremely difficult to predict. We apply the extreme event theory to the analysis of statistical properties of epileptic brain activity of rats with a genetically predisposition to absence epilepsy recorded with electroencephalography (EEG). These rats exhibit several hundred spontaneous SWD per day and have a high face and predictive validity to human condition26. The discovered well-pronounced extreme event features of the electrical brain activity provide a possibility for early prediction of epileptic seizures using clinical monitoring and real-time EEG processing. Since in humans the thalamic region is not easily accessible for EEG measuring, the epileptic early-warning signal can be recorded from the cortical area only. The animal models can be easily extrapolated to humans because the mechanisms for absence epilepsy in humans and rats are very similar27. Indeed, there exists a well-validated genetic animal model of absence epilepsy in WAG/Rij rats28,29, which can be easily supplied with intracranial electrodes to record epileptic brain activity.
## Methods
### Experimental procedure
The study was done in 5 male WAG/Rij rats, three of them aged 9 months, and two 11 months. Animals were born and raised at the Institute of Higher Nervous Activity (Moscow, Russian Federation). The experiments were conducted in accordance with the EU Directive 2016/63/EU for animal experiments and approved by the Ethical Committee of Institute of Higher Nervous Activity. Prior to surgery rats were housed in small groups with free access to food and water and were kept at natural lighting conditions. After surgery rats were housed individually. Distress and suffering of animals were minimal.
The recording EEG electrode was implanted epidurally over the frontal cortex (AP +2 mm and L 2.5 mm relative to bregma). Ground and reference electrodes were placed over the cerebellum. The EEG signal that was constantly recorded in freely moving rats during 24 h, was fed into a multi-channel differential amplifier via a swivel contact, filtered by a 0.5–200 Hz band-pass filter and digitized with 400 samples/s per channel.
After experimental procedure experienced neurophysiologist manually marked SWD onsets in recorded 24 h EEG signals of all five rats. Onset is defined as time moment, when the first well-developed “spike” appears. The “spikes” along with the “waves” are distinctive features of SWD. Each “spike” appears as single oscillation with frequency of 7–8 Hz, extremely high amplitude and well-pronounced asymmetry. Thus, appearance of the first “spike” marks onset of SWD and the last “spike” corresponds to offset of SWD.
### Time-frequency analysis
For description of pathological brain activity in terms of extreme behavior, we used a time-frequency representation of the original rat’s EEG via a continuous wavelet transform (CWT), a suitable tool for neurophysiological data analysis30. CWT convolves the EEG signal x(t) with the basic function ψ(η) as
$$W(f,{t}_{0})=\sqrt{f}{\int }_{-{\rm{\infty }}}^{+{\rm{\infty }}}x(t){\psi }^{\ast }(f(t-{t}_{0}))dt,$$
(1)
where ‘*’ stands for complex conjugation. As a basic complex function of CWT, we used the complex Morlet wavelet
$$\psi (\eta )=\frac{1}{\sqrt[4]{\pi }}{e}^{j{\omega }_{0}\eta }{e}^{-\frac{{\eta }^{2}}{2}},$$
(2)
where ω0 = 2π is the wavelet central frequency. In our statistical analysis, we deal with normalized wavelet energy Wn = W/W*, where W is an original value of wavelet energy obtained from (1), and W* is a 99.9th percentile of wavelet energy PDF during normal activity.
In our research we considered frequency range of 2–20 Hz. We chose this particular range since it includes all important SWD-related frequency components: main frequency of SWD (7–8 Hz), its first harmonics (14–16 Hz), possible preictal activity (2–4 and 5–9 Hz)30.
The wavelet analysis of EEG recordings was done using home written C/Cuda software for increasing computation performance31,32.
## Results and Discussion
SWDs are known to be an abnormal form of brain activity originated from hyper-synchronization in the cortico-thalamo-cortical neuronal network33,34. It is visually detected in long-term EEG recordings as an abrupt appearance of large-amplitude oscillations (Fig. 1(a)). Unlike typical extreme events, manifested as a short-term deviation of a measured variable from its normal state, individual SWD represents a regular sequence of spikes having a well-pronounced frequency (Fig. 1(b)). Thus, we refer SWD to as a single temporally distributed extreme event.
As can be seen from Fig. 1(c) which displays the time-frequency image of the corresponding EEG segment, SWD manifests itself as a sharp increase in the wavelet energy in the range of the main frequency (6–9 Hz) and its second harmonic (12–18 Hz). At the same time, the level of wavelet energy in the low-frequency range (<6 Hz) does not significantly change, as compared to the normal state. Due to the relation between wavelet energy in a particular frequency range and a size of neuronal population involved into particular rhythmic activity35,36, we concluded that during SWD, the majority of neurons located in the area of the EEG electrode implantation are in a synchronous bursting regime at 6–9 Hz and 12–18 Hz.
In the particular case presented in Fig. 1(a–c), the main SWD frequency is approximately 7 Hz. Considering the long-term time evolution of wavelet energy (Fig. 2(a)), one can note the difference between normal (left panel) and pathological (right panel) brain dynamics at the typical SWD frequency (7 Hz). Figure 2(b) displays the probability density functions (PDFs) of wavelet energy obtained from experimental data along with fitted distributions corresponding to normal (left panel) and pathological (right panel) behaviors. Surprisingly, we have uncovered a counter-intuitive fact that PDF of wavelet energy in case of normal EEG is not subject to Gaussian distribution with p < 0.01 via Pearson’s chi squared test (black curve in Fig. 2(b)). Instead, it is perfectly fitted by unimodal Weibull distribution (shape parameter b > 2)
$${f}_{W}(W|a,b)=\frac{b}{a}{(\frac{W}{a})}^{b-1}{e}^{-{(W/a)}^{b}}$$
(3)
with scale parameter a = 0.395, shape parameter b = 2.14 and p > 0.99 via Pearson’s chi squared test (yellow curve in Fig. 2(b)). Note that the perfect fitting of unimodal Weibull distribution to normal activity PDF is observed for each spectral component in the considered frequency range. It is known, that Weibull distribution well describes a particle size distribution obtained during fragmentation and performing geometric scale invariance (fractal properties)37. Due to this fact and taking into account the relation between wavelet energy and a size of synchronized neuronal population, we concluded that the process of formation and destruction of coherent clusters in brain cortex during normal activity is not random. On the contrary, it is likely to exhibit well-pronounced structural properties.
Compared to a normal state, pathological brain activity results in a well-developed heavy tail in the wavelet energy PDF. To validate the fact that the long tail is associated with the extreme behavior, we applied the extreme value theory, namely, the Pickands-Balkema-de Haan theorem38,39, and showed that the elongated tail can be fitted by the heavy-tailed Weibull distribution (shape parameter b < 1) with parameters a = 0.27, b = 0.73 and p > 0.99 via Pearson’s chi squared test (dark blue curve in Fig. 2(b)). Note that goodness of tail-fitting was tested in the range of Wn > 1. At the same time, the wavelet energy PDF corresponding to pathological brain activity is badly fitted by the unimodal Weibull distribution with p < 0.01 via Pearson’s chi squared test. In this case, the value of χ2 statistics provides the measure of the extremal behavior. Analyzing the dependence of χ2 on the oscillation frequency averaged over the group of participating rats (Fig. 3), one can observe two well-pronounced maxima marked with red dots. These maxima associated with the most extremal behavior correspond to the main SWD frequency (7 Hz) and its second harmonic (14 Hz). Notable, that for f < 6 Hz, 8 Hz <f < 10 Hz and f > 18 Hz the extremal properties are less pronounced. Thus, the statistical analysis demonstrates that abnormal brain activity related to absence epilepsy seizures exhibits well-pronounced properties of extreme dynamics. This type of behavior is localized in particular spectral ranges conditioned by the main frequency and its second harmonic.
It is known40 that during active behavior phases, arousal and deep slow-wave sleep SWDs are rare, because the characteristic interval between subsequent seizures lies in the range from several tens of minutes to one hour. On the contrary, sequences of SWDs with short return times are observed during a state of drowsiness or passive wakefulness (Fig. 4(a,b)). The problem of regularity and interrelation of absence seizures during these stages is of undoubted interest. Despite its close association with the vigilance state, SWD has long been thought as unpredictable in nature, occurred from apparently normal background EEG. Now, it is possible to assess these issues considering SWDs from the viewpoint of the extreme event theory.
To examine the clustering properties of absence epilepsy seizures, i.e., to find correlations in SWD sequences, we carried out a statistical analysis of the return time between adjacent discharges. Figure 4(c) shows the PDF of return intervals τ calculated for recording segments with dense SWD sequences observed in all 5 participating rats. In case of an uncorrelated SWD sequence, one expects that the return intervals are distributed according to the Poisson law. In turn, data correlation and long-term memory are determined by either stretched exponential or power law. As will be shown below, in a wide range of τ, the return time intervals of our data are power-law correlated (p ~ τγ) with γ = 3/2 (p > 0.99 via Pearson’s chi squared test). A good fit of the experimentally obtained return interval distribution by a power law is well-reproduced in the group of rats. This confirms that epileptic seizures during stages of a developed spiking behavior exhibit scaling properties and long-range correlations. Note, that our findings are in a good agreement with previous theoretical and experimental studies of intermittent behavior in epileptic brain41,42,43,44,45.
To prove the effect of data correlation observed in the PDF of return times, we apply the well-established technique, known as detrended fluctuation analysis (DFA)46, which allows studying a long-term evolution of wavelet energy W(t) in a wide range of frequencies. In long-term correlated data, the mean fluctuation F(s) of the signal in time window s obeys a power law logF(s) ~ α logs. Figure 4(d) displays the F(s) scaling, averaged over all experimental rats, observed in the wavelet energy time evolution of main frequency oscillations (7 Hz). It is seen that logF(s) is almost a straight line with the slope α = 0.945 in the log-log scale. As seen in the insert, the maximal slope, i.e., the maximal correlation occurs for 7-Hz and 14-Hz oscillations, that is well-correlated with the results of the statistical analysis presented in Fig. 3. Since these frequencies indicate dominant and subdominant SWD frequencies, the extreme behavior here is strongly pronounced. Thus, according to the return times analysis and DFA we can conclude that the rat’s epileptic brain exhibits a highly structural and self-organized behavior during dense spiking activity phases.
The uncovered properties of a long-range correlation in the epileptic behavior are inherent to systems in the vicinity of a critical point47, where the system amplifies any fluctuations due to increasing instability. This effect known as prebifurcation signal (noise) amplification has been observed in physical, ecological and biomedical systems (see, e.g.48,49,50,51,). Notably, this effect is completely unobvious from original EEG recording, but clearly seen when considering wavelet energy evolution in a particular spectral range is associated with the main SWD frequency (Fig. 5(a,b)). To reveal this phenomenon in long-term epileptic EEG records, we considered distributions of the wavelet energy amplitudes of 7-Hz oscillations assessed within 10 randomly chosen EEG fragments from single rat recordings. Each fragment contained SWDs, which onsets had been manually marked by expert-neurophysiologist. From these fragments we collected 1-s epochs of ictal activity (1 s after onset), preictal activity (1 s before onset) and interictal activity far before onset (10 s before). Afterwards, we constructed wavelet energy PDFs corresponding to each type of brain activity across collected epochs (Fig. 5(c–e)). As seen from this figure, each type of brain activity is characterized by a specific form of Weibull curve. During normal brain activity, the wavelet energy variation is very low and the scaling parameter a of fitted unimodal Weibull distribution is also small (Fig. 5(c)). However, when a seizure approaches, the fluctuations of the wavelet energy increase along with a. It is seen that preictal activity clearly differs from normal and ictal activities, characterized by the highest variance and the scaling parameter a (Fig. 5(c–e)). It follows that the transition from normal brain activity to the seizure does not occur abruptly; but it is preceded by a well-defined precursor with distinctive statistical properties. Thus, the time interval of SWD prediction can be measured in a following way. The distribution of wavelet energies is constructed in a floating 1 second window and when it is well-fitted with known preictal PDF (Fig. 5(d)) the precursor is detected. The goodness of fit is tested via Pearson’s chi squared test (p > 0.9). Therefore, the time interval from precursor detection to SWD onset is the prediction interval. To verify the predictability of absence epilepsy seizures we have checked the prediction intervals for 50 epileptic events collected over all 5 participating rats (10 seizures for each animal). Corresponding histogram of prediction intervals is presented in Fig. 5(f). One can see, that prediction intervals are of 1–7 s. At the same time, 5 of 50 seizures have been either poorly predicted (up to 0.5 s) or detected on the onset.
The obtained results are related to the important problem of early prediction of SWD seizures52. According to the Review by van Luijtelaar et al.53 a considerable success has been achieved in the field of absence seizures detection during last years, yet the research on their prediction is not so fruitful. However, there is a number of successful attempts in this area. In particular, Li et al.54 have provided a predictability analysis of absence seizures via permutation entropy approach. They considered experimental EEG dataset of 28 rat (GAERS) containing 314 seizures, from which 169 have been predicted with average anticipation time of 4.9 s. Van Luijtelaar et al.55 have analyzed the origin of SWDs in WAG/Rij rats and discovered that absence seizures are preceded by Δ (1–4 Hz) and θ (4.5–8 Hz) precursors. Afterwards, Maksimenko et al.34 have developed a system for real-time absence seizure control based on detecting Δ and θ precursors. This system allows 45% of seizures to be predicted with anticipation time of 1–2 s. Also, Sorokin et al.56 have demonstrated the correlation between SWDs and preictal changes in β oscillations (20–40 Hz) 1.5 s prior seizure onset robust across different recordings, which seems to be relevant in developing new predictive algorithms.
In the context, results of our research are in agreement with mentioned studies on seizure prediction. We suppose, that possibility to predict seizures for 7 s interval is exciting, since it opens a way to prevent ongoing seizure by optogenetic or electrical brain stimulation, where early prediction is highly demanded57,58.
## Conclusions
To summarize, we have studied epileptic brain dynamics using the extreme value theory. We have shown, for the first time to the best of our knowledge, that during periods of spiking activity, the epileptic brain exhibits statistical properties of extreme events in the range of spike-wave-discharge (SWD) characteristic frequencies. It is notable, that uncovered statistical properties of SWDs are more in line with classical definition of extreme events, than with its special type – dragon-king behavior – as one may expect from numerical modeling of neuronal systems59. The detailed analysis of epileptic brain EEG recordings from the viewpoint of extreme events revealed self-organization properties of the brain’s spiking activity. In particular, we have found that return intervals between epileptic seizures obey a power law behavior in long-range correlations in the brain. By considering the brain as a dynamical system, we detected an increase in the fluctuation amplitude near a critical point preceding a seizure. The presented results open a new possibility for early SWD prediction by real-time tracing of the variance of the wavelet energy PDF.
## References
1. 1.
Fedele, F., Brennan, J., De León, S. P., Dudley, J. & Dias, F. Real world ocean rogue waves explained without the modulational instability. Scientific Reports 6, 27715 (2016).
2. 2.
Goswami, B. N., Venugopal, V., Sengupta, D., Madhusoodanan, M. & Xavier, P. K. Increasing trend of extreme rain events over india in a warming environment. Science 314, 1442–1445 (2006).
3. 3.
Aloui, R., Aïssa, M. S. B. & Nguyen, D. K. Global financial crisis, extreme interdependences, and contagion effects: The role of economic structure? Journal of Banking and Finance 35, 130–141 (2011).
4. 4.
Helbing, D. Globally networked risks and how to respond. Nature 497, 51 (2013).
5. 5.
Nicolis, C., Balakrishnan, V. & Nicolis, G. Extreme events in deterministic dynamical systems. Phys. Rev. Lett. 97, 210602 (2006).
6. 6.
Kishore, V., Santhanam, M. & Amritkar, R. Extreme events on complex networks. Phys. Rev. Lett. 106, 188701 (2011).
7. 7.
Cavalcante, H. L. d. S., Oriá, M., Sornette, D., Ott, E. & Gauthier, D. J. Predictability and suppression of extreme events in a chaotic system. Phys. Rev. Lett. 111, 198701 (2013).
8. 8.
Kingston, S. L., Thamilmaran, K., Pal, P., Feudel, U. & Dana, S. K. Extreme events in the forced liénard system. Phys. Rev. E 96, 052204 (2017).
9. 9.
Chabchoub, A. et al. Observation of a hierarchy of up to fifth-order rogue waves in a water tank. Phys. Rev. E 86, 056601 (2012).
10. 10.
Liu, C. et al. Triggering extreme events at the nanoscale in photonic seas. Nature Physics 11, 358 (2015).
11. 11.
Montina, A., Bortolozzo, U., Residori, S. & Arecchi, F. Non-gaussian statistics and extreme waves in a nonlinear optical cavity. Phys. Rev. Lett. 103, 173901 (2009).
12. 12.
Bonatto, C. et al. Deterministic optical rogue waves. Phys. Rev. Lett. 107, 053901 (2011).
13. 13.
Dudley, J. M., Dias, F., Erkintalo, M. & Genty, G. Instabilities, breathers and rogue waves in optics. Nature Photonics 8, 755 (2014).
14. 14.
Walczak, P., Randoux, S. & Suret, P. Optical rogue waves in integrable turbulence. Phys. Rev. Lett. 114, 143903 (2015).
15. 15.
Selmi, F. et al. Spatiotemporal chaos induces extreme events in an extended microcavity laser. Phys. Rev. Lett. 116, 013901 (2016).
16. 16.
Albeverio, S., Jentsch, V. & Kantz, H. Extreme events in nature and society. (Springer Science and Business Media, 2006).
17. 17.
Field, C. B., Barros, V., Stocker, T. F. & Dahe, Q. Managing the risks of extreme events and disasters to advance climate change adaptation: Special report of the intergovernmental panel on climate change. (Cambridge University Press, 2012).
18. 18.
Boers, N. et al. Prediction of extreme floods in the eastern central andes based on a complex networks approach. Nature Communications 5, 5199 (2014).
19. 19.
Lehnertz, K. Epilepsy: Extreme events in the human brain. In Extreme Events in Nature and Society, 123–143 (Springer, 2006).
20. 20.
Osorio, I., Frei, M. G., Sornette, D., Milton, J. & Lai, Y.-C. Epileptic seizures: quakes of the brain? Phys. Rev. E 82, 021919 (2010).
21. 21.
Kuhlmann, L., Lehnertz, K., Richardson, M. P., Schelter, B. & Zaveri, H. P. Seizure prediction ready for a new era. Nature Reviews Neurology 1 (2018).
22. 22.
Moshé, S. L., Perucca, E., Ryvlin, P. & Tomson, T. Epilepsy: new advances. The Lancet 385, 884–898 (2015).
23. 23.
Mormann, F., Andrzejak, R. G., Elger, C. E. & Lehnertz, K. Seizure prediction: the long and winding road. Brain 130, 314–333 (2006).
24. 24.
Gadhoumi, K., Lina, J.-M., Mormann, F. & Gotman, J. Seizure prediction for therapeutic devices: A review. Journal of neuroscience methods 260, 270–282 (2016).
25. 25.
Bosnyakova, D. et al. Some peculiarities of time–frequency dynamics of spike–wave discharges in humans and rats. Clinical Neurophysiology 118, 1736–1743 (2007).
26. 26.
Depaulis, A. & van Luijtelaar, G. Chapter 18 - genetic models of absence epilepsy in the rat. In Pitkänen, A., Schwartzkroin, P. A. & Moshé, S. L. (eds) Models of Seizures and Epilepsy, 233–248, https://doi.org/10.1016/B978-012088554-1/50020-7 (Academic Press, Burlington, 2006).
27. 27.
Pitkänen, A., Buckmaster, P., Galanopoulou, A. S. & Moshé, S. L. Models of seizures and epilepsy. (Academic Press, 2017).
28. 28.
Coenen, A. & Van Luijtelaar, E. The wag/rij rat model for absence epilepsy: age and sex factors. Epilepsy research 1, 297–301 (1987).
29. 29.
Coenen, A. & Van Luijtelaar, E. Genetic animal models for absence epilepsy: a review of the wag/rij strain of rats. Behavior Genetics 33, 635–655 (2003).
30. 30.
Hramov, A. E., Koronovskii, A. A., Makarov, V. A., Pavlov, A. N. & Sitnikova, E. Wavelets in Neuroscience (Springer, 2016).
31. 31.
Maksimenko, V. A., Grubov, V. V. & Kirsanov, D. V. Use of parallel computing for analyzing big data in eeg studies of ambiguous perception. In Dynamics and Fluctuations in Biomedical Photonics XV, vol. 10493, 104931H (International Society for Optics and Photonics, 2018).
32. 32.
Grubov, V. & Nedaivozov, V. Stream processing of multichannel eeg data using parallel computing technology with nvidia cuda graphics processors. Technical Physics Letters 44, 453–455 (2018).
33. 33.
Jiruska, P. et al. Synchronization and desynchronization in epilepsy: controversies and hypotheses. The Journal of Physiology 591, 787–797 (2013).
34. 34.
Maksimenko, V. A. et al. Absence seizure control by a brain computer interface. Scientific Reports 7, 2487 (2017).
35. 35.
Hramov, A. E. et al. Analysis of the characteristics of the synchronous clusters in the adaptive kuramoto network and neural network of the epileptic brain. In Saratov Fall Meeting 2015: Third International Symposium on Optics and Biophotonics and Seventh Finnish-Russian Photonics and Laser Symposium (PALS), vol. 9917, 991725 (International Society for Optics and Photonics, 2016).
36. 36.
Maksimenko, V. A. et al. Macroscopic and microscopic spectral properties of brain networks during local and global synchronization. Physical Review E 96, 012316 (2017).
37. 37.
Brown, W. K. & Wohletz, K. H. Derivation of the weibull distribution based on physical principles and its connection to the rosin–rammler and lognormal distributions. Journal of Applied Physics 78, 2758–2763 (1995).
38. 38.
Balkema, A. A. & De Haan, L. Residual life time at great age. The Annals of Probability 2, 792 (1974).
39. 39.
Pickands, J. III. Statistical inference using extreme order statistics. The Annals of Statistics 3, 119 (1975).
40. 40.
Sarkisova, K. & van Luijtelaar, G. The wag/rij strain: a genetic animal model of absence epilepsy with comorbidity of depressiony. Progress in Neuro-Psychopharmacology and Biological Psychiatry 35, 854–876 (2011).
41. 41.
Hramov, A., Koronovskii, A. A., Midzyanovskaya, I., Sitnikova, E. & Van Rijn, C. On-off intermittency in time series of spontaneous paroxysmal activity in rats with genetic absence epilepsy. CHAOS: An Interdisciplinary Journal of Nonlinear Science 16, 043111 (2006).
42. 42.
Sitnikova, E., Hramov, A. E., Grubov, V. V., Ovchinnkov, A. A. & Koronovsky, A. A. On–off intermittency of thalamo-cortical oscillations in the electroencephalogram of rats with genetic predisposition to absence epilepsy. Brain research 1436, 147–156 (2012).
43. 43.
Goodfellow, M., Schindler, K. & Baier, G. Intermittent spike–wave dynamics in a heterogeneous, spatially extended neural mass model. Neuroimage 55, 920–932 (2011).
44. 44.
Maris, E., Bouwman, B. M., Suffczynski, P. & van Rijn, C. M. Starting and stopping mechanisms of absence epileptic seizures are revealed by hazard functions. Journal of neuroscience methods 152, 107–115 (2006).
45. 45.
Suffczynski, P. et al. Dynamics of epileptic phenomena determined from statistics of ictal transitions. IEEE Transactions on Biomedical Engineering 53, 524–532 (2006).
46. 46.
Kantelhardt, J. W., Koscielny-Bunde, E., Rego, H. H., Havlin, S. & Bunde, A. Detecting long-range correlations with detrended fluctuation analysis. Physica A 295, 441–454 (2001).
47. 47.
Bak, P., Tang, C. & Wiesenfeld, K. Self-organized criticality: An explanation of the 1/f noise. Phys. Rev. Lett. 59, 381 (1987).
48. 48.
Corbalán, R., Cortit, J., Pisarchik, A. N., Chizhevsky, V. N. & Vilaseca, R. Investigation of a co2 laser response to loss perturbation near period-doubling. Phys. Rev. A 118, 663–668 (1995).
49. 49.
Huerta-Cuellar, G., Pisarchik, A. N., Kir’yanov, A. V., Barmenkov, Y. O. & del Valle Hernández, J. Prebifurcation noise amplification in a fiber laser. Phys. Rev. E 79, 036204 (2009).
50. 50.
Pisarchik, A. N., Pochepen, O. N. & Pisarchyk, L. A. Increasing blood glucose variability is a precursor of sepsis and mortality in burned patients. PLoS One 7, e46582 (2012).
51. 51.
Stolbova, V., Surovyatkina, E., Bookhagen, B. & Kurths, J. Tipping elements of the indian monsoon: Prediction of onset and withdrawal. Geophysical Research Letters 43, 3982–3990 (2016).
52. 52.
Ovchinnikov, A. A., Luttjohann, A., Hramov, A. E. & Luijtelaar van, G. An algorithm for real-time detection of spike-wave discharges in rodents. Journal of Neuroscience Methods 194, 172–178 (2010).
53. 53.
Luijtelaar van, G. et al. Methods of automated absence seizure detection, interference by stimulation, and possibilities for prediction in genetic absence models. Journal of Neuroscience Methods 260, 144–158 (2016).
54. 54.
Li, X., Ouyang, G. & Richards, D. A. Predictability analysis of absence seizures with permutation entropy. Epilepsy research 77, 70–74 (2007).
55. 55.
van Luijtelaar, G., Sitnikova, E. & Luttjohann, A. On the origin and suddenness of absences in genetic absence models. Clinical EEG and Neuroscience 42, 83–97 (2011).
56. 56.
Sorokin, J. M., Paz, J. T. & Huguenard, J. R. Absence seizure susceptibility correlates with pre-ictal β oscillations. Journal of Physiology-Paris 110, 372–381 (2016).
57. 57.
Krook-Magnuson, E., Armstrong, C., Oijala, M. & Soltesz, I. On-demand optogenetic control of spontaneous seizures in temporal lobe epilepsy. Nature communications 4, 1376 (2013).
58. 58.
Paz, J. T. & Huguenard, J. R. Optogenetics and epilepsy: past, present and future. Epilepsy currents 15, 34–38 (2015).
59. 59.
Mishra, A. et al. Dragon-king-like extreme events in coupled bursting neurons. Physical Review E 97, 062311 (2018).
## Acknowledgements
This work has been supported by Russian Science Foundation (Grant 17-72-10183) in the part of big neurophysiological data processing. This paper was also developed within the scope of the IRTG 1740 /TRP 2015/50122-0, funded by the DFG/FAPESP.
## Author information
A.E.H., N.S.F., A.N.P. and J.K. conceived the study; E.Yu.S. collected EEG data; V.V.G., V.A.M., V.V.M. carried out time-frequency analysis of data set; N.S.F. and A.N.P. provided statistical analysis of data set; A.L. carried out prediction tests and gave biological interpretation. All authors wrote the manuscript.
Correspondence to Alexander E. Hramov.
## Ethics declarations
### Competing Interests
The authors declare no competing interests.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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2019-07-20 06:11:48
|
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https://cdsweb.cern.ch/collection/ATLAS%20Conference%20Slides?ln=no
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# CERN Accelerating science
The ATLAS Conference Slides collection contains transparencies on ATLAS that have been presented at conferences. More information about conferences can be found on the external ATLAS Talks at International Physics Conferences webpage.
SUBMIT ATLAS SLIDE
# ATLAS Conference Slides
Nyeste elementer:
2015-08-04
17:11
Electroweak Bosons in Heavy Ion Collisions with the ATLAS Detector / Hu, Qipeng (University of Science and Technology of China) Electroweak bosons processes (W, Z and photon) provide experimental controls over initial geometric and nuclear PDFs (nPDFs). The ATLAS has measured the production of all three bosons in Pb+Pb at 2.76 TeV and the production of Z bosons in p+Pb collisions at 5.02 TeV. [...] ATL-PHYS-SLIDE-2015-449.- Geneva : CERN, 2015 Fulltext: PDF; External link: Original Communication (restricted to ATLAS)
2015-08-04
15:35
Search for Higgs bosons decaying to aa in the mumu tautau final state in pp collisions at root(s) = 8 TeV with the ATLAS experiment / Kaplan, Benjamin (Department of Physics, New York University) A search for the decay to a pair of new particles of either the 125 GeV Higgs boson (ℎ) or a second CP-even Higgs boson (H) is presented. The dataset corresponds to an integrated luminosity of 20.3 fb−1 of pp collisions at √s= 8 TeV recorded by the ATLAS experiment at the LHC in 2012. [...] ATL-PHYS-SLIDE-2015-448.- Geneva : CERN, 2015 - 1 p. Fulltext: PDF; External link: Original Communication (restricted to ATLAS) In : European Physical Society Conference on High Energy Physics 2015, Vienna, Austria, 22 - 29 Jul 2015
2015-08-04
14:40
Search for diboson resonances with jets in 20 fb$^{−1}$ of pp collisions at $\sqrt(s) = 8$ TeV with the ATLAS detector / Picazio, Attilio (Section de Physique, Universite de Geneve) A search for narrow diboson resonances in a dijet final state is performed in 20.3 fb$−1$ of proton-proton collisions at a center-of-mass energy of $\sqrt(s) = 8$ TeV, collected in 2012 by the ATLAS detector at the Large Hadron Collider. The jet mass and jet substructure properties have been used to tag each jet as a boson. [...] ATL-PHYS-SLIDE-2015-447.- Geneva : CERN, 2015 - 1 p. Fulltext: PDF; External link: Original Communication (restricted to ATLAS) In : European Physical Society Conference on High Energy Physics 2015, Vienna, Austria, 22 - 29 Jul 2015
2015-08-04
14:39
Search for the Standard Model Higgs boson produced in association with top quarks and decaying into a bbbar-pair in pp collisions at sqrt(s)=8 TeV with the ATLAS detector / Serkin, Leonid (INFN Gruppo Collegato di Udine and ICTP, Trieste) A search for the Standard Model Higgs boson produced in association with a pair of top quarks (ttH) and decaying into a pair of bottom quarks (H→bb) is presented. The search is focused on the semileptonic decay of the tt system and exploits different topologies given by the jet and b-tagged jet multiplicities of the event. [...] ATL-PHYS-SLIDE-2015-446.- Geneva : CERN, 2015 - 1 p. Fulltext: PDF; External link: Original Communication (restricted to ATLAS) In : European Physical Society Conference on High Energy Physics 2015, Vienna, Austria, 22 - 29 Jul 2015
2015-08-04
13:15
ATLAS combinations, couplings and spin/CP / Polifka, Richard (Department of Physics, University of Toronto) Summary talk of latest ATLAS measurement of Higgs boson couplings and SpinCP properties ATL-PHYS-SLIDE-2015-445.- Geneva : CERN, 2015 - 12 p. Fulltext: PDF; External link: Original Communication (restricted to ATLAS)
2015-08-03
19:11
New results on two-particle correlations in proton-proton collisions at 13 TeV from ATLAS at the LHC / Arratia, Miguel (Cavendish Laboratory, University of Cambridge) - ATL-PHYS-SLIDE-2015-444.- Geneva : CERN, 2015 Fulltext: PDF; External link: Original Communication (restricted to ATLAS) In : European Physical Society Conference on High Energy Physics 2015, Vienna, Austria, 22 - 29 Jul 2015
2015-08-03
18:40
Jet results in heavy ion collisions with the ATLAS experiment at the LHC / Perepelitsa, Dennis (Brookhaven National Laboratory (BNL)) In relativistic collisions of heavy ions, a hot medium with a high density of unscreened color charges is produced, and jets propagating through this medium are known to suffer energy loss. This results in several distinct effects seen in central heavy ion collisions: the yield of inclusive jets measured via the nuclear modification factor is observed to be strongly suppressed; the yield of events with highly asymmetric dijet pairs is observed to be increased; the jet fragmentation is modified. [...] ATL-PHYS-SLIDE-2015-443.- Geneva : CERN, 2015 Fulltext: PDF; External link: Original Communication (restricted to ATLAS) In : European Physical Society Conference on High Energy Physics 2015, Vienna, Austria, 22 - 29 Jul 2015
2015-08-03
18:11
Search for squarks and gluinos in final state with jets and missing transverse momentum with the ATLAS detector / Ronzani, Manfredi (Albert-Ludwigs-Universitaet Freiburg, Fakultaet fuer Mathematik und Physik) Many extensions of the Standard Model (SM) include heavy coloured particles, such as the squarks and gluinos of supersymmetric (SUSY) theories, which could be accessible at the Large Hadron Collider (LHC) and detected by ATLAS. The current searches in the LHC Run-1 dataset have yielded sensitivity to TeV scale gluinos, as well as to squarks in the hundreds of GeV mass range. [...] ATL-PHYS-SLIDE-2015-442.- Geneva : CERN, 2015 - 1 p. Fulltext: PDF; External link: Original Communication (restricted to ATLAS) In : European Physical Society Conference on High Energy Physics 2015, Vienna, Austria, 22 - 29 Jul 2015
2015-08-03
14:15
Exotic Higgs Decays with photon(s), missing transverse momentum and forward jets / Bernius, Catrin (Department of Physics, New York University) A search is performed for Higgs-boson decays to neutralinos and/or gravitinos in events with at least one photon, missing transverse momentum (ETmiss) and two forward jets, a topology where vector boson fusion (VBF) production is enhanced. The analysis is based on a dataset of proton-proton collision data taken at √s = 8 TeV delivered by the Large Hadron Collider and recorded with the ATLAS detector, corresponding to an integrated luminosity of 20.3 fb−1. [...] ATL-PHYS-SLIDE-2015-441.- Geneva : CERN, 2015 - 1 p. Fulltext: PDF; External link: Original Communication (restricted to ATLAS) In : European Physical Society Conference on High Energy Physics 2015, Vienna, Austria, 22 - 29 Jul 2015
2015-08-03
13:47
Search for tt(H→bb) using the ATLAS detector at 8 TeV / Nackenhorst, Olaf (Georg-August-Universitat Goettingen, II. Physikalisches Institut) A search for the Standard Model Higgs boson produced in association with a pair of top quarks, $t\bar{t}H$, is presented. The analysis uses 20.3 fb$^{-1}$ of pp collision data at $\sqrt{s}$ = 8 TeV, collected with the ATLAS detector at the Large Hadron Collider during 2012. [...] ATL-PHYS-SLIDE-2015-440.- Geneva : CERN, 2015 Fulltext: Nackenhorst_Higgs15_v7 - PDF; ATL-PHYS-SLIDE-2015-440 - PDF; External link: Original Communication (restricted to ATLAS) In : 6th Workshop on Higgs Hunting, Orsay, France, 30 Jul - 01 Aug 2015
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2015-08-05 12:31:01
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http://www.optimization-online.org/DB_HTML/2019/05/7216.html
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- Hybrid methods for nonlinear least squares problems Ladislav Luksan(luksancs.cas.cz) Ctirad Matonoha(matonohacs.cas.cz) Jan Vlcek(vlcekcs.cas.cz) Abstract: This contribution contains a description and analysis of effective methods for minimization of the nonlinear least squares function $F(x) = (1/2) f^T(x) f(x)$, where $x \in R^n$ and $f \in R^m$, together with extensive computational tests and comparisons of the introduced methods. All hybrid methods are described in detail and their global convergence is proved in a unified way. Some proofs concerning trust region methods, which are difficult to find in the literature, are also added. In particular, the report contains an analysis of a new simple hybrid method with Jacobian corrections (Section~8) and an investigation of the simple hybrid method for sparse least squares problems proposed previously in [33] (Section~14). Keywords: Numerical optimization, nonlinear least squares, trust region methods, hybrid methods, sparse problems, partially separable problems, numerical experiments Category 1: Nonlinear Optimization (Nonlinear Systems and Least-Squares ) Category 2: Nonlinear Optimization (Unconstrained Optimization ) Citation: Technical Report V-1246, Institute of Computer Science AVCR, Prague, May 2019. Download: [Postscript][Compressed Postscript][PDF]Entry Submitted: 05/17/2019Entry Accepted: 05/17/2019Entry Last Modified: 05/17/2019Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.
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2020-01-22 10:46:01
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http://library.kiwix.org/ell.stackexchange.com_en_all_2020-11/A/question/178032.html
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## When to use "divides into"
3
I am trying to find out when the following phrase is correct and what is a better choice for the cases where it is not:
... (1) divides into ... (2)
# Examples
• Example 1: (1) is for example a country and (2) are states: "The US divides into 51 states"
• Example 2: (1) is a river which branches (gets split up) into several arms at one point, or a corridor: "The corridor starts in Bulgaria, crosses Turkey and then divides into two branches, one along the coast to Syria, Libya, Israel and Egypt, and the other through the Syrian and Jordanian plateaux" (source), "the river divides into several arms (furcation) and it begins to flow in bends" (source), more examples see linguee
• Example 3: (1) consists of several subparts, e.g. a software program consists of components: "The software divides into data integration and data import/export." (source)
# My assumptions
I think in general "divides into" is a bad choice if it means (1) is composed of (made up of) various parts.
It may be a good choice, if (1) is actually being split up at one point, like the river in example 2.
I think example 2 is ok, but the phrases in example 1 and 3 do not feel quite right.
# My question
Are my assumptions correct? What would be better?
• consists of?
• comprises?
1What's the 51st state? – Eddie Kal – 2018-08-30T14:21:21.823
3
I think that you are correct with your assumptions, and where you are getting hung up is that divide is commonly used in the passive voice. For examples 1 and 3, it is natural to use the passive voice, as you have the sentences constructed.
The US is divided into 50 states.
Here, the implicit actors are the statesmen who have carved out the states' boundaries over the past 200+ years. If, instead, you would like to make this be in the active voice, without referencing those statesmen, you could use something like:
50 states make up the US.
For the third example, we would say:
The software can be divided into data integration and data import/export.
Here, the implicit actors could be assumed to be the software developers who would have designed the software with those different segments. Here, if you want to use the active voice, it would be natural to use one of your suggestions:
The software consists of data integration and data import/export components.
In the second example, we ascribe a sort of sentience to nature, and so it is natural for us to say that the river divides, as though it has chosen to divide itself, just as we would say that cells divide themselves during mitosis.
1
Those statesmen would have carved out the state boundaries rather than carved them up. https://forum.wordreference.com/threads/carve-out-vs-carve-up.3375230/
– Ronald Sole – 2018-08-30T16:41:44.133
@RonaldSole Thanks. When I was typing it out, I was thinking of carving up the US, but obviously, I used the states' boundaries as my object, so they would have been carved out. Fixed that. – mathewb – 2018-08-30T19:13:01.843
Comprise would be better, e.g. The United States is comprised of 50 states or The software is comprised of data integration and data import/export components (although I rather don't mind consists of in the second example). – Jim MacKenzie – 2018-08-30T19:31:25.800
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2021-03-09 08:02:34
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https://proofwiki.org/wiki/Subgroup_of_Elements_whose_Order_Divides_Integer
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# Subgroup of Elements whose Order Divides Integer
Jump to navigation Jump to search
## Theorem
Let $A$ be an abelian group.
Let $k \in \Z$ and $B$ be a set of the form:
$\left\{{x \in A : x^k = e}\right\}$
Then $B$ is a subgroup of $A$.
## Proof
First note that the identity $e$ satisfies $e^k = e$ and so $B$ is non-empty.
Now assume that $a, b \in B$.
Then:
$\displaystyle \left({ab^{-1} }\right)^k$ $=$ $\displaystyle a^k \left({b^{-1} }\right)^k$ Power of Product in Abelian Group $\displaystyle$ $=$ $\displaystyle a^k \left({b^k}\right)^{-1}$ Powers of Group Elements $\displaystyle$ $=$ $\displaystyle ee^{-1}$ as $a^k = b^k = e \in B$ $\displaystyle$ $=$ $\displaystyle e$ Identity is Self-Inverse
Hence, by the One-Step Subgroup Test, $B$ is a subgroup of $A$.
$\blacksquare$
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2020-12-01 05:17:24
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https://www.intechopen.com/books/recent-development-in-optoelectronic-devices/infrared-sensors-for-autonomous-vehicles
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Open access peer-reviewed chapter
# Infrared Sensors for Autonomous Vehicles
By Rajeev Thakur
Submitted: June 21st 2017Reviewed: August 11th 2017Published: December 20th 2017
DOI: 10.5772/intechopen.70577
## Abstract
The spurt in interest and development of Autonomous vehicles is a continuing boost to the growth of electronic devices in the automotive industry. The sensing, processing, activation, feedback and control functions done by the human brain have to be replaced with electronics. The task is proving to be exhilarating and daunting at the same time. The environment sensors – RADAR (RAdio Detection And Ranging), Camera and LIDAR (Light Detection And Ranging) are enjoying a lot attention with the need for increasingly greater range and resolution being demanded by the “eyes” and faster computation by the “brain”. Even though all three and more sensors (Ultrasonic / Stereo Camera / GPS / etc.) will be used together; this chapter will focus on challenges facing Camera and LIDAR. Anywhere from 2 – 8 cameras and 1 – 2 LIDAR are expected to be part of the sensor suite needed by Autonomous vehicles – which have to function equally well in day and night. Near infrared (800 – 1000nm) devices are currently emitters of choice in these sensors. Higher range, resolution and Field of view pose many challenges to overcome with new electronic device innovations before we realize the safety and other benefits of autonomous vehicles.
### Keywords
• autonomous vehicles
• infrared
• sensors
• LIDAR
• camera
## 1. Introduction
The Federal Automated Vehicles Policy [2] document released by NHTSA in September 2016 states that 35,092 people died on US roadways in 2015 and 94% of the crashes were attributed to human error. Highly automated vehicles (HAVs) have the potential to mitigate most of these crashes. They also have such advantages as not being emotional, not fatiguing like humans, learning from past mistakes of their own and other HAVs, being able to use complementary technologies like Vehicle-to-Vehicle (V2V) and Vehicle-to-Infrastructure (V2I) – which could further enhance system performance. Add in the potential to save energy and reduce pollution (better fuel economy, ride sharing and electrification) – creating a huge impetus to implement autonomous vehicle technology as soon as possible.
On the other hand we have the consumer industry from Silicon Valley eyeing autonomous vehicles as a huge platform to engage, interact, customize and monetize the user experience. Think online shopping, watching a movie, doing your email or office work, video chats, customized advertisements based on user profile and location, etc. – all while our transport takes us to our destination. The innovation and business potential presented by the HAVs is only limited by imagination and savvy to overcome the challenges.
Among the various challenges to overcome are those of sensing the environment around and even inside the vehicle. Two of these sensing technologies are LIDAR and camera. Each of them are evolving fast to meet the industry demands. Levels 3–5 of autonomous vehicles as defined by NHTSA and SAE (Table 1) will need a high resolution and long range scanning LIDAR [3]. They will also need cameras which operate in infrared (and visible) spectrum to be able to function at night and low light conditions.
LevelNameNarrative definitionDynamic Driving Task (DDT)DDT fallbackOperational Design Domain (ODD)
Sustained lateral and longitudinal vehicle motion controlObject and Event Detection and Response (OEDR)
Driver performs part or all of the Dynamic Driving Task (DDT)
0No Driving AutomationThe performance by the driver of the entire DDT, even when enhanced by active safety systems.DriverDriverDrivern/a
1Driver AssistanceThe sustained and ODD-specific execution by a driving automation system of either the lateral or the longitudinal vehicle motion control subtask of the DDT (but not both simultaneously) with the expectation that the driver performs the remainder of the DDT.Driver and SystemDriverDriverLimited
2Partial Driving AutomationThe sustained and ODD-specific execution by a driving automation system of both the lateral and longitudinal vehicle motion control subtasks of the DDT with the expectation that the driver completes the OEDR subtask and supervises the driving automation system.SystemDriverDriverLimited
Automated Driving System (“System”) performs the entire DDT (while engaged)
3Conditional Driving AutomationThe sustained and ODD-specific performance by an ADS of the entire DDT with the expectation that the DDT fallback-ready user is receptive to ADS-issued requests to intervene, as well as to DDT performance-relevant system failures in other vehicle systems, and will respond appropriately.SystemSystemFallback ready user (Driver is fallback)Limited
4High Driving AutomationThe sustained and ODD-specific performance by an ADS of the entire DDT and DDT fallback without any expectation that a user will respond to a request to intervene.SystemSystemSystemLimited
5Full Driving AutomationThe sustained and unconditional (i.e., not ODD-specific) performance by an ADS of the entire DDT and DDT fallback without any expectation that a user will respond to a request to intervene.SystemSystemSystemUnlimited
### Table 1.
SAE J3016 – summary of levels of driving automation [3].
We will start with discussing the infrared spectrum, its advantages and disadvantages and then move onto LIDAR and Camera in some level of detail.
## 2. Infrared spectrum
The sun radiates electromagnetic energy in a wide spectrum from the shortest X-rays to radio waves. Figure 1 shows the portion visible to the human eye (~380–750 nm) and the infrared region [4]. The near infrared region (~750–1400 nm) is used in many sensing applications including the night vision camera and LIDAR. The active night vision cameras (use light from artificial sources) are different from the passive thermal imaging cameras which operate at higher wavelengths (8–15 μm) and use natural heat as sources of radiation. Figure 1 also shows the wide range of infrared radiation from 750 nm to 1 mm wavelength.
### 2.2. Sensitivity
Figure 2 shows the human eye and camera sensitivity to the visible – Near infrared (NIR) spectrum. The advantage and disadvantage for sensing applications primarily arises from the fact that infrared is mostly invisible in the far field. A fair amount of red color can be seen by most humans till 850 nm; beyond that lies a fair amount of subjectivity. The fact that the human eye is not very sensitive to NIR light allows cameras to be used unobtrusively (especially at night/poor lighting conditions). The disadvantage lies in the fact that silicon based image sensors have poor sensitivity in this wavelength (~35% QE at 850 and 10% at 940 nm). In addition these wavelengths can reach the retina of the eye – so the exposure has to be controlled to avoid damage.
The solar radiation outside the earth’s atmosphere and that reaching the surface is shown in Figure 3 [8].
Dips in the spectral irradiance at surface are primarily due to water in the atmosphere. In the infrared spectrum of interest they occur at 810, 935, 1130, 1380, 1880 nm and beyond. This means the ambient noise is lower at these specific wavelengths. However, wavelengths of many semiconductor devices shift with temperature (~0.3 nm/°C for Gallium arsenide and aluminum gallium arsenide materials used in infrared spectrum); for automotive applications this shift is ~44 nm from −40 to 105°C. Ideally we need a peak with flat ambient noise variation around it for good design.
Another observation from Figure 3 is the lower ambient noise as we go to the longer wavelengths. However, past ~1000 nm the material base for detectors changes from silicon to germanium or indium gallium arsenide – which can be expensive.
## 3. Light Detection And Ranging (LIDAR)
### 3.1. Need for LIDAR in automotive
LIDAR, RAdio Detection And Ranging (RADAR) and Camera are the environment sensors central to the autonomous car operation. They are used to detect and classify the objects around the car by location and velocity. Each of the sensors has limitations and the information obtained from them is fused together with confidence prior to making a decision on the vehicles trajectory.
Table 2 provides a brief summary of the above sensing technologies.
SensorTypical rangeHorizontal FOVVertical FOV2020 price rangeComments
24 GHz RADAR60 m [6]56° [6]~±20°<$100USA Bandwidth Max 250 MHz [7]Robust to snow/rain Poor angular resolution; sensitive to installation tolerances and materials 77 GHz RADAR200 m [6]18° [6]~±5°<$100Similar to 24 GHz RADAR with more bandwidth (600 MHz [7]); sensitive to installation tolerances and materials
Front Mono Camera50 m [6]36° [6]~±14°<$100Versatile sensor with high resolution; Poor depth perception; High processing needs; low range; sensitive to dirt/obstruction LIDAR (Flash)75 m140°~±5°<$100Better resolution than RADAR and more range than Camera. Eye safety limits; Poor in bad weather; sensitive to dirt/obstruction
LIDAR (Scanning)200 m360°~±14°<\$500Similar to Flash LIDAR with higher resolution and Cost; sensitive to dirt/obstruction
### Table 2.
RADAR – camera – LIDAR comparison.
### 3.2. Types
LIDAR sensors could be classified on any of its various key parameters:
• Operating principle: Time of Flight (ToF)/Frequency Modulated Continuous Wave (FMCW)
• Scanning technology: Mechanical/Micro-Mechanical-Mirror (MEMS)/Optical Phase Array (OPA)
• Scanning/Flash
• Solid State/Mechanical
• Wavelength: 905 /1550 nm
• Detection technology: Photodiode/Avalanche Photodiode/Single Photon Multiplier
• …and many other ways
### 3.3. Time of Flight LIDAR Operating Principle
The Time of Flight LIDAR operation can be explained using Figure 4.
A laser is used to illuminate or “FLASH” the field of view to be sensed. The laser pulse travels till it is reflected off a target and returned to a detector. The time taken for the pulse to travel back and forth provides the range. The location of the target is based off optics mapped over the field of view and detector array. Two or more pulses from the target provide the velocity. The angular resolution depends on the number of detector pixels which map the field of view. The more pixels we have – the better the resolution.
The same principle is used by 3D cameras or high resolution flash LIDAR. Higher power and more detector pixels are used.
### 3.4. Emitter and detector options
As shown in Figure 4, to increase the range by 2× – the needed power is 4×. As we increase the power – we start running into eye safety limits. Infrared light below 1400 nm can reach the retina of the eye. If the exposure limit is exceeded, permanent eye damage can occur.
There are many levers available to achieve the needed range – including better detectors, bigger lenses, and shorter pulse widths. Of course, the best option would be to use light above the 1400 nm wavelength. However, to use lasers and detectors in this wavelength region (>1400 nm) – we typically have to use more expensive materials (indium-gallium-arsenide—phosphide lasers and germanium-based detectors).
### 3.5. Eye safety
Sunlight on the earth’s surface is composed of ~52% infrared (>700 nm), ~43% visible (400–700 nm) and ~3% Ultraviolet (<400 nm) [9]. The intensity of infrared is low enough that it does not cause eye damage under normal exposure. When light is visible and bright, the eye has a natural blink response and we do not stare at it – helping to avoid eye damage. Infrared light is not visible and so can cause eye damage if exposure limits are not regulated.
The safe levels of infrared levels are regulated by IEC-62471 for Light Emitting Diodes and IEC-60825 (2014) for lasers. In USA, the equivalent federal standards are in 21 CFR 1040 (Code of Federal Regulations).
The standards have hazard exposure limits for the cornea of the eye, thermal hazard limit for skin and eye retinal thermal hazard exposure. For exposures above 1000 s, the irradiance limit is 100 W/m2 at room temperature and 400 W/m2 at 0°C. The retina exposure limits tend to be more stringent. The calculations are complex and depend on wavelength, size of the emitter, exposure time and other factors.
### 3.6. Signal processing challenges
As sensors demand higher resolution and faster response – it increase the computational needs. At the raw signal level, using the forward camera as an example:
Number of pixels to be processed = frames per seconds × horizontal field of view/resolution × vertical field of view/resolution.
Example: 30 fps camera, 40° HFOV, 40° VFOV, 0.1° resolution
• 30 × 400 × 400 = 4.8 Mpx/s
A similar amount of data needs to be processed by the LIDAR, RADAR and other sensors. At some level, this information has to be fused to recognize and classify objects and their trajectory.
As more and more sensing data is collected, processed and acted upon in real time (time between collection and use is extremely short), creating ways of storing/processing and updating data are being developed. For example – the 3 dimensional roadway maps needed for autonomous driving are stored in the cloud (remote server) and real time data is processed to look only for changes and updates; thus reducing the amount of data crunching to be done in real time. Another trend is to collect and process the raw analog signal when possible – thus reducing the downstream processing needs.
Security of data in autonomous vehicles is another growing concern and business opportunity for innovation. Automotive Information Sharing and Analysis Center (Auto-ISAC) (www.automotiveisac.com) was formed in 2015 by automakers to share the best practices related to cyber threats in the connected car.
## 4. Camera
Camera’s in automobiles continue to grow as their functional versatility is exploited with increasing innovation. They have become central to Advanced Driver Assistance Systems (ADAS) like adaptive cruise control, adaptive high beam, automatic emergency braking, lane departure warning, blind spot detection, driver monitoring, traffic sign detection and others.
The latest Tesla Model 3 is believed to have up to eight exterior cameras. Other OEM’s are also using interior driver monitoring and gesture recognition cameras. A presentation from IHS Markit [13] shows typically five exterior and one interior camera for Level 3; eight exterior cameras and 1 interior camera for Level 4 being planned by a number of Original Equipment Manufacturers.
### 4.1. Exterior infrared camera (night vision)
Cameras need light to illuminate the objects in its field of view. Currently most cameras used in ADAS functions work with visible light – which is fine for daytime operation. However, at night the prime source for visible light is usually the headlamps of the car. The visible light from the headlamps is strictly regulated by NHTSA with its Federal Motor Vehicles Safety Standard 108 (FMVSS 108). Figure 5 below shows a bird’s eye view of the permitted illumination region in the USA.
It can be observed that in essence, visible light can only be legally used for a limited range of ~60 m in front of the vehicle. Illumination outside the car lane and around the car is very limited (if any). These legal requirements are not expected to be changed anytime soon – since we will have cars driven by humans for at least another 20–30 years. This means to illuminate to longer and wider fields of view, the cameras have to work with infrared light (which is not regulated by FMVSS 108). As long as the infrared light is within eye safe limits, it can be used all around the car.
Figure 6 shows a graphic overview of the regions around the car that are covered by cameras. The forward camera needs to ideally sense as far as the RADAR and LIDAR to permit good sensor fusion.
The target range for RADAR and LIDAR is at least 200 m (Forward direction) and 50–100 m in all other directions.
### 4.2. Exterior camera illumination challenges
The spectral sensitivity of CMOS image sensors at 850 nm is ~35% compared to its peak at 550 nm (green). Further down at 940 nm, this reduces to ~10%. This means a larger number of infrared photons is needed to generate a clear image.
To illuminate targets at longer ranges and wider field of view more light is needed. In addition, different targets have different reflectivity – which can have a significant effect on the image quality. So while we put out more and more light to get a better signal – we need to ensure the intensity is still eye safe. We also start eating up more energy from the battery for illumination. Calculations show the amount of infrared flux needed could be anywhere from 6 W (100 m range, 12° FOV, 50% reflectivity, 850 nm, 0.15 μW/cm2, Lens F#1) to 1250 W (200 m range, 40° FOV, 10% reflectivity, 850 nm, 0.15 μW/cm2, Lens F#1) [10, 11].
A typical headlamp today may have 5 W of visible light used per lamp currently. Imagine the complexity of adding 100’s of more Watts to the headlamp. The self-driving eco system has not yet come to grasp the scope of challenge that it has to deal with here. The alternative would be to rely more on the LIDAR and RADAR sensors at the longer ranges and use the camera only in the short ranges. This option may not provide needed reliability – since all of these technologies have weakness (RADAR does not same resolution as camera at long ranges and LIDAR is more prone to poor performance in bad weather).
Potential solution options which have not been fully vetted are to use pulsed infrared lasers to illuminate the CMOS based cameras; use of infrared matrix lighting architectures where rows of LED’s are turned on in sequence with a rolling shutter camera more to come as we make progress.
### 4.3. Interior camera – market need
The need for an interior camera arises out of multiple market forces. The first is the introduction of self-driving cars which are autonomous only in certain driving conditions (highways/traffic Jams). The cars switch between the human driver and the computer as needed. To do this effectively, the human driver has to be monitored as part of the environment in and around the car. This is to ensure adequate warning is given to the driver to leave their current engagement and get ready to take over the task of driving.
The second market force is the increase of distracted driving. In 2014, 3179 (10% of Total) people were killed and an additional 431,000 (18% of total) were injured in collisions involving distracted drivers in the USA [10]. NHTSA has a blueprint to reduce accidents related to distracted driving – which encourages OEM’s to put in place measures to ensure the driver keeps their eyes on the road when the vehicle is moving. A definition of distraction in terms of driver gaze and time elapsed away from looking straight is provided in other-related NHTSA documents [12]. At a high level, looking more than 2 s in a direction 30° sideways of up-down when the vehicle speed is more than 5 mph would be classified as distracted. The increase in distracted driving is attributed to cell phone/smartphone/texting and related activities.
Additional benefits and applications are continuing to generate from the driver monitoring infrared camera system. It lends itself well to also catch drowsy drivers (eyelids shut or drowsy pupils); face recognition – not strong enough to be a biometric device, but enough to at least enable customized settings for different drivers in family and many more to come.
The auto industry is responding to these two needs (autonomous cars, distracted driving) by installing an infrared camera to monitor the gaze of the driver. Infrared illumination is needed – since we do not want to distract the driver at night with visible light. The wavelength for illumination is in the 850–950 nm range. The eye safety and camera sensitivity challenges of illumination in this spectrum were briefly discussed earlier sections. A few other challenges are discussed in the next section.
### 4.4. Interior camera illumination challenges
When we use an infrared camera facing the driver, the LED’s are shining the light right on our eyes and face. Light at 850 nm can be red enough to be seen easily by most people – especially at night. Measures to put in a dark filter and smudge the bright red LED spot with optics are partially successful. The problem arises from the fact that anything done to reduce the brightness will usually also reduce the illumination – which would result in poor image quality and failure to detect distraction in gaze by the software processing the image.
One solution is to go to higher wavelengths (940 nm) – the challenge here is lower camera sensitivity. This has been overcome by pulsing higher peak currents at lower duty cycle using a global shutter image sensor. The typical cameras used are 30 fps and these are fast enough – since gaze while driving does not change that often and fast.
On the eye safety side, measures are needed to ensure that when the eyes are too close to the Infrared LED (IRED) – then they either need to be shutoff or reduced in intensity. Typically the distance to the eye is estimated with the camera itself, as an added measure we can have proximity sensors.
Since these cameras work in infrared with a filter block for visible wavelengths, the biggest challenge for illumination tends to be during daytime under full sunlight. The IREDs have to typically overcome ambient noise from the sun. Polaroid sunglasses can also sometimes prevent function if the coating prevents the wavelength to pass through.
The last challenge worth mentioning is that of consumer acceptance and loss of privacy. From a legal perspective, if the camera is recording the driver’s face – the information can be pulled up in court if needed by a lawyer. NHTSA regulations mandate that any information needed for vehicle safety has to be stored for a short direction – essentially a black box (As used in aircrafts) to help reconstruct an accident. Will consumers trade a loss of privacy for safety and convenience (of automated driving) is yet to be seen. OEM’s may initially provide consumers with the option to turn off the camera (and related loss of function) to enable the transition.
### 4.5. Additional applications for interior camera
OEMs are evaluating the concept of using interior cameras to monitor all occupants in the car – to enable optimum deployment of airbags and other passive safety devices. At a basic level, if there is no occupant in the passenger seat (or just a cargo box) – do not deploy the airbag.
Another application is the use of gesture recognition. The idea is use gesture’s seamlessly and conveniently to open windows/sunroofs/turn on radio/change albums/etc. The successful combination of voice, touch and gesture to operate devices depend a lot on the age group (and resultant car design) and how well the technologies are implemented.
Face recognition and iris recognition are already making their way into smartphones. They are expected to penetrate the auto market. Even through the technologies are available and mature, the business case/consumer demand/willingness to pay for these functions is yet to be explored.
### 4.6. Signal processing
As cameras become ubiquitous around the car, the questions become how many cameras are enough and what should be the range and resolution of the cameras. The same question can be asked of LIDAR and RADAR also. However, signal processing tends to be more demanding the high resolution (comparatively) of cameras.
Assuming a VGA format for the image sensor, we get 480 (H) × 640 (W) pixels per frame; with typically 30 fps coming in for processing. The resolution we get from this VGA image sensor depends on the optical field of view it covers and the maximum range at which the smallest object has to be recognized and resolved for action. At 100 m and a 40° HFOV the width covered by the 640 pixels is ~7279 cm. This means each pixel covers 11.4 cm or ~4.5 in. Is this level of resolution good enough for self-driving cars? The next section digs a little deeper into this topic.
### 4.7. Exterior camera resolution requirement
What is the smallest object that can change the trajectory of the car? One could argue this could be as small as a nail or sharp object on the road. Maybe with the newer tires which can roll over nails, we can overlook this object (They would then become mandatory for self-driving cars). The next object I can think of would be a solid brick placed on the road which even though small, could change the trajectory of the car. Other such objects like tires, tin cans, potholes, etc. could be imagined that would have a similar impact.
The autonomous car machine vision has to detect such an object at a far enough distance to take appropriate measures (steer, brake/slow down or prepare for collision). With a speed of 100 mph and a dry road with friction of 0.7, a braking/sensing range of 190 m is calculated [13]. A modular USA brick with dimensions of 194 × 92 × 57 mm would subtend an angle of ~2 arc min (tan−1 65/100,000). This level of resolution would be outside the capability of a standard VGA camera.
After detection, the object has to be classified before an action can be taken on how to deal with it. The kinds of objects the car could come across on its path depends very much geo fenced location. Objects on the US road freeways and urban streets could be very different from those in India or china. This is the point where the admiration for the human senses and brain capacity start daunting current computer chips.
## 5. Sensor fusion
### 5.1. Need for sensor fusion
For self-driving cars to be accepted by society, they would have to demonstrate significantly lower probability of collision – when compared to human drivers. A 2016 study by Virginia Tech Transportation Institute [14] found that self-driving cars would be a comparable or a little better than humans for severe crashes, but significantly better at avoiding low severity level crashes (level 3). The level 3 crash rate was calculated at 14.4 crashes per million miles driven for humans and 5.6 crashes for self-driving cars.
To keep things in perspective, we could estimate an average person in USA to drive 900,000 miles in their lifetime (12,000 miles/year × 75 years). Also note that the above report uses only Google self-driving car data. These cars are known to have a full suite of sensors (Multiple LIDAR, RADAR, Cameras, Ultrasonic, GPS and other sensors).
The point is that just like the human driver, the car has to integrate the information from multiple sensors and make the best decision possible in the circumstance. On top of that, it has to be way better to get people to start adopting the technology. Knowing that each of the sensor technologies has some limitation, the need to fuse multiple inputs reliably is a daunting task. Incorrect or poor implementation of the sensor fusion could quickly take the car back to the dealer show room.
### 5.2. Challenges to sensor fusion
Figure 7 below illustrates the challenge of sensor fusion.
The objective sensor fusion is to determine the environment around the vehicle trajectory with enough resolution, confidence and latency to navigate the vehicle safely.
Figure 7 row 1 shows the ideal case when two sensors agree on an object and the object is detected early enough to navigate the car.
Figure 7 row 2 shows a case where each of the sensors classifies the object differently. In this case, the best option maybe to just agree that it is big enough object to avoid if possible.
Figure 7 row 3 similar situation where a person on a bicycle maybe identified as a person or a bicycle. Again, we could agree that it is an unidentified large moving object that needs to be avoided.
The last two rows shows smaller objects that pose difficult questions. Is it better to run over a small dog than to risk braking and getting rear-ended? Can the pothole be detected and classified early enough to navigate? Is the pothole or object small enough to run over?
These questions will take a longer time to resolve with improving technology in sensing, computing, public acceptance and legislation. The 80/20 Pareto principle would imply that the last 20% of the problems for self-driving cars will take 80% of the time it takes to bring it to mass market.
## 6. Conclusions
The exponential growth of electronics in the auto industry can be estimated by the number of sensors and electronic control units (ECUs) being added to each newer cars. From a 2003 VW golf (~35 ECUs, 30 sensors); a 2013 Ford Fusion (~70 ECUs, 75 Sensors) to a projection for automated car in 2030 (~120 ECUs, >100 Sensors) [1]. One could be forgiven for imagining the future car to be a supercomputer with wheels.
We are in the initial growth spurt for autonomous cars. A lot of technology still remains to be innovated and matured before regulation and standards kick-in. LIDAR technology is still evolving – range, resolution, eye safety, form factor and cost of the technology is improving rapidly. Camera hardware for medium range and VGA resolution has matured – but needs improvement in range (200 m target), resolution (>8 Megapixel) and performance under poor lighting or with infrared. Sensor fusion architectures can only be optimized after sensors needed are standardized or at least well understood. Real time operation with use of Artificial Intelligence – Neural networks is still in early stage. Society has still to debate and accept the safety performance with known behavior of these robots on wheels. What a great time for electronics and the Auto industry!
## More
© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
## How to cite and reference
### Cite this chapter Copy to clipboard
Rajeev Thakur (December 20th 2017). Infrared Sensors for Autonomous Vehicles, Recent Development in Optoelectronic Devices, Ruby Srivastava, IntechOpen, DOI: 10.5772/intechopen.70577. Available from:
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2019-11-20 13:26:03
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2600577473640442, "perplexity": 2343.7899770703793}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670558.91/warc/CC-MAIN-20191120111249-20191120135249-00446.warc.gz"}
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https://mathematica.stackexchange.com/questions/88049/handle-matrices-and-vectors-with-general-dimension
|
# Handle matrices and vectors with general dimension
I have a matrix of $n \times n$ dimension: $$K - \omega^2 M = \begin{pmatrix} 2\omega_0^2 - \omega^2 & - \omega_0^2 & 0 & \cdots & 0 \\ - \omega_0^2 & 2\omega_0^2 - \omega^2 & -\omega_0^2 & \cdots & 0 \\ 0 & -\omega_0^2 & 2\omega_0^2-\omega^2 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 2\omega_0^2-\omega^2 \end{pmatrix}$$
And I want a solution to the equation: $$\left( K - \omega^2 M \right) \cdot \mathbf{c} = 0, \quad \mathbf{c} = \left( c_1, c_2, \cdots, c_n \right)^T$$
The first problem is obviously the characteristic equation:
$$\det \left( K - \omega^2 M \right) = 0$$
which is too hard for Mathematica to handle (couldn't simplify even when I plugged in general eigenvalue), so I carried this manually and the eigenvalues are:
$$\omega_k = 2 \omega_0 \cos \frac{k \pi}{2(n+1)}, \quad 1 \leq k \leq n$$ My problem is to obtain eigenvectors for general case: $n$, which I can set to any integer value and have it evaluated. Definition of matrix is simaple so far:
M = Table[Table[KroneckerDelta[i, j] (2 - a^2) -
KroneckerDelta[i, j + 1] - KroneckerDelta[i, j - 1], {j, 1,
n}], {i, 1, n}]
Where I can set $n$ to any value ($n = 5$ for example). Notice that this is a nondimensionalised matrix with $\omega = a \omega_0$ and in sake of clarity $\omega_0$ was cancelled $n$-times.
Now here comes the problem: I will need a column vector of arbitrary length, but I can't write:
c = Table[ci, {i, 1, n}]
because Mathematica does not recognise "i" being a variable in "ci". Although this is desired result for $n = 5$:
c = {c1, c2, c3, c4, c5}
The next thing is solution to the problem with correct eigenvalues:
S1 = Solve[Dot[M/.a->2Cos[1 Pi/(2n+2)],c] == 0, c]
S2 = Solve[Dot[M/.a->2Cos[2 Pi/(2n+2)],c] == 0, c]
S3 = Solve[Dot[M/.a->2Cos[3 Pi/(2n+2)],c] == 0, c]
...
Sn = Solve[Dot[M/.a->2Cos[n Pi/(2n+2)],c] == 0, c]
Again, how can I rewrite this for some arbitrary eigenvalue, so I can handle it in general form?
The desired result is several lists of eigenvectors:
L1 = Flatten[{c1 /. S1, c2 /. S1, c3 /. S1, ..., cn /. Sn}]
L2 = Flatten[{c1 /. S2, c2 /. S2, c3 /. S2, ..., cn /. Sn}]
L3 = Flatten[{c1 /. S3, c2 /. S3, c3 /. S3, ..., cn /. Sn}]
...
Ln = Flatten[{c1 /. Sn, c2 /. Sn, c3 /. Sn, ..., cn /. Sn}]
And the final step is to plot all solutions:
Table[ListPlot[Li],{i,1,n}]
Now all of this is obtainable by simply invoking:
e = Eigenvectors[M]
Which is simply a matrix of eigenvectors (one can think of a basis in which $M$ is diagonal). The problem is, that Mathematica doesn't really know about the beauty and simplicity of eigenvalues of such a matrix. As a result, the eigenvalues for e.g. $n = 6$ are pretty nasty, involving complex numbers and such - it's because $\cos \frac{\pi}{7}$ is really not a nice closed-form expression. Then the problem is, that Mathematica cannot find eigenvectors for $n = 6$ in suitable form (a typical eigenvector is "2-a^2 - Root[...]" with strange things like #1) when the problem is obviously ONLY in eigenvalues (when plugging some eigenvalue manually I can obtain corresponding eigenvector).
My question is: how can I generalize those expressions for $\mathbf{c}$, $S_n$, $L_n$ and so on, or, alternatively, how can I obtain eigenvectors for every $n$ with Eigenvectors[M] without some time-consuming procedure involving #1 and Root[...] and so they are SORTED by corresponding eigenvalues?
P.S.: I know that eigenvectors are stationary waves.
• Since your matrix is Toeplitz, your eigenvector components are also expressible in terms of trigonometric functions. If you're interested, I can write up a solution. – J. M.'s ennui Jul 12 '15 at 13:14
• Yes, the eigenvector components are $\sin \frac{(n-k+1)\pi}{n+1} i$, $n$ is dimension of problem, $k$ is number of eigenvector a $i$ is it's the component. – user16320 Jul 12 '15 at 14:54
• So you know the result already; why not use that so you're assured that your eigenpairs are arranged in the manner you want? – J. M.'s ennui Jul 12 '15 at 14:55
• Oh, and another thing: look up SparseArray[] and Band[]. – J. M.'s ennui Jul 12 '15 at 14:57
As @GuessWhoItis mentioned, some or all of this can be done analytically. But I'll answer your question regarding Mathematica syntax, as it seems to be the goal of what you're trying to achieve here.
Mathematica does not recognise "i" being a variable in "ci".
The best way to deal with this is to define
cVector=Table[c[i],{i,n}];
or alternatively,
cVector=Array[c,n];
Then, treat each c[1] as c1, c[2] as c2 and so on. In some cases you can also use Subscript to do that, but this is discouraged and should be avoided.
Now, regarding your question about the eigenvectors. If you have a matrix $M$ and you already know that $\Lambda$ is an eigenvalue, what you are actually looking for is the NullSpace of the matrix $M-\Lambda I$. Therefore, you don't even need to define cVector. The following code does everything:
n = 8;
M = Table[Table[
KroneckerDelta[i, j] (2 - a^2) - KroneckerDelta[i, j + 1] -
KroneckerDelta[i, j - 1], {j, 1, n}], {i, 1, n}];
S = Table[NullSpace[M /. a -> 2 Cos[i Pi/(2 n + 2)]], {i, n}]
(I picked some value for n so that the thing would run)
• Yes, this is working great. Any idea on the problem if I would not know the eigenvalues? I noticed Mathematica can't really evaluate determinant with them to zero and those strange Root[...] and so on...Besides that, great answer, thank you. – user16320 Jul 12 '15 at 14:56
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2021-01-23 11:13:12
|
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|
http://abstract.ups.edu/aata/permute-sage-exercises.html
|
## Section5.5Sage Exercises
These exercises are designed to help you become familiar with permutation groups in Sage.
###### 1
Create the full symmetric group $S_{10}$ with the command G = SymmetricGroup(10).
###### 2
Create elements of G with the following (varying) syntax. Pay attention to commas, quotes, brackets, parentheses. The first two use a string (characters) as input, mimicking the way we write permuations (but with commas). The second two use a list of tuples.
• a = G("(5,7,2,9,3,1,8)")
• b = G("(1,3)(4,5)")
• c = G([(1,2),(3,4)])
• d = G([(1,3),(2,5,8),(4,6,7,9,10)])
1. Compute $a^3\text{,}$ $bc\text{,}$ $ad^{-1}b\text{.}$
2. Compute the orders of each of these four individual elements (a through d) using a single permutation group element method.
3. Use the permutation group element method .sign() to determine if $a,b,c,d$ are even or odd permutations.
4. Create two cyclic subgroups of $G$ with the commands:
• H = G.subgroup([a])
• K = G.subgroup([d])
List, and study, the elements of each subgroup. Without using Sage, list the order of each subgroup of $K\text{.}$ Then use Sage to construct a subgroup of $K$ with order 10.
5. More complicated subgroups can be formed by using two or more generators. Construct a subgroup $L$ of $G$ with the command L = G.subgroup([b,c]). Compute the order of $L$ and list all of the elements of $L\text{.}$
###### 3
Construct the group of symmetries of the tetrahedron (also the alternating group on 4 symbols, $A_4$) with the command A=AlternatingGroup(4). Using tools such as orders of elements, and generators of subgroups, see if you can find all of the subgroups of $A_4$ (each one exactly once). Do this without using the .subgroups() method to justify the correctness of your answer (though it might be a convenient way to check your work).
Provide a nice summary as your answer—not just piles of output. So use Sage as a tool, as needed, but basically your answer will be a concise paragraph and/or table. This is the one part of this assignment without clear, precise directions, so spend some time on this portion to get it right. Hint: no subgroup of $A_4$ requires more than two generators.
###### 4
The subsection Motion Group of a Cube describes the $24$ symmetries of a cube as a subgroup of the symmetric group $S_8$ generated by three quarter-turns. Answer the following questions about this symmetry group.
1. From the list of elements of the group, can you locate the ten rotations about axes? (Hint: the identity is easy, the other nine never send any symbol to itself.)
2. Can you identify the six symmetries that are a transposition of diagonals? (Hint: [g for g in cube if g.order() == 2] is a good preliminary filter.)
3. Verify that any two of the quarter-turns (above, front, right) are sufficient to generate the whole group. How do you know each pair generates the entire group?
4. Can you express one of the diagonal transpositions as a product of quarter-turns? This can be a notoriously difficult problem, especially for software. It is known as the “word problem.”
5. Number the six faces of the cube with the numbers $1$ through $6$ (any way you like). Now consider the same three symmetries we used before (quarter-turns about face-to-face axes), but now view them as permutations of the six faces. In this way, we construct each symmetry as an element of $S_6\text{.}$ Verify that the subgroup generated by these symmetries is the whole symmetry group of the cube. Again, rather than using three generators, try using just two.
###### 5
Save your work, and then see if you can crash your Sage session by building the subgroup of $S_{10}$ generated by the elements b and d of orders $2$ and $30$ from above. Do not submit the list of elements of N as part of your submitted worksheet.
What is the order of $N\text{?}$
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2018-01-22 10:24:12
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https://codegolf.stackexchange.com/questions/17096/helping-the-farmer/17217
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# Helping the Farmer
Farmer Jack is very poor. He wants to light his whole farm but with minimum of cost. A lamp can illuminate its own cell as well as its eight neighbors . He has arranged the lamps in his field but he needs your help in finding out whether or not he has kept any extra lamps.
Extra lamps : Lamps which on removing from the farm will make no difference to the number of cells lit. Also, the lamps you will point will not be removed one by one, but they will be removed simultaneously.
Note: The only action you can perform is to remove some lamps. You can neither rearrange nor insert lamps. Your final target is to remove maximum number of lamps such that every cell which was lit before is still lit.
Help Farmer Jack in spotting the maximum number of useless lamps so that he can use them elsewhere.
Input
You will be given in the first line dimensions of the field M and N.Next M lines follow containing N characters each representing the field.
'1' represents cell where lamp is kept.
'0' represents an empty cell.
Output
You have to output an integer containing number of useless lamps.
Sample Input:
3 3
100
010
001
Sample Output:
2
Winner:
Since it is code golf, winner will be the one who will successfully complete the task in least number of characters
• @PeterTaylor I have edited my post. Do you still have a confusion? – user2369284 Jan 1 '14 at 19:24
• Much better. Thanks. – Peter Taylor Jan 1 '14 at 19:33
• may we assume the input ends with a newline? – John Dvorak Jan 1 '14 at 19:51
• This looks like homework. – Johannes Jan 1 '14 at 23:34
• Are we guaranteed that the input lamps will light the whole farm? I'm going to guess no... – Keith Randall Jan 2 '14 at 5:04
# Mathematica 186 (greedy) and 224 (all combinations)
## Greedy Solution
t=MorphologicalTransform;n@w_:=Flatten@w~Count~1
p_~w~q_:=n[p~t~Max]==n[q~t~Max]
g@m_:=Module[{l=m~Position~1,r,d=m},While[l!={},If[w[m,r=ReplacePart[d,#-> 0]&
[l[[1]]]],d=r];l=Rest@l];n@m-n@d]
This turns off superfluous lights one by one. If the light coverage is not diminished when the light goes off, that light can be eliminated. The greedy approach is very fast and can easily handle matrices of 15x15 and much larger (see below). It returns a single solutions, but it is unknown whether that is optimal or not. Both approaches, in the golfed versions, return the number of unused lights. Un-golfed approaches also display the grids, as below.
Before:
After:
## Optimal Solutions using all combinations of lights (224 chars)
With thanks to @Clément.
### Ungolfed version using all combinations of lights
fThe morphological transform function used in sameCoverageQ treats as lit (value = 1 instead of zero) the 3 x3 square in which each light resides.When a light is near the edge of the farm, only the squares (less than 9) within the borders of the farm are counted.There is no overcounting; a square lit by more than one lamp is simply lit.The program turns off each light and checks to see if the overall lighting coverage on the farm is reduced.If it is not, that light is eliminated.
nOnes[w_]:=Count[Flatten@w,1]
sameCoverageQ[m1_,m2_]:=nOnes[MorphologicalTransform[m1,Max]]==
nOnes[MorphologicalTransform[m2,Max]]
(*draws a grid with light bulbs *)
h[m_]:=Grid[m/.{1-> Style[\[LightBulb],24],0-> ""},Frame-> All,ItemSize->{1,1.5}]
c[m1_]:=GatherBy[Cases[{nOnes[MorphologicalTransform[ReplacePart[Array[0&,Dimensions[m1]],
#/.{{j_Integer,k_}:> {j,k}-> 1}],Max]],#,Length@#}&/@(Rest@Subsets[Position[m1,1]]),
{nOnes[MorphologicalTransform[m1,Max]],_,_}],Last][[1,All,2]]
nOnes[matrix] counts the number of flagged cells. It is used to count the lights and also to count the lit cells
sameCoverageQ[mat1, mat2] tests whether the lit cells in mat1 equals the number of lit cells in mat2.MorphologicalTransform[[mat] takes a matrix of lights and returns a matrix of the cells they light up.
c[m1] takes all combinations of lights from m1 and tests them for coverage. Among those that have the maximum coverage, it selects those that have the fewest light bulbs. Each of these is an optimal solution.
Example 1:
A 6x6 setup
(*all the lights *)
m=Array[RandomInteger[4]&,{6,6}]/.{2-> 0,3->0,4->0}
h[m]
All optimal solutions.
(*subsets of lights that provide full coverage *)
h/@(ReplacePart[Array[0&,Dimensions[m]],#/.{{j_Integer,k_}:> {j,k}-> 1}]&/@(c[m]))
## Golfed version using all combinations of lights.
This version calculates the number of unused lights. It does not display the grids.
c returns the number of unused lights.
n@w_:=Flatten@w~Count~1;t=MorphologicalTransform;
c@r_:=n@m-GatherBy[Cases[{n@t[ReplacePart[Array[0 &,Dimensions[r]],#
/.{{j_Integer,k_}:> {j,k}-> 1}],Max],#,Length@#}&/@(Rest@Subsets[r~Position~1]),
{n[r~t~Max],_,_}],Last][[1,1,3]]
n[matrix] counts the number of flagged cells. It is used to count the lights and also to count the lit cells
s[mat1, mat2] tests whether the lit cells in mat1 equals the number of lit cells in mat2.t[[mat] takes a matrix of lights and returns a matrix of the cells they light up.
c[j] takes all combinations of lights from j and tests them for coverage. Among those that have the maximum coverage, it selects those that have the fewest light bulbs. Each of these is an optimal solution.
Example 2
m=Array[RandomInteger[4]&,{6,6}]/.{2-> 0,3->0,4->0};
m//Grid
Two lights can be saved while having the same lighting coverage. c[m]
2
• I don't have Mathematica at hand so I can't test this code, but I think your algorithm is incorrect — unless I misunderstood your explanations. If my understanding is correct, it relies on a greedy strategy that is dependent on the order in which light are processed: for example, starting from the middle lamp in your 3*3 test case would remove it and leave the two side lamps. I don't expect that the particular ordering that you use in the implementation makes it correct, but I don't have a counter-example right now. – Clément Jan 3 '14 at 23:10
• Your idea seems to be that it may be possible to have 2 superfluous lights, a, b, in the original setup, one of which is more superfluous than the other. So, it may be that there is better economy achieved if one is removed (first). I sense that this could not happen with 3 lights total, but it may indeed be possible with greater numbers of lights. I originally solved the problem by testing all combinations of lights. This is certainly optimal and thus ideal, but I found it impractical with a large set of lights. – DavidC Jan 3 '14 at 23:30
• @Clément I'm working on a solution that will test all possible combinations. Will take a while... – DavidC Jan 4 '14 at 0:34
• It sure will ;) But that's to be expected: as it stands this problem is an instance of the minimum set cover — which is NP. Whether the additional assumptions (almost all covering sets, except the lateral ones, have the same cardinality) allow for an efficient implementation is an interesting problem though. – Clément Jan 4 '14 at 2:26
• I strongly suspect the greedy solution is correct if you go sequentially by rows and columns, but I haven't proven it yet. – aditsu Jan 4 '14 at 6:18
## Python, 309 chars
import sys
X=len(I[0])
L=[]
m=p=1
for c in''.join(I):m|=('\n'!=c)*p;L+=('1'==c)*[p<<X+1|p<<X|p<<X-1|p*2|p|p/2|p>>X-1|p>>X|p>>X+1];p*=2
O=lambda a:m&reduce(lambda x,y:x|y,a,0)
print len(L)-min(bin(i).count('1')for i in range(1<<len(L))if O(L)==O(x for j,x in enumerate(L)if i>>j&1))
Works using bitmasks. L is a list of the lights, where each light is represented by an integer with (up to) 9 bits set for its light pattern. Then we exhaustively search for subsets of this list whose bitwise-or is the same as the bitwise-or of the whole list. The shortest subset is the winner.
m is a mask that prevents wraparound of the bits when shifting.
• Please try to provide a program which runs correctly.Java/C++ are safe to any kind of indentation or spacing but Python is not.Obfuscating or shortening code is another thing but providing a program which does not run is another. – user2369284 Jan 2 '14 at 9:01
• @user2369284 what are you talking about?! It works perfectly fine (with python 2) – aditsu Jan 2 '14 at 9:05
• @aditsu I have python 3. – user2369284 Jan 2 '14 at 9:08
• @user2369284 well, the print syntax is different so it fails in python 3 – aditsu Jan 2 '14 at 9:47
# Java 6 - 509 bytes
I made some assumptions about the limits and solved the problem as stated at this time.
import java.util.*;enum F{X;{Scanner s=new Scanner(System.in);int m=s.nextInt(),n=s.nextInt(),i=m,j,k=0,l=0,r=0,o,c,x[]=new int[30],y[]=x.clone();int[][]a=new
int[99][99],b;while(i-->0){String t=s.next();for(j=n;j-->0;)if(t.charAt(j)>48){x[l]=i;y[l++]=j;}}for(;k<l;++k)for(i=9;i-->0;)a[x[k]+i/3][y[k]+i%3]=1;for(k=1<<l;k-->0;){b=new
int[99][99];for(j=c=l;j-->0;)if((k&1<<j)>0)for(c--,i=9;i-->0;)b[x[j]+i/3][y[j]+i%3]=1;for(o=i=0;i++<m;)for(j=0;j++<n;)o|=a[i][j]^b[i][j];r=c-o*c>r?c:r;}System.out.println(r);}}
Run like this: java F <inputfile 2>/dev/null
• Not exactly short, but fits in a disk sector :p I may try a different language later. – aditsu Jan 2 '14 at 2:47
• @aditsu How to make this work on windows? – user2369284 Jan 2 '14 at 4:06
• @user2369284: I don't see how you can do 0011111100 with only 2 lamps. You need to cover 8 cells with light, and each lamp can do at most 3. – Keith Randall Jan 2 '14 at 5:07
• @user2369284 perhaps java F <inputfile 2>nul, if that fails then java F <inputfile and ignore the exception. Also it won't run with java 7. – aditsu Jan 2 '14 at 6:19
• @aditsu I'm really sorry.That was a typo error. Your program works correctly. – user2369284 Jan 2 '14 at 9:07
c++ - 477 bytes
#include <iostream>
using namespace std;int main(){
int c,i,j,m,n,p,q=0;cin>>m>>n;
int f[m*n],g[m*n],h[9]={0,-1,1,-m-1,-m,-m+1,m-1,m,m+1};
for(i=0;i<m*n;i++){f[i]=0;g[i]=0;do{c=getchar();f[i]=c-48;}while(c!='0'&&c!='1');}
for(i=0;i<m*n;i++)if(f[i])for(j=0;j<9;j++)if(i+h[j]>=0&&i+h[j]<m*n)g[i+h[j]]++;
for(i=0;i<m*n;i++)if(f[i]){p=0;for(j=0;j<9;j++)if(i+h[j]>=0&&i+h[j]<m*n)if(g[i+h[j]]<2)p++;if(p==0){for(j=0;j<9;j++)if(i+h[j]>=0&&i+h[j]<m*n)g[i+h[j]]--;q++;}}cout<<q<<endl;}
## Ruby, 303
[this was coded to answer a previous version of the question; read note below]
def b(f,m,n,r)b=[!1]*1e6;(n..n+m*n+m).each{|i|b[i-n-2,3]=b[i-1,3]=b[i+n,3]=[1]*3if f[i]};b[r*n+r+n+1,n];end
m,n=gets.split.map(&:to_i)
f=[!1]*n
m.times{(?0+gets).chars{|c|f<<(c==?1)if c>?*}}
f+=[!u=0]*n*n
f.size.times{|i|g=f.dup;g[i]?(g[i]=!1;u+=1if m.times{|r|break !1if b(f,m,n,r)!=b(g,m,n,r)}):0}
p u
Converting to arrays of Booleans and then comparing neighbourhoods for changes.
Limitation(?): Maximum farm field size is 1,000 x 1,000. Problem states "Farmer Jack is very poor" so I'm assuming his farm isn't larger. ;-) Limitation can be removed by adding 2 chars.
NOTE: Since I began coding this, it appears the question requirements changed. The following clarification was added "the lamps you will point will not be removed one by one, but they will be removed simultaneously". The ambiguity of the original question allowed me to save some code by testing individual lamp removals. Thus, my solution will not work for many test cases under the new requirements. If I have time, I will fix this. I may not. Please do not upvote this answer since other answers here may be fully compliant.
# APL, 97 chars/bytes*
Assumes a ⎕IO←1 and ⎕ML←3 APL environment.
m←{s↑⊃∨/,v∘.⊖(v←⍳3)⌽¨⊂0⍪0⍪0,0,s⍴⍵}⋄n-⌊/+/¨t/⍨(⊂m f)≡¨m¨(⊂,f)\¨t←⊂[1](n⍴2)⊤⍳2*n←+/,f←⍎¨⊃{⍞}¨⍳↑s←⍎⍞
Ungolfed version:
s ← ⍎⍞ ⍝ read shape of field
f ← ⍎¨ ⊃ {⍞}¨ ⍳↑s ⍝ read original field (lamp layout)
n ← +/,f ⍝ original number of lamps
c ← ⊂[1] (n⍴2) ⊤ ⍳2*n ⍝ all possible shutdown combinations
m ← {s↑ ⊃ ∨/ ,v ∘.⊖ (v←⍳3) ⌽¨ ⊂ 0⍪0⍪0,0, s⍴⍵} ⍝ get lighted cells given a ravelled field
l ← m¨ (⊂,f) \¨ c ⍝ map of lighted cells for every combination
k ← c /⍨ (⊂ m f) ≡¨ l ⍝ list of successful combinations
u ← ⌊/ +/¨ k ⍝ min lamps used by a successful comb.
⎕ ← n-u ⍝ output number of useless lamps
⎕ ← s⍴ ⊃ (⊂,f) \¨ (u= +/¨ k) / k ⍝ additional: print the layout with min lamps
I agree that more test cases would be better. Here's a random one:
Input:
5 5
10001
01100
00001
11001
00010
Output (useless lamps):
5
Layout with min lamps (not included in golfed version):
0 0 0 0 1
0 1 0 0 0
0 0 0 0 0
0 1 0 0 1
0 0 0 0 0
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
*: APL can be written in its own (legacy) single-byte charset that maps APL symbols to the upper 128 byte values. Therefore, for the purpose of scoring, a program of N chars that only uses ASCII characters and APL symbols can be considered to be N bytes long.
## C++ 5.806 Bytes
This is not optimized for size yet. But since there are few contestants i will leave it at that for now.
#pragma once
namespace FarmersLand
{
class FarmersField
{
private:
unsigned _m_size, _n_size;
int * _lamp, * _lumination;
char * _buffer;
void _illuminate(unsigned m, unsigned n);
void _deluminate(unsigned m, unsigned n);
void _removeLamp(unsigned m, unsigned n);
void _setLamp(unsigned m, unsigned n);
int _canRemoveLamp(unsigned m, unsigned n);
int _coordsAreValid(unsigned m, unsigned n);
int _getLuminationLevel(unsigned m, unsigned n);
int * _allocIntArray(unsigned m, unsigned n);
int _coordHelper(unsigned m, unsigned n);
public:
FarmersField(char * input[]);
FarmersField(const FarmersField & field);
~FarmersField(void);
int RemoveLamps(void);
char * Cstr(void);
};
}
FarmersField CPP:
#include "FarmersField.h"
#include <stdio.h>
namespace FarmersLand
{
void FarmersField::_illuminate(unsigned m, unsigned n)
{
if(this -> _coordsAreValid(m,n))
{
++this -> _lumination[this -> _coordHelper(m,n)];
}
}
void FarmersField::_deluminate(unsigned m, unsigned n)
{
if(this -> _coordsAreValid(m,n))
{
--this -> _lumination[this -> _coordHelper(m,n)];
}
}
void FarmersField::_removeLamp(unsigned m, unsigned n)
{
if(this -> _coordsAreValid(m,n))
{
unsigned mi_start = (m == 0) ? 0 : m - 1;
unsigned mi_end = m + 1;
unsigned ni_start = (n == 0) ? 0 : n - 1;
unsigned ni_end = n + 1;
for(unsigned mi = mi_start; mi <= mi_end; ++mi)
{
for(unsigned ni = ni_start; ni <= ni_end; ++ni)
{
this -> _deluminate(mi, ni);
}
}
--this -> _lamp[this -> _coordHelper(m,n)];
}
}
void FarmersField::_setLamp(unsigned m, unsigned n)
{
if(this -> _coordsAreValid(m,n))
{
unsigned mi_start = (m == 0) ? 0 : m - 1;
unsigned mi_end = m + 1;
unsigned ni_start = (n == 0) ? 0 : n - 1;
unsigned ni_end = n + 1;
for(unsigned mi = mi_start; mi <= mi_end; ++mi)
{
for(unsigned ni = ni_start; ni <= ni_end; ++ni)
{
this -> _illuminate(mi, ni);
}
}
++this -> _lamp[this -> _coordHelper(m,n)];
}
}
int FarmersField::_canRemoveLamp(unsigned m, unsigned n)
{
unsigned can = 1;
unsigned mi_start = (m == 0) ? 0 : m - 1;
unsigned mi_end = (m == (this->_m_size - 1)) ? m : m + 1;
unsigned ni_start = (n == 0) ? 0 : n - 1;
unsigned ni_end = (n == (this->_n_size - 1)) ? n : n + 1;
for(unsigned mi = mi_start; mi <= mi_end; ++mi)
{
for(unsigned ni = ni_start; ni <= ni_end; ++ni)
{
if( 1 >= this -> _getLuminationLevel(mi, ni) )
{
can = 0;
}
}
}
return can;
}
int FarmersField::_coordsAreValid(unsigned m, unsigned n)
{
return m < this -> _m_size && n < this -> _n_size;
}
int FarmersField::_getLuminationLevel(unsigned m, unsigned n)
{
if(this -> _coordsAreValid(m,n))
{
return this -> _lumination[this -> _coordHelper(m,n)];
}
else
{
return 0;
}
}
int * FarmersField::_allocIntArray(unsigned m, unsigned n)
{
int * a = new int[m * n];
for(unsigned i = 0; i < m*n; ++i)
{
a[i] = 0;
}
return a;
}
int FarmersField::_coordHelper(unsigned m, unsigned n)
{
return m * this -> _n_size + n;
}
int FarmersField::RemoveLamps(void)
{
int r = 0;
for(unsigned m = 0 ; m < this -> _m_size; ++m)
{
for(unsigned n = 0 ; n < this -> _n_size; ++n)
{
if(this -> _canRemoveLamp(m,n))
{
++r;
this -> _removeLamp(m,n);
}
}
}
return r;
}
char * FarmersField::Cstr(void)
{
unsigned size = this -> _m_size * this -> _n_size + _m_size ;
unsigned target = 0;
delete(this -> _buffer);
this -> _buffer = new char[ size ];
for(unsigned m = 0 ; m < this -> _m_size; ++m)
{
for(unsigned n = 0 ; n < this -> _n_size; ++n)
{
this -> _buffer[target++] = (0 == this -> _lamp[this -> _coordHelper(m,n)])? '0' : '1';
}
this -> _buffer[target++] = '-';
}
this -> _buffer[size - 1 ] = 0;
return this -> _buffer;
}
FarmersField::FarmersField(char * input[])
{
sscanf_s(input[0], "%u %u", &this -> _m_size, &this -> _n_size);
this -> _lamp = this -> _allocIntArray(this -> _m_size, this -> _n_size);
this -> _lumination = this -> _allocIntArray(this -> _m_size, this -> _n_size);
this -> _buffer = new char[1];
for(unsigned m = 0 ; m < this -> _m_size; ++m)
{
for(unsigned n = 0 ; n < this -> _n_size; ++n)
{
if('0' != input[m+1][n])
{
this -> _setLamp(m,n);
}
}
}
}
FarmersField::FarmersField(const FarmersField & field)
{
this -> _m_size = field._m_size;
this -> _n_size = field._n_size;
this -> _lamp = this -> _allocIntArray(this -> _m_size, this -> _n_size);
this -> _lumination = this -> _allocIntArray(this -> _m_size, this -> _n_size);
this -> _buffer = new char[1];
for(unsigned m = 0 ; m < this -> _m_size; ++m)
{
for(unsigned n = 0 ; n < this -> _n_size; ++n)
{
if(0 != field._lamp[this -> _coordHelper(m,n)])
{
this -> _setLamp(m,n);
}
}
}
}
FarmersField::~FarmersField(void)
{
delete(this -> _lamp);
delete(this -> _lumination);
delete(this -> _buffer);
}
}
And a set of tests to show that the code does what it was built to do:
#include "../../Utility/GTest/gtest.h"
#include "FarmersField.h"
TEST(FarmersField, Example1)
{
using namespace FarmersLand;
char * input[] = {"3 3", "100", "010", "001"};
FarmersField f(input);
EXPECT_STREQ("100-010-001", f.Cstr());
EXPECT_EQ(2, f.RemoveLamps());
EXPECT_STREQ("000-010-000", f.Cstr());
}
TEST(FarmersField, Example2)
{
using namespace FarmersLand;
char * input[] = {"3 6", "100000", "010000", "001000"};
FarmersField f(input);
EXPECT_STREQ("100000-010000-001000", f.Cstr());
EXPECT_EQ(1, f.RemoveLamps());
EXPECT_STREQ("000000-010000-001000", f.Cstr());
}
TEST(FarmersField, Example3)
{
using namespace FarmersLand;
char * input[] = {"6 3", "100", "010", "001", "000", "000", "000",};
FarmersField f(input);
EXPECT_STREQ("100-010-001-000-000-000", f.Cstr());
EXPECT_EQ(1, f.RemoveLamps());
EXPECT_STREQ("000-010-001-000-000-000", f.Cstr());
}
TEST(FarmersField, Example4)
{
using namespace FarmersLand;
char * input[] = {"3 3", "000", "000", "000",};
FarmersField f(input);
EXPECT_STREQ("000-000-000", f.Cstr());
EXPECT_EQ(0, f.RemoveLamps());
EXPECT_STREQ("000-000-000", f.Cstr());
}
TEST(FarmersField, Example5)
{
using namespace FarmersLand;
char * input[] = {"3 3", "111", "111", "111",};
FarmersField f(input);
EXPECT_STREQ("111-111-111", f.Cstr());
EXPECT_EQ(8, f.RemoveLamps());
EXPECT_STREQ("000-010-000", f.Cstr());
}
TEST(FarmersField, Example6)
{
using namespace FarmersLand;
char * input[] = {"6 6", "100001", "001010", "001001", "001010", "110000", "100001",};
FarmersField f(input);
EXPECT_STREQ("100001-001010-001001-001010-110000-100001", f.Cstr());
EXPECT_EQ(6, f.RemoveLamps());
EXPECT_STREQ("100011-001010-000000-000010-010000-000001", f.Cstr());
}
Perl 3420 bytes
Not a golf solution, but I found this problem interesting:
#!/usr/bin/perl
use strict;
use warnings;
{
package Farm;
use Data::Dumper;
# models 8 nearest neighbors to position i,j forall i,j
my $neighbors = [ [-1, -1], [-1, 0], [-1, +1], [ 0, -1], # current pos [ 0, 1], [+1, -1], [+1, 0], [+1, +1] ]; sub new { my ($class, %attrs) = @_;
bless \%attrs, $class; } sub field { my$self = shift;
return $self->{field}; } sub rows { my$self = shift;
return $self->{rows}; } sub cols { my$self = shift;
return $self->{cols}; } sub adjacents { my ($self, $i,$j) = @_;
NEIGHBORS:
for my $neighbor ( @$neighbors ) {
my ($imod,$jmod) = ($neighbor->[0] +$i, $neighbor->[1] +$j);
next NEIGHBORS
if $imod >=$self->rows || $jmod >=$self->cols;
# push neighbors
$self->field->[$imod]->[$jmod]; } return @adjs; } sub islit { my ($lamp) = @_;
return defined $lamp &&$lamp == 1;
}
sub can_remove_lamp {
my ($self,$i, $j) = @_; return scalar grep { islit($_) } $self->adjacents($i, $j); } sub remove_lamp { my ($self, $i,$j) = @_;
$self->field->[$i]->[$j] = 0; } sub remove_lamps { my ($self) = @_;
my $removed = 0; for my$i ( 0 .. @{ $self->field } - 1) { for my$j ( 0 .. @{ $self->field->[$i] } - 1 ) {
next unless islit( $self->field->[$i]->[$j] ); if($self->can_remove_lamp($i,$j) ) {
$removed++;$self->remove_lamp($i,$j);
}
}
}
return $removed; } 1; } { # Tests use Data::Dumper; use Test::Deep; use Test::More; { # 3x3 field my$farm = Farm->new( rows => 3,
cols => 3,
field => [ [1,0,0],
[0,1,0],
[0,0,1]
]
);
is( 2,
$farm->remove_lamps, 'Removed 2 overlapping correctly' ); is_deeply($farm->field,
[ [0,0,0],
[0,0,0],
[0,0,1],
],
'Field after removing lamps matches expected'
);
}
{ # 5x5 field
my $farm = Farm->new( rows => 5, cols => 5, field => [ [0,0,0,0,0], [0,1,0,0,0], [0,0,1,0,0], [0,0,0,0,0], [0,0,0,0,0] ] ); is( 1,$farm->remove_lamps,
'Removed 1 overlapping lamp correctly'
);
is_deeply( \$farm->field,
[ [0,0,0,0,0],
[0,0,0,0,0],
[0,0,1,0,0],
[0,0,0,0,0],
[0,0,0,0,0],
],
'Field after removing lamps matches expected'
);
}
}
(I/O was taken out so I could show concrete tests)
## Python - 305 bytes
import sys,itertools
w,h=map(int,input().split());w+=1
l=[i for i,c in enumerate(sys.stdin.read())if c=="1"]
f=lambda l:{i+j for i in l for j in(0,1,-1,w-1,w,w+1,1-w,-w,-w-1)if(i+j+1)%w}&set(range(w*h))
for n in range(1,len(l)):
for c in itertools.combinations(l,n):
if f(c)^f(l):print(len(l)-n);exit()
|
2019-09-15 12:46:00
|
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|
https://archive.lib.msu.edu/crcmath/math/math/c/c897.htm
|
## Cyclotomic Equation
The equation
where solutions are the Roots of Unity sometimes called de Moivre Numbers. Gauß showed that the cyclotomic equation can be reduced to solving a series of Quadratic Equations whenever is a Fermat Prime. Wantzel (1836) subsequently showed that this condition is not only Sufficient, but also Necessary. An irreducible'' cyclotomic equation is an expression of the form
where is Prime. Its Roots satisfy .
See also Cyclotomic Polynomial, de Moivre Number, Polygon, Primitive Root of Unity
References
Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 99-100, 1996.
Scott, C. A. The Binomial Equation .'' Amer. J. Math. 8, 261-264, 1886.
Wantzel, M. L. Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas.'' J. Math. pures appliq. 1, 366-372, 1836.
|
2021-11-27 01:58:17
|
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|
http://www.netlib.org/utk/people/JackDongarra/etemplates/node50.html
|
Next: Equivalences (Similarities) Up: Non-Hermitian Eigenproblems J. Demmel Previous: Eigenvalues and Eigenvectors Contents Index
Invariant Subspaces
A (right) invariant subspace of satisfies for all . We also write this as . The simplest example is when is spanned by a single eigenvector of . More generally an invariant subspace may be spanned by a subset of the eigenvectors of , but since some matrices do not have eigenvectors, there are invariant subspaces that are not spanned by eigenvectors. For example, the space of all possible vectors is clearly invariant, but it is not spanned by the single eigenvector of in (2.3). This is discussed further in §2.5.4 below.
A left invariant subspace of analogously satisfies for all , and may be spanned by left eigenvectors of .
Susan Blackford 2000-11-20
|
2016-09-25 00:20:21
|
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|
https://aptitude.gateoverflow.in/6378/nielit-2016-dec-scientist-b-section-a-54
|
344 views
$A$ & $B$ together have ₹$1210$. If $4/15$ of $A's$ amount is equal to $2/5$ of $B's$ amount, how much amount does $B$ have?
1. ₹$664$
2. ₹$550$
3. ₹$484$
4. ₹$460$
Given that $,A + B = ₹1210$
If $\dfrac{4}{15}$ of A′s amount is equal to $\dfrac{2}{5}$ of B′s amount.
$\implies \dfrac{4}{15}A = \dfrac{2}{5}B$
$\implies \dfrac{2}{3}A = B$
$\implies A = \dfrac{3}{2}B$
Now, $A+B = ₹1210$
$\implies \dfrac{3}{2}B + B = ₹1210$
$\implies \dfrac{3B + 2B}{2} =₹1210$
$\implies 5B = ₹2420$
$\implies B = ₹484.$
So, the correct answer is $(C).$
12.1k points
1
335 views
1 vote
|
2022-12-06 01:37:35
|
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|
https://www.techwhiff.com/issue/python-code-only-given-a-variable-current-members-that--38480
|
# PYTHON CODE ONLY Given: a variable current_members that refers to a list, and a variable member_id that has been defined. Write some code that assigns True to a variable is_a_member if the value associated with member_id can be found in the list associated with current_members, but that otherwise assigns False to is_a_member. Use only current_members, member_id, and is_a_member.
###### Question:
PYTHON CODE ONLY
Given:
a variable current_members that refers to a list, and
a variable member_id that has been defined.
Write some code that assigns True to a variable is_a_member if the value associated with member_id can be found in the list associated with current_members, but that otherwise assigns False to is_a_member. Use only current_members, member_id, and is_a_member.
### Solve each equation by completing the square. If necessary, round to the nearest hundredth. 19. X²-18x=19 20. 4 a²+8a-20=0
Solve each equation by completing the square. If necessary, round to the nearest hundredth. 19. X²-18x=19 20. 4 a²+8a-20=0...
### What was the effect that the Homestead Act had on the American economy in the late 1800's?
what was the effect that the Homestead Act had on the American economy in the late 1800's?...
### Consider the phrase: a flat Earth. This phrase... Group of answer choices a. Has 1 referent, but more than 1 sense b. Has 1 sense, but more than 1 referent c. Has sense but no referent d. Has a referent but no sense
Consider the phrase: a flat Earth. This phrase... Group of answer choices a. Has 1 referent, but more than 1 sense b. Has 1 sense, but more than 1 referent c. Has sense but no referent d. Has a referent but no sense...
### Christa wrote 1 page in 2/3 hour. Which is Christa's writing rate per hour?
Christa wrote 1 page in 2/3 hour. Which is Christa's writing rate per hour?...
### Find the value of x
Find the value of x...
### Whats 2x2+34x24+3x0+1x400x100x456x0
whats 2x2+34x24+3x0+1x400x100x456x0...
### Sold the inequality X/(-12) < -12
Sold the inequality X/(-12) < -12...
### Illustrative of functional fixedness, people are more likely to solve the candle problem if
Illustrative of functional fixedness, people are more likely to solve the candle problem if...
### Which sentence from the story BEST illustrates how Jeannette feels about her family’s choice of vacation?
Which sentence from the story BEST illustrates how Jeannette feels about her family’s choice of vacation?...
### For ethyl alcohol, C2H5OH, the enthalpy of fusion is 108.9 J/g, and the entropy of fusion is 31.6 J/mol •K. The enthalpy of vaporization at the boiling point is 837 J/g, and the molar entropy of vaporization is 109.9 J/mol •K.
For ethyl alcohol, C2H5OH, the enthalpy of fusion is 108.9 J/g, and the entropy of fusion is 31.6 J/mol •K. The enthalpy of vaporization at the boiling point is 837 J/g, and the molar entropy of vaporization is 109.9 J/mol •K....
### In the United States in 2007, real GDP was $14 comma 029 billion and the GDP price index was 103.2. What is the value of nominal GDP in 2007? The value of real GDP in 2007 is$ __ billion.
In the United States in 2007, real GDP was $14 comma 029 billion and the GDP price index was 103.2. What is the value of nominal GDP in 2007? The value of real GDP in 2007 is$ __ billion....
### Find m Angle DEH and m Angle FEH.
Find m Angle DEH and m Angle FEH....
### What main goal did all the European countries wanting to settle in North America have in common? *
What main goal did all the European countries wanting to settle in North America have in common? *...
### 5) A partir de chaque mot, écris un adverbeCourageux= ? Intelligent = ? Long= ? Tendre= ?
5) A partir de chaque mot, écris un adverbeCourageux= ? Intelligent = ? Long= ? Tendre= ? ...
### I need help A,b,c, or d
I need help A,b,c, or d...
|
2022-10-07 18:11:40
|
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|
https://www.ias.ac.in/listing/bibliography/pram/A_SAXENA
|
• A SAXENA
Articles written in Pramana – Journal of Physics
• Elastic and inelastic scattering of 270 MeV3He particles from58Ni,90Zr,116Sn and208Pb
Differential cross-section angular distributions for the elastic scattering of 270 MeV3He particles from58Ni,90Zr,116Sn and208Pb have been measured. Optical model analysis of the cross-sections has yielded the optical model parameters for3He particles at 270 MeV. Angular distributions have also been measured for the inelastic excitation of the low-lying levels in the above mentioned nuclei. A collective model analysis using the distorted wave Born approximation (DWBA) of these cross-sections with the distorted waves generated by the optical model parameters determined from the elastic scattering analysis, has yielded the reduced transition probability (B(EL)) values consistent with those reported in the literature.
• Giant resonances in90Zr and116Sn
The giant resonance region in90Zr and116Sn excited by 270 MeV helions has been measured up to about 35 MeV excitation energy. The low and the high energy octupole resonances are seen prominently in addition to the quadrupole and the monopole resonances. The angular distribution data for the various multipoles are satisfactorily explained by the collective model calculations. The percentatge energy weighted sum rule strengths have been determined for all the prominent resonances.
• Prompt neutron emission spectra and multiplicities in the thermal neutron induced fission of235U
The emission spectra of prompt fission neutrons from mass and kinetic energy selected fission fragments have been measured in235U(nth,f). Neutron energies were determined from the measurement of the neutron time of flight using a NE213 scintillation detector. The fragment energies were measured by a pair of surface barrier detectors in one set of measurements and by a back-to-back gridded ionization chamber in the second set of measurements. The data were analysed event by event to deduce neutron energy in the rest frame of the emitting fragment for the determination of neutron emission spectra and multiplicities as a function of the fragment mass and total kinetic energy. The results are compared with statistical model calculations using shell and excitation energy dependent level density formulations to deduce the level density parameters of the neutron rich fragment nuclei over a large range of fragment masses.
• Sub-barrier fission fragment angular distributions for the system19F+232Th
The measurements of fission fragment angular distributions for the system19F+232Th have been extended to the sub-barrier energies of 89.3, 91.5 and 93.6 MeV. The measured anisotropies, within errors are nearly the same over this energy region. However, the deviation of the experimental values of anisotropies from that of standard statistical model predictions increases as the bombarding energy is lowered.
• Mass asymmetry dependence of fusion time-scales in11B+237Np and12C,16O,19F+232Th reactions in a dynamical trajectory model
Dynamical trajectory calculations were carried out for the reactions of11B+237Np and12C,16O and19F+232Th, having mass asymmetries on either side of the Businaro-Gallone critical mass asymmetry αBG, in order to examine the mass asymmetry dependence of fusion reactions in these systems. The compound nucleus formation times were calculated as a function of the partial wave of the reaction for all the systems. This study brings out that for systems with α<αBG, the formation times are significantly larger than for α>αBG, which is caused by the dynamical effects involved in the large scale shape changes taking place in the fusion process as well as due to the interplay between the thermal and the collective motion during the collision process. The calculated time scales are comparable to the experimental values derived from the pre-fission neutron multiplicity measurements.
• Deep inelastic collisions of32S +27Al reaction at 130 MeV bombarding energy
The32S +27Al reaction was studied to investigate the deep inelastic collisions at a bombarding energy of 130 MeV which is well above the Coulomb barrier. The energy distributions of the binary decay products of 6⩽Z⩽10 were determined using a large area position sensitive ionization chamber. The average kinetic energies of the reaction products indicate that the exit shapes correspond to highly stretched scission configurations in the deep-inelastic processes.
• Search for $^{12}$C+$^{12}$C clustering in $^{24}$Mg ground state
In the backdrop of many models, the heavy cluster structure of the ground state of $^{24}$Mg has been probed experimentally for the first time using the heavy cluster knockout reaction $^{24}$Mg($^{12}$C, $^{212}$C)$^{12}$C in thequasifree scattering kinematic domain. In the ($^{12}$C, $^{212}$C) reaction, the direct $^{12}$C-knockout cross-section was found to be very small. Finite-range knockout theory predictions were much larger for ($^{12}$C, 212C) reaction,indicating a very small $^{12}$C−$^{12}$C clustering in $^{24}$Mg(g.s.). Our present results contradict most of the proposed heavy cluster ($^{12}$C+$^{12}$C) structure models for the ground state of $^{24}$Mg.
• # Pramana – Journal of Physics
Volume 94, 2020
All articles
Continuous Article Publishing mode
• # Editorial Note on Continuous Article Publication
Posted on July 25, 2019
|
2020-09-20 21:53:23
|
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|
https://math.meta.stackexchange.com/questions/19188/proving-that-e-pi-pi20-without-using-a-calculator/19189
|
# Proving that $e^\pi-\pi<20$ Without Using a Calculator
No, I did not mistake the meta for the main site. :-) I'm just curious how you would recommend to approach such utterly intractable questions. Apparently, simply answering them in the negative is not accepted. The reason invoked is that one cannot simply make a statement without offering some proof: which is mainly a true and sound advice, however, the trouble with such questions lies precisely the fact that one cannot actually prove their negative. Does the site already have a policy in place for dealing with such absurd situations ?
• Merely leaving a comment leaves the question open, thus stuffing the Unanswered Questions queue. Why do I care about the Unanswered Questions queue ? Because in the very first few months of active use of the site, I had to sift through about $100$ pages of such questions, only from the tags that I follow.
• Closing it will automatically trigger the opposite reaction from many, since such questions concerning mathematical coincidences are interesting to many people, and a vicious never-ending cycle of opening-and-closing-then-reopening soon ensues.
Any other ideas ? :-\
• Your answer in particular starts with "If...". In this case, it is rather better to ask the OP if they intend to include "hand calculations" in the term "calculator"; and in such a case, then answer (or not). The starting words make your answer more into a comment (in some people's eye) and hence such action was taken. – Pedro Tamaroff Jan 9 '15 at 17:21
• Bit of an aside, but isn't the particular "question" in the title reasonably tractable, even if tedious? There exist numerous methods to bound $\pi$. Then use the Taylor series for $\exp$, and bound all but the first however many terms by some geometric series. – epimorphic Jan 9 '15 at 18:23
• @epimorphic: Answers using such an approach are usually downvoted. – Lucian Jan 9 '15 at 18:46
• Do you know of examples of such incidents? – epimorphic Jan 9 '15 at 18:48
• I remember one happening recently, but I can't seem to locate it. – Lucian Jan 9 '15 at 19:02
• I rather disagree with the decision to delete your answer, and if I had 20k rep would vote to undelete. – user7530 Jan 9 '15 at 19:44
• So, you think the question should be off-topic, but are willing to let it stay as long as it stays off the Unanswered list? – Aryabhata Jan 9 '15 at 21:14
• @epimorphic There is a, perhaps under-specified, expectation of a "readable" proof. Sure, everything that an electronic calculator does can also be done by a human calculator performing the same arithmetical operations. So it's not surprising that an answer that is a numerical method in disguise would be downvoted. On the other hand, if the requirement "without a calculator" is never made precise, the question itself can be closed as unclear. – user147263 Jan 9 '15 at 21:20
• @Aryabhata: That is the main concern, yes. That, and someone taking the time to explain to the users posting the questions why it is not reasonable to expect an elegant solution: And I refer here strictly to those that belong in the realm of sheer coincidence, not about those that are completely legitimate and fully justified. – Lucian Jan 9 '15 at 21:39
• @Lucian: I have to agree with achille hui. Unless we see such questions to be a problem, we should allow them. Yes, it is probably likely to be a coincidence (especially if the context is "I was playing with a calculator"), so what? For instance, what if I had asked, Is there a good proof $22/7$ is closer to $\pi$ than $3.14$? Would you close the question? – Aryabhata Jan 9 '15 at 21:51
• @Aryabhata: I didn't vote to close the question that you are referring to. Also, the one in your example seems reasonable. Others, however, aren't. – Lucian Jan 9 '15 at 22:02
• @Lucian: What is reasonable is entirely subjective, and the reason you see the close-reopen wars. One could of course, argue that the whole question is subjective ("not tedious", "elegant" etc). If you look at the past history of such questions, you will see that they are actually quite welcome on MSE (highly upvoted). There aren't that many, and some of them have interesting (IMO) answers. – Aryabhata Jan 9 '15 at 22:47
• It's not absurd. It's just xkcd. – John Jan 15 '15 at 2:45
• One fool can ask more questions than seven wise men can answer. (Proverb.) – Myself Jan 15 '15 at 22:16
• While these sorts of questions have some ambiguity (what exactly does 'without a calculator' mean), it is often the case that with some transformation/trick that the computation is entirely reasonable. – copper.hat Jan 23 '15 at 4:44
I find this sort of Question mildly interesting, susceptible of Answers backed by mathematical reasoning, and hence opportunities for learning something new.
While this doesn't guarantee all or even most such "utterly(?) intractable" problems will receive a well-supported Answer here, I believe it augurs against allowing unsupported Answers merely for a potential to cross them off the Unanswered list.
A positive answer can of course be supported by a demonstration. If one wants to argue for a negative answer, a start would be establishing how narrow is the "numerical coincidence". A difference in the 30th least significant digit is bound to be more difficult to resolve without "calculator" than (as here) one in the fifth significant digit:
$$e^\pi - \pi = 19.9990999\ldots \lt 20$$
Of course the Question invites ways to make such differences wider and more apparent. In any particular case it may be of interest to learn a technique for doing this, as well as the techniques for making a rational or other concise approximation adequate to making such close comparisons.
This situation is similar to Are questions of the form "has this ever been studied?" appropriate? I quote the top voted answer there, by David Speyer:
Questioners should understand, though, that if the true answer is "no" then the question will probably never be answered.
I don't have a problem with $1$ extra post in the Unanswered list. The Unanswered list is not a "queue" that one has to go through sequentially, it's a searchable database of questions that have not been resolved yet. There is nothing wrong with it being large as long as the posts therein indeed have not been resolved. (Saying "I don't think so" is not a resolution.)
Is there a danger that the unanswered list will be bloated with "Can one prove $A<B$ without a calculator"? At present there isn't: one question does not matter on the scale of $75000$. In the unlikely event that such questions become repetitive to the point of being bothersome, there is a self-correcting mechanism: when a pattern of questions becomes repetitive, they begin to draw downvotes. A negatively scored question without answers is automatically deleted when it's 30 days old or more.
• If the question becomes repetitive and they are sufficiently close to each other. Instead of downvotes, we always have the alternative to close them as abstract duplicate. Of course, we first need someone to answer one of these question in a sufficiently generic and meaningful manner. – achille hui Jan 9 '15 at 17:22
• Searchable indeed... But since I was interested in any non-trivial question involving integration, the search quickly turned into a browsing session. :-$)$ – Lucian Jan 9 '15 at 17:41
• Even after giving a "No, probably not possible" answer, the question might still stay on the unanswered list... – Aryabhata Jan 9 '15 at 18:38
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2020-02-19 21:53:15
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https://en.academic.ru/dic.nsf/enwiki/112893
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# Weak acid
Weak acid
A weak acid is an acid that does not completely donate all of its hydrogens when dissolved in water. These acids have higher pKa compared to strong acids, which release all of their hydrogens when dissolved in water.
While strong acids are generally assumed to be the most corrosive, this is not always true . The carborane superacid (H(CHB11Cl11), which is one million times stronger than sulfuric acid, is entirely non-corrosive, whereas the weak acid hydrofluoric acid (HF) is extremely corrosive and can dissolve, among other things, glass and all metals except iridium.
Explanation
Weak acids do not ionize in a solution to a significant extent; that is, if the acid was represented by the general formula "HA", then in aqueous solution a significant amount of undissociated HA still remains. Weak acids in water dissociate as
$mathrm\left\{ HA_\left\{\left(aq\right)\right\} , leftrightarrow , H^+,_\left\{\left(aq\right)\right\} +, A^-,_\left\{\left(aq\right)\right\} \right\}.$
The equilibrium concentrations of reactants and products are related by the Acidity constant expression, (Ka):
$mathrm\left\{ K_a, =, frac \left\{ \left[H^+,\right] \left[A^-,\right] \right\}\left\{ \left[HA\right] \right\} \right\}$
The greater the value of Ka, the more the formation of H+ is favored, and the lower the pH of the solution. The Ka of weak acids varies between 1.8×10-16 and 55.5. Acids with a Ka less than 1.8×10-16 are weaker acids than water. Acids with a Ka of greater than 55.5 are strong acids and almost totally dissociate when dissolved in water.
Examples
The vast majority of acids are weak acids. Organic acids are a large subset of weak acids. However, there are some mineral acids in this field.
*acetic acid
*citric acid
*boric acid
*phosphoric acid
*hydrofluoric acid
ee also
* Strong acid
* Weak base
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2019-10-21 14:41:48
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https://www.numerade.com/questions/in-exercises-27-30-integrate-f-over-the-given-curve-fx-yxy-quad-c-x2y24-in-the-first-quadrant-from-2/
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### In Exercises $27-30,$ integrate $f$ over the give…
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YZ
University of Illinois at Urbana-Champaign
Problem 29
# In Exercises $27-30,$ integrate $f$ over the given curve.$f(x, y)=x+y, \quad C : x^{2}+y^{2}=4$ in the first quadrant from (2, 0) to (0, 2)
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## Video Transcript
Oh, has function this to assign See to co sign t 40 from zero to well, your pie. So the march of the derivative would be too. Using the identity of the children is a function. We're the nine to go f oversee. Forget the data. We're gonna have zero to have high. I'm sorry. 0 to 1 0/4 Bi, You saw that? We have for sun t squared on this tickles nt Did she? Tom's too. So the answer is plying minus two minus to Times Square to
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2020-10-31 11:12:42
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https://www.ann-geophys.net/36/205/2018/
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Journal cover Journal topic
Annales Geophysicae An interactive open-access journal of the European Geosciences Union
Journal topic
Ann. Geophys., 36, 205-211, 2018
https://doi.org/10.5194/angeo-36-205-2018
Special issue: Space weather connections to near-Earth space and the...
Ann. Geophys., 36, 205-211, 2018
https://doi.org/10.5194/angeo-36-205-2018
Regular paper 09 Feb 2018
Regular paper | 09 Feb 2018
# Cross-correlation and cross-wavelet analyses of the solar wind IMF Bz and auroral electrojet index AE coupling during HILDCAAs
IMF-Bz–AE coupling during HILDCAAs
Adriane Marques de Souza1, Ezequiel Echer1, Mauricio José Alves Bolzan2, and Rajkumar Hajra3 Adriane Marques de Souza et al.
• 1National Institute for Space Research (INPE), São José dos Campos, Brazil
• 2Federal University of Jataí, Jataí, Brazil
• 3Laboratoire de Physique et Chimie de l'Environnement et de l'Espace, CNRS, Orléans, France
Abstract
Solar-wind–geomagnetic activity coupling during high-intensity long-duration continuous AE (auroral electrojet) activities (HILDCAAs) is investigated in this work. The 1 min AE index and the interplanetary magnetic field (IMF) Bz component in the geocentric solar magnetospheric (GSM) coordinate system were used in this study. We have considered HILDCAA events occurring between 1995 and 2011. Cross-wavelet and cross-correlation analyses results show that the coupling between the solar wind and the magnetosphere during HILDCAAs occurs mainly in the period 8 h. These periods are similar to the periods observed in the interplanetary Alfvén waves embedded in the high-speed solar wind streams (HSSs). This result is consistent with the fact that most of the HILDCAA events under present study are related to HSSs. Furthermore, the classical correlation analysis indicates that the correlation between IMF Bz and AE may be classified as moderate (0.4–0.7) and that more than 80 % of the HILDCAAs exhibit a lag of 20–30 min between IMF Bz and AE. This result corroborates with Tsurutani et al. (1990) where the lag was found to be close to 20–25 min. These results enable us to conclude that the main mechanism for solar-wind–magnetosphere coupling during HILDCAAs is the magnetic reconnection between the fluctuating, negative component of IMF Bz and Earth's magnetopause fields at periods lower than 8 h and with a lag of about 20–30 min.
Keywords. Magnetospheric physics (solar-wind–magnetosphere interactions)
1 Introduction
The main mechanism of energy/momentum transfer from the solar wind to the Earth's magnetosphere is magnetic reconnection (Dungey, 1961; Akasofu, 1981). When the interplanetary magnetic field (IMF) lines are southwardly oriented, that is, antiparallel to the lines of the geomagnetic field, the frozen-in plasma condition is broken in a small region in the magnetopause. When this happens, the IMF and the geomagnetic field connect in this region, known as the diffusion region. Once connected, the IMF lines are drawn into the magnetosphere tail position, where they reconnect again (Dungey, 1961). This reconnection allows the penetration of the solar wind plasma flow into the inner magnetosphere (Cowley, 1995). Thus, the entry of energy from the solar wind into the inner magnetosphere is mainly controlled by the orientation of the IMF lines, mostly with their southward component. However, the enhanced energy transfer takes place mainly during the geomagnetic storms and substorms, when such condition is achieved (e.g., Gonzalez et al., 1994).
In addition to magnetic storms and substorms, another kind of geomagnetic activity, known as the high-intensity long-duration continuous AE (auroral electrojet) activity (HILDCAA), was identified by Tsurutani and Gonzalez (1987). They suggested four criteria for characterizing the HILDCAA events: (i) the peak AE index must be 1000 nT at least once during the event, (ii) the AE index should not decrease below 200 nT for longer than 2 h at a time, (iii) the event must continue for a minimum of 2 days, and (iv) the event must occur outside of the main phase of a geomagnetic storm.
The main cause of the HILDCAAs was suggested to be the magnetic reconnection between the magnetopause field and the southward IMF Bz in the Alfvén waves embedded in the high-speed solar wind streams (HSSs) (Tsurutani and Gonzalez, 1987; Tsurutani et al., 1990b). The HSSs are emanated from the coronal holes located in the polar regions of the Sun (Sheeley et al., 1976) and are embedded with Alfvénic fluctuations (Belcher and Davis, 1971). During the descending phase of the solar cycle, the HILDCAA events are more frequently observed owing to the higher chance of the Earth encountering the HSSs as the coronal holes are shifted to the solar equatorial regions during this phase (Tsurutani et al., 1995; Hajra et al., 2013, 2014c, 2017; Mendes et al., 2017).
Although HILDCAAs can occur after geomagnetic storms caused by coronal mass ejections (CMEs) (Guarnieri, 2006), it was observed that more than 94 % of HILDCAAs occurred after co-rotating interaction regions (CIRs) (Hajra et al., 2013). The long duration of the recovery phase of the geomagnetic storms followed by HILDCAAs (Tsurutani and Gonzalez, 1987) was explained by Soraas et al. (2004) as being due to precipitation of particles in the ring current during HILDCAA events. Such particle precipitation prevents the decay of the ring current, which delays the Dst (disturbance storm time) recovery. Comparing the intensity of energy that enters into inner magnetosphere during the HILDCAAs and during geomagnetic storms, Guarnieri (2006) showed that the HILDCAA events can be more “geoeffective” than some geomagnetic storms, since HILDCAA events generally continue for longer durations (Hajra et al., 2014a).
Due to the injection of 10–100 keV electrons during the HILDCAAs, these events can lead to the acceleration of relativistic ( MeV) electrons in the outer Van Allen radiation belt (Hajra et al., 2014b, 2015a, b; Tsurutani et al., 2016). The relativistic “killer” electrons can cause rapid degradation of semiconductors and satellite sensors in orbits in this region (Guarnieri, 2005; Hajra et al., 2014b, 2015a, b).
Ionospheric effects of the HILDCAAs were studied by several authors (Sobral et al., 2006; Wei et al., 2008; Kelley and Dao, 2009; Koga et al., 2011; Silva et al., 2017). Koga et al. (2011) showed that the interplanetary electric field (IEF) is correlated with the variation of the F2-layer peak height in São Luís (44.6 W, 2.33 S), Brazil, during the HILDCAAs. Penetration of the IEF was observed during the events.
During the HILDCAAs, 6.3 × 1016 J of kinetic energy is transferred from the solar wind to the magnetosphere–ionosphere system (Hajra et al., 2014a). It was observed that the major part of the energy is dissipated as Joule heating (67 %), and the rest is dissipated as the auroral precipitation ( 22 %) and the ring current energy ( 11 %).
In a previous work (Souza et al., 2016) we have determined the main periodicities in the solar wind and in the AE index parameters during the HILDCAA events occurring between 1975 and 2011 for the AE index and between 1995 and 2011 for the IMF Bz. It was noted that during the HILDCAAs the main periods of the AE index are generally between 4 and 12 h, which corresponds to 50 % of the total periods identified. For the Bz component the main periods are found to be 8 h. In this work, the cross-wavelet analysis was applied between the IMF Bz component and the AE index during the HILDCAAs which occurred between 1995 and 2011 in order to identify the periods where solar-wind–magnetosphere coupling is more efficient. Further, the classical correlation analysis was applied in order to obtain the correlation coefficients and time lags between the IMF Bz and the AE index.
In order to perform this work, we used the AE index and IMF Bz for the 52 HILDCAAs, which occurred between 1995 and 2011, compiled by Hajra et al. (2013). The AE index was obtained from the World Data Center for Geomagnetism, Kyoto, Japan (http://wdc.kugi.kyoto-u.ac.jp/aedir/). The solar wind and interplanetary data were obtained from the OMNI database (http://omniweb.gsfc.nasa.gov/). These are a compilation of observations from various spacecraft near the Earth. Data from the solar wind are propagated from observation points up to the position of the “nose” of the bow shock of the Earth.
We have used IMF Bz data in the geocentric solar magnetospheric (GSM) coordinate system. The GSM system is centered in the Earth, with its x axis pointing in the Earth–Sun direction, the y axis perpendicular to the Earth's dipole, and the z axis being the projection of the dipole, in such a manner that the xz plane contains the dipole axis and z is positive towards the north (Russell, 1971).
3 Methodology
This work is based on the cross-wavelet transform (XWT) and the classical cross-correlation techniques applied between the IMF Bz and AE index. Thus, it is important to describe a simple introduction of these mathematical tools.
## 3.1 Cross-wavelet analysis
The wavelet functions are generated by expansions, ψ(t)→ψ(t), and translations, $\mathit{\psi }\left(t\right)\to \mathit{\psi }\left(t+\mathrm{1}\right)$, from a simple generating function over the time (t), the mother wavelet given by the following equation: ${\mathit{\psi }}_{a,b}\left(t\right)=\frac{\mathrm{1}}{\sqrt{a}}\mathit{\psi }\left(\frac{t-b}{a}\right)$. Here a represents the scale associated with expansion and contraction from the wavelet, and b is the time localization. In this paper, the Morlet wavelet will be used (Torrence and Compo, 1998), which is given as follows:
$\begin{array}{}\text{(1)}& \mathit{\psi }\left(t\right)={e}^{i{\mathit{\xi }}_{\mathrm{0}}t}{e}^{\frac{-{t}^{\mathrm{2}}}{\mathrm{2}}},\end{array}$
where ξ0 is a dimensionless frequency.
Figure 1Solar wind parameters and geomagnetic indices during the HILDCAA event occurring from 17:11 UT on 24 April to 16:46 UT on 27 April 1998. From top to bottom, the panels show the solar wind speed; density; temperature; IMF components Bx (red), By (black), and Bz (green); the IMF magnitude; the AE index; and the Dst index.
The wavelet transform (WT) applied on f(t) time series is defined as
$\begin{array}{}\text{(2)}& \text{TW}\left(a,b\right)=\int f\left(t\right){\mathit{\psi }}_{a,b}^{\ast }\left(t\right)\mathrm{d}t,\end{array}$
where f(t) is a time series, ψa,b(t) is the wavelet function, and ${\mathit{\psi }}_{a,b}^{\ast }\left(t\right)$ represents the complex conjugate of the wavelet function ψa,b(t).
We used the cross-wavelet transform to obtain the common periods between two time series and, also, to study the temporal variability of the main periods found (Bolzan et al., 2012). The XWT is given by (Grinsted et al., 2004)
$\begin{array}{}\text{(3)}& {W}^{yx}\left(a,b\right)={W}^{y}\left(a,b\right){W}^{x}\left(a,b{\right)}^{\ast },\end{array}$
where Wy and Wx represent the WT applied on the time series y(t) and x(t), respectively, and () represents the complex conjugate of the transform.
We also used the global wavelet spectrum (GWS) that is used to identify the most energetic periods present on the cross-wavelet analysis. The GWS is given by
$\begin{array}{}\text{(4)}& \text{GWS}\phantom{\rule{0.25em}{0ex}}=\int \mathrm{|}\text{TW}\left(a,b\right){\mathrm{|}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}\mathrm{d}b.\end{array}$
## 3.2 Classical cross correlation
The cross correlation between two series provides the degree of similarity between them, along with the displacement between them in time (lag). The correlation between two series, X and Y, is given by
$\begin{array}{}\text{(5)}& r=\frac{\sum \left({X}_{i}-\stackrel{\mathrm{‾}}{X}\right)\cdot \sum \left({Y}_{i}-\stackrel{\mathrm{‾}}{\stackrel{\mathrm{‾}}{Y}}\right)}{\sqrt{\sum \left({X}_{i}-\stackrel{\mathrm{‾}}{X}{\right)}^{\mathrm{2}}\sqrt{\sum \left({Y}_{i}-\stackrel{\mathrm{‾}}{Y}{\right)}^{\mathrm{2}}}}},\end{array}$
where r is the correlation coefficient.
The correlation coefficient defines how well correlated the two series are, varying from 1 to 1. When the correlation coefficient is less than zero it means that the correlation is negative, with 1 being the maximum negative correlation value, known as the perfect negative correlation. When the correlation coefficient is greater than zero, we have the positive correlation, with 1 being the perfect positive correlation. When the correlation coefficient is zero, it means that there is no correlation between the two series.
The classical correlation is calculated by the displacement of one series relative to the other by units of time (t), which provides the lag of the correlation (Davis, 1986).
4 Results
Figure 1 shows the behavior of the solar wind parameters during a HILDCAA event that occurred from 17:11 UT on 24 April to 16:46 UT on 27 April 1998. The HILDCAA interval is marked by two vertical lines. The top panel shows the solar wind speed V. It increases from a value of 420 to > 530 km s−1 during this interval. The latter represents a HSS. The proton density is shown in the second panel. At the beginning of the event, the density decays from 15 protons cm−3 in the first 7 h to 7 protons cm−3. It keeps oscillating between 7 and 15 protons cm−3 until 08:00 UT on 26 April, when a jump is observed, and the density is enhanced to 27 protons cm−3, followed by a decay, reaching a value of 4 protons cm−3. After this, the density is more or less constant until the end of the event. The third panel presents the solar wind proton temperature that varies from 2.8 × 104 to 2.65 × 105 K.
Figure 2(a) Time series of the IMF Bz component. (b) The AE index. (c) Cross-wavelet spectrum periodogram during the HILDCAA event from 17:11 UT on 24 April to 16:46 UT on 27 April 1998. (d) The global wavelet spectrum shows the main periods of correlation.
Table 1Main periods of higher correlation between the IMF Bz component and AE index.
The fourth panel shows the IMF components: Bx (red), By (black), and Bz (green). The Bz component exhibits oscillations between 8 and 7 nT, caused by the Alfvén waves. The IMF magnitude (fifth panel) decays in the initial hours and shows some variations until about the 00:00 UT of 27 April. These variations are in the range of 3–11 nT, which are in the range of the typical IMF intensity (5–10 nT) observed near the orbit of the Earth (Baumjohann and Nakamura, 2007).
The AE index (sixth panel) fulfills the HILDCAA criteria. The bottom panel shows the Dst index, which increased slowly from 40 to 20 nT. Particle precipitation in the ring current during the HILDCAA event can be responsible for the slow variations of the Dst index (Soraas et al., 2004).
In order to study the solar-wind–magnetosphere coupling during HILDCAA events, the cross-wavelet analysis was applied to the IMF Bz (considered the cause of events) and to the AE index (consequence). From the GWS results, the distribution of the correlated major periods between these two variables was also studied. In addition to the cross-wavelet analysis, the classical correlation technique was also applied in order to analyze the correlation between those two series and to obtain the time delay (lag) between them.
Figure 2a and b show the temporal variations of the IMF Bz and the AE index, respectively. Figure 2c and d show the cross-wavelet analysis between Bz and AE, as well as the GWS. These are for the HILDCAA event shown in Fig. 1. Three periods of higher correlation can be observed: these are at 5.30, at 11.56, and at 18.04 h (Fig. 2c).
Table 1 shows a summary of the results for all of the 52 HILDCAA events between 1995 and 2011. It was observed that the interval between 4 and 8 h represented most of the periods of highest correlation, with 27.3 %. More than 53 % of the events presented high correlations in periods shorter than 8 h. Further, 85 % of HILDCAAs showed higher cross-correlation power for periods 16 h.
Figure 3Classical cross correlation between IMF Bz and AE index during the HILDCAA event from 17:11 UT on 24 April to 16:46 UT on 27 April 1998. Maximum cross-correlation coefficient (r=0.74) is found at a lag =30 min.
Table 2Classification and distribution of the classical cross-correlation coefficients between IMF Bz and AE index during HILDCAA events that occurred between 1995 and 2011.
The periods observed here are similar to the periods (< 10 h) of Alfvén waves in the polar region of the Sun (Smith et al., 1995). Among all 52 HILDCAA events studied, only 4 are associated to ICMEs (interplanetary CMEs); the other 48 events are related to CIRs and HSSs. As mentioned earlier, the HSSs emanate from the solar coronal holes and are embedded with Alfvénic fluctuations. Thus, the efficient solar-wind–magnetosphere coupling during the HILDCAAs is associated with the IMF Bz Alfvén fluctuations with southward IMFs leading to the reconnection with the geomagnetic field. This is considered to be the main cause of the HILDCAA-related geomagnetic activity (Tsurutani and Gonzalez, 1987).
As mentioned previously, the classical correlation allows determining the correlation and time lag between two time series. Both the IMF Bz and the AE index had 1 min resolution, but an average of 10 min was used for the calculation of the classical cross correlation, due to the presence of noise observed for an average of 1 min.
Classical cross correlation between Bz and the AE index during the HILDCAA event shown in Fig. 1 is presented in Fig. 3. We can observe that the Bz and AE are highly anticorrelated, with a considerably high correlation coefficient of 0.74 at a time lag of 30 min. This can be interpreted as the response time of the AE index to the perturbations that occurs in the IMF Bz component. The negative correlation coefficient (anticorrelation) occurs because more energy would be transferred from solar wind to the magnetosphere when Bz is more negative, giving as a response a higher AE index. The horizontal axis of Fig. 3 presents the lag between these two time series. We can see that the lag is negative, which occurs because in the computational algorithm used to calculate the correlation between the time series; the AE index was supplied first to the IMF Bz. The positive time lag has no physical meaning, because it would mean that the AE, which is considered to be the geomagnetic consequence, happened before the Bz (cause).
Table 2 presents the cross-correlation results. In this table are shown the correlation classification intervals and the percentage distribution of the events for which the correlation was estimated. The cross correlation was moderate for 47.1 % of the events, for 33.3 % the correlation was weak, and 19.6 % of the events have high correlation. Thus, 66.7 % of HILDCAA events exhibited a moderate–strong correlation ( 0.4).
In order to find the best lag where the cross correlation between the two time series is the higher, we applied this procedure for chosen 5 lag intervals. Table 3 shows the 5 lag intervals and the number of events where we obtained the maximum correlation between the AE and IMF Bz of IMF time series. We can see that the lag where we have the maximum number of events was 30 min with 51 % of the events. Furthermore, it is possible to observe that 84.4 % of the events have the lag between 20 and 30 min, which is similar to the value observed by Tsurutani et al. (1990). They reported time lags of 20–25 min during HILDCAAs.
Table 3Distribution of the intervals of lag between IMF Bz component and the AE index.
Bargatze et al. (1985) studied the relationship between the solar wind and magnetic activity using a solar wind input function and the auroral AL index and observed two pulse peak responses from the magnetosphere in different time lags (20 and 60 min). The peak with a lag of 20 min was associated with magnetospheric activity driven directly by the solar wind coupling. The second pulse, with a lag of 60 min, was related to the magnetospheric activity driven by the release of energy stored in the magnetotail. In the present work, only a lag of 20 min was observed, and no lag of 60 min was found. This result can be explained as follows. During the HILDCAA events the AE index exhibits high values, implying strong geomagnetic and auroral activities. These strong geomagnetic HILDCAA intervals seem to be directly driven, associated with a 20 min time lag as reported by Bargatze et al. (1985). The peak of 60 min is possibly dominant in the case of moderate geomagnetic and auroral activities (not applicable to HILDCAAs).
5 Conclusions
In this work, we studied the solar-wind–magnetosphere coupling during HILDCAA events. We have identified the main periodicities of the IMF Bz and AE index during these events using the cross-wavelet (XWT) analysis, and we also applied the classical cross-correlation analysis to study the correlation and time lag between them.
In the present work, we have shown that the solar-wind–magnetosphere coupling during HILDCAA events is most efficient in periods equal to or shorter than 8 h. These are in the same range as the periodicities observed in the interplanetary Alfvén waves in the HSSs (Smith et al., 1995). This result corroborates the fact that the reconnection between the Alfvénic fluctuations in the IMF Bz and the geomagnetic fields in the magnetopause is the main cause of the HILDCAA events.
Through the classical correlation analysis technique, moderate correlation (0.4–0.7) was obtained between the AE and the IMF Bz. The time lag between them is mostly 20–30 min. This is close to the time lag (20–25 min) reported by Tsurutani et al. (1990). The correlation coefficients between IMF Bz and AE observed in the present work (0.4–0.7) are also consistent with the value (0.62) reported by them. This represents moderate correlation between the geomagnetic activity (AE) index and the interplanetary parameter (IMF Bz).
Thus we may conclude that the solar-wind–magnetosphere coupling during HILDCAAs is mainly due to magnetic reconnection between southward IMF Bz and magnetopause fields. This mechanism is more efficient at periods of 8 h or less, with a 20–30 min time lag between IMF variations and magnetosphere and auroral response.
Data availability
Data availability.
The AE index and IMF Bz data used in this manuscript are public available at http://wdc.kugi.kyoto-u.ac.jp/dstae/index.html (WDC, 2018) and https://omniweb.gsfc.nasa.gov/form/sc_merge_min1.html (GSFC, 2018).
Competing interests
Competing interests.
The authors declare that they have no conflict of interest.
Special issue statement
Special issue statement.
This article is part of the special issue “Space weather connections to near-Earth space and the atmosphere”. It is a result of the 6o Simpósio Brasileiro de Geofísica Espacial e Aeronomia (SBGEA), Jataí, Brazil, 26–30 September 2016.
Acknowledgements
Acknowledgements.
AMSF would like thank the FAPESP agency for support (project 2016/10794-2). MJA was supported by the Goiás Research Foundation (FAPEG) (grant no. 201210267000905) and CNPq (grants no. 302330/2015-1). EE thanks the CNPq agency for support (project CNPq/PQ 302583/2015-7). The work of RH was supported by ANR under financial agreement ANR-15-CE31-0009-01 at LPC2E/CNRS.
The topical editor, Alisson Dal Lago, thanks Gurbax S. Lakhina and one anonymous referee for help in evaluating this paper.
References
Akasofu, S.-I.: Energy coupling between the solar wind and the magnetosphere, Space Sci. Rev., 28, 121–190, 1981.
Bargatze, L. F., Baker, D. N., McPherron, R. L., Hones Jr., E. Wl.: Magnetospheric inpulse response for many levels of geomagnetic activity, J. Geophys. Res., 90, 6387–6394, 1985.
Baumjohann, W. and Nakamura, R.: Magnetospheric Contributions to the Terrestrial Magnetic Field, Space Research Institute, Austrian Academy of Sciences, Graz, Austria, 77–91, 2007.
Belcher, J. W. and Davis Jr., L.: Large-amplitude Alfvén waves in the interplanetary medium, 2, J. Geophys. Res., 76, 3534–3563, 1971.
Bolzan, M. J. A. and Rosa, R. R.: Multifractal analysis of interplanetary magnetic field obtained during CME events, Ann. Geophys., 30, 1107–1112, https://doi.org/10.5194/angeo-30-1107-2012, 2012.
Cowley, S. W. H.: The Earth's magnetosphere: a brief beginner's guide, EOS T. Am. Geophys. Un., 76, 525–532, 1995.
Davis, J. C.: Statistics and Data Analysis in Geology, John Wiley & Sons, New York, NY, USA, 1986.
Dungey, J. W.: Interplanetary magnetic field and auroral zones, Phys. Rev. Lett., 6, 47–48, 1961.
Gonzalez, W. D., Joselyn, J. A., Kamide, Y., Kroehl, H. W., Rostoker, G., Tsurutani, B. T., and Vasyliunas, V. M.: What is a geomagnetic storm?, J. Geophys. Res., 99, 5771–5792, 1994.
Grinsted, A., Moore, J. C., and Jevrejeva, S.: Application of the cross wavelet transform and wavelet coherence to geophysical time series, Nonlin. Processes Geophys., 11, 561–566, https://doi.org/10.5194/npg-11-561-2004, 2004.
GSFC: IMF Bz data, available at: https://omniweb.gsfc.nasa.gov/form/sc_merge_min1.html, last access: 1 February 2018.
Guarnieri, F. L.: A study of the interplanetary and solar origin of high intensity long duration and continuous auroral activity events, Ph.D. thesis, Inst. Nac. Pesqui. Espaciais, Sao Jose dos Campos, Brazil, 2005.
Guarnieri, F. L.: The nature of auroras during high-intensity long-duration continuous AE activity (HILDCAA) events, 1998 to 2001, in: Recurrent Magnetic Storms: Corotating Solar Wind Streams, edited by: Tsurutani, B. T., Mcpherron, R. L., Gonzalez, W. D., Lu, G., Sobral, J. H. A., and Gopalswamy, N., Geophys. Monogr., Am. Geophys. Univ. Press, Washingtom, DC, 167, 235 p., 2006.
Hajra, R., Echer, E., Tsurutani, B. T., and Gonzalez, W. D.: Solar cycle dependence of High-Intensity Long-Duration Continuous AE Activity (HILDCAA) events, relativistic electron predictors?, J. Geophys. Res.-Space, 118, 5626–5638, https://doi.org/10.1002/jgra.50530, 2013.
Hajra, R., Echer, E., Tsurutani, B. T., and Gonzalez, W. D.: Solar wind-magnetosphere energy coupling efficiency and partitioning: HILDCAAs and preceding CIR storms during solar cycle 23, J. Geophs. Res., 119, 2675–2690, 2014a.
Hajra, R., Echer, E., Tsurutani, B. T., and Gonzalez, W. D.: Relativistic electron acceleration during high-intensity, long-duration, continuous AE activity (HILDCAA) events: solar cycle phase dependences, Geophys. Res. Lett., 41, 1876–1881, 2014b.
Hajra, R., Echer, E., Tsurutani, B. T., and Gonzalez, W. D.: Superposed epoch analyses of HILDCAAs and their interplanetary drivers: solar cycle and seasonal dependences, J. Atmos. Sol.-Terr. Phy., 121, 24–31, 2014c.
Hajra, R., Tsurutani, B. T., Echer, E., Gonzalez, W. D., Brum, C. G. M., Vieria, L. E. A., and Santolik, O.: Relativistic electron acceleration during HILDCAA events: are CIR magnetic storms important?, Earth Planets Space, 61, 1–11, 2015a.
Hajra, R., Tsurutani, B. T., Echer, E., Gonzalez, W. D., and Santolik, O.: Relativistic (E> 0.6, > 2.0, and > 4.0 MeV) electron acceleration at geosynchronous orbit during high-intensity, long-duration, continuous ae activity (HILDCAA) events, Astrophys. J., 799, 39, https://doi.org/10.1088/0004-637X/799/1/39, 2015b.
Hajra, R., Tsurutani, B. T., Brum, C. G. M., and Echer, E.: High-speed solar wind stream effects on the topside ionosphere over Arecibo: a case study during solar minimum, Geophys. Res. Lett., 44, 7607–7617, https://doi.org/10.1002/2017GL073805, 2017.
Kelley, M. C. and Dao, E.: On the local time dependence of the penetration of solar wind-induced electric fields to the magnetic equator, Ann. Geophys., 27, 3027–3030, https://doi.org/10.5194/angeo-27-3027-2009, 2009.
Koga, D., Sobral, J. H. A., Gonzalez, W. D., Arruda, S. C. S., Abdu, M. A., Castilho, V. M., Mascarenhas, M., Gonzalez, A. C., Tsurutani, B. T., Denardini, C. M., and Zamlutti, C. J.: Electrodynamic coupling process between the magnetosphere and the equatorial ionosphere during a 5-day HILDCAA event, J. Atmos. Sol.-Terr. Phy., 73, 148–155, 2011.
Mendes, O., Domingues, M. O., Echer, E., Hajra, R., and Menconi, V. E.: Characterization of high-intensity, long-duration continuous auroral activity (HILDCAA) events using recurrence quantification analysis, Nonlin. Processes Geophys., 24, 407–417, https://doi.org/10.5194/npg-24-407-2017, 2017.
Russell, C.: Geophysical coordinate transformations, Cosmic Electrodynamics, 2, 184–196, 1971.
Sobral, J. H. A., Abdu, M. A., Gonzalez, W. D., Clua De Gonzalez, A. L., Tsurutani, B. T., Da Silva, R. R. L., Barbosa, I. G., Arruda, D. C. S., Denardini, C. M., Zamlutti, C. J., and Guarnieri, F.: Equatorial ionospheric responses to high-intensity long-duration auroral electrojet activity (HILDCAA), J. Geophys. Res., 111, A07S02, https://doi.org/10.1029/2005JA011393, 2006.
Soraas, F., Aarsn, K., Oksavik, K., Sandanger, M. I., Evans, D. S., and Greer, M. S.: Evidence for particle injection as the cause of Dst reduction during HILDCAA events, J. Atmos. Sol.-Terr. Phy., 66, 177–186, 2004.
Souza, A. M., Echer, E., Bolzan, M. J. A., and Hajra, R.: A study on the main periodicities in interplanetary magnetic field Bz component and geomagnetic AE index during HILDCAA events using wavelet analysis, J. Atmos. Sol.-Terr. Phy., 149, 81–86, https://doi.org/10.1016/j.jastp.2016.09.006, 2016.
Sheeley Jr., N. R., Harvey, J. W., and Feldman, W. C.: Coronal holes, solar wind streams, and recurrent geomagnetic disturbances: 1973–1076, Sol. Phys., 49, 271–278, 1976.
Silva, R. P., Sobral, J. H. A., Koga, D., and Souza, J. R.: Evidence of prompt penetration electric fields during HILDCAA events, Ann. Geophys., 35, 1165–1176, https://doi.org/10.5194/angeo-35-1165-2017, 2017.
Smith, E. J., Balogh, A., Neugebauer, M., and McComas, D.: Ulysses observations of Alfvén waves in the southern and northern solar hemispheres, Geophys. Res. Lett., 22, 3381–3384, 1995.
Torrence, C. and Compo, G. P.: A practical guide to wavelet analysis, B. Am. Meteorol. Soc., 79, 61–78, https://doi.org/10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2, 1998.
Tsurutani, B. T. and Gonzalez, W. D.: The cause of High-Intensity Long-Duration Continuous AE Activity (HILDCAAS) interplanetary alfven-wave trains, Planet. Space Sci., 35, 405–412, 1987.
Tsurutani, B. T., Goldstein, B. E., Smith, E. J., Gonzalez, W. D., Tang, F., Akasofu, S. I., and Anderson, R. R.: The interplanetary and solar causes of geomagnetic activity, Planet. Space Sci., 38, 109–126, https://doi.org/10.1016/0032-0633(90)90010-N, 1990.
Tsurutani, B. T., Gonzalez, W. D., Gonzalez, A. L. C., Tang, F., Arballo, J. K., and Okada, M.: Interplanetary origin of geomagnetic activity in the declining phase of the solar cycle, J. Geophys. Res., 100, 21717–21733, 1995.
Tsurutani, B. T., Hajra, R., Tanimori, T., Takada, A., Bhanu, R., Mannucci, A. J., Lakhina, G. S., Kozyra, J. U., Shiokawa, K., Lee, L. C., Echer, E., Reddy, R. V., and Gonzalez, W. D.: Heliospheric plasma sheet (HPS) impingement onto the magnetosphere as a cause of relativistic electron dropouts (REDs) via coherent EMIC wave scattering with possible consequences for climate change mechanisms, J. Geophys. Res., 121, 10130–10156, https://doi.org/10.1002/2016JA022499, 2016.
WDC: AE index, available at: http://wdc.kugi.kyoto-u.ac.jp/dstae/index.html, last access: 1 February 2018.
Wei, Y., Hong, M., Wan, W., Du, A., Lei, J., Zhao, B., Wang, W., Ren, Z., and Yue, X.: Unusually long lasting multiple penetration of interplanetary electric field to equatorial ionospheric under oscillating IMF Bz, J. Geophys. Res., 35, L02102, https://doi.org/10.1029/2007GL032305, 2008.
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2019-06-26 08:02:38
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https://math.stackexchange.com/questions/1680980/using-mean-value-theorem-on-an-example
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# Using Mean Value Theorem on an example
I thought of just using the Mean Value Theorem and plugging in the interval values in it for (a) and (b). Although it seems too easy, am I missing something? Also how would I do (c)?
Suppose $f:(-\infty,0] \rightarrow \mathbb R$ is continuous everywhere and differentiable on (-$\infty,0$). Suppose also that $\lim_{x\to -\infty} f(x)=0$.
(a) Show that there exists a $c \in (-1,0)$ such that $f'(c)=f(0)-f(-1).$
(b) Formulate and prove a similar statement in the interval $(-2,-1)$.
(c) Suppose further that $\lim_{x\to -\infty} f'(x)=R$ for some $R \in \mathbb R$. Prove that $R=0$.
• Part a doesn't make sense; $f$ isn't defined at $0$ – Michael Harrison Mar 3 '16 at 4:24
• @MichaelHarrison A typo, fixed it, thanks for pointing out. – NeoXx Mar 3 '16 at 4:28
• Ah, okay. In that case, you are right, it's a direct application of MVT. For part c, is it intuitively clear? – Michael Harrison Mar 3 '16 at 4:31
By the Mean Value Theorem, for every positive integer $n$ there is a $c_n$ between $-(n+1)$ and $-n$ such that $$\frac{f(-n)-f(-(n+1))}{1}=f'(c_n).\tag{1}$$
Since $\lim_{x\to-\infty} f(x)=0$, it follows that the limit of the left-hand side of (1) is $0$, and therefore $\lim_{n\to\infty} f'(c_n)=0$.
Since $\lim_{x\to\infty}=R$, it follows that $R=0$.
Remark: You are right, there is nothing much to (a) and (b), it is all in preparation for (c).
HINT:
$$f'(\xi)=\frac{f(-n)-f(-n-1)}{-n-(-n-1)}=f(-n)-f(-n-1)$$
and $\lim_{x\to -\infty}f(x)=0$
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2019-12-06 18:28:32
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https://rdoodles.rbind.io/tags/fake-data/
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# What is the consequence of a Shapiro-Wilk test-of-normality filter on Type I error and Power?
Set up Normal distribution Type I error Power Right skewed continuous – lognormal What the parameterizations look like Type I error Power This 1990-wants-you-back doodle explores the effects of a Normality Filter – using a Shapiro-Wilk (SW) test as a decision rule for using either a t-test or some alternative such as a 1) non-parametric Mann-Whitney-Wilcoxon (MWW) test, or 2) a t-test on the log-transformed response.
# What is the bias in the estimation of an effect given an omitted interaction term?
Some background (due to Sewall Wright’s method of path analysis) Given a generating model: $$$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3$$$ where $$x_3 = x_1 x_2$$; that is, it is an interaction variable. The total effect of $$x_1$$ on $$y$$ is $$\beta_1 + \frac{\mathrm{COV}(x_1, x_2)}{\mathrm{VAR}(x_1)} \beta_2 + \frac{\mathrm{COV}(x_1, x_3)}{\mathrm{VAR}(x_1)} \beta_3$$. If $$x_3$$ (the interaction) is missing, its component on the total efffect is added to the coefficient of $$x_1$$.
# GLM vs. t-tests vs. non-parametric tests if all we care about is NHST -- Update
Update to the earlier post, which was written in response to my own thinking about how to teach stastics to experimental biologists working in fields that are dominated by hypothesis testing instead of estimation. That is, should these researchers learn GLMs or is a t-test on raw or log-transformed data on something like count data good enough – or even superior? My post was written without the benefit of either [Ives](Ives, Anthony R.
# The statistical significance filter
1 Why reported effect sizes are inflated 2 Setup 3 Exploration 1 4 Unconditional means, power, and sign error 5 Conditional means 5.1 filter = 0.05 5.2 filter = 0.2 1 Why reported effect sizes are inflated This post is motivated by many discussions in Gelman’s blog but start here When we estimate an effect1, the estimate will be a little inflated or a little diminished relative to the true effect but the expectation of the effect is the true effect.
# Paired line plots
load libraries make some fake data make a plot with ggplot ggplot scripts to draw figures like those in the Dynamic Ecology post Paired line plots (a.k.a. “reaction norms”) to visualize Likert data load libraries library(ggplot2) library(ggpubr) library(data.table) make some fake data set.seed(3) n <- 40 self <- rbinom(n, 5, 0.25) + 1 others <- self + rbinom(n, 3, 0.5) fd <- data.table(id=factor(rep(1:n, 2)), who=factor(rep(c("self", "others"), each=n)), stigma <- c(self, others)) make a plot with ggplot The students are identified by the column “id”.
# A simple ggplot of some measure against depth
set up The goal is to plot the measure of something, say O2 levels, against depth (soil or lake), with the measures taken on multiple days library(ggplot2) library(data.table) First – create fake data depths <- c(0, seq(10,100, by=10)) dates <- c("Jan-18", "Mar-18", "May-18", "Jul-18") x <- expand.grid(date=dates, depth=depths) n <- nrow(x) head(x) ## date depth ## 1 Jan-18 0 ## 2 Mar-18 0 ## 3 May-18 0 ## 4 Jul-18 0 ## 5 Jan-18 10 ## 6 Mar-18 10 X <- model.
#### R doodles. Some ecology. Some physiology. Much fake data.
Thoughts on R, statistical best practices, and teaching applied statistics to Biology majors.
Jeff Walker, Professor of Biological Sciences
University of Southern Maine, Portland, Maine, United States
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2019-09-20 23:09:44
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https://johannesbader.ch/blog/project-euler-problem-435-polynomials-of-fibonacci-numbers/
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# Project Euler Problem 435 - Polynomials of Fibonacci Numbers
Spoiler Alert! This blog entry gives away the solution to problem 435 of Project Euler. Please don’t read any further if you have yet to attempt to solve the problem on your own. The information is intended for those who failed to solve the problem and are looking for hints or test data to help them track down bugs.
First I provide references of the major theoretical background that you probably need to know to solve the problem and test data to validate your own algorithm. The last section presents the solution from start to finish.
The post reflects my approach to the problem. Even though the final outcome was accepted by Project Euler doesn’t mean the information is correct or elegant. Algorithms won’t necessarily be the most efficient ones, but they are guaranteed to run within the time limit of one minute.
## Test Data
Edit: This post originally listed wrong values. Thanks, Michael_Foo_Bar, for pointing out the errors.
$f_{10^15} \text{ mod } 15!$ 36651874875 $\sum_{x=0}^{3} F_3(x)$ 92 $\sum_{x=0}^{10} F_{10}(x) \text{ mod } 27$ 14 $F_{10^15}(2) \text{ mod } 15!$ 960038235750
## Solution
### Simplify the polynomial
The generating function
$$$F_n(x) = f_1 x + f_2 x^2 + f_3 x^3 + f_4 x^4 + \cdots + f_n x^n$$$
can be simplified the same way the Sum of geometric progression is derived. First, multiply (1) by $x$ , and by $x^2$ respectively to obtain the following set of equations:
\begin{align} F_n(x) &= f_1 x + f_2 x^2 + f_3 x^3 + f_4 x^4 + \cdots + f_n x^n \label{eq:a} \\ x\cdot F_n(x) &= \phantom{f_1 x +}\; f_1 x^2 + f_2 x^3 + f_3 x^4 + f_4 x^5 + \cdots + f_n x^{n+1} \label{eq:b} \\ x^2\cdot F_n(x) &=\phantom{f_1 x + f_2 x^2 + }\; f_1 x^3 + f_2 x^4 + f_3 x^5 + f_4 x^6 + \cdots + f_n x^{n+2} \label{eq:c} \end{align}
Now subtract the last two equations from the first to get:
$$$\begin{split} F_n(x)(1-x-x^2) =& f_1 x + (f_2-f_1) x^2 + (f_3 - f_2 - f_1) x^3 \\ &+ (f_4 - f_3 - f_2) x^4 + \cdots + (f_n - f_{n-1} - f_{n-2}) x^n \\ &- f_n x^{n+1} - f_{n-1} x^{n+1} - f_n x^{n+2} \end{split}$$$
Most of the terms like $(f_3 - f_2 - f_1)$ drop out because $$f_n = f_{n-1} + f_{n-2}$$, and $$F_n(x)$$ is:
$$$F_n(x) = \frac{ f_n x^{n+2} + f_{n+1} x^{n+1} - x}{x^2+x-1}$$$
### Calculate the Fibonacci terms
Equation (2) still requires Fibonacci terms for very large n. The closed-form won’t work. Instead, the matrix form turns out to be useful:
$$$f_n = (1, 0) \begin{pmatrix}1 & 1 \\1 & 0\end{pmatrix}^n \begin{pmatrix}1\\0\end{pmatrix}$$$
Exponentiation by squaring helps to calculate the nth power of the matrix – and the powers of x – fast.
### Apply modular arithmetic
Since Fibonacci numbers grow large very fast we will only be able to calculate the modulus. Unfortunately we can’t just determine all values in (2) modulo 15! and end up with the desired result, because the following does not hold:
$$$\frac{a}{b} \text{ mod } m \neq \frac{a \text{ mod } m}{b \text{ mod } m}$$$
One way to do division is to use the modular multiplicative inverse, but those aren’t always defined in our instance. The Chinese Remainder Theorem would help fix that, but there is a simpler solution:
$$$\frac{a}{b} \text{ mod } m = \frac{a \text{ mod } m\cdot b}{b}, \text{ if } a \text{ mod } b \equiv 0$$$
If we apply that to Equation (2):
$$$\begin{split} F_n(x) \text{ mod } m = \frac{(f_n x^{n+2} + f_{n+1} x^{n+1} - x) \text{ mod } \big(m(x^2+x-1)\big)}{ (x^2+x-1)} \end{split}$$$
The numerator only uses multiplication and addition which allows to calculate the Fibonacci terms as well as the powers of x modulo $m\cdot(x^2+x-1)$ .
### Putting it all together
def mat_mult(A, B, m):
C = [[0,0],[0,0]]
for i in range(2):
for j in range(2):
for k in range(2):
C[i][j] += A[i][k]*B[k][j]
C[i][j] %= m
return C
def mat_pow(A, p, m):
if p == 1:
return A
if p % 2:
return mat_mult(A, mat_pow(A, p-1, m), m)
X = mat_pow(A, p//2, m)
return mat_mult(X, X, m)
def fib(n, m):
T = [[1,1], [1,0]]
if n == 1:
return 1
T = mat_pow(T, n-1, m)
return T[0][0]
def mpow(x, p, m):
if p == 1:
return x
if p % 2:
return x*mpow(x, p-1, m) % m
r = mpow(x, p//2, m)
return r*r % m
def F(n, x, m):
b = (x*x + x - 1)
f_nn = fib(n+1, m*b)
f_n = fib(n, m*b)
a = (f_nn*mpow(x, n+1, m*b) + f_n*mpow(x, n+2, m*b) - x) % (m*b)
return a/b
def calc(n,m,x_range):
R = 0
for x in x_range:
R += F(n, x, m)
R %= m
return R
if __name__ == "__main__":
n = pow(10,15)
m = 1307674368000
x_range = range(0, 101)
print(calc(n, m, x_range))
Note: I removed the Disqus integration in an effort to cut down on bloat. The following comments were retrieved with the export functionality of Disqus. If you have comments, please reach out to me by Twitter or email.
Sep 14, 2013 19:43:21 UTC
Well, checking this with pencil and paper, Sum_x=0,...,3 F_3(x) seems not to be 26, but 92. And the next sum does not match my calculation, too. Can you please check your test data?
Sep 14, 2013 20:09:29 UTC
You are absolutely right. I got tricked by the upper limit of Python's range function. The value should now be correct.
Mar 08, 2016 05:43:48 UTC
It's actually good that you have shared about this kind of information in order for those people to know some effective strategies on how are they going to promote some good solution in case if they encounter that certain kind of problem.
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2019-11-14 12:43:25
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https://adm.ebu.io/tutorial/content_part.html
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# Content part
The ADM can be divided in the content and the format part. The format part can exist without the content part, but not the other way around. Here we'll go through the content part with some simple examples.
## Single Programme
In many situations the ADM file you'll be generating will be for a single programme. A programme is the top level of the ADM, and the audioProgramme element is used to describe it. As with the other ADM main elements, we can give the audioProgramme a name, an ID, some time-related information and some other useful parameters. But lets start with the most basic settings, with name and ID set (both are mandatory):
<audioProgramme audioProgrammeName="Documentary"
audioProgrammeID="APR_1001">
</audioProgramme>
This doesn't really tell us much, but it provides an entry point into the ADM, from which further things can be referenced. What we can add at this stage is the start and duration of the programme, so let's make it 30 minutes long:
<audioProgramme audioProgrammeName="Documentary"
audioProgrammeID="APR_1001"
start="00:00:00.00000" duration="00:30:00.00000">
</audioProgramme>
## Describing the Content
So, what have we got in our programme? In this example, we've got some narration, sound effects and background music. These are not described in the audioProgramme, but in the next element: audioContent.
For example, we can generate three audioContent elements, and give them some suitable names and IDs. Another thing we can add to each of these three elements is some information about whether the audio is dialogue or not:
<audioContent audioContentName="Narration"
audioContentID="ACO_1001">
<dialogue dialogueContentKind="1">1</dialogue>
</audioContent>
<audioContent audioContentName="SoundFX"
audioContentID="ACO_1002">
<dialogue nonDialogueContentKind="2">0</dialogue>
</audioContent>
<audioContent audioContentName="BgMusic"
audioContentID="ACO_1003">
<dialogue nonDialogueContentKind="1">0</dialogue>
</audioContent>
Now that we've defined these three audioContent elements, we need the audioProgramme element to be able to see them. This is done by adding some ID references to the audioProgramme element:
<audioProgramme audioProgrammeName="Documentary"
audioProgrammeID="APR_1001"
start="00:00:00.00000" duration="00:30:00.00000">
<audioContentIDRef>ACO_1001</audioContentIDRef>
<audioContentIDRef>ACO_1002</audioContentIDRef>
<audioContentIDRef>ACO_1003</audioContentIDRef>
</audioProgramme>
So our elements are now structured like this:
## Connecting the Content Description to the Audio
We've now got three audioContent elements that each describe part of our programme, but these content descriptions need some actual audio connected to them. This is where the 'audioObject' element comes in. This element references audio tracks and the format description for those tracks, and can be referenced from the audioContent element.
Let's make some audioObject element for our example, we'll generate three of them, one for each audioContent element:
<audioObject audioObjectName="Narration"
audioObjectID="AO_1001">
<audioPackFormatIDRef>AP_00031001</audioPackFormatIDRef>
<audioTrackUIDRef>ATU_00000001</audioTrackUIDRef>
</audioObject>
<audioObject audioObjectName="SoundFX"
audioObjectID="AO_1002">
<audioPackFormatIDRef>AP_00010003</audioPackFormatIDRef>
<audioTrackUIDRef>ATU_00000002</audioTrackUIDRef>
<audioTrackUIDRef>ATU_00000003</audioTrackUIDRef>
<audioTrackUIDRef>ATU_00000004</audioTrackUIDRef>
<audioTrackUIDRef>ATU_00000005</audioTrackUIDRef>
<audioTrackUIDRef>ATU_00000006</audioTrackUIDRef>
<audioTrackUIDRef>ATU_00000007</audioTrackUIDRef>
</audioObject>
<audioObject audioObjectName="BgMusic"
audioObjectID="AO_1003">
<audioPackFormatIDRef>AP_00010002</audioPackFormatIDRef>
<audioTrackUIDRef>ATU_00000008</audioTrackUIDRef>
<audioTrackUIDRef>ATU_00000009</audioTrackUIDRef>
</audioObject>
In each object there is an audioPackFormatIDRef sub-element, and this is a reference to an audioPackFormat element that describes the format of the group of channels the audio has. There are also some audioTrackUIDRef sub-elements, and these are references to the actual track of audio. So for each of the three objects we have these references:
• Narration (AO_1001)
• Pack: AP_00031001 - 'Object' type containing a single channel
• Track UID: ATU_00000001 - Single track
• SoundFX (AO_1002)
• Pack: AP_00010003 - 'DirectSpeakers' type containing a 5.1 set of channels
• Track UIDs: ATU_00000002 to ATU_00000007 - Six tracks
• BgMusic (AO_1003)
• Pack: AP_00010002 - 'DirectSpeakers' type containing a stereo pair of channels
• Track UIDs: ATU_00000008 and ATU_00000009 - Two tracks
We can now go back and connect our audioContent elements to the audioObject elements:
<audioContent audioContentName="Narration"
audioContentID="ACO_1001">
<dialogue dialogueContentKind="1">1</dialogue>
<audioObjectIDRef>AO_1001</audioObjectIDRef>
</audioContent>
<audioContent audioContentName="SoundFX"
audioContentID="ACO_1002">
<dialogue nonDialogueContentKind="2">0</dialogue>
<audioObjectIDRef>AO_1002</audioObjectIDRef>
</audioContent>
<audioContent audioContentName="BgMusic"
audioContentID="ACO_1003">
<dialogue nonDialogueContentKind="1">0</dialogue>
<audioObjectIDRef>AO_1003</audioObjectIDRef>
</audioContent>
So, we have now generated a description of our content and connected to the format description via the audioPackFormatIDRef sub-elements within audioObject. The audioObject element also contains other parameters to allow setting of time restrictions (more can be read on the Timing page), interactivity and mutual exclusivity.
Our elements are now structured like this:
You should have already seen the audioPackFormat elements in the format part page, so you can now see how the format part connects with the content part.
## Track UIDs and Actual Audio Tracks
In the audioObject element you'll see the audioTrackUIDRef sub-elements, which reference audioTrackUID elements. The audioTrackUID element represents part of, or a complete, audio track in the file. In its simplest form it doesn't have to carry any other information, but it can include the sample-rate and bit-depth used if wanted.
In a BW64 the 'chna' chunk carries the relationship between audioTrackUIDs and the actual tracks in the file like this:
TrackNum audioTrackUID audioTrackFormatID audioPackFormatID
1 ATU_00000001 AT_00031001_01 AP_00031001
2 ATU_00000002 AT_00010001_01 AP_00010003
3 ATU_00000003 AT_00010002_01 AP_00010003
4 ATU_00000004 AT_00010003_01 AP_00010003
5 ATU_00000005 AT_00010004_01 AP_00010003
6 ATU_00000006 AT_00010005_01 AP_00010003
7 ATU_00000007 AT_00010006_01 AP_00010003
8 ATU_00000008 AT_00010001_01 AP_00010002
9 ATU_00000009 AT_00010002_01 AP_00010002
So, you can see how the 9 audio tracks in the file are each given an audioTrackUID ID, as well as audioTrackFormatID and audioPackFormatIDs which describe the format of each tracks.
|
2021-09-20 20:58:03
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|
https://guitarknights.com/guitar-open-tuning-guitar-kits-usa.html
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After you’re feeling more comfortable with the transitions, plug in this progression to your Uberchord. You should find that it’s much easier to play along with the progressions. Even with chords you aren’t yet comfortable with. The key to playing cleanly and precisely is training yourself to pay attention to the movement of your fingers. You’ll find that this heightened awareness translates into every new chord you learn.
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"Open" chords get their name from the fact that they generally include strings played open. This means that the strings are played without being pushed down at a fret, which makes chords including them easier to play for beginners. When you start to learn chords, you have to focus on using the right fingers to press down each note and make sure you're pressing the strings down firmly enough.
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Unlike a piano or the voices of a choir, the guitar (in standard tuning) has difficulty playing the chords as stacks of thirds, which would require the left hand to span too many frets,[40] particularly for dominant seventh chords, as explained below. If in a particular tuning chords cannot be played in closed position, then they often can be played in open position; similarly, if in a particular tuning chords cannot be played in root position, they can often be played in inverted positions. A chord is inverted when the bass note is not the root note. Additional chords can be generated with drop-2 (or drop-3) voicing, which are discussed for standard tuning's implementation of dominant seventh chords (below).
When you’re learning a new chord, make the shape and leave it on the guitar for about thirty seconds. Then remove your hand, shake it out, and make the chord shape again. It may take some time for you to make the chord shape again, but that’s okay because you’re working on your muscle memory. Repeating this process a few times is a great way of memorizing your chords.
The top, back and ribs of an acoustic guitar body are very thin (1–2 mm), so a flexible piece of wood called lining is glued into the corners where the rib meets the top and back. This interior reinforcement provides 5 to 20 mm of solid gluing area for these corner joints. Solid linings are often used in classical guitars, while kerfed lining is most often found in steel string acoustics. Kerfed lining is also called kerfing because it is scored, or "kerfed"(incompletely sawn through), to allow it to bend with the shape of the rib). During final construction, a small section of the outside corners is carved or routed out and filled with binding material on the outside corners and decorative strips of material next to the binding, which are called purfling. This binding serves to seal off the end grain of the top and back. Purfling can also appear on the back of an acoustic guitar, marking the edge joints of the two or three sections of the back. Binding and purfling materials are generally made of either wood or plastic.
An open tuning allows a chord to be played by strumming the strings when "open", or while fretting no strings. The base chord consists of at least three notes and may include all the strings or a subset. The tuning is named for the base chord when played open, typically a major triad, and each major-triad can be played by barring exactly one fret.[60] Open tunings are common in blues and folk music,[59] and they are used in the playing of slide and lap-slide ("Hawaiian") guitars.[60][61] Ry Cooder uses open tunings when he plays slide guitar.[59]
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The main purpose of the bridge on an acoustic guitar is to transfer the vibration from the strings to the soundboard, which vibrates the air inside of the guitar, thereby amplifying the sound produced by the strings. On all electric, acoustic and original guitars, the bridge holds the strings in place on the body. There are many varied bridge designs. There may be some mechanism for raising or lowering the bridge saddles to adjust the distance between the strings and the fretboard (action), or fine-tuning the intonation of the instrument. Some are spring-loaded and feature a "whammy bar", a removable arm that lets the player modulate the pitch by changing the tension on the strings. The whammy bar is sometimes also called a "tremolo bar". (The effect of rapidly changing pitch is properly called "vibrato". See Tremolo for further discussion of this term.) Some bridges also allow for alternate tunings at the touch of a button.
The ratio of the spacing of two consecutive frets is {\displaystyle {\sqrt[{12}]{2}}} (twelfth root of two). In practice, luthiers determine fret positions using the constant 17.817—an approximation to 1/(1-1/ {\displaystyle {\sqrt[{12}]{2}}} ). If the nth fret is a distance x from the bridge, then the distance from the (n+1)th fret to the bridge is x-(x/17.817).[15] Frets are available in several different gauges and can be fitted according to player preference. Among these are "jumbo" frets, which have much thicker gauge, allowing for use of a slight vibrato technique from pushing the string down harder and softer. "Scalloped" fretboards, where the wood of the fretboard itself is "scooped out" between the frets, allow a dramatic vibrato effect. Fine frets, much flatter, allow a very low string-action, but require that other conditions, such as curvature of the neck, be well-maintained to prevent buzz.
Learn the C chord. The first chord we will cover is a C chord—one of the most basic chords in music. Before we do, let's break down just what that means. A proper chord, whether played on a piano, a guitar, or sung by well-trained mice, is simply three or more notes sounded together. (Two notes is called a "diad," and while musically useful, is not a chord.) Chords can also contain far more than three notes, but that's well beyond the scope of this article. This is what a C chord looks like on the guitar:
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For the second note of the A minor 7 chord, place your second finger on the second fret of the D string. This is the second of the two notes you need to fret to play this chord. Make sure you’re on the tip of your finger and right behind the fret. Now that you have both notes in place, strum the top five strings, remembering to leave the low E string out.
As with most chords in this list, a clear G major chord depends on curling your first finger so the open fourth string rings clearly. Strum all six strings. Sometimes, it makes sense to play a G major chord using your third finger on the sixth string, your second finger on the fifth string, and your fourth (pinky) finger on the first string. This fingering makes the move to a C major chord much easier.
For example, in the guitar (like other stringed instruments but unlike the piano), open-string notes are not fretted and so require less hand-motion. Thus chords that contain open notes are more easily played and hence more frequently played in popular music, such as folk music. Many of the most popular tunings—standard tuning, open tunings, and new standard tuning—are rich in the open notes used by popular chords. Open tunings allow major triads to be played by barring one fret with only one finger, using the finger like a capo. On guitars without a zeroth fret (after the nut), the intonation of an open note may differ from then note when fretted on other strings; consequently, on some guitars, the sound of an open note may be inferior to that of a fretted note.[37]
"Open" chords get their name from the fact that they generally include strings played open. This means that the strings are played without being pushed down at a fret, which makes chords including them easier to play for beginners. When you start to learn chords, you have to focus on using the right fingers to press down each note and make sure you're pressing the strings down firmly enough.
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Do you play a warm-up exercise when you practice guitar? Guitar teacher Kirk R. shares three guitar exercises that are perfect for players at all levels... There are literally thousands of exercises and studies for the guitar. There are some that are great for beginners who are just getting used to having their fingers on the guitar, and some that are designed to challenge and grow the technique of seasoned players. But who has time to learn thousands of guitar exercises, even over man
Kyser®'s 92/8 phosphor bronze acoustic strings quickly settle in to give your guitar a warm, bright, and well balanced tone. They are precision wound with a corrosion resistant blend of 92% copper and 8% tin phosphide onto a carefully drawn hex shaped high carbon steel core. The result is a long lasting, even tone, with excellent intonation. Click the individual string images to view more gauge information.
Português: Tocar Acordes de Guitarra, Español: tocar acordes de guitarra, Deutsch: Akkorde auf der Gitarre spielen, Italiano: Suonare gli Accordi con la Chitarra, Français: jouer des accords à la guitare, Русский: играть аккорды на гитаре, 中文: 弹吉他和弦, Nederlands: Gitaarakkoorden spelen, हिन्दी: गिटार बजाएँ, Tiếng Việt: Chơi Hợp âm Ghita, ไทย: จับคอร์ดกีตาร์, 日本語: ギターでコードを弾く, العربية: عزف كوردات الجيتار
What's the best way to learn guitar? No matter which method you choose, or what style of music you want to play, these three rules from guitar teacher Sean L. are sure to put you on the road to success... Learning guitar can be a daunting task when first approached. For many it is seen as only for the musically adept, but in reality anyone can learn guitar. By following these three simple rules, anyone can become a great guitarist. 1. Set Goals There is no one path to take for learning
In music, a guitar chord is a set of notes played on a guitar. A chord's notes are often played simultaneously, but they can be played sequentially in an arpeggio. The implementation of guitar chords depends on the guitar tuning. Most guitars used in popular music have six strings with the "standard" tuning of the Spanish classical-guitar, namely E-A-D-G-B-E' (from the lowest pitched string to the highest); in standard tuning, the intervals present among adjacent strings are perfect fourths except for the major third (G,B). Standard tuning requires four chord-shapes for the major triads.
Guitars have been played since the Renaissance era, after descending from the ancient Greek instrument, the kithara. Or, at least, that's how the theory goes: the only real proof for this is the similarities between the Greek word and the Spanish word, quitarra. Early guitars often had four strings, evolving into the six-string version we know today in the late 1600s. Anton Stradivari, the famous violin-maker, also had a hand in making guitars. There's now only one Stradivarius guitar left in existence.
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2019-05-26 22:15:48
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https://en.wikibooks.org/wiki/Statistical_Thermodynamics_and_Rate_Theories/Translational_energy
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# Statistical Thermodynamics and Rate Theories/Translational energy
A gas particle can move through space in three independent directions: x, y, and z. Each of these directions is a distinct translational degree of freedom. For ideal gases, these particles can move freely through space in any of these directions until it collides with the wall of its container. For isolated systems, collisions with the wall of the container are assumed to be elastic, meaning no energy is lost upon impact.
## Particle in a 1D Box
The first quantum mechanical model used for describing translation is a particle in a simple 1D box. It is free to move anywhere along one axis (usually assigned to be the x axis) between the arbitrarily assigned boundary limits 0 and a. At distances smaller than 0 and greater than “a” the potential energy function immediately rises to infinity. The particle cannot move past these points. 0 and a represent the walls of the 1D box. We can mathematically assign boundary conditions using this information which allows the derivation of the wave function for a particle in a 1D box. The resulting piece-wise function, wave function and energy level equation are given as follows,
${\displaystyle {\mathcal {V}}(x)={\begin{cases}\infty ,x<0\\0,0{\leq }x{\leq }a\\\infty ,x>a\end{cases}}}$
${\displaystyle \Psi (x)={\sqrt {\frac {2}{a}}}\sin {\left({\frac {n\pi x}{a}}\right)}}$
${\displaystyle n=1,2,3,...}$
As well as the corresponding energy levels of the system.
${\displaystyle \epsilon _{n}={\frac {h^{2}n^{2}}{8ma^{2}}}}$
Note: In the equations above m represents the mass of the gas particle, and h is Planck's constant.
## Zero-Point Energy
The energy levels for translation within the box are quantized, only discrete energy levels can be occupied by the particle at any given time. Each of these energy levels are defined by a single quantum number n. n can take on any integer values starting from 1 and up to a hypothetical infinity. The n = 0 state for a particle in a box does not exist. The particle, in accordance with the Heisenberg uncertainty principle cannot be motionless. If this were to be the case then both the momentum and position of the particle could be determined simultaneously, which is a violation of the principle. The energy of a particle in a 1D box at the lowest translational energy level is therefore non-zero (i.e., n = 1).
## Probability Density Plots and the Correspondence Principle
It is possible to construct a probability density plot of the particle in a 1D box wave function. The plot is characterized by sizable “humps” at low values of n. For example, at n=1 there is one large hump which spans from a minimum at x = 0, a maximum somewhere in the center and a second minimum at x = a. These minimums represent regions of zero probability density. In other words, the particle will not be found in these regions.
As n increases the spacing between humps of the probability density plot becomes smaller and smaller until a near continuum is achieved at sufficiently high values of n. It must also be noted that the magnitude of the energy spacing’s between translational energy levels is extremely small relative to the amount of energy available to a particle under normal conditions. At room temperature ideal gas particles will occupy very high energy levels. As we increase the quantum number to a large enough value the behavior of the system will begin to reproduce that of classical mechanics in that the particle is essentially equally likely to be found anywhere in the box rather than within discrete hump regions seen at lower values of n. This is what is known as the correspondence principle.
## Particle in a 3D Box
The correspondence principle applies to a 3D box as well. For a particle in a 3D box, the particle is able to move in any direction along the x, y and z axes. Since its motion now incorporates a combination of three possible directions the system must include two additional quantum numbers to compensate for the difference. nx, ny, and nz for the x, y, and z dimensions, respectively. Both the wave function and the equation for the energy levels must be adjusted to compensate for the new quantum numbers. These are given as follows,
${\displaystyle \Psi _{n_{x},n_{y},n_{z}}={\sqrt {\frac {8}{abc}}}\sin \left({\frac {n_{x}{\pi }x}{a}}\right)\sin \left({\frac {n_{y}{\pi }y}{b}}\right)\sin \left({\frac {n_{z}{\pi }z}{c}}\right)}$
${\displaystyle \epsilon _{n_{x},n_{y},n_{z}}={\frac {h^{2}}{8m}}\left({\frac {{n_{x}}^{2}}{a^{2}}}+{\frac {{n_{y}}^{2}}{b^{2}}}+{\frac {{n_{z}}^{2}}{c^{2}}}\right)}$
a, b, and c represent the length of each corresponding side of the box. If the box were a cube then the energy level equation can be simplified because sides a, b, and c of the cube are all equal. The resulting equation would take on the form,
${\displaystyle \epsilon _{n_{x},n_{y},n_{z}}={\frac {h^{2}}{8m{a^{2}}}}({{n_{x}}^{2}}+{{n_{y}}^{2}}+{{n_{z}}^{2}})}$
For a particle in a cube some combinations of quantum numbers will give the same energy level. Energy levels with a different set of quantum numbers but having the same energy are said to be degenerate and unless all three of the quantum numbers are identical there is always another combination of the three quantum numbers that will give rise to a degenerate state.
## Example
Calculate the energy difference, ${\displaystyle \Delta E}$, for the translation of N2 from the ground state to the first excited state. Assume the box is a cube with an edge length of 10 cm.
### Solution
The ground state, where ${\displaystyle n_{x}=n_{y}=n_{z}=1}$, has a degeneracy of 1. The first excited state has a degeneracy of 3, involving the different combinations of quantum numbers ${\displaystyle n=1,1,}$ and 2. Can be determined by using the following equation:
${\displaystyle \epsilon _{n_{x},n_{y},n_{z}}={\frac {h^{2}}{8m}}\left({\frac {{n_{x}}^{2}}{a^{2}}}+{\frac {{n_{y}}^{2}}{b^{2}}}+{\frac {{n_{z}}^{2}}{c^{2}}}\right)}$
However because all edges of the cube is 10 cm, then a=b=c=10 cm = 0.1 m. Then the equation becomes:
${\displaystyle \epsilon _{n_{x},n_{y},n_{z}}={\frac {h^{2}}{8m{a^{2}}}}\left({{n_{x}}^{2}}+{{n_{y}}^{2}}+{{n_{z}}^{2}}\right)}$
The mass of N2 can be determined with the following equation:
${\displaystyle \mu ={\cfrac {m_{N}m_{N}}{m_{N}+m_{N}}},\!\,}$
${\displaystyle \mu ={\cfrac {m_{N}^{2}}{2m_{N}}},\!\,}$
${\displaystyle \mu ={\cfrac {m_{N}}{2}},\!\,}$
${\displaystyle \mu ={\cfrac {14.0067u}{2}},\!\,}$
${\displaystyle \mu =(7.00335u)(1.660549\times 10^{-27}kg/u),\!\,}$
${\displaystyle \mu =1.16294\times 10^{-26}kg,\!\,}$
Using the mass the ${\displaystyle \Delta E}$ can now be determined.
${\displaystyle \epsilon _{n_{x},n_{y},n_{z}}={\frac {h^{2}}{8m{a^{2}}}}({{n_{x}}^{2}}+{{n_{y}}^{2}}+{{n_{z}}^{2}})-({{n_{x}}^{2}}+{{n_{y}}^{2}}+{{n_{z}}^{2}})}$
${\displaystyle \epsilon _{n_{x},n_{y},n_{z}}={\frac {({6.626\times 10^{-34}})^{2}Js}{8(1.16294\times 10^{-26}kg){0.1m^{2}}}}({{2}^{2}}+{{1}^{2}}+{{1}^{2}})-({{1}^{2}}+{{1}^{2}}+{{1}^{2}})}$
${\displaystyle \epsilon _{n_{x},n_{y},n_{z}}={\frac {{4.3904\times 10^{-67}}J^{2}s^{2}}{9.30352\times 10^{-28}kgm^{2}}}(6-3)}$
${\displaystyle \epsilon _{n_{x},n_{y},n_{z}}=1.4157\times 10^{-39}J}$
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2020-07-10 14:01:14
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http://www.reddit.com/r/math/comments/10ip69/completeness_of_power_set/?sort=random
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[–] 1 point2 points (3 children)
sorry, this has been archived and can no longer be voted on
...ZF? Isn't this pretty much the definition of k <= |X|?
[–][S] 1 point2 points (2 children)
sorry, this has been archived and can no longer be voted on
Not quite. For any Y in P(X), there exists an injection from Y to X; hence for any Y in P(X), |Y| <= |X| by the definition of <= for cardinal numbers. My question is whether the power set is in some sense "complete", i.e. whether every smaller cardinal is "represented" in the power set.
[–] 4 points5 points (1 child)
sorry, this has been archived and can no longer be voted on
Let k <= |X|. Let Z be any representative of k, i.e. |Z|=k. Then |Z| <= |X|. By your definition of <=, there exists an injection from Z to X. Let Y be the image of this injection. Then Y is in P(X) and |Y| = |Z| = k.
[–][S] 0 points1 point (0 children)
sorry, this has been archived and can no longer be voted on
Ah yes, clearly I was being short-sighted. Thanks.
[–] 0 points1 point (0 children)
sorry, this has been archived and can no longer be voted on
With choice, cardinals have a total order : Let A, B be two sets. Apply Zorn's lemma to the set X of partial bijections between A and B (the set of subsets R of AxB such that for any (a,b) in R, there is no other(a,b') nor (a',b) in R). Order X with the subset relation. It is an inductive set because the reunion of any increasing chain in X is still in X. So by Zorn's lemma, there is a maximal partial bijection between A and B, and then it is easy to see that it has to use all of A or all of B (if not, it wouldn't be a maximal partial bijection). Thus there is an injection from A to B or an injection from B to A (or both, but in this case, there is a bijection by the Cantor Bernstein theorem)
So yes, for any set X, P(X) contains sets for all the cardinalities up to X.
[–] 0 points1 point (2 children)
sorry, this has been archived and can no longer be voted on
Intuitively, what choice gets you is that with choice, every set has a cardinality. But in your question this doesn't matter, because when you talk about cardinals and |X|, you're only dealing with sets that have cardinality already.
You're basically asking, of the sets that DO have a cardinality, does this intuitively obvious property hold? So the fact that there are some bizzare sets out there that maybe don't have a cardinality doesn't matter, because you're ignoring those from the start.
That's why you only need ZF.
[–][S] 0 points1 point (1 child)
sorry, this has been archived and can no longer be voted on
My issue didn't actually have anything to do with choice, but rather with how you get at the weird cardinals (which are still cardinals) that are implied to exist by the negation of GCH (and, more importantly for standard mathematics, by the negation of CH). But yes, in hindsight I see that this is actually rather obvious.
[–]Logic 0 points1 point (0 children)
sorry, this has been archived and can no longer be voted on
how you get at the weird cardinals (which are still cardinals) that are implied to exist by the negation of GCH
Usually, by forcing.
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2015-05-04 00:21:55
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https://www.groundai.com/project/multivariate-covariance-generalized-linear-models/
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[
# [
## Abstract
We propose a general framework for non-normal multivariate data analysis called multivariate covariance generalized linear models (McGLMs), designed to handle multivariate response variables, along with a wide range of temporal and spatial correlation structures defined in terms of a covariance link function combined with a matrix linear predictor involving known matrices. The method is motivated by three data examples that are not easily handled by existing methods. The first example concerns multivariate count data, the second involves response variables of mixed types, combined with repeated measures and longitudinal structures, and the third involves a spatio-temporal analysis of rainfall data. The models take non-normality into account in the conventional way by means of a variance function, and the mean structure is modelled by means of a link function and a linear predictor. The models are fitted using an efficient Newton scoring algorithm based on quasi-likelihood and Pearson estimating functions, using only second-moment assumptions. This provides a unified approach to a wide variety of different types of response variables and covariance structures, including multivariate extensions of repeated measures, time series, longitudinal, spatial and spatio-temporal structures.
Generalized Kronecker product; Linear covariance model; Matrix linear predictor; Non-normal data; Pearson estimating function; Quasi-likelihood; Spatio-temporal data
McGLM]Multivariate Covariance Generalized Linear Models Bonat, W.H and Jørgensen, B.]Wagner Hugo Bonat Bonat, W.H and Jørgensen, B.]Bent Jørgensen
## 1 Introduction
The analysis of non-normal multivariate data currently involves a choice between a considerable array of different modelling frameworks, ranging from, say, generalized estimating equations (GEE) and time-series models to generalized linear mixed models and model-based geostatistics. Each framework allows the modelling of a specific type of dependence or correlation structure, without fitting into any clear overall pattern. Current software implementations have, as we shall see below, limited capacity in terms of the complexity and size of data that can be handled.
This situation stands in sharp contrast to the univariate case, where Nelder and Wedderburn’s (1972) generalized linear models (GLMs) provide a unified and versatile approach to regression modelling of normal and non-normal data, implemented in an efficient fitting algorithm. A further advantage of the GLM approach is that estimation and inference for GLMs require only second-moment assumptions.
In order to obtain a multivariate modelling framework of comparable range and versatility, we shall propose the class of multivariate covariance generalized linear models (McGLMs), which, following Pourahmadi (1999), are specified via separate link functions and linear predictors for the mean vector and covariance matrix, respectively. This allows a unified approach to analysis of multivariate correlated data, taking into account response variable of mixed types, and allowing a wide range of covariance structures for repeated measures, longitudinal, spatial and spatio-temporal data. The models are fitted by means of quasi-likelihood and Pearson estimating functions, based on second-moment assumptions, and implemented in an efficient Newton scoring algorithm.
The idea of modelling a function of the covariance matrix by a linear structure goes back at least as far as Anderson (1973), followed later by Chiu et al. (1996), who used the matrix logarithm as covariance link function. More recently, the idea was extended in several different ways by Pourahmadi (1999, 2011), Pan and Mackenzie (2003) and Zhang et al. (2015), among others. These authors consider mainly the multivariate normal distribution, whereas we shall use a variance function to take non-normality into account in the style of Liang and Zeger (1986). Contrary to the latter authors we shall, however, emphasize the need to model the covariance structure explicitly, rather than treating it as a nuisance parameter.
The availability of standard software is an indicator for which kind of statistical methods are in currently use by the statistical and scientific communities. It is hence interesting to note that well-established R packages such as lme4 (Bates et al., 2014) and nlme (Pinheiro et al., 2013) do not deal with multivariate response variables. In the Bayesian context the flexible packages INLA (Rue et al., 2014) and MCMCpack (Martin et al., 2011) do not deal with multivariate response variables, judging from the package documentation. In R, there are at least two generalized linear mixed models packages that can deal with multivariate response variables, namely MCMCglmm (Hadfield, 2010), which uses Monte Carlo Markov Chain (MCMC) methods in the Bayesian framework, and the package SabreR (Crouchley, 2012), which uses marginal likelihood, but is limited to dealing with at most three response variables. The modelling of the covariance structure is currently restricted to making a selecting from a short list of pre-specified covariance structures, such as autoregression or compound symmetry. We were not able to find any R packages for fitting joint mean-covariance models, not even in the multivariate normal case. In SAS the GLIMMIX procedure for generalized linear mixed models (GLMMs) deals with multivariate response variables, but is limited to the exponential family of distributions and a few pre-determined covariance structures (SAS Institute, 2011). Other software platforms for fitting generic random effects models via MCMC, such as JAGS (Plummer, 2003) or WinBUGS (Lunn et al., 2000), can deal with multivariate response variables, but carry substantial overheads in terms of computational times and convergence checks, while being restricted to a small set of pre-specified covariance structures and probability distributions. These limitations on current software availability for joint mean-covariance modelling of multivariate response variables may reflect either a lack of interest on the part of software users, or a lack of sufficiently flexible modelling frameworks. In any case, we will use the latter as motivation for developing the new class of McGLMs.
We now present three correlated data examples along with a short review of currently available methods for each type of data. The examples were selected in order to highlight some of the limitations of current methodology, while illustrating the range of different problems that may be handled by the McGLM method.
### 1.1 Data set 1: Australian health survey
The first data set is from the Australian Health Survey for 1987–1988 (Deb and Trivedi, 1997; Cameron and Trivedi, 1998). We selected the following five count response variables for our analysis: number of consultations with a doctor or specialist (Ndoc) or with health professionals (Nndoc); total number of prescribed and non prescribed medications used in the past two days (Nmed); number of nights in a hospital during the most recent admission (Nhosp) and number of admissions to a hospital, psychiatric hospital, nursing or convalescence home in the past months (Nadm). The data set had nine covariates concerning social conditions (see Appendix for details). There were respondents and no missing data.
This example illustrates the fairly common situation of a multivariate regression problem with non-normal (discrete) response variables. The histograms in Figure 1 suggest that the five error distributions may not be identical, and hint at potential problems with excess of zeroes and under/overdispersion. These problems may, in turn, reflect on the solution to the main questions of the analysis, namely assessing the effects of the covariates on each outcome, and determining the residual correlation structure.
Given currently available software, it is a daunting task to select a suitable marginal error distribution for each of the five response variables. Besides the classical Poisson and negative binomial distributions, other distributions such as the Neyman Type A (Dobbie and Welsh, 2001) or the Poisson-inverse Gaussian (PIG) (Holla, 1967) may be relevant. Different distributions may have to be fitted by separate software packages, each of which comes with its own set of problems due to badly behaved likelihood function etc.
If we decide to use formal methods of model selection, we are faced with the choice of selection criterion, such as the Akaike or Bayesian information criterion in the likelihood framework, or the deviance information criterion in the Bayesian framework. The Bayesian case involves additional work due to the need for choosing suitable prior distributions. These problems persist in the special case where all error distributions belong to the same family. One option is the multivariate Poisson regression (Tsionas, 1999), which is suitable for multivariate count data, but is restricted to positive correlations and equidispersed data. A second option is the multivariate negative binomial distribution proposed by Shi and Valdez (2014). Such models are not easy to fit, and require careful attention to the implementation of algorithms and starting values. The assumption of a common error distribution required for these models may, however, not be satisfied in practice, and methods for handling the case of unequal marginal distributions do not seem to be easily available.
A different approach for correlated data is the family of generalized linear mixed models (GLMM) (Breslow and Clayton, 1993; Fong et al., 2010), which is based on specifying a GLM conditionally on a multivariate latent distribution, often the multivariate normal. A specific example of a GLMM for multivariate count data was presented by Rodrigues-Motta et al. (2013). GLMMs are computationally demanding, and many different algorithms have been proposed in the past three decades, see McCulloch (1997) and Fong et al. (2010) for reviews and further references.
A further aspect of GLMMs that gives rise to concern is the general lack of a closed-form expression for the likelihood and the marginal distribution of the data vector. This makes model selection even more complicated than for the marginal models discussed above. A related question is the special interpretation of parameters inherent from the construction of GLMMs. Thus, the covariate effects are conditional on the latent variables, whereas the correlation structure is marginal for the latent variables rather than for the response variables. An interesting discussion of random-effects and marginal models may be found in Lee and Nelder (2004).
Additional methods for specifying models for multivariate response variables include the copula models (Krupskii and Joe, 2013) and the class of hierarchical generalized linear models (Lee and Nelder, 1996). The fact that several different approaches are available for multivariate regression modelling, none of which is particularly easy to use, amplifies our call for a universal multivariate modelling framework, preferably one that facilitates model selection and allows marginal interpretation of parameters.
### 1.2 Data set 2: Respiratory physiotherapy on premature newborns
We consider some aspects of a prospective study to assess the effect of respiratory physiotherapy on the cardiopulmonary function of ventilated preterm newborn infants with birth weight lower than 1500 g. The study had three response variables: respiratory rate (RR), heart rate (HR) and oxygen saturation (OSat). The HR and OSat data were collected by electronic monitoring and RR by means of a stopwatch. Response variables were taken three times: before starting the physiotherapy (Evaluation 1), immediately after finishing (Evaluation 2), and five minutes after finishing the physiotherapy (Evaluation 3). Sixteen newborns were evaluated in consecutive sessions by two therapists at the neonatal unit. The number of evaluation days varied between and days. The data set has covariates concerning health conditions and there are cases (see the Appendix). Figure 2 shows the individual and average trajectories by outcome and evaluation.
The main goal of the investigation was to assess the effect of respiratory physiotherapy on the outcome variables, while taking into account the effects of covariates and the correlation induced by the repeated measures and the longitudinal structures. A special feature of these data is that the outcome variables are of mixed types. Thus, the variables HR and RR are continuous, whereas the oxygen saturation variable OSat takes values in the unit interval, including about exact ones, making it hard to propose a suitable probability distribution for this variable. We may, of course, use for example the beta (Bonat et al., 2015) or the simplex distribution (Zhang, 2014) with some ad hoc method for dealing with the exact ones. A better option may be to use the beta distribution inflated with ones (Ospina and Ferrari, 2010), but this model is complicated to fit and interpret. It may hence be preferable in this situation to use a quasi-likelihood method based on second-moment assumptions, which is easier to fit and interpret.
Similar to what we saw in Example , the literature may be divided into two main approaches: marginal models, mostly based on the GEE approach (O’Brien and Fitzmaurice, 2004; Rochon, 1996; Gray and Brookmeyer, 2000), and random-effects models based on GLMMs, see Verbeke et al. (2014). These authors also provide an extensive review of models for response variables of mixed type, whereas Fieuws et al. (2007) reviewed random-effects models for multivariate repeated measures. The question of how to model the covariance structure for repeated measures and longitudinal data is often solved by choosing from a short list of options, such as compound symmetry, autoregressive, banded and unstructured (Diggle et al., 2002). Such choices are, however, not suitable for the combination of repeated measures and longitudinal data found in the present data, thereby motivating the development of a more general and flexible approach for covariance modelling in multivariate data analysis.
### 1.3 Data set 3: Venezuelan rainfall data
This example concerns monthly rainfall data from stations in the Venezuelan state of Guaárico for a period of years ( months). The data set has cases with missing data. We also have the spatial coordinates (latitude and longitude) of the stations available, along with the covariate height (height above sea level). The data were previously analyzed by Sansó and Guenni (1999) using Bayesian MCMC methods, based on a censored and transformed multivariate normal distribution.
The statistical modelling of rainfall data involves a number of challenges, such as the need for simultaneous modelling of seasonal and geographical variation, the complicated nature of the spatio-temporal correlation structure, the special form of the marginal distribution (having a discrete component at zero), and the possible influence of the sampling scale on the form of the analysis (Dunn, 2004). The plots shown in Figure 3 illustrate some of these features for the Venezuelan rainfall data. In particular, the histogram in panel D highlights the right-skewed distribution and the considerable proportion of exact zeroes (around ), whereas the approximate linearity of the Taylor plot in Panel C suggests a variance function of power form.
A simple model for the marginal distribution of total rainfall over a certain time period is to write , where is the number of rainfall episodes, assumed to be Poisson distributed, and the i.i.d. variables are the amounts of rain for each episode, with the convention for , corresponding to a discrete component at zero. A special case of this compound Poisson model is the Tweedie family (Jørgensen, 1997) (where the are gamma distributed), with power variance functions, in agreement with the Taylor plot of Figure 3. The Tweedie model has been successfully applied to rainfall data by Dunn (2004) and Hasan and Dunn (2010, 2012). These authors, however, assume independent data, which is not realistic for the present data set.
A popular approach for analyzing rainfall data (Chandler and Wheater, 2002; Sigrist et al., 2012) is to use separate models for the discrete component, indicating the number of wet periods, and the continuous component, indicating the amount of rain for wet periods (Stern and Coe, 1984; Wilks, 1990). A variety of distributions have been proposed for modelling the continuous component of rainfall under the independence assumption, including the log-normal, Weibull, generalized log-normal, gamma and mixed gamma distributions (Hasan and Dunn, 2010, 2012). While these distributions may have their merits for analyzing rainfall data, the above compound Poisson model seems more natural, and the Tweedie family is flexible enough to mimic many of the shapes of other distributions.
Turning now to the question of spatio-temporal modelling of rainfall data, one possibility is to use models based on marked point processes (Wheater et al., 2000; Cowpertwait et al., 2006), which may be useful for detailed simulation studies. Another approach is to follow the conventional geostatistical paradigm, assuming a parametric covariance function (Diggle and Ribeiro, 2007). There are several parametric families available for modelling the joint space-time covariance structure (Cressie and Huang, 1999; Gneiting, 2002), although there are issues with their interpretability and computational complexity, making it difficult to handle large data sets with this approach.
A different approach to spatio-temporal modelling is to take into account the fundamental difference between the spatial and temporal dimensions, the latter obeying a natural ordering which is not present in the spatial dimension. It may hence be natural to assume a dynamic temporal evolution model in combination with spatially correlated errors, see Sansó and Guenni (1999, 2004); Sigrist et al. (2012) and the monograph by Cressie and Wikle (2011). While providing a flexible form of spatio-temporal modelling, this method is also computationally demanding, and handles response variables with a discrete component at zero by means of a censored multivariate normal distribution, which does not provide as reasonable an interpretation as the Tweedie model.
A significant simplification may be obtained by assuming that the spatial domain is discrete, rather than being continuous as in the last two methodologies discussed above. This approach is used for example in disease mapping (Besag et al., 1991), where the covariance structure is determined by a neighborhood matrix. This is computationally less demanding, because for a given neighborhood structure we may specify the inverse covariance (or precision) matrix. The precision matrix, in turn, contains information about the structure of conditional independence of the data (Rue and Held, 2005). The proposed simplification may hence be seen as a reasonable compromise between model complexity and the capacity to model real data sets, achieveable by modelling the covariance structure using a linear combination of neighborhood matrices. To accommodate rainfall data, such a modelling strategy should allow for Tweedie distributed response variables with power variance functions. Section 2 presents the class of McGLMs, and Section 3 considers the Newton scoring algorithm. The three data examples presented here are analyzed in Section 4 using McGLMs. The results are discussed in Section 5, including some directions for future investigations.
## 2 Multivariate covariance generalized linear models
In this Section we will present the McGLM approach as an extension of GLMs. Let be an response vector, an design matrix and a regression parameter vector. A GLM can be written in the following form:
E(Y) = μ=g−1(Xβ) Var(Y) = Σ=V(μ;p)12(τ0I)V(μ;p)12 (1)
where is the link function, , is a diagonal matrix whose main entries are given by the variance function applied elementwise to the vector . Finally and are the power and dispersion parameters, respectively, and denotes the identity matrix.
The success enjoyed by the GLM framework comes from its ability to deal with a wide range of non-normal data using just two separate functions, namely the link and variance functions. The variance function plays an important role for GLMs, since different choices imply different assumptions about the response variable distribution. The power variance functions are a frequent choice in the GLM framework. It characterizes the Tweedie family of distributions, whose most important special cases are the normal , Poisson , Gamma and inverse Gaussian distributions (Jørgensen, 1987, 1997). But in spite of its flexibility the GLM approach has some limitations: it deals only with independent and univariate response variables, and the variance function is assumed to be known.
Our main objectives are to extend the GLM approach to deal with first non-independent data and second multivariate response variables. A third objective is to estimate the power parameter, which works as automatic model selection.
The Tweedie family is quite flexible for handling continuous response variables, but it is less flexible for discrete response variables. Therefore, we propose to use the Poisson-Tweedie family to deal with discrete data (El-Shaarawi et al., 2011). The Poisson-Tweedie family has variance function , and many important models for count data are special cases, for example the Hermite , Neyman Type A , negative binomial and Poisson-inverse Gaussian , see Jørgensen and Kokonendji (2014). When using the Poisson-Tweedie family, the matrix in (2) takes the special form because the dispersion parameter appears only in the second term. Another important case is when the response variable is binary, bounded, or the number of successes within a given number of trials. In that case the binomial variance function may be useful.
It is important to emphasize that by using just these three sets of variance functions we can deal with most frequently occurring types of response variables. Such flexibility is very useful, for example when analysing data set , where the choice of count distribution for each response variable is not obvious. Using the Poisson-Tweedie variance function we can deal with zero-inflation and overdispersion, such as that observed in data set . A similar situation appears for data set , where we have a bounded response variable with exact ones, which can be well modelled using the binomial variance function. The Tweedie family, through its power variance function, can model zero-inflated and right skewed response variables, such as the monthly rainfall data.
In Eq. (2) it is easy to see where the assumption of independent observations appears in the covariance matrix, which in turn suggests how to introduce dependence between observations. It is enough to change the identity matrix to a non-diagonal matrix . This approach is similar to the idea of a working correlation matrix in the Generalized Estimation Equation (GEE) framework (Liang and Zeger, 1986; Zeger et al., 1988). Our approach differs from GEE in that we propose to model in terms of a linear combination of known matrices, following the ideas of Anderson (1973) and Pourahmadi (2000), i.e.
h(Ω(τ))=τ0Z0+⋯+τDZD. (2)
Here is the covariance link function, with are known matrices reflecting the covariance structure, and is a parameter vector. This structure is a natural analogue of the linear predictor of the mean structure, and we call it a matrix linear predictor. Plugging the matrix linear predictor (2) into Eq. (2), we obtain a so-called covariance generalized linear model.
Two new issues appear here, concerning how to specify the covariance link function and how to define the matrices . The first issue was discussed by Pinheiro and Bates (1996) and Pourahmadi (2011). In this paper we will focus on well-known covariance link functions, such as the identity and the inverse functions. In Section 4 we show how to specify the matrices in order to obtain some well-known models for time series, spatial and space-time data.
Many authors claim that a suitable covariance link function must provide an unrestricted and interpretable parametrization. While laudable, such a goal is probably over-optimistic, and does not seem to have been achieved yet, at least not for the general case (Pourahmadi, 2000; Pinheiro and Bates, 1996). The modified Cholesky decomposition proposed by Pourahmadi et al. (2007) presents both features, but is restricted to the case where there is a natural ordering of the observations. In general, the identity and inverse covariance link functions allow for simple interpretations, but these covariance link functions do not provide unrestricted parametrizations. In fact it is quite hard to define the parameter space for . In Section 3 we propose the so-called reciprocal likelihood algorithm where we use a tuning constant in order to control the step length of the algorithm and avoid unrealistic values for the parameter vector . From an algorithmic point of view, there is hence no need to require an unrestricted parametrization.
The second main contribution of this paper is to extend the covariance generalized linear model to deal with multivariate response variables. Let be a response variable matrix and let denote the corresponding matrix of expected values. To indicate that each response variable has its own covariance matrix we use the notation . It is important to emphasize that this matrix models the covariances within each response variable. We introduce the correlation matrix to model the correlation between response variables. To specify the joint covariance matrix for all response variables, we adopt the generalized Kronecker product proposed by Martinez-Beneito (2013) in the context of multivariate disease mapping. We hence define the McGLM by
E(Y) = M={g−11(X1β1),…,g−1R(XRβR)} Var(Y) = C=ΣRG⊗Σb (3)
where is the generalized Kronecker product. The matrix denotes the lower triangular matrix of the Cholesky decomposition of . The operator denotes a block diagonal matrix and denotes an identity matrix.
## 3 Estimation and inference
In this Section we describe the estimating function approach used to estimate the model parameters (Jørgensen and Knudsen, 2004). We divide the set of parameters into two subsets, . In this notation denotes a vector containing all regression parameters. Similarly, we let be a vector of all dispersion parameters.
To simplify the discussion, let be the stacked vector of the response variable matrix by columns. Similarly, let be the stacked vector of the expected values matrix by columns.
We adopt the following quasi-score function for the regression parameters:
ψβ(β,λ)=D⊤C−1(Y−M),
where is an matrix, and denotes the gradient operator. The matrix
Sβ=E(∇βψβ)=−D⊤C−1D (4)
is the sensitivity matrix of and the matrix
Vβ=Var(ψβ)=D⊤C−1D (5)
is the variability matrix of .
Similarly, we adopt the Pearson estimating function, defined by the components
ψλi(β,λ)=tr(Wλi(r⊤r−C))fori=1,…,Q, (6)
where and . Details on how to compute these weight matrices are given in Section 3.1.
The entry of the sensitivity matrix of is given by,
Sλij=E(∂∂λiψλj)=−tr(WλiCWλjC). (7)
We may show using results about characteristic functions of linear and quadratic forms of non-normal variables (Knight, 1985), that the entry of the variability matrix of is given by
Vλij=Cov(ψλi,ψλj)=2tr(WλiCWλjC)+NR∑l=1k(4)l(Wλi)ll(Wλj)ll, (8)
where denotes the fourth cumulant of to be discussed below, see Eq. (14). To take into account the covariance between the vectors and , we compute the cross-sensitivity and -variability matrices. The entry of the cross-sensitivity matrix between and is given by
Sβiλj=E(∂∂λjψβi)=0. (9)
In a similar way the entry of the cross-sensitivity matrix between and is given by
Sλiβj=E(∂∂βjψλi)=−tr(WλiCWβjC). (10)
We can show that the entry of the cross-variability matrix between and is given by
Vλiβj=E[NR∑k=1NR∑l=1NR∑m=1W(lm)λiA(j)krkrlrm], (11)
where and denotes the column of . In a similar way denotes the entry of the matrix . Furthermore, the joint sensitivity matrix of and is given by
Sθ=(SβSβλSλβSλ),
whose entries are defined by (4), (7), (9) and (10). Finally, the joint variability matrix of and is given by
Vθ=(VβV⊤λβVλβVλ),
whose entries are defined by (5), (8) and (11).
Let be the estimating function estimator of . Then the asymptotic distribution of is
^θ∼N(θ,J−1θ)
where is the inverse of Godambe information matrix,
J−1θ=S−1θVθS−Tθ,
where .
Jørgensen and Knudsen (2004) proposed the modified chaser algorithm to solve the system of equations and , defined by
β(i+1) = β(i)−S−1βψβ(β(i),λ(i)) λ(i+1) = λ(i)−S−1λψλ(β(i+1),λ(i)). (12)
The modified chaser algorithm uses the insensitivity property (9), which allows us to use two separate equations to update and . This procedure was implemented in R (R Core Team, 2015) and some generic functions are made available in the supplement material. The modified chaser algorithm is often quite efficient, but it does not have any way to control the step length. Thus, based on ideas from Jensen et al. (1991) we propose the reciprocal likelihood algorithm involving an additional tuning constant to control the step length. The reciprocal likelihood algorithm replaces the second equation of (3) by
λ(i+1) = λ(i)−[αψλ(β(i+1),λ(i))Tψλ(β(i+1),λ(i))V−1λSλ+Sλ]−1ψλ(β(i+1),λ(i)). (13)
The strategy for choosing used in this paper consists of starting the algorithm with , and continuing with as long as the proposed value of corresponds to a positive-definite covariance matrix. In the opposite case, we increase the value of by a small quantity (e.g. ) and try again until the covariance matrix becomes positive-definite, after which we return to , corresponding to the modified chaser algorithm. Compared with conventional step length methods, our method is adaptive in the sense that directions where the estimating function is far from zero are being less penalized.
To compute the variance of the dispersion parameter estimators we used the empirical fourth cumulants, i.e.
k(4)l=(yl−^μl)4−3^C2ll. (14)
The empirical third central moment was computed based on equation (11), ignoring the expectation. The main advantage of using empirical third and fourth moments is that the resulting method depends on second-moment assumptions only. The additional variability induced by the use of empirical moments implies, however, increased variability of the asymptotic covariance of the dispersion parameter estimators, in particular for small sample sizes.
The Pearson estimating function (6) is unbiased only if the vector of regression parameters is known. Jørgensen and Knudsen (2004) proposed a bias-correction for the Pearson estimating function. The th bias-correction term is given by
bλi=−tr(J(λi)βJ−1β), (15)
where denotes the Godambe information matrix for and . The corrected Pearson estimating function may be solved using the same algorithm as for the Pearson estimating function. The variability matrix does not depend on the bias-correction term. This is not the case for the sensitivity matrix, but the contribution of the correction term to the sensitivity is so small that it can be ignored.
### 3.1 Derivatives of the covariance matrix
The key calculation in relation to the fitting algorithm is to compute the derivative of the covariance matrix . In this Section we will provide details of this calculation for the model presented in Eq. (2). Let for denote the correlation parameters. We use the convention to stack the lower triangle of the correlation matrix by columns. Let be an vector of power parameters. Finally, let be a vector of dispersion parameters. To denote a specific element we use the notation for and .
The weight matrix is defined by
Wλi=−∂C−1∂λi=C−1∂C∂λiC−1.
The partial derivative of with respect to the element is given by
∂C∂ρi=Bdiag(~Σ1,…,~ΣR)(∂Σb∂ρi⊗I)Bdiag(~Σ1,…,~ΣR).
Using elementary matrix calculus the partial derivative of with respect to the element is given by
∂C∂pr=Bdiag(0,…,∂~Σr∂pr,…,0)(Σb⊗I)Bdiag(~ΣT1,…,~ΣTR) (16)
Similar equation may be obtained with respect to the elements in the vector . Given the block diagonal structure of Eq. (3.1), it is enough to compute the derivatives and insert in Eq. (3.1). Based on the results from Särkkä (2013) the partial derivative of with respect to and are given by
∂~Σr∂pr=~ΣrΦ(~Σ−1r∂Σr∂pr~Σ−1r),
and
∂~Σr∂τrd=~ΣrΦ(~Σ−1r∂Σr∂τrd~Σ−1r),
respectively, where the function returns the lower triangular part of the argument and half of its diagonal. Now, recalling that , we may hence see that the partial derivative with respect to and are given by
∂Σr∂pr=∂Vr(μ;p)12∂prΩr(τr)Vr(μ;p)12+Vr(μ;p)12Ωr(τr)∂Vr(μ;p)12∂pr, (17)
and
∂Σr∂τrd=Vr(μ;p)12∂Ωr(τr)∂τrdVr(μ;p)12,
respectively, where
∂Ωr(τr)∂τrd=∂h−1(U)∂UZrd (18)
and where . The derivative in Eqs. (17) and (18) depends on the derivative of the variance function and covariance link function, respectively, and it should be evaluated accordingly.
## 4 Data analyses
### 4.1 Results from data set 1
In this section we apply the McGLM approach to analyze the multivariate count data set presented in Section 1.1. We adopted the log link function, the Poisson-Tweedie variance function and the identity covariance link function for the five count response variables. The matrix linear predictor is composed of an identity matrix, since we have independent respondents. The linear predictor is composed of nine covariates plus the intercept for each response variable. The covariance structure is described by five power parameters, five dispersions and ten correlation parameters. We fitted this model using the modified chaser algorithm (3) and the correction term in (15). Table 1 shows the estimates and standard errors for the power and dispersion parameters.
The results in Table 1 show that for the response variables Ndoc, Nndoc and Nmed the suggested distribution is the Neyman Type A (), which indicates zero inflation relative to the Poisson distribution. Regarding the response variable Nhosp the model indicates that the Pólya-Aeppli distribution () is suitable. Finally, the model indicates that for Nadm both Neyman Type A, Pólya-Aeppli and negative binomial () distributions are suitable. This result is obtained because the dispersion parameter is not different from zero; hence the response variable Nadm is equidispersed and all these distributions work well, including the Poisson. In this case, we do not have enough information in the data to distinguish between these distributions. Therefore, we suggest to opt for the simplest possibility, i.e. the Poisson model. The dispersion estimates show weak overdispersion for the response variables Ndoc, Nndoc and Nmed and high overdispersion for Nhosp. In order to compare the regression coefficients with a conventional model, Figure 4 shows the estimates and confidence intervals for McGLM and a conventional Poisson log-linear model for each response variable. The intercept is not shown in order to avoid scale issues.
The results in Figure 4 show that the two approaches agree in terms of estimates, but differ in terms of standard errors. The differences may be explained by the covariance structure. The Poisson model assumes equidispersion, whereas the McGLM models allow for a flexible modelling of the covariance structure, allowing in particular various degrees of overdispersion and zero-inflation. For the response variable Nadm, the model shows that equidispersion is suitable, making the McGLM and Poisson confidence intervals similar. On the other hand, for the response variable Nhosp, where the overdispersion is strong, the McGLM confidence intervals are about five times wider than the Poisson ones. In a similar way, the McGLM confidence intervals for Nndoc, Ndoc and Nmed are on average , and wider than the corresponding Poisson intervals. These results highlight the importance of modelling the covariance structure even when the main interest is in the regression parameters, because the covariance structure controls the standard errors for the regression parameters.
An additional feature of McGLM is that we can estimate the correlation between response variables. It is important to emphasize that the estimation of the correlation matrix does not inflate the standard errors for the regression coefficients, due to the insensitivity of the quasi-score function with respect to the covariance parameters. The estimates and standard errors for the entries of the matrix were as follows:
^Σb=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣10.1066(0.0161)10.1708(0.0156)0.0601(0.0144)10.0905(0.0164)0.0679(0.0156)0.0478(0.0144)10.1503(0.0160)0.0688(0.0147)0.0699(0.0140)0.5464(0.0510)1⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦.
All correlations are significantly different from zero, but only the correlation between Nhosp and Nadm is substantial in size. The standard errors are all of a similar magnitude, which is natural since all are computed using the same sample size. Furthermore, these correlations take into account the effect of all covariates, zero inflation and overdispersion. We know of no other statistical method that allows estimation of correlations taking into account all these important features.
### 4.2 Results from data set 2
In this section, we apply the McGLM approach to analyze data set 2 from Section 1.2, which has response variables of mixed types. There are three response variables, namely HR, RR and OSat, the first two being continuous and the last being confined to the unit interval, having exact zeroes. We adopted the constant variance function, identity link function, and identity covariance link function for HR and RR, reflecting a belief that HR and RR are normally distributed. For OSat we adopted the logit link function combined with the binomial variance function and identity covariance link function. We fitted the model using the modified chaser algorithm (3) and the correction term (15).
The matrix linear predictor is composed of a diagonal matrix (intercept) combined with two sets of matrices to model the longitudinal and repeated measures structures. The longitudinal structure is modeled by a compound symmetry matrix (of ones), the reciprocal of Euclidean distances and reciprocal of Euclidean distances squared. The repeated measures structure is described by an unstructured covariance matrix. Since we have three Evaluations to represent this structure we need three matrices. Therefore, the matrix linear predictor is a linear combination of seven known matrices and it is described by dispersion parameters (seven for each outcome). Details of the matrix linear predictor is available in the supplementary material. In this example we have no power parameters and the matrix contains three parameters. Table 2 shows the estimates and standard errors for the dispersion parameters. The parameters ( and ) and ( and ) are associated to the repeated measures and longitudinal structures, respectively.
The results in Table 2 show that the longitudinal structure is not significant for all response variables. The repeated measures structure is significant for RR and HR. For the outcome RR the estimate of is not significant, which means that the covariance between Evaluation 1 and Evaluation 3 is not different from zero. For the outcome OSat there are no significant dispersion coefficients, so we may assume independent observations. The final model is composed of the repeated measures structure for the response variables RR and HR and independent structure (only ) for OSat.
We have a set of covariates entering the linear predictor, and we used a stepwise procedure to select the most significant set of covariates. This procedure selected a different set of covariates for each outcome. After completing this procedure, we included the covariate of particular interest, namely treat, which is a factor with two levels. Our goal is to assess whether or not the treatment has an effect on each response variable.
In order to evaluate the effects of the covariance structure on the regression coefficients, Figure 5 shows estimates and confidence intervals obtained from the final McGLM and a quasi GLM using the same link and variance functions as for the McGLM. The linear predictors for the outcome RR, HR and OSat are composed of , and regression coefficients, respectively. The intercept is not shown, in order to avoid scaling issues. It is important to emphasize that the last two regression coefficients (numbered 9–10, 12–13 and 9–10, respectively) measure the treatment effects.
The results in Figure 5 show that in general the confidence intervals from McGLM are wider than the corresponding ones based on quasi GLM. For the outcome RR and HR the standard errors from McGLM are on average and greater than the corresponding quasi GLM ones, respectively. These results are as expected, because correlation within response variables generally implies less information in the data on the regression coefficients. It is hence interesting to note that, in contras to the other regression coefficients, the two treatment coefficients for each response variable have smaller standard errors using McGLM than those obtained using quasi GLM. We attribute this effect to the fact that the treatment covariate is also used for modelling the covariance structure, which apparently improves on the estimation of the treatment effect, although we are uncertain if this is a general feature, or if it is specific to this data set.
Regarding the outcome OSat the standard errors from McGLM are on average smaller than those obtained by quasi GLM. This may be explained by the difference between the estimates of , which are and for McGLM and quasi binomial, respectively. This small difference seems to be due to the use of the corrected Pearson estimator in our model.
Regarding the treatment effects, the final model shows that for RR the Evaluation 1 differs from Evaluation 3, but not from Evaluation 2. On the other hand, for HR the model shows that the Evaluation 1 differs from Evaluation 2, but does not differ from Evaluation 3. This result contrasts with the quasi GLM analysis, which does not show any significant difference between Evaluation 1 and Evaluation 2. Finally, our final model shows that for OSat both constrasts are significant. The estimated correlation matrix between response variables was as follows:
^Σb=⎡⎢⎣10.1682(0.0607)1$−$0.0482(0.0607)$−$0.0733(0.0608)1⎤⎥⎦
We observe that there is a significant but weak correlation between RR and HR, whereas the other two correlation estimates are not significant.
### 4.3 Results from data set 3
In this section we apply the McGLM approach to the space-time data set presented in Section 1.3. The response variable monthly rainfall is right-skewed with a positive probability at zero. We hence adopted the log link function, the power variance function and the inverse covariance link function. The linear predictor is expressed in terms of Fourier harmonics (seasonal variation) and B-splines (general trend),
g(μtj)=β0+β1cos(2πt/12)+β2sin(2πt/12)+4∑k=1βk+2Bk(j)
where indexes months, and indexes years. The form a B-spline basis with four degree of freedom. We used only the first term of the Fourier harmonics, since the second and third terms were not significant.
The main challenge in the analysis was to model the covariance structure suitably, in order to take into account the spatial and temporal autocorrelation and, if necessary, the interaction between space and time. We propose to model the space-time structure using a linear combination of neighborhood matrices. Let us motivate our approach using the Conditional Autoregressive (CAR) model. The CAR model specifies the inverse of the covariance matrix by
Ω−1(τ,ρ)=τ(D−ρW)
where is a neighborhood matrix and is a diagonal matrix with the number of neighbors in the main diagonal. The model is parametrized by the precision () and autocorrelation () parameters. The matrices and can model both space, time and space-time interaction in a straightforward way, by using different neighborhood matrices.
For the Venezuelan rainfall data, we used the spatial coordinates (latitude and longitude) to build a Voronoi tessellation (see Figure 3B). The tessellation structure helps us to specify a neighborhood structure. Let us denote this structure by and . Temporal neighbors are naturally specified by the time structure. Let us denote these matrices by and . In the space-time case we have replicates of the space and time structures, so assuming independent replicates, the full neighborhood matrix is block-diagonal. Finally, the interaction between space and time is described by the Kronecker product between the space and time neighborhood structures. Let us denote these matrices by and . Thus, the matrix linear predictor is given by
Ω−1(τ,ρ)=τt(Dt+ρtWt)+τs(Ds+ρsWs)+τst(Dst+ρstWst),
which may be written as a linear combination of known matrices as follows:
Ω−1(τ)=τ0Z0+τ1Z1+τ2Z2+τ3Z3+τ4Z4+τ5Z5, (19)
where , , , , and . In a similar way, we find that , and . Finally, , and . In practical situations, fitting this model may be hard if the autocorrelation parameters are near . In the Bayesian context, it is common to fix the autocorrelation parameter at the value 1, which is the so-called Intrinsic Conditional Autoregressive (ICAR) model.
In order to investigate the space-time structure we fitted three models, cf. Table 3. The first (Model 1) considers time effects. The second (Model 2) considers space effects and the third (Model 3) the space-time interaction effects. After this procedure we decided that the autocorrelation parameters associated with space and interaction effects must be fixed at the value . We then fitted the final model (Model 4) with all three components time, space and space-time interaction. All models were fitted using the reciprocal likelihood algorithm (13). Table 3 presents estimates and standard errors for the power and dispersion parameters for each of the four models.
The results in Table 3 show a moderate temporal autocorrelation, but high spatial and space-time interaction autocorrelations. All the power parameter estimates are in the interval , suggesting a compound Poisson distribution, as expected, since the response variable is continuous with exact zeros. The fitted values and confidence interval are shown in Figure 3A above.
The model allows us to make prediction using the Best Linear Unbiased Predictor (BLUP). Furthermore, we may obtain predictions for different response variable measures, such as the mean number of precipitations events per month, the avarage amount of precipitation per event, or the probability of no precipitation. Such extensions are straightforward and will be presented elsewhere.
## 5 Discussion
In this paper we have developed a comprehensive statistical modelling framework for correlated data, obtained by using separate pairs of link functions and linear predictors for the mean and covariance structures in the style of Pourahmadi (2011). Motivated by three data examples, we have shown that the McGLM framework can deal with a wide variety of correlation structures where existing modelling approaches have difficulties. Following Nelder and Wedderburn (1972), there are obvious pedagogical advantages to the modular specification of models in McGLM, incentivating the researcher to think constructively about the covariance structure, while drawing on previous experiences from GLM modelling. The generalized Kronecker product facilitates the specification of the covariance structure for multivariate response variables.
The main features of the McGLM framework include the ability to deal with most common types of response variables, such as continuous, count and binary. Characteristics such as symmetry/asymmetry, excess of zeros and overdispersion are easily handled due to the flexibility of the model class. We can model many different types of dependencies, such as repeated measures, longitudinal, time series, spatial and space-time data. All of these features extend to multivariate response variables, including the case of mixed response variable types, while maintaining the population average interpretations of regression parameters as well as for covariance parameters. This gives a very flexible modelling of the covariance structure compared with for example current GEE implementations, where the researcher must choose from a small set of pre-defined covariance models.
The main technical advantage of the McGLM framework is the simplicity of the fitting method, which amounts to finding the root for a set of non-linear equations. Based on second-moment assumptions, we use a quasi-score function for the regression parameters and a Pearson estimating function for the covariance parameters. The modified chaser algorithm of Jørgensen and Knudsen (2004) requires an approximate derivative matrix in the form of the sensitivity matrix, which is usually relatively simple. The new reciprocal likelihood algorithm requires the additional calculation of the variability matrix in order to stabilize the covariance parameter update, resulting in a very efficient algorithm, although the sensitivity matrix may be hard to calculate for big data. In such cases a numerical approximation for the sensitivity matrix may be used. For both algorithms a careful choice of initial values is required. In any case, we avoid using computationally more intensive methods such as MCMC, numerical optimization or numerical integration, which are common in the context of random effects models.
An important feature of the McGLM fitting method is that while the mean parameter estimators depend relatively little on the form of the covariance structure, this is not the case for the standard errors of the mean parameter estimators, which depend directly on the choice of covariance structure. A related matter is that the discussion of the efficiency of the mean and covariance parameter estimators is difficult due to the lack of a fully specified probability model.
The current version of the fitting algorithm (available in the supplementary material) is a preliminary implementation of the McGLM method. We plan to develop a full McGLM R package with a GLM-style interface that takes full advantage of the modular specification of the models. There are many possible extensions to the basic model discussed in the present paper, including for example facilities for censored data in survival analysis and other special types of data, or new estimating functions to handle data not missing at random. The specification of the matrix linear predictor is one of the key points of the McGLM approach. While we have used some simple and easily interpretable matrices here, there is wide scope for further research on the proper choice of the matrix linear predictor. Similarly, other covariance link functions, such as the matrix-logarithm Chiu et al. (1996) may also be explored. It is also possible to incorporate penalized splines into the mean and covariance structures, and to use regularization for high-dimensional data, with important applications in genetics. Furthermore, McGLMs may be scaled to test for a common exposure effect in the style of Roy et al. (2003).
## Appendix - Data sets description
Data set 1: Australian health survey
Response variables: Ndoc - Number of consultations with a doctor or specialist. Nndoc - Number of consultations with health professionals. Nmed - Total number of prescribed and non prescribed medications used in the past two days. Nhosp - Number of nights in a hospital during the most recent admission. Nadm - Number of admissions to a hospital, psychiatric hospital, nursing or convalescence home in the past 12 months.
Covariates: sex - factor, two levels (0-Male; 1-Female). age - respondent’s age in years divided by 100. income - respondent’s annual income in Australian dollars divided by 1000. levyplus - factor, two levels (1- if respondent is covered by private health insurance fund for private patients in public hospital (with doctor of choice); 0 - otherwise). freepoor - factor, two levels (1 - if respondent is covered by government because low income, recent immigrant, unemployed; 0 - otherwise). freerepa - factor, two levels (1 - if respondent is covered free by government because of old-age or disability pension, or because invalid veteran or family of deceased veteran; 0 - otherwise). illness - number of illnesses in past 2 weeks, with 5 or more weeks coded as 5. actdays - number of days of reduced activity in the past two weeks due to illness or injury. hscore - respondent’s general health questionnaire score using Goldberg’s method; high score indicates poor health. chcond1 - factor, two levels (1 - if respondent has chronic condition(s) but is not limited in activity; 0 - otherwise). chcond2 - factor, two levels (1 if respondent has chronic condition(s) and is limited in activity; 0 - otherwise). id - respondent’s index.
Data set 2: Respiratory physiotherapy on premature newborns
Response variables
: RR - Respiratory rate (continuous). HR - Heart rate (continuous). O2Sat - Oxygen saturation (bounded).
Covariates: Sex - factor, two levels (Female; Male). GA - Gestational age (weeks). BW - Birth weight (mm). APGAR1M - APGAR index in the first minute of life. APGAR5M - APGAR index in the fifth minute of life. PRE - factor, two levels (Premature: yes; no). HD - factor, two levels (Hansen’s disease, yes; no). SUR - factor, two levels (Surfactant, yes; no). JAU - factor, two levels (Jaundice, yes; no). PNE - factor, two levels (Pneumonia, yes; no). PDA - factor, two levels (Persistence of ductus arteriosus, yes; no). PPI - factor, two levels (Primary pulmonary infection, yes; no). OTHERS - factor, two levels (Other diseases, yes; no). DAYS - Age (days). AUX - factor, two levels (Type of respiratory auxiliary, HOOD; OTHERS). TREAT - factor, three levels (Respiratory physiotherapy, Evaluation 1; Evaluation 2; Evaluation 3). UNIT - Unit sample code. TIME - Days of treatment.
Data set 3: Venezuelan rainfall data
Response variable
: rainfall - monthly rainfall (mm)
Covariates: month - month code. Longitude - Longitude (UTM). Latitude - Latitude (UTM). height - height above sea level (m).
Supplement material web page
## Acknowledgements
The first author is supported by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) - Brazil.
### References
1. Anderson, T. W. (1973). Asymptotically efficient estimation of covariance matrices with linear structure, The Annals of Statistics 1(1): 135–141.
2. Bates, D., Maechler, M., Bolker, B. and Walker, S. (2014). lme4: Linear mixed-effects models using Eigen and S4. R package version 1.1-6.
3. Besag, J., York, J. and Mollié, A. (1991). Bayesian image restoration with two applications in spatial statistics, Annals of Institute of Statistical Mathematics 43(1): 1–59.
4. Bonat, W. H., Ribeiro Jr, P. J. and Zeviani, W. M. (2015). Likelihood analysis for a class of beta mixed models, Journal of Applied Statistics 42(2): 252–266.
5. Breslow, N. E. and Clayton, D. G. (1993). Approximate inference in generalized linear mixed models, Journal of the American Statistical Association 88(421): 9–25.
6. Cameron, A. C. and Trivedi, P. K. (1998). Regression Analysis of Count Data, Econometric society monographs, Cambridge University Press, Cambridge (UK).
7. Chandler, R. E. and Wheater, H. S. (2002). Analysis of rainfall variability using generalized linear models: A case study from the west of Ireland, Water Resources Research 38(10): 1–11.
8. Chiu, T. Y. M., Leonard, T. and Tsui, K. (1996). The matrix-logarithmic covariance model, Journal of the American Statistical Association 91(433): 198–210.
9. Cowpertwait, P. S. P., Lockie, T. and Davis, M. D. (2006). A stochastic spatial-temporal disaggregation model for rainfall, Journal of Hydrology (New Zealand) 45(1): 1–12.
10. Cressie, N. and Huang, H. (1999). Classes of nonseparable, spatio-temporal stationary covariance functions, Journal of the American Statistical Association 94(448): 1330–1339.
11. Cressie, N. and Wikle, C. K. (2011). Statistics for Spatio-Temporal Data, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., Hoboken, NJ.
12. Crouchley, R. (2012). sabreR: Multivariate Generalized Linear Mixed Models. R package version 2.0.
13. Deb, P. and Trivedi, P. K. (1997). Demand for medical care by the elderly: A finite mixture approach, Journal of Applied Econometrics 12(3): 313–36.
14. Diggle, P. J., Heagerty, P., Liang, K.-Y. and Zeger, S. L. (2002). Analysis of Longitudinal Data, Oxford Statistical Science Series, Oxford.
15. Diggle, P. and Ribeiro, P. (2007). Model-based Geostatistics, Springer Series in Statistics, Springer-Verlag New York.
16. Dobbie, M. J. and Welsh, A. H. (2001). Models for zero-inflated count data using the Neyman Type A distribution, Statistical Modelling 1(1): 65–80.
17. Dunn, P. K. (2004). Occurrence and quantity of precipitation can be modelled simultaneously, International Journal of Climatology 24(10): 1231–1239.
18. El-Shaarawi, A. H., Zhu, R. and Joe, H. (2011). Modelling species abundance using the Poisson-Tweedie family, Environmetrics 22(2): 152–164.
19. Fieuws, S., Verbeke, G. and Molenberghs, G. (2007). Random-effects models for multivariate repeated measures, Statistical Methods in Medical Research 16(5): 387–397.
20. Fong, Y., Rue, H. and Wakefield, J. (2010). Bayesian inference for generalized linear mixed models, Biostatistics 11(3): 397–412.
21. Gneiting, T. (2002). Nonseparable, stationary covariance functions for space-time data, Journal of the American Statistical Association 97(458): 590–600.
22. Gray, S. M. and Brookmeyer, R. (2000). Multidimensional longitudinal data: Estimating a treatment effect from continuous, discrete, or time-to-event response variables, Journal of the American Statistical Association 95(450): 396–406.
23. Hadfield, J. D. (2010). MCMC methods for multi-response generalized linear mixed models: The MCMCglmm R package, Journal of Statistical Software 33(2): 1–22.
24. Hasan, M. M. and Dunn, P. K. (2010). A simple Poisson-gamma model for modelling rainfall occurrence and amount simultaneously, Agricultural and forest meteorology 150(10): 1319–1330.
25. Hasan, M. M. and Dunn, P. K. (2012). Understanding the effect of climatology on monthly rainfall amounts in Australia using Tweedie GLMs, International Journal of Climatology 32(7): 1006–1017.
26. Holla, M. (1967). On a Poisson-inverse Gaussian distribution, Metrika 11(1): 115–121.
27. Jensen, S. T., Johansen, S. and Lauritzen, S. L. (1991). Globally convergent algorithms for maximizing likelihood function, Biometrika 78(4): 867–877.
28. Jørgensen, B. (1987). Exponential dispersion models, Journal of the Royal Statistical Society. Series B (Methodological) 49(2): 127–162.
29. Jørgensen, B. (1997). The Theory of Dispersion Models, Chapman & Hall.
30. Jørgensen, B. and Knudsen, S. J. (2004). Parameter orthogonality and bias adjustment for estimating functions, Scandinavian Journal of Statistics 31(1): 93–114.
31. Jørgensen, B. and Kokonendji, C. C. (2014). Discrete dispersion models and their Tweedie asymptotics, ArXiv e-prints .
32. Knight, J. L. (1985). The joint characteristic function of linear and quadratic forms of non-normal variables, Sankhyā: The Indian Journal of Statistics, Series A (1961-2002) 47(2): 231–238.
33. Krupskii, P. and Joe, H. (2013). Factor copula models for multivariate data, Journal of Multivariate Analysis 120(0): 85–101.
34. Lee, Y. and Nelder, J. A. (1996). Hierarchical generalized linear models, Journal of the Royal Statistical Society. Series B (Methodological) 58(4): 619–678.
35. Lee, Y. and Nelder, J. A. (2004). Conditional and Marginal models: Another view, Statistical Science 19(2): 219–238.
36. Liang, K.-Y. and Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models, Biometrika 73(1): 13–22.
37. Lunn, D. J., Thomas, A., Best, N. and Spiegelhalter, D. (2000). Winbugs: A bayesian modelling framework: Concepts, structure, and extensibility, Statistics and Computing 10(4): 325–337.
38. Martin, A. D., Quinn, K. M. and Park, J. H. (2011). MCMCpack: Markov Chain Monte carlo in R, Journal of Statistical Software 42(9): 22.
39. Martinez-Beneito, M. A. (2013). A general modelling framework for multivariate disease mapping, Biometrika 100(3): 539–553.
40. McCulloch, C. E. (1997). Maximum likelihood algorithms for generalized linear mixed models, Journal of the American Statistical Association 92(437): 162–170.
41. Nelder, J. A. and Wedderburn, R. W. M. (1972). Generalized linear models, Journal of the Royal Statistical Society. Series A 135(3): 370–384.
42. O’Brien, L. M. and Fitzmaurice, G. M. (2004). Analysis of longitudinal multiple-source binary data using generalized estimating equations, Journal of the Royal Statistical Society: Series C (Applied Statistics) 53(1): 177–193.
43. Ospina, R. and Ferrari, S. L. P. (2010). Inflated beta distributions, Statistical Papers 51(1): 111–126.
44. Pan, J. and Mackenzie, G. (2003). On modelling mean-covariance structures in longitudinal studies, Biometrika 90(1): 239–244.
45. Pinheiro, J., Bates, D., DebRoy, S., Sarkar, D. and R Core Team (2013). nlme: Linear and Nonlinear Mixed Effects Models. R package version 3.1-113.
46. Pinheiro, J. C. and Bates, D. M. (1996). Unconstrained parametrizations for variance-covariance matrices, Statistics and Computing 6(3): 289–296.
47. Plummer, M. (2003). JAGS: a program for analysis of Bayesian graphical models using Gibbs sampling, Proceedings of the 3rd International Workshop on Distributed Statistical Computing.
48. Pourahmadi, M. (1999). Joint mean-covariance models with applications to longitudinal data: Unconstrained parameterisation, Biometrika 86(3): 677–690.
49. Pourahmadi, M. (2000). Maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix, Biometrika 87(2): 425–435.
50. Pourahmadi, M. (2011). Covariance estimation: The GLM and regularization perspectives, Statistical Science 26(3): 369–387.
51. Pourahmadi, M., Daniels, M. J. and Park, T. (2007). Simultaneous modelling of the Cholesky decomposition of several covariance matrices, Journal of Multivariate Analysis 98(3): 568–587.
52. R Core Team (2015). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0.
53. Rochon, J. (1996). Analyzing bivariate repeated measures for discrete and continuous outcome variables, Biometrics 52(2): 740–750.
54. Rodrigues-Motta, M., Pinheiro, H. P., Martins, E. G., Araújo, M. S. and dos Reis, S. F. (2013). Multivariate models for correlated count data, Journal of Applied Statistics 40(7): 1586–1596.
55. Roy, J., Lin, X. and Ryan, L. M. (2003). Scaled marginal models for multiple continuous outcomes, Biostatistics 4(3): 371–383.
56. Rue, H. and Held, L. (eds) (2005). Gaussian Markov Random Fields: Theory and Applications, Chapman & Hall, London.
57. Rue, H., Martino, S., Lindgren, F., Simpson, D., Riebler, A. and Krainski, E. T. (2014)). INLA: Functions which allow to perform full Bayesian analysis of latent Gaussian models using Integrated Nested Laplace Approximaxion. R package version 0.0-1404466487.
58. Sansó, B. and Guenni, L. (1999). Venezuelan rainfall data analysed by using a Bayesian space-time model, Journal of the Royal Statistical Society: Series C (Applied Statistics) 48(3): 345–362.
59. Sansó, B. and Guenni, L. (2004). A Bayesian approach to compare observed rainfall data to deterministic simulations, Environmetrics 15(6): 597–612.
60. Särkkä, S. (2013). Bayesian Filtering and Smoothing, Cambridge University Press, London. Cambridge Books Online.
61. SAS Institute (2011). SAS/STAT 9.3 User’s Guide: The GLIMMIX Procedure (Chapter), North Carolina, USA.
62. Shi, P. and Valdez, E. A. (2014). Multivariate negative binomial models for insurance claim counts, Insurance: Mathematics and Economics 55(0): 18–29.
63. Sigrist, F., Künsch, H. R. and Stahel, W. A. (2012). A dynamic nonstationary spatio-temporal model for short term prediction of precipitation, The Annals of Applied Statistics 6(4): 1452–1477.
64. Stern, R. and Coe, R. (1984). A model fitting analysis of daily rainfall data, Journal of the Royal Statistical Society. Series A (General) 1(147): 1–34.
65. Tsionas, E. G. (1999). Bayesian analysis of the multivariate Poisson distribution, Communications in Statistics - Theory and Methods 28(2): 431–451.
66. Verbeke, G., Fieuws, S., Molenberghs, G. and Davidian, M. (2014). The analysis of multivariate longitudinal data: A review, Statistical Methods in Medical Research 23(1): 42–59.
67. Wheater, H. S., Isham, V. S., Cox, D. R., Chandler, R. E., Kakou, A., Northrop, P. J., Oh, L., Onof, C. and Rodriguez-Iturbe, I. (2000). Spatial-temporal rainfall fields: modelling and statistical aspects, Hydrology and Earth System Sciences 4(4): 581–601.
68. Wilks, D. S. (1990). Maximum likelihood estimation for the gamma distribution using data containing zeros, Journal of Climate 3(12): 1495–1501.
69. Zeger, S. L., Liang, K.-Y. and Albert, P. S. (1988). Models for longitudinal data: A generalized estimating equation approach, Biometrics 44(4): 1049–1060.
70. Zhang, P., Q. Z. (2014). Regression analysis of proportional data using simplex distribution, Scientia Sinica Mathematica 44(1): 89–104.
71. Zhang, W., Leng, C. and Tang, C. Y. (2015). A joint modelling approach for longitudinal studies, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77(1): 219–238.
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# ATLAS Notes
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2017-12-15
09:57
Impact of Alternative Inputs and Grooming Methods on Large-R Jet Reconstruction in ATLAS During Run 1 of the LHC, the optimal reconstruction algorithm for large-$R$ jets in ATLAS, characterized in terms of the ability to discriminate signal from background and robust reconstruction in the presence of pileup, was found to be anti-$k_{t}$ jets with a radius parameter of 1.0, formed from locally calibrated topological calorimeter cell clusters and groomed with the trimming algorithm to remove contributions from pileup and underlying event. [...] ATL-PHYS-PUB-2017-020. - 2017. Original Communication (restricted to ATLAS) - Full text
2017-12-15
09:49
Search for direct pair production of higgsinos by the reinterpretation of the disappearing track analysis with 36.1 fb$^{-1}$ of $\sqrt{s}=13$ TeV data collected with the ATLAS experiment This note presents a search for direct production of higgsinos in which a chargino is nearly mass-degenerate with a stable neutralino. [...] ATL-PHYS-PUB-2017-019. - 2017. - 10 p. Original Communication (restricted to ATLAS) - Full text
2017-12-14
23:39
Search for $W' \rightarrow tb$ in the hadronic final state with the ATLAS Detector in $\sqrt{s} = 13$ TeV pp collisions A search for a $W'$ boson in the $W' \rightarrow t\bar{b} \rightarrow q\bar{q}' b\bar{b}$ final state is presented using \lumi\ of 13~\TeV\ proton--proton collision data collected by the ATLAS detector at the Large Hadron Collider in 2015 and 2016. [...] ATLAS-CONF-2017-082. - 2017. Original Communication (restricted to ATLAS) - Full text
2017-12-14
23:19
Search for pair production of higgsinos in final states with at least 3 $b$-tagged jets using the ATLAS detector in $\sqrt{s} = 13$ TeV $pp$ collisions A search for pair production of the supersymmetric partners of the Higgs boson (higgsinos $\tilde{H}$ ) in gauge-mediated scenarios is reported. [...] ATLAS-CONF-2017-081. - 2017. Original Communication (restricted to ATLAS) - Full text
2017-12-14
23:04
Search for photonic signatures of gauge-mediated supersymmetry in 13 TeV $pp$ collisions with the ATLAS detector A search is presented for photonic signatures motivated by generalized models of gauge-mediated supersymmetry breaking. [...] ATLAS-CONF-2017-080. - 2017. Original Communication (restricted to ATLAS) - Full text
2017-12-14
19:31
Upgrade of the ATLAS hadronic Tile Calorimeter for the High luminosity LHC / Rodriguez Bosca, Sergi (Instituto de Fisica Corpuscular (IFIC), Centro Mixto Universidad de Valencia - CSIC) The Tile Calorimeter is the hadronic calorimeter covering the central region of the ATLAS detector at the Large Hadron Collider. [...] ATL-TILECAL-PROC-2017-024. - 2017. - 5 p. Original Communication (restricted to ATLAS) - Full text
2017-12-14
00:19
Search for top squarks decaying to tau sleptons in $pp$ collisions at $\sqrt s$ = 13 TeV with the ATLAS detector A search for direct pair production of top squarks in final states with two tau leptons, $b$-jets, and missing transverse momentum is presented, based on 36.1 fb$^{-1}$ of proton$-$proton collision data recorded at $\sqrt{s}$ = 13 TeV with the ATLAS detector at the Large Hadron Collider in 2015 and 2016. [...] ATLAS-CONF-2017-079. - 2017. - 34 p. Original Communication (restricted to ATLAS) - Full text
2017-12-12
22:38
Modeling Radiation Damage Effects in 3D Pixel Digitization for the ATLAS Detector / Wallangen, Veronica (Stockholm University) Silicon Pixel detectors are at the core of the current and planned upgrade of the ATLAS detector. [...] ATL-INDET-PROC-2017-005. - 2017. - 6 p. Original Communication (restricted to ATLAS) - Full text
2017-12-09
23:37
Top pair and single top production in ATLAS / Fabbri, Federica (Universita e INFN, Bologna) Measurements of inclusive and differential top-quark production cross sections in proton-proton collisions at a center of mass energy of 8 TeV and 13 TeV at the Large Hadron Collider using the ATLAS detector are presented. [...] ATL-PHYS-PROC-2017-268. - 2017. - 4 p. Original Communication (restricted to ATLAS) - Full text
2017-12-09
16:00
Hunting New Physics with ATLAS / Mitsou, Vasiliki A. (Instituto de Fisica Corpuscular (IFIC), Centro Mixto Universidad de Valencia - CSIC) Highlights from recent new physics searches with the ATLAS detector at the CERN LHC are presented in this paper. [...] ATL-PHYS-PROC-2017-267. - 2017. - 10 p. Original Communication (restricted to ATLAS) - Full text
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2017-12-15 12:02:04
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https://stacks.math.columbia.edu/tag/08IM
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## 73.20 Cohomology and base change, IV
This section is the analogue of Derived Categories of Schemes, Section 36.22.
Lemma 73.20.1. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. For $E$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ and $K$ in $D_\mathit{QCoh}(\mathcal{O}_ Y)$ we have
$Rf_*(E) \otimes _{\mathcal{O}_ Y}^\mathbf {L} K = Rf_*(E \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*K)$
Proof. Without any assumptions there is a map $Rf_*(E) \otimes _{\mathcal{O}_ Y}^\mathbf {L} K \to Rf_*(E \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*K)$. Namely, it is the adjoint to the canonical map
$Lf^*(Rf_*(E) \otimes _{\mathcal{O}_ Y}^\mathbf {L} K) = Lf^*(Rf_*(E)) \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*K \longrightarrow E \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*K$
coming from the map $Lf^*Rf_*E \to E$. See Cohomology on Sites, Lemmas 21.18.4 and 21.19.1. To check it is an isomorphism we may work étale locally on $Y$. Hence we reduce to the case that $Y$ is an affine scheme.
Suppose that $K = \bigoplus K_ i$ is a direct sum of some complexes $K_ i \in D_\mathit{QCoh}(\mathcal{O}_ Y)$. If the statement holds for each $K_ i$, then it holds for $K$. Namely, the functors $Lf^*$ and $\otimes ^\mathbf {L}$ preserve direct sums by construction and $Rf_*$ commutes with direct sums (for complexes with quasi-coherent cohomology sheaves) by Lemma 73.6.2. Moreover, suppose that $K \to L \to M \to K[1]$ is a distinguished triangle in $D_\mathit{QCoh}(Y)$. Then if the statement of the lemma holds for two of $K, L, M$, then it holds for the third (as the functors involved are exact functors of triangulated categories).
Assume $Y$ affine, say $Y = \mathop{\mathrm{Spec}}(A)$. The functor $\widetilde{\ } : D(A) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ is an equivalence by Lemma 73.4.2 and Derived Categories of Schemes, Lemma 36.3.5. Let $T$ be the property for $K \in D(A)$ that the statement of the lemma holds for $\widetilde{K}$. The discussion above and More on Algebra, Remark 15.58.13 shows that it suffices to prove $T$ holds for $A[k]$. This finishes the proof, as the statement of the lemma is clear for shifts of the structure sheaf. $\square$
Definition 73.20.2. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X$, $Y$ be algebraic spaces over $B$. We say $X$ and $Y$ are Tor independent over $B$ if and only if for every commutative diagram
$\xymatrix{ \mathop{\mathrm{Spec}}(k) \ar[d]_{\overline{y}} \ar[dr]_{\overline{b}} \ar[r]_-{\overline{x}} & X \ar[d] \\ Y \ar[r] & B }$
of geometric points the rings $\mathcal{O}_{X, \overline{x}}$ and $\mathcal{O}_{Y, \overline{y}}$ are Tor independent over $\mathcal{O}_{B, \overline{b}}$ (see More on Algebra, Definition 15.60.1).
The following lemma shows in particular that this definition agrees with our definition in the case of representable algebraic spaces.
Lemma 73.20.3. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X$, $Y$ be algebraic spaces over $B$. The following are equivalent
1. $X$ and $Y$ are Tor independent over $B$,
2. for every commutative diagram
$\xymatrix{ U \ar[d] \ar[r] & W \ar[d] & V \ar[d] \ar[l] \\ X \ar[r] & B & Y \ar[l] }$
with étale vertical arrows $U$ and $V$ are Tor independent over $W$,
3. for some commutative diagram as in (2) with (a) $W \to B$ étale surjective, (b) $U \to X \times _ B W$ étale surjective, (c) $V \to Y \times _ B W$ étale surjective, the spaces $U$ and $V$ are Tor independent over $W$, and
4. for some commutative diagram as in (3) with $U$, $V$, $W$ schemes, the schemes $U$ and $V$ are Tor independent over $W$ in the sense of Derived Categories of Schemes, Definition 36.22.2.
Proof. For an étale morphism $\varphi : U \to X$ of algebraic spaces and geometric point $\overline{u}$ the map of local rings $\mathcal{O}_{X, \varphi (\overline{u})} \to \mathcal{O}_{U, \overline{u}}$ is an isomorphism. Hence the equivalence of (1) and (2) follows. So does the implication (1) $\Rightarrow$ (3). Assume (3) and pick a diagram of geometric points as in Definition 73.20.2. The assumptions imply that we can first lift $\overline{b}$ to a geometric point $\overline{w}$ of $W$, then lift the geometric point $(\overline{x}, \overline{b})$ to a geometric point $\overline{u}$ of $U$, and finally lift the geometric point $(\overline{y}, \overline{b})$ to a geometric point $\overline{v}$ of $V$. Use Properties of Spaces, Lemma 64.19.4 to find the lifts. Using the remark on local rings above we conclude that the condition of the definition is satisfied for the given diagram.
Having made these initial points, it is clear that (4) comes down to the statement that Definition 73.20.2 agrees with Derived Categories of Schemes, Definition 36.22.2 when $X$, $Y$, and $B$ are schemes.
Let $\overline{x}, \overline{b}, \overline{y}$ be as in Definition 73.20.2 lying over the points $x, y, b$. Recall that $\mathcal{O}_{X, \overline{x}} = \mathcal{O}_{X, x}^{sh}$ (Properties of Spaces, Lemma 64.22.1) and similarly for the other two. By Algebra, Lemma 10.155.14 we see that $\mathcal{O}_{X, \overline{x}}$ is a strict henselization of $\mathcal{O}_{X, x} \otimes _{\mathcal{O}_{B, b}} \mathcal{O}_{B, \overline{b}}$. In particular, the ring map
$\mathcal{O}_{X, x} \otimes _{\mathcal{O}_{B, b}} \mathcal{O}_{B, \overline{b}} \longrightarrow \mathcal{O}_{X, \overline{x}}$
is flat (More on Algebra, Lemma 15.45.1). By More on Algebra, Lemma 15.60.3 we see that
$\text{Tor}_ i^{\mathcal{O}_{B, b}}(\mathcal{O}_{X, x}, \mathcal{O}_{Y, y}) \otimes _{\mathcal{O}_{X, x} \otimes _{\mathcal{O}_{B, b}} \mathcal{O}_{Y, y}} (\mathcal{O}_{X, \overline{x}} \otimes _{\mathcal{O}_{B, \overline{b}}} \mathcal{O}_{Y, \overline y}) = \text{Tor}_ i^{\mathcal{O}_{B, \overline{b}}}( \mathcal{O}_{X, \overline{x}}, \mathcal{O}_{Y, \overline{y}})$
Hence it follows that if $X$ and $Y$ are Tor independent over $B$ as schemes, then $X$ and $Y$ are Tor independent as algebraic spaces over $B$.
For the converse, we may assume $X$, $Y$, and $B$ are affine. Observe that the ring map
$\mathcal{O}_{X, x} \otimes _{\mathcal{O}_{B, b}} \mathcal{O}_{Y, y} \longrightarrow \mathcal{O}_{X, \overline{x}} \otimes _{\mathcal{O}_{B, \overline{b}}} \mathcal{O}_{Y, \overline y}$
is flat by the observations given above. Moreover, the image of the map on spectra includes all primes $\mathfrak s \subset \mathcal{O}_{X, x} \otimes _{\mathcal{O}_{B, b}} \mathcal{O}_{Y, y}$ lying over $\mathfrak m_ x$ and $\mathfrak m_ y$. Hence from this and the displayed formula of Tor's above we see that if $X$ and $Y$ are Tor independent over $B$ as algebraic spaces, then
$\text{Tor}_ i^{\mathcal{O}_{B, b}} (\mathcal{O}_{X, x}, \mathcal{O}_{Y, y})_\mathfrak s = 0$
for all $i > 0$ and all $\mathfrak s$ as above. By More on Algebra, Lemma 15.60.6 applied to the ring maps $\Gamma (B, \mathcal{O}_ B) \to \Gamma (X, \mathcal{O}_ X)$ and $\Gamma (B, \mathcal{O}_ B) \to \Gamma (X, \mathcal{O}_ X)$ this implies that $X$ and $Y$ are Tor independent over $B$. $\square$
Lemma 73.20.4. Let $S$ be a scheme. Let $g : Y' \to Y$ be a morphism of algebraic spaces over $S$. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Consider the base change diagram
$\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y }$
If $X$ and $Y'$ are Tor independent over $Y$, then for all $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$ we have $Rf'_*L(g')^*E = Lg^*Rf_*E$.
Proof. For any object $E$ of $D(\mathcal{O}_ X)$ we can use Cohomology on Sites, Remark 21.19.3 to get a canonical base change map $Lg^*Rf_*E \to Rf'_*L(g')^*E$. To check this is an isomorphism we may work étale locally on $Y'$. Hence we may assume $g : Y' \to Y$ is a morphism of affine schemes. In particular, $g$ is affine and it suffices to show that
$Rg_*Lg^*Rf_*E \to Rg_*Rf'_*L(g')^*E = Rf_*(Rg'_* L(g')^* E)$
is an isomorphism, see Lemma 73.6.4 (and use Lemmas 73.5.5, 73.5.6, and 73.6.1 to see that the objects $Rf'_*L(g')^*E$ and $Lg^*Rf_*E$ have quasi-coherent cohomology sheaves). Note that $g'$ is affine as well (Morphisms of Spaces, Lemma 65.20.5). By Lemma 73.6.5 the map becomes a map
$Rf_*E \otimes _{\mathcal{O}_ Y}^\mathbf {L} g_*\mathcal{O}_{Y'} \longrightarrow Rf_*(E \otimes _{\mathcal{O}_ X}^\mathbf {L} g'_*\mathcal{O}_{X'})$
Observe that $g'_*\mathcal{O}_{X'} = f^*g_*\mathcal{O}_{Y'}$. Thus by Lemma 73.20.1 it suffices to prove that $Lf^*g_*\mathcal{O}_{Y'} = f^*g_*\mathcal{O}_{Y'}$. This follows from our assumption that $X$ and $Y'$ are Tor independent over $Y$. Namely, to check it we may work étale locally on $X$, hence we may also assume $X$ is affine. Say $X = \mathop{\mathrm{Spec}}(A)$, $Y = \mathop{\mathrm{Spec}}(R)$ and $Y' = \mathop{\mathrm{Spec}}(R')$. Our assumption implies that $A$ and $R'$ are Tor independent over $R$ (see Lemma 73.20.3 and More on Algebra, Lemma 15.60.6), i.e., $\text{Tor}_ i^ R(A, R') = 0$ for $i > 0$. In other words $A \otimes _ R^\mathbf {L} R' = A \otimes _ R R'$ which exactly means that $Lf^*g_*\mathcal{O}_{Y'} = f^*g_*\mathcal{O}_{Y'}$. $\square$
The following lemma will be used in the chapter on dualizing complexes.
Lemma 73.20.5. Let $g : S' \to S$ be a morphism of affine schemes. Consider a cartesian square
$\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ S' \ar[r]^ g & S }$
of quasi-compact and quasi-separated algebraic spaces. Assume $g$ and $f$ Tor independent. Write $S = \mathop{\mathrm{Spec}}(R)$ and $S' = \mathop{\mathrm{Spec}}(R')$. For $M, K \in D(\mathcal{O}_ X)$ the canonical map
$R\mathop{\mathrm{Hom}}\nolimits _ X(M, K) \otimes ^\mathbf {L}_ R R' \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _{X'}(L(g')^*M, L(g')^*K)$
in $D(R')$ is an isomorphism in the following two cases
1. $M \in D(\mathcal{O}_ X)$ is perfect and $K \in D_\mathit{QCoh}(X)$, or
2. $M \in D(\mathcal{O}_ X)$ is pseudo-coherent, $K \in D_\mathit{QCoh}^+(X)$, and $R'$ has finite tor dimension over $R$.
Proof. There is a canonical map $R\mathop{\mathrm{Hom}}\nolimits _ X(M, K) \to R\mathop{\mathrm{Hom}}\nolimits _{X'}(L(g')^*M, L(g')^*K)$ in $D(\Gamma (X, \mathcal{O}_ X))$ of global hom complexes, see Cohomology on Sites, Section 21.35. Restricting scalars we can view this as a map in $D(R)$. Then we can use the adjointness of restriction and $- \otimes _ R^\mathbf {L} R'$ to get the displayed map of the lemma. Having defined the map it suffices to prove it is an isomorphism in the derived category of abelian groups.
The right hand side is equal to
$R\mathop{\mathrm{Hom}}\nolimits _ X(M, R(g')_*L(g')^*K) = R\mathop{\mathrm{Hom}}\nolimits _ X(M, K \otimes _{\mathcal{O}_ X}^\mathbf {L} g'_*\mathcal{O}_{X'})$
by Lemma 73.6.5. In both cases the complex $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, K)$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 73.13.10. There is a natural map
$R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, K) \otimes _{\mathcal{O}_ X}^\mathbf {L} g'_*\mathcal{O}_{X'} \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, K \otimes _{\mathcal{O}_ X}^\mathbf {L} g'_*\mathcal{O}_{X'})$
which is an isomorphism in both cases Lemma 73.13.11. To see that this lemma applies in case (2) we note that $g'_*\mathcal{O}_{X'} = Rg'_*\mathcal{O}_{X'} = Lf^*g_*\mathcal{O}_ X$ the second equality by Lemma 73.20.4. Using Derived Categories of Schemes, Lemma 36.10.4, Lemma 73.13.3, and Cohomology on Sites, Lemma 21.44.5 we conclude that $g'_*\mathcal{O}_{X'}$ has finite Tor dimension. Hence, in both cases by replacing $K$ by $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, K)$ we reduce to proving
$R\Gamma (X, K) \otimes ^\mathbf {L}_ A A' \longrightarrow R\Gamma (X, K \otimes ^\mathbf {L}_{\mathcal{O}_ X} g'_*\mathcal{O}_{X'})$
is an isomorphism. Note that the left hand side is equal to $R\Gamma (X', L(g')^*K)$ by Lemma 73.6.5. Hence the result follows from Lemma 73.20.4. $\square$
Remark 73.20.6. With notation as in Lemma 73.20.5. The diagram
$\xymatrix{ R\mathop{\mathrm{Hom}}\nolimits _ X(M, Rg'_*L) \otimes _ R^\mathbf {L} R' \ar[r] \ar[d]_\mu & R\mathop{\mathrm{Hom}}\nolimits _{X'}(L(g')^*M, L(g')^*Rg'_*L) \ar[d]^ a \\ R\mathop{\mathrm{Hom}}\nolimits _ X(M, R(g')_*L) \ar@{=}[r] & R\mathop{\mathrm{Hom}}\nolimits _{X'}(L(g')^*M, L) }$
is commutative where the top horizontal arrow is the map from the lemma, $\mu$ is the multiplication map, and $a$ comes from the adjunction map $L(g')^*Rg'_*L \to L$. The multiplication map is the adjunction map $K' \otimes _ R^\mathbf {L} R' \to K'$ for any $K' \in D(R')$.
Lemma 73.20.7. Let $S$ be a scheme. Consider a cartesian square of algebraic spaces
$\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y }$
over $S$. Assume $g$ and $f$ Tor independent.
1. If $E \in D(\mathcal{O}_ X)$ has tor amplitude in $[a, b]$ as a complex of $f^{-1}\mathcal{O}_ Y$-modules, then $L(g')^*E$ has tor amplitude in $[a, b]$ as a complex of $f^{-1}\mathcal{O}_{Y'}$-modules.
2. If $\mathcal{G}$ is an $\mathcal{O}_ X$-module flat over $Y$, then $L(g')^*\mathcal{G} = (g')^*\mathcal{G}$.
Proof. We can compute tor dimension at stalks, see Cohomology on Sites, Lemma 21.44.10 and Properties of Spaces, Theorem 64.19.12. If $\overline{x}'$ is a geometric point of $X'$ with image $\overline{x}$ in $X$, then
$(L(g')^*E)_{\overline{x}'} = E_{\overline{x}} \otimes _{\mathcal{O}_{X, \overline{x}}}^\mathbf {L} \mathcal{O}_{X', \overline{x}'}$
Let $\overline{y}'$ in $Y'$ and $\overline{y}$ in $Y$ be the image of $\overline{x}'$ and $\overline{x}$. Since $X$ and $Y'$ are tor independent over $Y$, we can apply More on Algebra, Lemma 15.60.2 to see that the right hand side of the displayed formula is equal to $E_{\overline{x}} \otimes _{\mathcal{O}_{Y, \overline{y}}}^\mathbf {L} \mathcal{O}_{Y', \overline{y}'}$ in $D(\mathcal{O}_{Y', \overline{y}'})$. Thus (1) follows from More on Algebra, Lemma 15.65.13. To see (2) observe that flatness of $\mathcal{G}$ is equivalent to the condition that $\mathcal{G}[0]$ has tor amplitude in $[0, 0]$. Applying (1) we conclude. $\square$
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# Lexicographic Order Calculator
public class Sort {. Forrester worked with Microsoft customers to uncover the key benefits of Dynamics 365 Field Service and develop this calculator. This QTc calculator estimates the corrected QT interval expressed in seconds or milliseconds and based on patient's heart rate in beats per minute. With respect to the lexicographic order based on theusual less than relation: a)find all pairs in S x S less than (2,3). O (n2 + log n) D. CTP Calculator Child-Turcotte-Pugh (CTP) Calculator. For example, given 13, return: [1,10,11,12,13,2,3,4,5,6,7,8,9]. It is possible to implicitly convert a string to bool in a filter as in user_label. The lexicographic order on the resulting sequences induces thus an order on the subsets, which is also called the lexicographical order. An ordering (or order) is a method for sorting elements. A Numeric Algorithm for Generating Permutations in Lexicographic Order with a C Implementation 1. println(s2 + "follows" + s3 + " in lexicographic ordering"). 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Acknowledgment: This web page contains JROCFIT and JLABROC4, JavaScript programs for calculating receiver operating characteristic (ROC) curves. ) I continue to the second character, R. Define an order on. htm 1 10 10% 25. Lexicographic Order Algorithms 1. Knuth's Binary Lexicographic Strings. Strings can be thought of as using alphabetic order, although that doesn't capture the whole truth. The worst case running time of this computation is A. Translation of: Liberty BASIC * Permutations 26/10/2015 PERMUTE CSECT USING PERMUTE,R15 set. Online calculator for quick calculations, along with a large collection of calculators on math Calculator. Turkington (2000 and 2002) has used the notation devecX = (vecX0)0 to denote Xr and. 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On the other hand, the interval of real numbers $[0,1]$ with the natural order is not well-ordered. The lexicographical order assigns a rank number to each permutation. Saliu1999Nov28. bbs/messages/77. Following are the steps to print the permutations lexicographic-ally. Lexicographical ordering | Order of permutation Engineering Mathematics. (In the above, statements, < denotes the usual ordering on the rational numbers we know and love!). println(s2 + "follows" + s3 + " in lexicographic ordering"). See full list on bernardosulzbach. definition of - senses, usage, synonyms, thesaurus. Click to see our best Video content. The software can define and graph relations and also draw the transitive, symmetric, and reflexive closure of a relation. The asymptote calculator takes a function and calculates all asymptotes and also graphs the function. Potter's Herbal Cigarettes are a smoking preparation made from stra- monium. O (n2 log n) C. What is the Fee Calculator? This tool will ask questions to help determine your fee; however, it does not store answers to the questions or any other personal information. The order 3 context ANA has been seen once before, followed by N. (ii) Compute the directional derivative of fat (1; 1) in the direction u = 4 5; 3 5. In mathematics, the lexicographic or lexicographical order (aka lexical order, dictionary order or Finding the combination by its lexicographical index. $$\pi: \{1,\ldots , n\} \mapsto \{1,\ldots , n\}$$ One way to get permutations in lexicographic order is based on the algorithm successor which fi. Example using this calculator. Use it for solving word puzzles, scrambles and for writing poetry, lyrics for your song or coming up with rap verses. The lexicographic order algorithm was developed by B. In mathematics, the lexicographic or lexicographical order (also known as lexical order, dictionary order, alphabetical order or lexicographic(al) product) is a generalization of the alphabetical order. to code generation, lexicographic optimization, dependence analysis, transitive closures and the symbolic computation of upper bounds and sums of piecewise quasipolynomials over their domains. Kindly click on the ads, thanks! | Home | Comment For the notes drawn in the videos, please refer to this github repo. Generate fair value prices and Greeks for any of CME Group's options on futures contracts or price up a generic option with our universal calculator. This is the total number of calories you need in order to maintain your current weight. Online exponents calculator with negative numbers support and steps. , by observing that. Translation of: Liberty BASIC * Permutations 26/10/2015 PERMUTE CSECT USING PERMUTE,R15 set. Lists use lexicographic order. The ALCON® Online Toric IOL Calculator With the barrett toric algorithm. After explaining the nature of my quest, they told me about their own favourite legal high. Lexicographical order. Kolmogorov-Chaitin complexity) (or $K$) and Algorithmic Probability for. The solution to problems can be submitted in over 60 languages including C, C++, Java, Python, C#, Go, Haskell, Ocaml, and F#. Long Run Production-- Production activity where all factors of production may vary in quantity. Includes all the functions Easy to use and 100% Free! We also have several other calculators. Given an integer n, return 1 - nin lexicographical order. The class probabilities of the input samples. Relative risk calculator. ► Lexicographic order (1 C, 9 F). radius_neighbors (X = None, radius = None, return_distance = True, sort_results = False) [source] ¶ Finds the neighbors within a given radius of a point or points. Given lexicographic preferences, both u(x’1, 2) and u(x’1, 1) must be larger than u(x1, 2): therefore for any rational number r(x’1) in the interval [u(x’1, 2), u(x’1, 1)] we must have r. Lexicographic ordering is defined to be the cartesian product of two, or more, posets. Barrett Toric Calculator K Index 1. The FRAX® tool has been developed to evaluate fracture risk of patients. Topsoil Calculator Use this topsoil calculator to find out how much topsoil you need to buy for your yard. This order is what the compareTo() method of class String uses. It’s just a table, which shows glyphs position to encoding system. Hello! I’m Nayuki, a magical girl software developer in Toronto, Canada. The first permutation is always the string sorted in non-decreasing order. APRI Calculator AST to Platelet Ratio Index (APRI) Calculator. It is also known as lexical order, dictionary order and alphabetical order. Password requirements: 6 to 30 characters long; ASCII characters only (characters found on a standard US keyboard); must contain at least 4 different symbols;. My interests are in computer science and mathematics, and I strive to write clean, simple code for practical applications. The ALCON® Online Toric IOL Calculator With the barrett toric algorithm. The name comes from the order used in a dictionary, where strings are. * The phrase lexicographic order' means in '''alphabetical order. Viewed 504 times 2 $\begingroup$ As far as my. Algorithm Visualizations. org/w/index. Ask Question Asked 3 years, 9 months ago. Any subset of a well-ordered set is itself well-ordered. The lexicographic order is an order on the Cartesian product of two or more partially ordered sets. INTRODUCTION The polyhedral model [8] is a powerful formalism for an-alyzing and transforming program fragments that meet cer-tain requirements. The teacher told us to assume the elements of the array were in ascending order. For example, the company provided internet banking, mobile banking for quick fund transfer. Sets use subset. The symbol we want to code is different so we code an "escape" symbol. Combinatorial Coding and Lexicographic Ordering (https. We have discussed a program to print all permutations in this post, but here we must print the permutations in increasing order. Lexicographic order These notes are about an example that isn't directly relevant to the material in Definition 2. Simplify exponential expressions using algebraic rules step-by-step. The calculator can find horizontal, vertical, and slant asymptotes. To use our own sorting order, we need to supply a function as the All tests and numeric conversions are done in the calculate method. 11 and it’s the same result as we received by calculating distances. Bibliography. We hope that the following list of synonyms for the word dandy will help you to finish your crossword today. In this particular example, sorting by the whole line wouldn’t matter, but we’re here to learn!. Type in your sum to see how to solve it step by step. Many translated example sentences containing "in lexicographic order" - Russian-English dictionary and search engine for Russian translations. But how may I simply compare two strings to determine which is first (by dictionary order) ? This must be a constituent part of the sort routine, but I cannot find a way to do it. Tax credits. This calculator uses algorithm described by. Generate fair value prices and Greeks for any of CME Group's options on futures contracts or price up a generic option with our universal calculator. In order to choose a value for x crude oil, we need to establish an order of priorities. In order to make something with a bit more potential, we need to create a special object in our Python code called a WSGIApplication. The implication (2. The other type is numerical ordering. Which of the following case does not exist in complexity theory? A. , an electronic health record) may open the web. If we fix a term order ≺, then every polynomial fhas a unique initial term in≺(f)=xa. Oct 7th 2012, 23:04 GMT. s i = i // Print the first r-combination 4. (mathematics) Formally, given two partially ordered sets A and B, the order ≤ on the Cartesian product A × B such that (a,b) ≤ (a′,b′) if and only if a < a′ or (a = a′ and b ≤ b′). The worst case running time of this computation is A. To calculate oxidation numbers of elements in the chemical compound, enter it's formula and click 'Calculate' (for example: Ca2+, HF2^-, Fe4[Fe(CN)6]3, NH4NO3, so42-, ch3cooh, cuso4*5h2o). Pricing Calculator. A collection of C/C++ software developer interview questions. After the first person is executed, a certain number of people are skipped and one person is executed. Knuth's Binary Lexicographic Strings. com wishes everyone to BE WELL, STAY WELL, GET WELL. INTRODUCTION The polyhedral model [8] is a powerful formalism for an-alyzing and transforming program fragments that meet cer-tain requirements. It will explore it recursively and it uses the lexicographical order on names for sorting its content before performing the hash computation. 11011 is less than the middle. With respect to the lexicographic order based on theusual less than relation: a)find all pairs in S x S less than (2,3). Lexicographic order calculator Case 2: i ≥ 2 n and j ≥ 2 n. Please use one of the browsers below. Combination Lexicographic Order, Rank, Index: The Comprehensive and Fast Algorithm. Order-Preserving Verbal Names. Month names when sorted in lexicographic order (even when abbreviated to three characters) are not in chronological order. Algorithm Visualizations. Average case D. How to say LEXICOGRAPHIC ORDER in other languages? See comprehensive translations to 40 different langugues "LEXICOGRAPHIC ORDER translations. We need to force the order of operation so that we are not waiting on multiplication before returning. ► Lexicographic order (1 C, 9 F). Calculate the maturity amount and interest earned for any Fixed Deposit. Sets use subset. Windows vista / 7 users may get a security. Calculate the margin required on F&O trading now only at 5paisa. * and below cannot use negative subscripts to address array indexes relative to the highest-numbered index. The order 5 context ANANA has not been seen previously. Two algorithms in software to calculate the combination lexicographical order, or rank, or index; generate combination for a given lexicographic order or rank. Combinations, arrangements and permutations is. Combination Lexicographic Order, Rank, Index: The Comprehensive and Fast Algorithm. Quickly put information in alphabetical order using this super duper free online tool. In Part 2 of this Coding Challenge, I discuss Lexicographic Ordering (aka Lexical Order) and demonstrate one algorithm to iterate over all the permutations. It is similar to the way we search for any word in the dictionary. The sorted function sorts it for you. Given an integer n, return 1 - nin lexicographical order. Proposition (lexicographic order induced by total orders is total). This free online math web site will help you learn mathematics in a easier way. • S a set • Sn is the set of all n-tuples whose entries are. We write the largest degree terms of our answer first, breaking ties by lexicographic order ignoring the leading coefficient of the term. Can calculate the volume of aquarium regardless of shape. Use this online PASI calculator to calculate PASI score for your patients. The step-by-step calculation help parents to assist their kids studying 3rd, 4th or 5th grade to verify the steps and answers of ascending and descending order homework and assignment problems in pre-algebra or in number and operations in base ten (NBT) of common. Potter's Herbal Cigarettes are a smoking preparation made from stra- monium. The basis in -dimensional space is called the ordered system of linearly independent vectors. Lexicographic Order. Lexicographic sorting in C++ Sorting a list lexicographically, means sorting the contents of list so that the resultant list have its contents in dictionary order. We hope that the following list of synonyms for the word dandy will help you to finish your crossword today. O (n2 + log n) D. Take A Sneak Peak At The Movies Coming Out This Week (8/12) Here’s your soundtrack for Black History Month. Subscribe to updates on RSS. Lexicographical Order, Index, Rank of Permutations, Combinations. , by observing that. On this graph, draw the gradient of the function at (1; 1). Lexicographic Preferences--Preferences that can be strictly ranked --usually applies in situations where only one good in a bundle is preferred by the consumer. When calculating lists, minus and multiply will be converted to add after preprocessing. SAT / ACT Prep Online Guides and Tips. Then is said to be the lexicographic succesor of if and only if. Proposition (lexicographic order induced by posets is poset): Whenever. The first permutation is always the string sorted in non-decreasing order. Therefore, in the following, we will consider only orders on subsets of fixed cardinal. The order 2 context NA was seen once, followed by N. For instance the identity permutation (L=[0,1,2]) could have rank 1. Monomial order, specified as the comma-separated pair of 'MonomialOrder' and one of the values 'degreeInverseLexicographic', 'degreeLexicographic', or 'lexicographic'. Show all output here on this page. Ask Question Asked 3 years, 9 months ago. Bibliography. Calculate the maturity amount and interest earned for any Fixed Deposit. The first mismatching element defines which range is lexicographically less or greater than the other. Booleans use the order than False is less than True. The method is based on the fact that it is simple to verify if Zero is the first element representing the Mth lexicographic ordering of a. The most sophisticated and comprehensive graphing calculator online. Kolmogorov-Chaitin complexity) (or $K$) and Algorithmic Probability for. The Calculator app for Windows 10 is a touch-friendly version of the desktop calculator in previous versions of Windows. CoLex order is obtained by reflecting all tuples, applying Lex order, and reflecting the tuples again. Ion Saliu | Philosopher, Mathematician, Software Developer, Writer, Web Publisher - 153 Followers, 6 Following, 174 pins. Algorithm Visualizations. Take A Sneak Peak At The Movies Coming Out This Week (8/12) Here’s your soundtrack for Black History Month. With strings, the usual order is Lexicographic Order. The lexicographic or lexicographical order (also known as lexical order, dictionary order Lexicographically next permutation of the string ABCD is ABDC, for string ABDC is ACBD, and for. Lexicographic and colexicographic order. Monomial order, specified as the comma-separated pair of 'MonomialOrder' and one of the values 'degreeInverseLexicographic', 'degreeLexicographic', or 'lexicographic'. Once you file your claim , the EDD will verify your eligibility and wage information to determine your weekly benefit amount (WBA). 5: Well Ordering and Lexicographical Orde r We are often interested in sets of objects that are equipped with an order relation that satisfies certain properties. Signup for detailed step-by-step solutions. Your browser does not support the HTML 5 Canvas. Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios Exponents Calculator. Proposition (lexicographic order induced by posets is poset): Whenever. What is the Fee Calculator? This tool will ask questions to help determine your fee; however, it does not store answers to the questions or any other personal information. Alternatively referred to as an alphabetic sort, a lexicographic sort is a method of sorting data in alphabetical order (A to Z). 4 True Pick-3 lottery software is free to run, win 385. For example, given 13, return: [1,10,11,12,13,2,3,4,5,6,7,8,9]. LexicographicSubsets[l] gives all subsets of set l in lexicographic order. For example, if the set of polynomials has a finite number of solutions and we use the lexicographic ordering, then the Gröbner basis will contain a univariate polynomial in one of the variables, which can be solved using standard univariate polynomial solving techniques. The software can define and graph relations and also draw the transitive, symmetric, and reflexive closure of a relation. In Part 2 of this Coding Challenge, I discuss Lexicographic Ordering (aka Lexical Order) and demonstrate one algorithm to iterate over all the permutations. Free functions calculator - explore function domain, range, intercepts, extreme points and asymptotes step-by-step. From Wikipedia, the free encyclopedia. Knuth's Binary Lexicographic Strings. : Lexicographic generation of ordered trees. The order 3 context ANA has been seen once before, followed by N. The applications of Lexicographic Order in English (dictionary applications) and Mathematics (Cartesian product of two. Active 3 years, 9 months ago. Let Owens Corning Roofing help you calculate exactly how much ventilation you will need for a healthy and balanced attic, with our 4-step ventilation calculator. STANDS4 LLC, 2020. Note: There will be different way of operations integrated into Standard and Scientific calculators. You will see what the calculator thinks you entered (which may be a little different to what you typed), and then a. Access this free calculator to estimate your savings and ROI when you: Increase work order efficiency. If you are sedentary, multiply your BMR (1745) by 1. Ion Saliu | Philosopher, Mathematician, Software Developer, Writer, Web Publisher - 153 Followers, 6 Following, 174 pins. For it is necessary to define binary lexicographical preference. A lexicographical comparison is the kind of comparison generally used to sort words alphabetically in dictionaries; It involves comparing sequentially the elements that have the same position in both ranges against each other until one element is not equivalent to the other. Now, we will create a calculator program in C using functions. Animation Speed: w: h: Algorithm Visualizations. What is the Fee Calculator? This tool will ask questions to help determine your fee; however, it does not store answers to the questions or any other personal information. It has become conventional to denote Xc by vecX. Lybanonto determine the combinationfor a given rank (index or lexicographic order). Patient Data. In order to choose a value for x crude oil, we need to establish an order of priorities. In this program, you'll learn to sort the element words in lexicographical order using a for loop and if Example: Program to Sort Strings in Dictionary Order. Windows vista / 7 users may get a security. 5paisa margin calculator is an online tool to help you calculate comprehensive span margin requirements for option. The ntheory module gives us numtoperm and permtonum commands similar to Pari/GP, though in the preferred lexicographic order. Alternatively referred to as an alphabetic sort, a lexicographic sort is a method of sorting data in alphabetical order (A to Z). Free online factoring calculator that factors an algebraic expression. In mathematics, the lexicographic or lexicographical order, (also known as lexical order, dictionary order, alphabetical order or lexicographic(al) product), is a generalization of the way the. The calculator leverages Power BI’s What If parameters, bookmarks, buttons, conditional formatting, and a few simple DAX expressions to deliver interactive scenario building. Calculate the maturity amount and interest earned for any Fixed Deposit. The algorithm is included in the Association for Computing Machinery (ACM algorithm #515, published in 1977). Lexicographical Order, Index, Rank of Permutations, Combinations. Read more on this subject below the form. Hope the above information helps. 2010 Mathematics Subject Classification: Primary: 06A [MSN][ZBL]. Permutation Generator. Print the desired output. This step-by-step online fraction calculator will help you understand how to convert rectangular form Using this online calculator, you will receive a detailed step-by-step solution to your problem, which. The ALCON® Online Toric IOL Calculator With the barrett toric algorithm. Algorithms: Generating Combinations #100DaysOfCode, Generate combinations of n numbers taken r at a time. the lexicographic product 2 γ and the usual Tychonoff product 2 γ are homeomorphic, (2) the identity map from the lexicographic product 2 γ onto the usual Tychonoff product 2 γ is a homeomorphism, (3) the lexicographic product 2 γ is homeomorphic to the usual Tychonoff product 2 Λ for some Λ, (4) γ ≤ ω. The subset in the lexicographic order. •For simplicity, we will discuss n-tuples of natural numbers. Combinatorial Coding and Lexicographic Ordering (https. In sequential order, with the first step at the top, list the steps taken by consumers using the elimination-by-aspects decision rule to evaluate alternatives to a recognized problem 1. Compositions In order to demonstrate the rules of matrix composition, let us consider the matrix equation (29) Y = AXB0, which can be construed as a mapping from X to Y. Online exponents calculator with negative numbers support and steps. 2 Iterative, lexicographical order; 104 XPL0; 105 zkl; 360 Assembly. You do not have a sample of mRNA from any of the missing persons you think may be the victim, but you do. The lexicographic order algorithm was developed by B. Lexicographic and colexicographic order. Pastebin is a website where you can store text online for a set period of time. The Calculator app for Windows 10 is a touch-friendly version of the desktop calculator in previous versions of Windows. Lexicographic Order. Permutations[list] generates a list of all possible permutations of the elements in list. Use the slider to scroll through the permutations of objects in lexicographic order. Calculate Your Fees. The function f (x,y)=xy (1−9x−3y) has 4 critical points. The first permutation is always the string sorted in non-decreasing order. In mathematics, the lexicographic or lexicographical order (aka lexical order, dictionary order or Finding the combination by its lexicographical index. Enumeration of all Permutations (Recursion, Single Swap, and in Lexicographic Order), and Combinations. An ordering for the Cartesian product of any two sets and with order relations and , respectively, such that if and both belong to , then iff either 1. The optimization process starts minimizing the most important objective and proceeds according to the assigned order of importance of the criteria. Given 13, return: [1,10,11,12,13,2,3,4,5,6,7,8,9]. The lexicographic order algorithm was developed by B. The worst case running time of this computation is A. In order to provide better and efficient service to the customers, Credit Union Bank keeps their tune and changes their technologies more efficiently. See full list on bernardosulzbach. Generation in lexicographic order There are many ways to systematically generate all permutations of a given sequence. Implicit bool conversion. Please use one of the browsers below. WSGI stands for Web Server Gateway Interface and is a way to allow Python to communicate with the web server in a better way than simply “printing” a single chunk of information back as a response. Just enter your p-value, which must be between 0 and 1, and. Not the answer you're looking for? Browse other questions tagged string sorting lexicographic or ask your own. Select Scientific calculator. Axial Length: Ant Ch Depth: K1: @ K2: @ TargRx: PICK IOL MODEL Bausch li61AO SofPort Alcon SN60AT or SA60AT Alcon SA60T3/4/5 Toric. \$calculator = new RankingCalculator( new Alpha36TokenSet(), new. 0: Algorithms, software, source code calculate lexicographic order of combinations; compiled program and source code are included. There are many different sorting algorithms, each has its own advantages and limitations. The leading coefficient of the term is placed directly to the left with an asterisk separating it from the variables (if they exist. In this program, the user has the choice for operation, and it will continue until the user doesn’t want to exit from the. The Inflation Calculator uses monthly consumer price index (CPI) data from 1914 to the present to show changes in the cost of a fixed "basket" of consumer purchases. List them and specify whether they are minimum, maximum or a saddle point. ** To find the exponent from the base and the exponentation result, use: Logarithm calculator ►. An ordering for the Cartesian product of any two sets and with order relations and , respectively, such that if and both belong to , then iff either 1. Calls are charged at local rates from landline and mobiles. Knuth's Binary Lexicographic Strings. NAME_ASCENDING) annotation. (ii) The two derivatives are equal as the order in which derivatives are computed is unimpor-tant. Online calculator for quick calculations, along with a large collection of calculators on math Calculator. I know the sort builtin function will sort cell arrays of strings in ascii dictionary order. Lexicographical order. Lexicographical comparison is a operation with the following properties: Two ranges are compared element by element. Calculator for Factorial Base and Permutations Factorials are exact up to 500! which has 1135 digits. Worst case C. To use our own sorting order, we need to supply a function as the All tests and numeric conversions are done in the calculate method. Order-Preserving Verbal Names. Let Owens Corning Roofing help you calculate exactly how much ventilation you will need for a healthy and balanced attic, with our 4-step ventilation calculator. Our online molarity calculator makes calculating molarity and normality for common acid and base stock solutions easy with most common values pre-populated. A list of n strings, each of length n, is sorted into lexicographic order using the merge-sort algorithm. Proposition (lexicographic order induced by total orders is total). println(s2 + "follows" + s3 + " in lexicographic ordering"). Online based tool to ascending or descending each lines using lexicographically sorting algorithm. The function f (x,y)=xy (1−9x−3y) has 4 critical points. The manual method of multiplication procedure involves a large number of calculations especially when it comes to higher order of matrices, whereas a program in C can carry out the operations with short, simple and understandable codes. Lexicographic order calculator Case 2: i ≥ 2 n and j ≥ 2 n. -k1,1 means “sort by only the first whitespace-separated column”. Sorting is often an important first step in algorithms that solves more complex problems. As with all risk calculators, calculated risk numbers are +/- 5% at best. You cannot order travel money in store, online or by phone. We define the lexicographic order by saying that ( p , q ) < L ( p ′ , q ′ ) if and only if one of the following two statements is the case: * p < p'; or * p = p' and q < q'. If at any time E prints a string wi that comes after w in the lexicographic ordering of strings in ∑*, then. Ultimately, we are trying to find the most efficient production chain for the items we want to make, where efficiency is defined as using as few resources as possible. (The wikipedia. INTRODUCTION The polyhedral model [8] is a powerful formalism for an-alyzing and transforming program fragments that meet cer-tain requirements. org/w/index. Proposition (lexicographic order induced by posets is poset): Whenever. integers, floating-point numbers, strings, etc) of an array (or a list) in a certain order (increasing, non-decreasing, decreasing, non-increasing, lexicographical, etc). (ii) Compute the directional derivative of fat (1; 1) in the direction u = 4 5; 3 5. Given an integer n, return 1──►n (inclusive) in lexicographical order. Given two partially ordered sets A and B, the lexicographical order on the Cartesian product A × B is defined as (a,b) ≤ (a′,b′) if and only if a < a′ or (a = a′ and b ≤ b′). If you want to convert another number, just type over the original. The function f(x,y)=xy(1−9x−3y) has 4 critical points. Given a string, print all permutations of it in sorted order. STANDS4 LLC, 2020. As an example of a total order permutations can be listed in lexicographic order. In order to make something with a bit more potential, we need to create a special object in our Python code called a WSGIApplication. Computer dictionary definition for what lexicographic sort means including related links, information Alternatively referred to as an alphabetic sort, a lexicographic sort is a method of sorting data in. This list of Best Free Online Applications now includes 240 items in multiple categories. From Wikiversity. Can calculate the volume of aquarium regardless of shape. public class Sort {. DEFAULT parameter in this annotation. We hope that the following list of synonyms for the word dandy will help you to finish your crossword today. Initializing live version. Don't forget the dot at the end (every statement ends with it) You didn't write a constraint which uses your newly defined x. Munafo Alphabetic PT System. NAME_ASCENDING) annotation. STANDS4 LLC, 2020. * and below cannot use negative subscripts to address array indexes relative to the highest-numbered index. Hope the above information helps. Lexicographic order calculator Case 2: i ≥ 2 n and j ≥ 2 n. See full list on nayuki. Online calculator for quick calculations, along with a large collection of calculators on math Calculator. Online aquarium volume calculator. Fraction calculator to add, subtract, divide, and multiply fractions with a step-by-step explanation The calculator performs basic and advanced operations with fractions, expressions with fractions. This program takes 10 words from the user and sorts them in lexicographical order. After the first person is executed, a certain number of people are skipped and one person is executed. This free online math web site will help you learn mathematics in a easier way. The lexicographic order algorithm was developed by B. Please use one of the browsers below. The applications of Lexicographic Order in English (dictionary applications) and Mathematics (Cartesian product of two. As an example of a total order permutations can be listed in lexicographic order. Enumeration of all Permutations (Recursion, Single Swap, and in Lexicographic Order), and Combinations. Problem 24 of Project Euler is about permutations, which is in the field of combinatorics. You can use an annotation to define that the test methods are sorted by method name, in lexicographic order. The lexicographic order algorithm was developed by B. MIT License. (and sort is builtin so I cannot inspect it). Click ‘Convert’ to convert. Theoretically accounts for posterior corneal astigmatism. Run enumerator E and let w1, w2, w3, … be the strings printed by E. For example, the company provided internet banking, mobile banking for quick fund transfer. Please use another device. To put items in order, there must be a way to compare two items. Most accurate preoperative prediction of residual astigmatism1-3. Once you open, click on menu button (Three horizontal lines at top left corner). Use the slider to scroll through the permutations of objects in lexicographic order. The binary tree is maintained so that it is always both a search tree relative to the suffix lexicographic ordering, and a max-heap for the dictionary position [10] (in other words, the root is always the most recent string, and a child cannot have been added more recently than its parent): assuming all strings are lexicographically ordered. If at any time E prints a string wi that comes after w in the lexicographic ordering of strings in ∑*, then. Calculate Your Fees. Make fewer service calls. The Student class should have instance variables (with getters and setters) for a student's last name (String), first name (String), student id number (int), an array for up to 15 project scores that are represented as double values, named projects, and an array for up to 10 quiz scores that are represented as double values, named quizzes. Here it is. Example III. Lexicographic rank of the string BDAC is 11 A simple solution would to use std::next_permutation that generates the next greater lexicographic permutation of a string. Reduce field dispatch. [46] One classic, simple, and flexible algorithm is based upon finding the next permutation in lexicographic ordering , if it exists. Please pick the appropriate. (and sort is builtin so I cannot inspect it). We have discussed a program to print all permutations in this post, but here we must print the permutations in increasing order. Frequently Asked Questions. To put items in order, there must be a way to compare two items. STANDS4 LLC, 2020. From Java 5 onwards java. be preordered sets, where. The lexicographic order algorithm was developed by B. Once you open, click on menu button (Three horizontal lines at top left corner). The estimates are calculated using data from a large number of patients who had a surgical Risk Calculator Permitted Use: An external platform (e. Combinatorial Coding and Lexicographic Ordering (https. It is possible to implicitly convert a string to bool in a filter as in user_label. Barrett Toric Calculator K Index 1. Online Dictionaries: Definition of Options|Tips Options|Tips. (In the above, statements, < denotes the usual ordering on the rational numbers we know and love!). Clinical Calculators. An alternative is to randomly select an objective when there is no more rank available. SPOJ (Sphere Online Judge) is an online judge system with over 315,000 registered users and over 20000 problems. Lexicographic order of numbers in c example program. 11 and it’s the same result as we received by calculating distances. Calculator for Factorial Base and Permutations Factorials are exact up to 500! which has 1135 digits. In mathematics, the lexicographic or lexicographical order (also known as lexical order, dictionary order, alphabetical order or lexicographic(al) product) is a generalization of the alphabetical order. INTRODUCTION The polyhedral model [8] is a powerful formalism for an-alyzing and transforming program fragments that meet cer-tain requirements. Jump to navigation Jump to search. Don't forget the dot at the end (every statement ends with it) You didn't write a constraint which uses your newly defined x. The teacher told us to assume the elements of the array were in ascending order. Number with positive (bad) outcome. It is implied that this order is lexicographical. Returns true if the range [first1,last1) compares lexicographically less than the range [first2,last2). What is the nth lexicographical permutation of a given string? Instead of finding all…. Social Security Quick Calculator. (and sort is builtin so I cannot inspect it). The calculator can find horizontal, vertical, and slant asymptotes. In the index notation, this is written as (30) (y kle l k) = (a kie i k)(x. From Wikiversity. In order to make something with a bit more potential, we need to create a special object in our Python code called a WSGIApplication. The base factorial representation of an index number n in the lexicographic order is easily changed into the permutation itself. Let's see how we can use FMS as a calculator. org entry does mention this just before disappearing into set theory). It can be called as follows : DirHash. An ordering for the Cartesian product of any two sets and with order relations and , respectively, such that if and both belong to , then iff either 1. Unicode is a computing standard for the consistent encoding symbols. The most sophisticated and comprehensive graphing calculator online. Lines are open 24 hours a day, 7 days a week (except Christmas Day). The subset in the lexicographic order. org/w/index. Following the discussion in "Generating the mth Lexicographical Element of a Mathematical Combination", I present here another algorithm for the Mth lexicographic ordering based on a linear search procedure. Open the Calculator through Start menu. 332 +ve Cylinder -ve Cylinder. O (n2 + log n) D. -k1,1 means “sort by only the first whitespace-separated column”. Hello! I’m Nayuki, a magical girl software developer in Toronto, Canada. Fraction calculator to add, subtract, divide, and multiply fractions with a step-by-step explanation The calculator performs basic and advanced operations with fractions, expressions with fractions. I know the sort builtin function will sort cell arrays of strings in ascii dictionary order. Use the slider to scroll through the permutations of objects in lexicographic order. Now we formulate lexicographical structure on the set E. Let's see how we can use FMS as a calculator. ** To find the exponent from the base and the exponentation result, use: Logarithm calculator ►. Section III. Definition (lexicographic order): Let. It allows them in providing speedy service. Sending completion. 3375 K Index 1. O (n2 + log n) D. Technically this calculates the lexicographical index of all permutations, but if you only give it To use the squashed order to get the lexicographic order in the set of k-subsets of {1,,n) is by taking. lexicographical order is alphabetical order. Not the answer you're looking for? Browse other questions tagged string sorting lexicographic or ask your own. See evaluation order for details. The lexicographic order on words is the relation defined by X < Y if X comes (strictly). print(s1, s2, , s r) 5. The lexicographic order is an order on the Cartesian product of two or more partially ordered sets. Binary trees Left-child sequences Lexicographic order Ranking algorithms Unranking algorithms Amortized cost. The leading coefficient of the term is placed directly to the left with an asterisk separating it from the variables (if they exist. Jump to navigation Jump to search. Calculate the maturity amount and interest earned for any Fixed Deposit. Permutations in Lexicographic Order. The order of bases is below. Cardinal Means in Lexicographic (or Lexicographical) Direction and Indexing, Algorithms, Software You can bargain here two algorithms to calculate the combination lexicographical disposition, or station, or index; reversely, fabricate the combination for a prone lexicographic classification or grade. Lexicographic order calculator, useful for persisting ordered lists. Sort the given string in non-decreasing order and print it. Section III. 4 Generate Leads. Includes all the functions Easy to use and 100% Free! We also have several other calculators. Ask Question Asked 3 years, 9 months ago. Now we formulate lexicographical structure on the set E. The input size may be as large as 5,000,000. Calculate the company car tax charge based on a car's taxable value and CO2 rating. If w = wi for some i, then ACCEPT w and halt. Now type 1+2*3, it will give the answer as 7. Scanner class can be used to read file in Java. how to calculate index of lexicographical order. Best case B. The order 5 context ANANA has not been seen previously. calculators. O (n2 log n) C. To calculate oxidation numbers of elements in the chemical compound, enter it's formula and click 'Calculate' (for example: Ca2+, HF2^-, Fe4[Fe(CN)6]3, NH4NO3, so42-, ch3cooh, cuso4*5h2o). Now we formulate lexicographical structure on the set E. Worst case C. The order 2 context NA was seen once, followed by N. Lexicographic order is the way of ordering of words based on the alphabetical order of their component letters. Above is the source code for C Program to Sort strings Lexicographically (Dictionary Order) which is successfully compiled and run on Windows System. Lexicographic order calculator, useful for persisting ordered lists. The algorithm is included in the Association for Computing Machinery(ACM algorithm #515, published in 1977). A permutation is a bijection from a set to itself. An ordering (or order) is a method for sorting elements. This program asks the user to enter Array Size and array elements. Please don’t do this problem In 1918, during the bad old days when students drank beer, a fraternity had six barrels of beer in the basement, just like this:. In the lexicographic ordering, D, E, and O come before R. We will help you to choose most appropriate processor and graphic card for your PC. Translation of: Liberty BASIC * Permutations 26/10/2015 PERMUTE CSECT USING PERMUTE,R15 set. Sort the given string in. Following are the steps to print the permutations lexicographic-ally. LexicographicSubsets[l] gives all subsets of set l in lexicographic order.
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2021-02-25 05:13:57
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https://collegemathteaching.wordpress.com/2010/03/06/the-principle-of-mathematical-induction-why-it-works/
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# College Math Teaching
## March 6, 2010
### The Principle of Mathematical Induction: why it works
Filed under: induction, logic, transfinite induction, Uncategorized — collegemathteaching @ 11:03 pm
I am writing this post because I’ve seen that there is some misunderstanding of what mathematical induction is and why it works.
• What is mathematical induction? It is a common proof technique. Basically, if one wants to show that a statement is true in generality and that one can index the set of statements via the integers (or by some other appropriate index set), then one can use induction.
Here is a common example: suppose one wants to show that
$1 + 2 + 3 + ....+ k = (1/2)*(k)*(k+1)$ for all positive integers $k$
(for example, $1 + 2 + 3 + 4 + 5 = (1/2)*(5)*(5+1) = 15$).
Initial step: $1 = (1/2)*(1)*(1+1)= 1$ so the statement is true for $k = 1$.
Inductive step: assume that the formula holds for some integer $k$.
Finish the proof: show that if the formula holds for some integer $k$, then it holds for $k+1$ as well.
So $1 + 2 + 3 +....+ k + (k+1) = (1 + 2 + ...+ k) + (k+1) =(1/2)*(k)*(k+1) + (k + 1)$
(why? because we assumed that $k$ was an integer for which
$1 + 2 + 3 + ....+ k = (1/2)*(k)*(k+1)$. )
so $1 + 2 + 3 +....+ k + (k+1) = (1/2)*(k)*(k+1) + (k + 1) = ((1/2)*k + 1)*(k+1)$ (factor out a k+1 term)
$= (1/2)*2*((1/2)*k + 1)*(k+1) = (1/2)*(k+2)*(k+1)=(1/2)*(k+1)*(k+2)=(1/2)*(k+1)*(k+1 + 1)$
which is what we needed to show. So the proof would be done.
• Why does induction “prove” anything? Mathematical induction is equivalent to the so called “least positive integer” principle in mathematics.
• What is the least positive integer principle? It says this: “any non-empty set of positive integers has a smallest element”. That statement is taken as an axiom; that is, it isn’t something that can be proved.
Notice that this statement is false if we change some conditions. For example, is is NOT true that, say, any set of positive numbers (or even rational numbers) has a smallest element. For example, the set of all numbers between 0 and 1 (exclusive; 0 is not included) does NOT have a least element (not according to the “usual” ordering induced by the real number line; it is an easy exercise to see that the rationals can be ordered so as to have a least element). Why? Let $b$ be a candidate to be the least element. Then $b$ is between 0 and 1. But then $(1/2)b$ is greater than zero but is less than $b$; hence $b$ could not have been the least element. Neither could any other number.
Note that the set of negative integers has no least element; hence we need the condition that the integers are positive.
Notice also that there could be groups of positive integers with no greatest element. For example, Let $x$ be the largest element in the set of all even integers. But then $2x$ is also even and is bigger than $x$. Hence it is impossible to have a largest one.
• What does this principle have to do with induction? This is what: an induction proof is nothing more than a least integer argument in disguise. Lets return to our previous example for a demonstation; that is, our proof that $1 + 2 + ....+ n = (1/2)n(n+1)$
We start by labeling our statements: $1 = (1/2)(1)(1+1)$ is statement P(1),
$1 + 2 = (1/2)(2)(2+1)$ is statement P(2), …$1+2+...+ 5 = (1/2)(5)(5+1)$ is statement P(5) and so on.
We assume that the statement is false for some integer. The set of integers for which the statement is false has a least element by the least element principle for positive integers.
We assume that the first integer for which the statement is false is $k+1$. We can always do this, because we proved that the statement is true for $k = 1$, so the first possible false statement is $k = 2$ or some larger integer, and these integers can always be written in the form $k + 1$.
That is why the anchor statement (the beginning) is so important.
We now can assume that the statement is true for $n = 1,....n = k$ since $k+1$ is the first time the statement fails.
Now when we show “if statement P($k$) is true then P($k+1$) is also true (this is where we did the algebra to add up $1 + 2 + .....+ k + (k+1) = ((1/2)k(k+1)) + (k+1) )$. This contradicts that statement P($k+1$) is false.
Hence the statement cannot be false for ANY positive integer $k$.
• Weak versus strong induction. As you can see, the least positive integer principle supposes that the statement is true for all statements P(1) through P($k$), so in fact there is no difference (when inducting on the set of positive integers) between weak induction (which assumes the induction hypothesis for some integer $k$) and strong induction (which assumes the induction hypothesis for $n = 1$ through $n = k$).
• Other index sets: any index set that one has to induct on has to have the “least element principle” to its subsets. Also, if there is a cardinal w that has no immediate predecessor, then one must “reanchor” the induction hypothesis as $k = w$ prior to proceeding.
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2017-03-24 17:48:12
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https://nm.dev/courses/introduction-to-data-science/lessons/finding-roots-of-system-of-equations/topic/introduction-to-system-of-non-linear-equations/
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Now that we have learnt about system of linear equations and how to solve it, let’s move on to non-linear system of equations.
Non-linear system of equations
Non-linear equations are equations that cannot be linearly plotted, and hence cannot be plotted on a graph as a straight line. The highest degree of the variables inside a non-linear equation is higher than 1. An example of a non-linear equation is
$y=x^2+1$
A system of non-linear equations refers to a system of equations in which a non-linear equation is present, though not all equations inside the system needs to be non-linear. It also differs from a linear system as it can also have more than one solution. The following is an example of a non-linear system of equations with 2 equations with variables $x$ and $y$.
$y=x+1$
$y=x^2+1$
The concept of solving a non-linear system of equations is similar to that of a linear system of equation, as we are trying to find the point of intersection for all of the graphs. Using the above system as an example, by plotting it in a graph,
And from the 2 points of intersection, we can deduce the 2 solutions, which are
$x=0$
$y=1$
and
$x=1$
$y=2$
We are also able to use substitution to obtain our results. Using the previous system as an example, we can use the equation
$x=y-1$
and substitute it into
$y=x^2+1$
$y=(y-1)^2+1$
$y=y^2-2y+1+1$
$0=y^2-3y+2$
By completing the root, we will get
$0=(y-2)(y-1)$
### Real World Examples!
Did you know? One of the best known and most used case of non-linear system of equations is calculating alternating current power-flow model in an electric power system, as they use non-linear equations to calculate power for each transmission line.
Solving this system gives the flow of active power in an electrical system. It is the same flow of power that powers your home and keeps the computer charging, and is very important for analysing power grids. The system of equations is expressed as: $\begin{bmatrix}\Delta \theta \\\Delta |V|\end{bmatrix}=-J^{-1}\begin{bmatrix}\Delta P\\\Delta Q\end{bmatrix}$
%use s2
//f1:y=x+1
val f1:BivariateRealFunction = object : AbstractBivariateRealFunction() {
//Using AbstractBivariateRealFunction as we are only using 2 variables
override fun evaluate(x: Double, y: Double): Double {
//if the values are correct, it returns 0
return x + 1 - y
}}
//f2:y=x^2+1
val f2:BivariateRealFunction = object : AbstractBivariateRealFunction() {
//Using AbstractBivariateRealFunction as we are only using 2 variables
override fun evaluate(x: Double, y: Double): Double {
//if the values are correct, it returns 0
return x*x + 1 - y
}}
println(f1.evaluate(1.0,2.0))
println(f2.evaluate(1.0,2.0))
As systems of nonlinear equations becomes more complex, it will be harder to obtain solutions by substitutions or graphing. In the next chapter, we will be looking at the theory behind Newton’s method, as well as how to implement it in NMdev, to be able to find solutions to more complex non-linear systems.
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2022-01-23 16:03:41
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https://cs6505.wordpress.com/schedule/notes-923-space-time-nondeterminism/
|
# Space & Time hierarchy
Space & Time hierarchy theorems
Complexity of an algorithm measures how much of a certain resource the algorithm uses, such as time, space, or number of random bits.
Does more space increase the power of a TM? For instance, for every decidable language, can we construct a TM with a constant size tape that decides this language? The answer is no, because a constant size tape TM can be written as a finite state machine, and we know there are languages that can be recognized by a TM that an FSM cannot.
How much power is granted to a machine by additional resources? Perhaps we get a hierarchy of languages, where each level of the hierarchy is the set of languages that are recognizable by a TM with a certain amount of space.
First, we define formally the notion of the space requirement of computing a function.
Definition. A function $f: \mathbb{N} \rightarrow \mathbb{N}$ is said to be space-constructible if there exists a TM $M$ that on input $1^n$ outputs $f(n)$ and uses $O(f(n))$ space. We assume that $f(n) \ge \log n$.
Theorem (Space Hierarchy). For any space-constructible function $f(n)$, there exists a language $L_f$ such that $L_f$ can be decided by a TM using $O(f(n))$ space and there does not exist a TM that decides $L_f$ using $o(f(n))$ space.
Recall the difference between “little-o” and “big-O” for bounding a function. $f=O(g)$ means that $f$ grows at most the rate of $g$, up to constants; $f=o(g)$ means that $f$ grows strictly slower than $g$ asymptotically. More precisely,
$f(n) = O(g(n))$ means $\exists n_0, c>0 : \forall n \ge n_0, g(n) \le c \cdot f(n)$
$f(n) = o(g(n))$ means $\forall c>0$, $\exists n_0 : \forall n \ge n_0, f(n) \le c\cdot g(n)$
For example, the Space Hierarchy theorem implies that a language is recognizable by a TM using $n^2$ space, but no TM exists that recognizes the language and uses $n^{1.99}$ space.
$SPACE(s(n)) =$ languages accepted by TM’s that use at most $s(n)$ space.
$TIME(t(n)) =$ languages accepted by TM’s that run for at most $t(n)$ steps.
Theorem $SPACE(s(n)) \subseteq TIME(C^{s(n)})$.
Proof:
Consider the current “configuration” of a TM, specified by $(q,$ head position, tape contents$)$. Therefore, with space $s(n)$, we can bound the number of configurations as:
\begin{aligned} \text{\# possible configurations} &\le |Q| \cdot s(n) \cdot |\Gamma|^{s(n)} \\ &= c_1 \cdot s(n) \cdot c_2^{s(n)} \\ &\le c_2^{s(n) + \log n} \\ &\le c^{s(n)}\end{aligned}
Thus we can simulate the NTM using a DTM in $c^{s(n)}$ time by visiting every allowable configuration and checking if it is an accept state.
We can now prove the Space Hierarchy theorem.
Proof: (of Space Hierarchy theorem)
Define $L_f$ as the set of TM’s M and strings $1^n$ such that $M$ has a tape alphabet of constant size and does not accept $(\langle M\rangle,1^n)$ and using space $\le f(n)$ and time $\le 2^{f(n)}$. The proof then follows from the following two claims.
Claim 1: There exists a TM $D$ that decides $L_f$ using space $O(f(n))$.
Proof: We construct the following TM $D$ on input $(\langle M\rangle, 1^n)$ (if the input is not in this form, reject):
1. Mark $f(n)$ space on the tape.
2. Simulate $M$ on $(\langle M\rangle,1^n)$
3. If $M$ tries to use more space, reject.
4. If time used is $\ge 2^{f(n)}$, reject.
5. Else
1. If $M$ accepts, reject.
2. If $M$ rejects, accept.
Claim 2: No TM using $o(f(n))$ space can decide $L_f$.
Proof: Suppose there exists such a TM $M$. Consider whether $M\in L_f$? If $M\in L_f$, when $M$ run on $(\langle M\rangle, 1^n)$, $M$ is supposed to accept it using $o(f(n))$ space since $M$ decide $L_f$, which is contradict to the fact that $M\in L_f$. And if $M\notin L_f$ the case will be similar. Thus no such TM could exist.
Note that such a contradiction doesn’t happens on $D$ since $D$ uses either $>f(n)$ space or $>2^{f(n)}$ time when simulating itself.
The following theorem gives a similar result as the Space Hierarchy Theorem for time, but with a $\log$ factor.
Definition: We say that a function $f: \mathbb{N} \rightarrow \mathbb{N}$, where $f(n) \ge n \log n$, is time-constructible if the function that maps $1^n$ to the binary representation of $f(n)$ is computable in time $O(f(n))$.
Theorem (Time Hierarchy Theorem): For any time-constructible function $f(n)$, there exists a language $L$ such that $L$ can be decided by a TM using $O(f(n))$ time and cannot be decided by any TM using $o(\frac{f(n)}{log(f(n))})$ time.
Proof (of Time Hierarchy Theorem): Consider the following TM $B$ that, in time $O(f(n))$, decides a language $L$ which we will show is not decidable in $o(\frac{f(n)}{\log f(n)})$ time.
$B(w)$:
1. Let $n$ be the length of $w$.
2. Compute $f(n)$ using time constructibility, and store the value $\lceil f(n)/\log f(n)\rceil$ in a binary counter. Decrement this counter before each step used in stages 3, 4, and 5. If the counter ever hits $0$, REJECT.
3. If $w$ is not of the form $\langle M \rangle 10^*$ for some TM $M$, REJECT.
4. Simulate $M$ on $w$.
5. If $M$ accepts, then REJECT. If $M$ rejects, then ACCEPT.
We first need to show that $B$ runs in $O(f(n))$ steps. Stages 1-3 can be completed in time $O(f(n))$ by the definition of time-constructibility.
For stages 4 and 5, we need to simulate the operation of $M$. To simulate $M$, we store the current state of the machine, the current tape, and the transition function. These objects will need to be constantly updated throughout the simulation, and need to be performed efficiently. We separate $B$‘s tape into tracks (e.g. even and odd positions), where on one track we store $M$‘s tape, and on another track, we store the current state of $M$ along with a copy of $M$‘s transition function.
To keep the simulation efficient, we maintain that the description of $M$ remains near the head of the TM. Whenever the position of $M$‘s head moves, we move the entire description of $M$ along the second track. Because we consider the size of $M$ to be a constant (not dependent on the input size provided the $M$), this only adds a constant overhead. Thus, we can simulate the operation of $M$ with only constant overhead: if $M$ runs in $g(n)$ time, then $B$ can simulate it in $O(g(n))$ time.
When $B$ simulates the operation of $M$, it must also keep track of the counter at each step. We use a third track to store and update the counter. The length of this counter will be $O(\log\big[f(n)/\log f(n)\big]) = O(\log f(n))$. Thus updating this counter will require a $O(\log f(n))$ multiplicative overhead to the simulation of $M$, which means that the language $L$ we constructed is decidable in $O(f(n))$ time.
Finally we need to show that no TM can decide $L$ in time $o(f(n)/\log f(n))$. Suppose for contradiction there existed a TM $T$ that decided $L$ in time $o(f(n)/\log f(n))$. Since both $T$ and $B$ decide the same language, they should agree on all inputs. Run the TM $B$ on input $\langle T \rangle 10^k$ for $k$ sufficiently large. $B$ will simulate $T(\langle T \rangle 10^k)$ in $o(f(n))$ steps, and will disagree with $T$ on the same input. Thus the TM $T$ cannot exist.
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2017-10-23 13:14:51
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http://aube-management.com/cnjh54ab/numericals-based-on-concentration-of-solution-f8eabd
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# numericals based on concentration of solution
A solution can be qualitatively described as. This video explains colloids, suspensions and the Tyndall effect. It is abbreviated as ‘N’ and is sometimes referred to as the equivalent concentration of a solution. 0. Multiple Choice Questions (Type-I) Solution: Percentage by mass of HCl = 24.8%. … Raoult's Law: Vapor Pressure and Volatile Solutes (in Ideal Solutions) Raoult's Law: Vapor Pressure and Volatile Solutes (in Non-ideal Solutions) Problem Sets. Solution — Molar mass (Molecular mass in gram) of CaCO3 = 40+12+3×16 = 100 g No. 3. define and explain molarity with examples. 100+ Free tests will be provided in this package. 5. define and explain mole fraction with examples. Q3. Calculate the concentration in terms of volume by volume percentage of the . b) What will be the absorbance if the solution is 5 M? 52) The area of base of a cylindrical vessel is 300 cm^2. Thus, if one gram molecule of a solute is present in 1 litre of the solution, the concentration of the solution is said to be one molar. Use significant figures. This collection of ten chemistry test questions deals with … Question #17770. Find the mole fraction of HCl in a solution of HCl containing 24.8 % of HCl by mass. Expressing Concentration. Cor S = $\frac{\text{Weight of solute in grams}}{\text{Volume in litres}}$ We will also see other methods on how to calculate concentration of a solution based on the different methods of expressing concentrations. Each successive unit change in pH represents a tenfold change in hydrogen ion concentration. Reading Time: 2min read 0. concentration = 37g / (100g solution) * 1180g / (1L solution) * (1mol HCL) /36.46g . Classes: 6 days in a week. Solution: click this link for solution Q52. Each method of expressing concentration of the solutions has its own merits and demerits. Share on Facebook Share on Twitter. This is because volume depends on temperature and the mass does not. Given: Percentage by mass = 24.8%. Save. by Anand Meena. 4) A solution contains 50g of sugar in 350g of water. Therefore, the change in the freezing point of the water is -3.8 o C. The freezing point of the solution is, therefore, -3.8 o C. Problem : A solution of 0.5 g of an unknown nonvolatile, nonelectrolyte solute is added to 100 mL of water and then placed across a semipermeable membrane from a volume of pure water. Q2. This webinar is hosted in English. This video is unavailable. Eligibility : 12th pass/Appeared. Concentration of solutions 1. Question #a373a. How much solution do I need to add increase the concentration by 2ppm? Question #d42c0. Water (density = 1000 kg /m^3) is poured into it up to a depth of 6 cm. Learning Objectives. The concentration of a solution is a macroscopic property, represents the amount of solute dissolved in a unit amount of solvent or of solution, and can be expressed in a variety of ways (qualitatively and quantitatively). Express the amount of solute in a solution in various concentration units. Amit Goyal. It is given that out of 100 atoms, 93 atoms are Hydrogen and 7 atoms are Helium. WE will discuss numerical s based on the density of solution , mixing of two solutions, dilution of the solutions to calculate the molarity of the solutions Calculate the masses of sugar and water required to prepare 250 g of 25% solution of cane sugar by mass. to make a saturated solution 36 gram of sodium chloride is dissolved in 100 grams of water at 293K. Concentration in Parts per Million. It is mainly used as a measure of reactive species in a solution and during titration reactions or particularly in situations involving acid-base chemistry. Share. A 10% mass by volume solution means that 10 gm solute is present in 100 mL of solution. Solutions can be classified into three types based on the concentration of the solvent in two solvents (in a beaker and a cell in it), in the solution, • Hypertonic Solutions: Hypertonic Solutions are those types of solutions in which the concentration of the solute in a beaker is higher than that of in the cell, so water comes out of the cell making the cell to plasmolyze/ shrink. Zostavax. Calculate: (a) the pressure and (b) the thrust of water on the base. Given H = 1, Cl = 35.5. To Find: Mole fraction of HCl =? Molarity. pH of a solution, therefore, is defined as the negative logarithm, to the base 10, of the hydrogen ion concentration [H +] in moles per liter. of molecules/Avogadro constant = 6.022 × 1023/ 6.022 × 1023 = 1 mole… Heterogeneous Mixtures. Watch Queue Queue Learning Outcomes: Students will be able to 1. define concentration of a solution. 4. define and explain m_oJalj1y with examples. Normality in chemistry is one of the expressions used to measure the concentration of a solution. This lesson includes numerical based on concentration of solution. April 22, 2019. in 12th Class, Class Notes. Qualitative Expressions of Concentration . 14.6: Acid–Base Titration: A Way to Quantify the Amount of Acid or Base in a Solution; 14.7: Strong and Weak Acids and Bases; 14.8: Water: Acid and Base in One ; 14.9: The pH and pOH Scales: Ways to Express Acidity and Basicity; 14.10: Buffers: Solutions That Resist pH Change . Numerical based on concentration of solution. Q1. Numerical problems based On Mole Concept Question 1. Question #ab0ea. Question #40a6a. A diluent such as sterile water is added to the drug to create the desired concentration. An aqueous solution has a mass of #490# grams containing #8.5 xx 10^"-3"# gram of calcium ions. Note: The density of water is 1 g/mL. Solubility of a substance is its maximum amount that can be dissolved in a specified amount of solvent at a … A solution contains 40 ml of ethanol mixed with 100 ml of water. of moles of CaCO3 = No. The concentration of yeast t-RNA in an aqueous solution is 10 M. The absorbance is found to be 0.209 when this Solution is placed in a 1.00 cm cuvette and 258 nm radiations are passed through it. The concentration of solution formula is given as follows. Let us consider 100 g of HCl solution. 1. Concentration is the amount of a substance in a predefined volume of space. When the pH is 7 ([H +] = 10-7 mol/liter): Number of hydrogen ions precisely equals the number of hydroxide ions (OH –) And the solution is neutral - … ( Molarity, Normality, Molality and mole fraction) (Hindi) Numerical of Solution for IIT JEE. (g = 10 m/ s^2). Thanks for helping out! 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https://www.zama.ai/post/tfhe-deep-dive-part-1
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# TFHE Deep Dive - Part I - Ciphertext types
By
Ilaria Chillotti
Published on:
May 4, 2022
in
Engineering
This blog post is part of a series of posts dedicated to the Fully Homomorphic Encryption scheme called TFHE (also known as CGGI, from the names of the authors Chillotti-Gama-Georgieva-Izabachène). Each post will allow you to go deeper into the understanding of the scheme. The subject is challenging, we know, but don’t worry, we will dive into it little by little.
Disclaimer: If you have watched the video TFHE Deep Dive, you might find some minor differences in this series of blog posts. That’s because here there is more content and a larger focus on examples. All the dots will connect in the end.
TFHE is a Fully Homomorphic Encryption (FHE) scheme. That means it is an encryption scheme that allows you to perform computations over encrypted data. To know more about FHE schemes in general, take a look at this blog post.
TFHE was initially proposed as an improvement of the scheme FHEW, and then it started developing in a broader direction. The security of the scheme is based on a hard lattice problem called Learning With Errors, or LWE in short, and its variants, such as Ring LWE (RLWE). In fact, the majority of FHE schemes used nowadays are LWE based and use noisy ciphertexts. TFHE is, however, distinguished from the others because it proposes a special bootstrapping which is very fast and able to evaluate a function at the same time as it reduces the noise.
A few blog posts will be necessary before we talk in detail about bootstrapping, so no rush for now in trying to understand how it works. Let’s start from “the beginning” by describing the ciphertexts used in TFHE.
Some notation
A few mathematical objects will be needed to understand this blog post series.
• $\mathcal{R} = \mathbb{Z}[X]/(X^N +1)$ the ring of integer polynomials modulo the cyclotomic polynomial $X^N + 1$, with $N$ power of 2. In practice, it contains integer polynomials up to degree $N-1$.
• $\mathcal{R}_q = (\mathbb{Z}/q\mathbb{Z})[X]/(X^N +1)$, i.e., the same ring of integers $\mathcal{R}$ as above, but this time the coefficients are modulo $q$. Observe that we often note $\mathbb{Z}/q\mathbb{Z}$ as $\mathbb{Z}_q$.
• Our modular reductions are centered around zero. As an example, when reducing modulo $8$, we use the congruence classes $\{ -4, -3, -2, -1, 0, 1, 2, 3 \}$.
• $\chi_{\mu, \sigma}$ is a Gaussian probability distribution with mean $\mu$ and standard deviation $\sigma$. If $\mu = 0$, we will simply note $\chi_\sigma$.
• We will use small letters for (modular) integers $(a, b, m, s, \ldots)$, we will use capital letters for polynomials $(A, B, M, S, \ldots)$.
• We will note the list of integer elements from $a\in \mathbb{Z}$ to $b\in \mathbb{Z}$ included, as $[a..b]$.
• We use the abbreviations MSB and LSB for Most Significant Bit and Least Significant Bit respectively.
• We denote with $\lfloor \cdot \rceil$ the rounding operation to the nearest integer value.
## TFHE Ciphertexts
What is the first thing you talk about when you give a recipe? The ingredients, of course! In our case, the main ingredients are the ciphertexts.
In TFHE, we mainly use three types of ciphertexts: LWE, RLWE, and RGSW ciphertexts. Why we need three different types of ciphertexts, you might wonder. Long story short, the reason is that all of them have different properties which will be useful in the homomorphic operations that we will describe in the following blog posts. All of them have security that relies on the LWE problem or its variants. To know more about LWE security, please take a look at this blog post.
In this blog post we will give you more general definitions in order to help you understand the objects we manipulate.
These ciphertext are not only used in TFHE, but also in other LWE-based FHE schemes.
• GLWE (General LWE) - a generalization for both LWE and RLWE ciphertexts;
• GGSW (General GSW) - a generalization for RGSW ciphertexts;
• GLev - an intermediate ciphertext type that will be very useful to better understand GGSW ciphertexts and that we will largely use in the following blog posts.
Let’s start!
## GLWE
If you’ve already heard about LWE based FHE schemes, you have also probably heard about LWE and RLWE ciphertexts.
In this section we will use a generalization that includes both of them, called General LWE, or GLWE in short.
To generate any kind of ciphertext, we first need a secret key. With GLWE ciphertexts, the secret key is a list of $k$ random polynomials from $\mathcal{R}$:
$$\vec{S} = (S_0, \ldots, S_{k-1}) \in \mathcal{R}^k.$$
In particular, the coefficients of the $\mathcal{R}$ elements can be sampled from a uniform binary distribution, a uniform ternary distribution, a Gaussian distribution, or a uniform distribution.
Please note that for any of these types of secret keys, we can find parameters to archive a desired security level.
In this series of blog posts, as in the original TFHE description, we will assume that our secret keys are sampled from uniform binary distributions.
Now let’s see how to encrypt messages. Let $p$ and $q$ be two positive integers, such that $p\leq q$ and let’s define $\Delta = q/p$. In TFHE, $q$ and $p$ are often chosen to be powers of two: if they are not, a rounding at the moment of encoding of messages should be applied. We will call $q$ ciphertext modulus, $p$ plaintext modulus, and $\Delta$ scaling factor. Let’s consider a message $M \in \mathcal{R}_p$. A GLWE ciphertext encrypting the message $M$ under the secret key $\vec{S}$ is a tuple:
$$(A_0, \ldots, A_{k-1}, B) \in GLWE_{\vec{S}, \sigma}(\Delta M) \subseteq \mathcal{R}_q^{k+1}$$
where the elements $A_i$ for $i\in [0..k-1 ]$ are sampled uniformly random from $\mathcal{R}_q$, and $B = \sum_{i=0}^{k-1} A_i \cdot S_i + \Delta M + E \in \mathcal{R}_q$, and $E \in \mathcal{R}_q$ has coefficients sampled from a Gaussian distribution $\chi_{\sigma}$.
We often call $(A_0, \ldots, A_{k-1})$ the mask and $B$ the body. The polynomial $\Delta M$ is what we sometimes call an encoding of $M$. Observe that, to compute $\Delta M$, we lift the message $M$ as an element of $\mathcal{R}_q$. Also, every time we encrypt a message, we sample new randomness (mask and noise error), so every encryption (even of the same message) is different from the other. This is essential for security. The set of GLWE encryptions of the same encoding $\Delta M$, under the secret key $\vec{S}$, with Gaussian noise with standard deviation $\sigma$, will be noted $GLWE_{\vec{S}, \sigma}(\Delta M)$.
Now, if we have a ciphertext $(A_0, \ldots, A_{k-1}, B) \in GLWE_{\vec{S}, \sigma}(\Delta M) \subseteq \mathcal{R}_q^{k+1}$ encrypted under the secret key $\vec{S} = (S_0, \ldots, S_{k-1}) \in \mathcal{R}^k$, then we can decrypt it by computing:
1. $B - \sum_{i=0}^{k-1} A_i \cdot S_i = \Delta M + E \in \mathcal{R}_q$,
2. $M = \lfloor (\Delta M + E)/\Delta \rceil$.
Observe that the message $M$ is in the MSB part of $\Delta M + E$ (thanks to the multiplication by $\Delta$) while $E$ is in the LSB part. If $|E|<\Delta/2$ (so if every coefficient $e_i$ of $E$ is $|e_i|<\Delta/2$), then the second step of the decryption returns $M$ as expected. If the error does not respect the condition, the decryption is incorrect. In the following image we represent the i-th coefficients of $\Delta M + E$.
#### Toy example
To better understand GLWE ciphertexts, let’s make a toy example where we use parameters that are totally insecure, just to fix ideas.
Let’s choose $q=64$, $p=4$ so $\Delta = q/p = 16$. Let’s choose $N=4$ and $k =2$.
We sample the secret key with uniform binary distribution as $k$ polynomials of degree smaller than $N$:
$$\vec{S} = (S_0, S_1) = (0 + 1\cdot X+1\cdot X^2 + 0\cdot X^3, 1 + 0\cdot X+1\cdot X^2 + 1\cdot X^3) \in \mathcal{R}^2.$$
Let’s encrypt a message $M \in \mathcal{R}_p$, which is a polynomial of degree smaller than $N$ with coefficients in $\{ -2, -1, 0, 1 \}$, say:
$$M = - 2 + 1\cdot X+ 0\cdot X^2 - 1\cdot X^3.$$
In order to encrypt the message, we need to sample a uniformly random mask with coefficients in $\{ -32, -31, \ldots,$ $-1, 0, 1, 2, \ldots, 30, 31 \}$:
$$\vec{A} = (A_0, A_1) = (17 -2\cdot X-24\cdot X^2 + 9\cdot X^3, -14 + 0\cdot X-1\cdot X^2 + 21\cdot X^3) \in \mathcal{R}_q^2$$
and a discrete Gaussian Error (small coefficients):
$$E = -1 + 1\cdot X+0\cdot X^2 + 1\cdot X^3 \in \mathcal{R}_q.$$
To encrypt, we need to compute the body as :
$$B = A_0 \cdot S_0 + A_1 \cdot S_1 + \Delta M + E \in \mathcal{R}_q.$$
When we compute in $\mathcal{R}_q$, we do polynomial operations modulo $X^N+1$ and modulo $q$. To reduce modulo $X^N+1$, you can observe that $X^N = X^4 \equiv -1 \mod X^4 +1$. So:
\begin{aligned} A_0 \cdot S_0 &= (17 -2 X -24 X^2 + 9 X^3)\cdot (X+X^2) \\ &= 17 X + (17 - 2 ) X^2 + (-2 -24) X^3 + (-24 +9) X^4 + 9 X^5 \\ &= 17 X + 15 X^2 -26 X^3 + 15 - 9 X \\ &= 15 + 8 X + 15 X^2 -26 X^3 \in \mathcal{R}_q. \end{aligned}
In the same way:
$$A_1 \cdot S_1 = (-14 -X^2 + 21 X^3) \cdot (1 + X^2 + X^3) = -13 - 20 X +28X^2 +7 X^3 \in \mathcal{R}_q.$$
Observe that
$$\Delta M = -32 + 16\cdot X+ 0\cdot X^2 - 16\cdot X^3 \in \mathcal{R}_q.$$
Then:
$$B = A_0 \cdot S_0 + A_1 \cdot S_1 + \Delta M + E = -31 +5 X -21 X^2 +30 X^3 \in \mathcal{R}_q.$$
So the encryption is:
$$(A_0, A_1, B) = (17 -2 X-24 X^2 + 9 X^3, -14 -X^2 + 21 X^3, -31 +5 X -21 X^2 +30 X^3) \in \mathcal{R}^3_q.$$
When we decrypt, by computing $B - \sum_{i=0}^{k-1} A_i \cdot S_i \in \mathcal{R}_q$ we find
$$31 +17 X -15 X^3.$$
Then:
$$\lfloor (31 +17 X -15 X^3)/16 \rceil = -2 + X -X^3 \in \mathcal{R}_p$$
which is the message we encrypted. Decryption worked fine because the error coefficients were all smaller (in absolute value) than $\Delta/2 = 8$.
#### Trivial GLWE ciphertexts
In the next blog posts we will sometimes use what we call trivial GLWE ciphertexts. Those ciphertexts are not true encryptions, in the sense that they hide information, but must be seen more as placeholders: they have in fact the shape of a GLWE ciphertext but the message is in clear. A trivial ciphertext of a message $M$ has all the $A_i$ set to $0$ and the $B$ equal to $\Delta M$:
$$(0, \ldots, 0, \Delta M) \in \mathcal{R}_q^{k+1}.$$
No worries, we never use these ciphertexts to encrypt sensitive information of course! In the next blog posts we will show how to use them to inject publicly known data in homomorphic computations.
#### LWE and RLWE
Now you might wonder how we can obtain LWE and RLWE from GLWE ciphertexts.
When we instantiate GLWE with $k = n \in \mathbb{Z}$ and $N = 1$ we get LWE. Observe that $\mathcal{R}_q$ (resp. $\mathcal{R}$) is actually $\mathbb{Z}_q$ (resp. $\mathbb{Z}$) when $N = 1$.
When we instantiate GLWE with $k = 1$ and $N$ a power of 2 we get RLWE.
#### Public key encryption
In the previous section we showed you how to do secret key encryption. It is also possible to encrypt by using a public key. In practice, a public key would be a list of encryptions of zero (i.e. $M=0$). To encrypt a message, it is sufficient to take a random combination of these encryptions of zero and add the desired message $\Delta M$. Since we will not use public key encryption in this blog post series, we will not go into more details. But if you are curious about the subject, check this paper.
## GLev
GLev ciphertexts have been used in FHE for a long time: in one of the following blog posts you will see them being largely used in some crucial FHE leveled operations. The name GLev was used for the first time in CLOT21 to be able to identify an intermediate type of ciphertext between GLWE and GGSW ciphertexts and make at the same time GGSW ciphertexts easier to understand. GLev can be seen as a generalization of the well known Powers of 2 encryptions used in BGV.
A GLev ciphertext contains redundancy: it is composed by a list of GLWE ciphertexts encrypting the same message $M$ with different, and very precise, scaling factors $\Delta$. Two parameters are necessary to define these special $\Delta$’s: a base $\beta$, generally a power of 2, and a number of levels $\ell \in \mathbb{Z}$:
$$\left( GLWE_{\vec{S}, \sigma}\left(\frac{q}{\beta^1} M\right) \times \ldots \times GLWE_{\vec{S}, \sigma}\left(\frac{q}{\beta^\ell} M\right) \right) = GLev^{\beta, \ell}_{\vec{S}, \sigma}(M) \subseteq \mathcal{R}_q^{\ell \cdot (k+1)}.$$
If $\beta$ and $q$ are not powers of 2, a rounding should be applied at the moment of encoding. The secret key is the same as for GLWE ciphertexts. To decrypt, it is sufficient to decrypt one of the GLWE ciphertexts with the corresponding scaling factor. The set of GLev encryptions of the same message $M$, under the secret key $\vec{S}$, with Gaussian noise with standard deviation $\sigma$, with base $\beta$ and level $\ell$, will be noted $GLev^{\beta,\ell}_{\vec{S}, \sigma}(M)$.
#### Lev and RLev
In the same way we saw that GLWE was a generalization for both LWE and RLWE, we can observe that GLev can be specialized into Lev and RLev, by following the same rules.
## GGSW
Now that we know what GLWE and GLev ciphertext are, GGSW will be very easy to understand. Let’s put it this way.
• A GLWE ciphertext is a vector of elements from $\mathcal{R}_q$ (or a 1 dimensional matrix),
• A GLev ciphertext is a vector of GLWE ciphertexts (or a 2 dimensional matrix of elements from $\mathcal{R}_q$),
• A GGSW ciphertext is a vector of GLev ciphertexts (or a 3 dimensional matrix of elements from $\mathcal{R}_q$, or a 2 dimensional matrix of GLWE ciphertexts).
With GGSW ciphertexts we once again add some redundancy thanks to a 3rd dimension in the structure.
In particular, in a GGSW, each GLev ciphertext encrypts the product between $M$ and one of the polynomials of the secret key $-S_i$. The last GLev in the list just encrypts the message $M$:
$$\left( GLev^{\beta, \ell}_{\vec{S}, \sigma}(-S_0 M) \times \ldots \times GLev^{\beta, \ell}_{\vec{S}, \sigma}(-S_{k-1} M) \times GLev^{\beta, \ell}_{\vec{S}, \sigma}(M) \right) = GGSW^{\beta, \ell}_{\vec{S}, \sigma}(M) \subseteq \mathcal{R}_q^{(k+1) \times \ell (k+1)}.$$
The secret key is the same as for GLWE and GLev ciphertexts. To decrypt, it is sufficient to decrypt the last GLev ciphertext. The set of GGSW encryptions of the same message $M$, under the secret key $\vec{S}$, with Gaussian noise with standard deviation $\sigma$, with base $\beta$ and level $\ell$, will be noted $GGSW^{\beta,\ell}_{\vec{S}, \sigma}(M)$.
#### GSW and RGSW
In the same way we already saw for both GLWE and GLev, we can observe that GGSW can be specialized into GSW and RGSW, by following the same rules presented before.
Curious to know how we use these ciphertexts to build FHE operations? Read part II to get one step forward in the comprehension of the TFHE scheme by showing you how to use different encodings and how to use some basic homomorphic operations.
And a special thank you to
Damien Ligier for the valuable feedback and editing of this blog post.
At zama, we are building products to make fully homomorphic encryption accessible for all.
Related articles
### TFHE Deep Dive - Part II - Encodings and linear leveled operations
This second blog post of the series shows you how to perform operations on the ciphertexts used in TFHE.
### TFHE Deep Dive - Part III - Key switching and leveled multiplications
This third blog post of the TFHE series describes some more leveled homomorphic operations and building blocks.
### TFHE Deep Dive - Part IV - Programmable Bootstrapping
This fourth and last part of the blog post of the TFHE series is dedicated to bootstrapping.
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2023-02-04 21:41:34
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http://www.numdam.org/item/CM_1962-1964__15__34_0/
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On a function, which is a special case of Meijer’s $G$-function
Compositio Mathematica, Volume 15 (1962-1964), p. 34-63
@article{CM_1962-1964__15__34_0,
author = {Boersma, J.},
title = {On a function, which is a special case of Meijer's $G$-function},
journal = {Compositio Mathematica},
publisher = {Kraus Reprint},
volume = {15},
year = {1962-1964},
pages = {34-63},
zbl = {0100.06505},
mrnumber = {132847},
language = {en},
url = {http://www.numdam.org/item/CM_1962-1964__15__34_0}
}
Boersma, J. On a function, which is a special case of Meijer’s $G$-function. Compositio Mathematica, Volume 15 (1962-1964) pp. 34-63. http://www.numdam.org/item/CM_1962-1964__15__34_0/
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2019-08-23 14:29:16
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https://infocom.spbstu.ru/en/article/2011.22.11/
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# Mathematical modelling and evaluation of strength of linear elastic body in the vicinity of a corner indent
Authors:
Abstract:
An asymptotic solution of a theory of elasticity problem in the vicinity of a corner indent has been constructed. An algorithm for calculating stress intensity factors on the basis of the reciprocity theorem and the finite element method was developed. A force strength criterion for the case of mechanical and thermal impact is proposed.
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2021-12-01 06:35:53
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https://math.stackexchange.com/questions/1670794/doubts-on-inverse-power-method
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# Doubts on inverse power method
I found written that if matrix A is real and you use the Power method to find eigenvalues then "If the matrix and starting vector are real then the power method can never give a result with an imaginary part." reference.
Is it also true for the inverse power method used to find a better approximation of the eigenvalue given a initial approximation? I've written a simple MATLAB program and I think it's false but I need some clarification.
What about the initial approximation of complex eigenvalue? Should it be complex in order to converge?
Inverse iteration will also stay real along the way. Finding complex eigenvalues is tricky; either your method needs to make a block matrix like $\begin{bmatrix} a & -b \\ b & a \end{bmatrix}$ show up by itself, or else it needs to give complex eigenvalues "a shot", by looking for the eigenvalues of $A-\lambda I$ for some complex number $\lambda$ or by looking at complex starting vectors.
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2021-04-23 01:33:30
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https://en.m.wikibooks.org/wiki/Quantum_Chemistry/Example_9
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# Quantum Chemistry/Example 9
Write a question and its solution that calculates the locations of the nodes of an electron in a 2s orbital.
## The Question
The question is to find the location of the radial node in a 2s electron for a hydrogen atom. To find the node one can start by analyzing, generally, how many nodes one should expect to see in a 2s electron system. There are two equations that give the number of nodes present in an orbital, the angular node equation and radial node equations:
1. Radial Nodes:
${\displaystyle n-\ell -1=n_{r}}$
2. Angular Nodes:
${\displaystyle \ell =n_{a}}$
Therefore ℓ must be determined, and based on the table 1 data one can determine ℓ is equal to 0. And the n is equal to the ${\displaystyle n}$ which is 2, this comes from the number before the orbital type which tells you the principle quantum number.
Table 1: Orbitals and Quantum Number
Orbital Angular
Momentum
Quantum
Number (ℓ)
s 0
p 1
d 2
f 3
So how many nodes are there?
First analyze the number of angular nodes:
${\displaystyle \ell =n_{a}}$
${\displaystyle 0=n_{a}}$
Therefore, the number of angular nodes is 0.
Radial nodes:
${\displaystyle n_{r}=n-\ell -1}$
${\displaystyle n_{r}=2-0-1}$
${\displaystyle n_{r}=1}$
Therefore there is one radial node present in a 2s orbital, resulting in the question becoming where is the location of that radial node?
The Wavefunction
Now the next main step is to determine what wavefunction describes the wavefunction this scenario of the 2s electron. The wavefunction can be found online which is${\displaystyle :^{1}}$
${\displaystyle \psi _{2s}={\frac {1}{4(2\pi )^{\frac {1}{2}}}}\left({\frac {1}{a_{0}}}\right)^{3/2}\left(2-{\frac {r}{a_{0}}}\right){\text{e}}^{-r/(2a_{0})}}$
In the equation the ${\displaystyle \psi _{2s}}$ is the wavefunction for the 2s electron, 𝒓 is the radius, and ${\displaystyle a_{0}}$ is the Bohr radius.
Based on the equation we can then solve for the position of the electron. The best way to do this is to find where is the equation going to be equal to zero and what term that contains the position causes this. The first part ${\displaystyle {\frac {1}{4(2\pi )^{\frac {1}{2}}}}}$ is a constant thus won't change with the radius, the position, of the electron. So the only place that will change with the position of the electron are the 𝒓 terms. With ${\displaystyle {\text{e}}^{-r/(2a_{0})}}$ the 𝒓 term can be any number and the term won't be zero unless the 𝒓 is approaching infinity, while the ${\displaystyle \left(2-{\frac {r}{a_{0}}}\right)}$ can potentially be equal to zero since it has 2 subtracted by the position term. Therefore one can set this term equal to zero and solve for 𝒓.
${\displaystyle 0=2-{\frac {r}{a_{0}}}}$
${\displaystyle 2={\frac {r}{a_{0}}}}$
${\displaystyle 2a_{0}=r}$
Therefore, we get the solution to the position of the radial node which is 𝒓 ${\displaystyle =2a_{0}}$ , so when 𝒓 ${\displaystyle =2a_{0}}$ the probability of the electron being there is 0 all around the nucleus creating a node. The ${\displaystyle a_{0}}$ has a length of 52.9 pm which means the node is 105.8 pm in radius away from the nucleus.
In Conclusion
In conclusion the approach to the problem of finding nodes for an electron in an orbital boils down to first finding the number of theoretical nodes, then determining the wavefunction, analyzing the wavefunction's variables and solving for 0. After this is all done you will have the solution to the radial node locations. Further problems can be solved as well, because they are follow up questions that are made easier to solve after finding the nodes, like the position of the electron in it's most probable state. This problems follows the solution of arranging the P=${\displaystyle \psi _{2s}}$ ${\displaystyle \left(\psi _{2s}^{*}\right)}$ ${\displaystyle \left(r^{2}\right)}$ then finding the derivative of the wavefunction and then simplifying it. The final step is to find the zero points, where the 𝒓 is equal to zero which will give the most probable locations of the electron. The practical applications of finding the nodal locations can help with understanding how orbitals work which can help with making molecular orbital diagrams and SALCs that can be used to determine the way atoms and molecules bond. Other applications include understanding the energy levels of the bonds and orbitals to predict possible interactions between molecules and atoms, for research purposes and chemical engineering.
Reference
1) Branson, J. The Radial Wavefunction Solutions. https://quantummechanics.ucsd.edu/ph130a/130_notes/node233.html (accessed November 16, 2021).
By Dmitry Ivanov
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2023-03-25 20:07:12
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https://math.stackexchange.com/questions/2982593/given-x-y-x-z-y-z-recover-the-tree-write-it-in-usual-nota
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# Given $x\ y\ +\ x\ z\ +\ *\ y\ z\ *\ +$ recover the tree, write it in usual notation and simplify
Given the boolean expression given in reverse Polish notation $$x\ y\ +\ x\ z\ +\ *\ y\ z\ *\ +$$ recover the tree, write it in usual notation and simplify.
The usual notation is
$$\begin{array}{ll} &x\ y\ +\ x\ z\ +\ *\ y\ z\ *\ +\\ \iff&(x+y)\ x\ z\ +\ *\ y\ z\ *\ +\\ \iff&((x+y)+x)\ z\ *\ y\ z\ *\ +\\ \iff&(((x+y)+x)*z)\ y\ z\ *\ +\\ \iff&(((x+y)+x)*z)\ (y*z)\ +\\ \iff&(((x+y)+x)*z)+(y*z)\\ \end{array}$$
The recovery tree is
Finally, the simplification is
$$\begin{array}{ll} &(((x+y)+x)*z)+(y*z)\\ \iff&(2*x+y)*z+y*z\\ \iff&2*x*z+y*z+y*z\\ \iff&2*z*(x+y) \end{array}$$
Is that correct? Is it possible to write $$2*x\equiv2x$$ and so on?
Thanks!
• Your third line is incorrect: $x~z~+$ translates to $(x+z)$ before it is multiplied by $(x+y)$. – Fabio Somenzi Nov 3 '18 at 6:11
• @FabioSomenzi oh, thanks! So it would be $(x+y)*(x+y)+(y*z)$? – manooooh Nov 3 '18 at 6:18
• The second $x+y$ is actually $x+z$, and then you can simplify a bit. – Fabio Somenzi Nov 3 '18 at 6:24
• It's a Boolean expression, isn't it? So, $+$ is OR and $*$ is AND. – Fabio Somenzi Nov 3 '18 at 6:35
• Right. You should post the answer, because it's your solution. I just gave a little nudge. – Fabio Somenzi Nov 3 '18 at 17:35
No, it is not correct. As @FabioSomenzi said in comments, the expression must be $$(x+y)*(x+z)+(y*z)$$, which has as a tree
and after applying some properties ends up with $$x\vee(y\wedge z)$$.
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2019-12-08 21:25:40
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https://stats.stackexchange.com/questions/218548/performing-a-t-test-with-discrete-currency-data
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# Performing a t-test with discrete (currency) data
I want to perform a 2 sample t-test assuming unequal variances, however my variable is currency. Currency is discrete, however when checking the assumptions of the t-test, I see that the data should be continuous.
Technically, they aren't continuous, but I guess it's closer to a ratio scale (maybe interval). Is this assumption violated? What else should I check?
• How much data do you have? Jun 12 '16 at 15:52
• n=24 for both variables. Jun 12 '16 at 15:53
• What do they look like (within the groups)? Are they vaguely normal looking? Jun 12 '16 at 15:54
• I ran a JB test, they both appear normal. Jun 12 '16 at 15:55
• What do you precisely mean by your "variable is currency." As in, your variable is measured in dollars and there cannot be fractions of a cent (discrete units are $1/100$ of a dollar)? And you're trying to compare the mean of two samples? Jun 12 '16 at 17:32
Discrete isn't continuous, so technically the assumption of the t-test is not met, and that's that. However, the t-test is fairly robust and having $N=48$ with equal groups is a decent sample, so it might be OK. After all, in practice all data are discrete at some level because we don't record data to infinite decimal places.
• Worth pointing out too that with more data, he could lean on asymptotic arguments. Under certain regularity conditions, the sample mean $\frac{1}{n}\sum_i x_i$ would converge to a normally distributed random variable even if $x$ were discrete. Jun 12 '16 at 17:44
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2022-01-26 08:50:54
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http://grephysics.net/ans/9277/64
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GR 8677927796770177 | # Login | Register
GR9277 #64
Problem
GREPhysics.NET Official Solution Alternate Solutions
\prob{64}
If an electric field is given in a certain region by $E_x=0,E_y=0,E_z=kz$, where k is a nonzero constant, which of the following is true?
1. There is a time-varying magnetic field.
2. There is charge density in the region.
3. The electric field cannot be constant in time.
4. The electric field is impossible under any circumstances.
5. None of the above.
Electromagnetism$\Rightarrow$}Gauss Law
Gauss Law gives $\nabla \cdot \vec{E} = \rho/\epsilon_0$. Since the divergence of E in Cartesian coordinates is non-zero, there is a charge density in the region. QED
Alternate Solutions
There are no Alternate Solutions for this problem. Be the first to post one!
NoPhysicist3
2017-03-23 12:31:21
The words \"certain region\" are VERY confusing. However, when choosing between B and E, one should keep in mind that it is unlikely for ETS to consider a correct answer containing ultimate statements. Therefore B is the only correct.
Naismith
2011-10-10 04:30:08
What do they mean by "in a certain region" ? In my opinion, it is always possible to find a region small enough so that it doesn't contain any charges, therefore charge density. The charge then will be outside the region...
h.fei102012-11-04 07:42:12 That's not possible. The electric field pevades this certain region, so does the charge.
calcuttj2014-09-03 17:57:04 Think about the field inside a cylinder of constant charge density. The cylinder has radius R, constrain r < R such that |E|*$\pi$r^2d = $\frac{4}{3}$$\pi$r^3d$\frac{\rho}{\epsilon_0}$ (d is the length of our Gaussian cylinder) |E| = $\frac{4}{3}$r$\frac{\rho}{\epsilon_0}$ The region could be the z axis inside the cylinder Not necessarily the only charge distribution to create E =kz, and this definitely doesn't prove there is ALWAYS a distribution to create a field like this, but it definitely disproves that the field is impossible Now think about this. If there wasn't a charge density in the region. shouldn't the field be decreasing (i.e. E=k/z)?
calcuttj2014-09-10 16:36:07 I made a mistake in my last comment, ignore it.
r10101
2007-10-27 16:32:37
Why does a small region of vacuum near the surface of an infinite charged plate (with constant $\vec{E}$ = E$\hat{n}$ normal to the surface) not satisfy this question, making answer (E) correct?
panos852007-10-31 05:12:32 It says $E_z=kz$, not $E_z=k\hat{z}$. The electric field near the surface of a conductor is constant, while the field in this problem is not.
tonyhong2008-10-25 01:54:24 this is a trap...
sharpstones
2006-12-01 10:25:11
how could you possibly construct a charge density that would make such an E field?
mhas0352007-04-04 23:58:56 Remember that it says that the field is only in a certain region. We just need the region to be small with a relatively large charged plane of constant charge density.
evanb2008-06-24 11:56:52 How about a uniform-density infinite-plane slab. So, it would be thick and the region of interest would be from the middle of the slab to the edge of the slab.
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## Bare Basic LaTeX Rosetta Stone
LaTeX syntax supported through dollar sign wrappers $, ex.,$\alpha^2_0$produces $\alpha^2_0$. type this... to get...$\int_0^\infty$$\int_0^\infty$$\partial$$\partial$$\Rightarrow$$\Rightarrow$$\ddot{x},\dot{x}$$\ddot{x},\dot{x}$$\sqrt{z}$$\sqrt{z}$$\langle my \rangle$$\langle my \rangle$$\left( abacadabra \right)_{me}$$\left( abacadabra \right)_{me}$$\vec{E}$$\vec{E}$$\frac{a}{b}\$ $\frac{a}{b}$
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2019-02-20 21:20:51
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https://sara-github.readthedocs.io/en/latest/features/normalizing_transform.html
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# Normalizing Transform of a Feature¶
Let us remark the following proposition which relates the normalizing transform $$\bT_x$$ to the feature shape $$\bSigma_x$$.
Important
Let $$L$$ be an invertible linear transformation in $$\mathbb{R}^2$$ whose matrix is denoted by $$\bL$$. For any point $$\x$$ in the zero-centered unit circle in $$\mathbb{R}^2$$, its transformed point by $$L$$ is in the ellipse defined by
$\left\{ \z \in \mathbb{R}^{2} | \z^T (\bL^{T})^{-1} \bL^{-1} \z = 1 \right\}$
Note
This note provides a proof of the proposition above.
Fix a point $$\begin{bmatrix} \cos(t) \\ \sin(t) \end{bmatrix}$$ of the unit circle in $$\mathbb{R}^2$$. We write its transformed point by $$L$$ as
$\begin{bmatrix} u \\ v \end{bmatrix} = \bL \begin{bmatrix} \cos(t) \\ \sin(t) \end{bmatrix}.$
Since $$\bL$$ is invertible
$\bL^{-1} \begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} \cos(t) \\ \sin(t) \end{bmatrix}$
The squared Euclidean norm of the equality yields
$\begin{bmatrix} u & v \end{bmatrix} (\bL^{-1})^T \bL^{-1} \begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} \cos(t) & \sin(t) \end{bmatrix} \begin{bmatrix} \cos(t) \\ \sin(t) \end{bmatrix} = 1$
We recognize the equation of an ellipse, which concludes the proof of proposition.
## Geometric interpretation of the QR factorization¶
Consider the shape matrix $$\bSigma_x$$. Recall that $$\bSigma_x$$ defines the elliptic shape $$\Shape_x$$. We want to retrieve the transformation $$L_x$$ that satisfies
(11)$\bSigma_x = (\bL_x^{-1})^T \bL_x^{-1}.$
Observe from the QR factorization $$\bL_x = \bQ \bR$$ that $$L_x$$ can be decomposed uniquely in two specific transformations $$\bQ$$ and $$\bR$$. The upper triangular matrix $$\bR$$ encodes a transformation that combines of shear and scaling transforms. The orthonormal matrix $$\bQ$$ encode a rotation. This geometric interpretation is illustrated in Figure [Geometric interpretation of the QR factorization of linear transform matrix \bL_x.].
Fig. 4 Geometric interpretation of the QR factorization of linear transform matrix $$\bL_x$$.
Unless $$L_x$$ involves no rotation, $$\bL_x$$ is an upper triangular matrix. Then, because Equation (11) is a Cholesky decomposition, $$\bL_x$$ can be identified by unicity of the Cholesky decomposition.
In general, $$\bL_x$$ is not upper triangular. Orientations $$\bo_x$$ of elliptic shape $$\bSigma_x$$ are provided from feature detectors. In SIFT, $$\bo_x$$ corresponds to a dominant local gradient orientation.
Thus, introducing $$\theta_x \eqdef \angle \left( \begin{bmatrix}1\\0\end{bmatrix}, \bo_x \right)$$, we have
$\bQ = \begin{bmatrix} \cos(\theta_x) & -\sin(\theta_x) \\ \sin(\theta_x) & \cos(\theta_x) \end{bmatrix}$
and expanding Equation (11) yields
\begin{aligned} \bSigma_x &= (\bL_x^{-1})^T \bL_x^{-1} \\ &= \bQ (\bR^{-1})^T \bR^{-1} \bQ^{T} \quad \text{since}\ \bQ^T = \bQ^{-1}\\ \bQ^T \bSigma_x \bQ &= (\bR^{-1})^T \bR^{-1} \end{aligned}
We recognize the Cholesky decomposition of matrix $$\bQ^T \bSigma_x \bQ$$ which is the rotated ellipse as shown in Figure [Geometric interpretation of the QR factorization of linear transform matrix \bL_x.], in which case $$\bL_x$$ can be determined completely.
Finally, the affinity that maps the zero-centered unit circle to ellipse $$\Shape_x$$ is of the form, in homogeneous coordinates
$\displaystyle \bT_x = \begin{bmatrix} \bL_x & \x \\ \mathbf{0}_2^T & 1 \end{bmatrix}.$
## Calculation of the Normalizing Transform¶
The algorithm below summarizes how to compute $$\bT_x$$.
Important
• Calculate the angle
$\theta_x := \mathrm{atan2}\left( \left\langle \bo_x, \begin{bmatrix}0\\1\end{bmatrix}\right\rangle, \left\langle \bo_x, \begin{bmatrix}1\\0\end{bmatrix}\right\rangle \right)$
• Form the rotation matrix
$\bQ := \begin{bmatrix} \cos(\theta_x) & -\sin(\theta_x) \\ \sin(\theta_x) & \cos(\theta_x) \end{bmatrix}$
• Decompose the ellipse matrix $$\bM := \mathrm{Cholesky}(\bQ^T \bSigma_x \bQ)$$
• $$\bM$$ is a lower triangular matrix such that
• $$\bM \bM^T = \bQ^T \bSigma_x \bQ$$
• $$\bR := (\bM^T)^{-1}$$
• $$\bL := \bQ \bR$$
• $$\bT_x := \begin{bmatrix} \bL & \x \\ \mathbf{0}_2^T & 1 \end{bmatrix}$$
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2022-11-28 02:15:27
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https://stats.stackexchange.com/questions/495341/if-a-random-variable-y-converges-in-distribution-can-we-use-the-parameters-of
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# If a random variable $Y$ converges in distribution, can we use the parameters of the asymptotic distribution as if they are the parameters of $Y$?
Let $$Y_n$$ be a sequence of random variable such that $$\sqrt{n}(Y_n-\mu) \stackrel{d}{\to} \mathcal{N}(0, \sigma^2),$$ and thus we can say $$Y_n$$ is asymptotically normally distributed as $$Y_n \stackrel{a}{\sim} \mathcal{N}\bigg(\mu, \frac{\sigma^2}{n}\bigg).$$ Now suppose want to approximate $$E[f(Y_n)]$$. By a Taylor series expansion we have $$E[f(Y_n)] \approx f(E[Y_n]) + \frac{f''(E[Y_n])}{2}\text{Var}(Y_n).$$
It seems that this means that as $$n\to \infty$$ we can make use of the asymptotic distribution of $$Y_n$$, i.e., we are allowed to say: $$E[f(Y_n)] \approx f(\mu) + \frac{f''(\mu)}{2}\frac{\sigma^2}{n}, \quad \quad \text{as} \ n \to \infty.$$
Does this expression hold? Do we need some assumptions before we can say it? One of the reasons I am not certain is that I read here that convergence in distribution just means the CDFs of the random variable is becoming closer to the limit CDF, but the actual values of the random variable may not be becoming closer together to the values of the limiting random variable. We need convergence in probability for the values to become closer.
• The symbol $\approx$ being vague, what do you mean by it [in a mathematical sense] ? – Xi'an Nov 6 '20 at 14:26
• Even when a sequence of random variables converges in distribution, the corresponding sequence of expectations needn't converge at all. A standard example is the sequence $X_n=nY_n,$ $n=1,2,3,\ldots,$ where $Y_n$ has a Bernoulli$(p(n))$ distribution and $p(n)$ is chosen to converge to $0$ (so that $X_n$ converges to $0$ in distribution) in such a way that $E[X_n]=np(n)$ does not converge; e.g., $p(n)=1/\sqrt{n}.$ – whuber Nov 6 '20 at 14:35
• @whuber In general they need not converge, but my situation is more specific. I have a sequence of random variables that are asymptotically normal. Maybe in my case the convergence in distribution can pass over to the convergence in expectation? – Bertus101 Nov 6 '20 at 15:09
• It doesn't work that way: for instance, we could start with a sequence of random variables like yours whose expectations do converge and add my sequence to them. The new sequence is still asymptotically Normal but its expectation diverges. – whuber Nov 6 '20 at 15:47
• That would work provided you use the absolute value of $f$, as required by that theorem. The Taylor series approach does not necessarily work: you need to adduce additional conditions to justify omitting the remainder term in the Taylor expansion and you need to assume that sufficiently high moments of $Y_n$ are bounded. – whuber Nov 17 '20 at 17:10
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2021-01-18 03:44:52
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https://homework.cpm.org/category/CCI_CT/textbook/pc3/chapter/7/lesson/7.1.2/problem/7-21
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### Home > PC3 > Chapter 7 > Lesson 7.1.2 > Problem7-21
7-21.
Sketch the graph of $f(x)=x^2+3$. What is $\lim\limits_{x\rightarrow2}f(x)$? Homework Help ✎
$\lim\limits_{x\rightarrow2}f(x)=f(2)$
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2019-11-17 13:59:13
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https://math.stackexchange.com/questions/1021389/prove-that-langle-v-w-rangle-a-bf-v-cdot-a-bf-w-defines-an-inner
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# Prove that $\langle v,w \rangle = (A{\bf v}) \cdot (A{\bf w})$ defines an inner product on $\mathbb{R}^m$ iff $\ker(A)=\{{\bf 0}\}$
My instructor showed the proof of this result by proving the three axioms of inner product on the given proposed inner product, then using positive definiteness to show the $\ker(A)=\{{\bf 0}\}$. I was wondering if someone could offer an alternative proof that proves both directions of the "if and only if" statement. That is, first assume that $\langle v,w \rangle = (A{\bf v}) \cdot (A{\bf w})$ defines an inner product, why does this mean the $\ker(A) = \{{\bf 0}\}$? Then assume $\ker(A) = \{{\bf 0}\}$, why does this mean $\langle v,w \rangle = (A{\bf v}) \cdot (A{\bf w})$ defines an inner product.
1. \begin{align}\langle\alpha v1+\beta v_2,w\rangle&=(A(\alpha v1+\beta v_2))\cdot(Aw)\\&=\alpha(Av_1)\cdot(Aw)+\beta(Av_2)\cdot(Aw)\\&=\alpha\langle v_1,w\rangle+\beta\langle v_2,w\rangle\end{align}
2. $$\langle v,w\rangle=(Av)\cdot(Aw)=(Aw)\cdot(Av)=\langle w,v\rangle$$
1. If $$0=\langle v,v\rangle=(Av)\cdot(Av)$$
we have $Av=0$. Since the kernel is zero we get $v=0$
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2019-12-14 03:11:01
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http://www.gradesaver.com/textbooks/math/calculus/calculus-early-transcendentals-2nd-edition/chapter-8-sequences-and-infinite-series-8-1-an-overview-8-1-exercises-page-605/43
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## Calculus: Early Transcendentals (2nd Edition)
$a_n = n^2-n$ $a_1 = 0$ $a_2 = 2$ $a_3 = 6$ $a_4 = 12$ $a_5 = 20$ $a_6 = 30$ $a_7 = 42$ $a_8 = 56$ $a_9 = 72$ $a_{10} = 90$ The terms seem to be increasing without a bound, meaning the sequence diverges.
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2018-04-21 00:52:15
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https://math.stackexchange.com/questions/1523461/what-is-the-intuition-behind-differential-forms
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# What is the intuition behind differential forms?
I am comfortable with the way physicists use differentials as elements of area/volume. I know the (algebraic) formal definition of differential forms, but it makes no intuitive sense, especially since it is not immediately compatible (to me) with the physicist POV. How do the two fit in?
• I tried (with only little success IMHO) to give physical intuition (in the form of mechanical work) for 1-forms in this answer a while back. – user137731 Nov 11 '15 at 2:19
• Check the book by Edwards called "Advanced Calculus: a Differential Form Approach" and also that by Hubber called "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach", where you may find what you want. :) Good luck. – Megadeth Nov 11 '15 at 2:59
• I think they can be motivated by trying to extend the idea of line and surface integrals to higher dimensional manifolds. "Differential forms exist to be integrated." Chop up a $k$ - manifold into tiny pieces, each of which is spanned by $k$ tiny vectors. Plug those $k$ vectors into a differential form to get the contribution of that piece. Add up all the contributions to get the total integral over the manifold. If you think about chopping more finely, we see the thing we're integrating should have certain linearity properties. – littleO Nov 11 '15 at 3:09
• The point is that under changes of coordinates, integrals change by the determinant of the Jacobian. Differential n-forms on an n-manifold are defined precisely so that this is how they change under change of coordinates. (Of course this doesn't explain where the other forms come from, but out this idea together with, say, a justification for the definition of n-forms.) – user98602 Nov 11 '15 at 3:14
Let's start with a euclidean space for a moment, but impose upon this a general curvilinear coordinate system.
The tangent vectors to the coordinate lines through a given point define the usual basis vectors, which are called various names. They constitute vector fields, at any rate, and for the purposes of this answer, I'll call them only the tangent basis vectors (with fields being implied).
These tangent basis vectors are not necessarily orthogonal, and as a result, one can form hyperplanes from $n-1$ of them and find the normal vectors to those hyperplanes. You can choose a particular normalization of these vectors so that a particular tangent vector and its normal counterpart (defined by the normal to the hyperplane formed by all other tangent basis vectors) have unit inner product. These particular normal vectors are called, variously, the dual basis vectors, or the cotangent basis vectors.
All of the above applies in a setting with a metric--in particular, the normal vector requires a metric to be defined as orthogonal to all those vectors in the hypersurfaces.
The leap forward is to consider the case when you don't have a metric; you can still define linear functionals, or forms, such that the form applied to a particular tangent basis vector yields 1, and these forms still span their own vector space, the dual vector space.
Now I'll stop right there, actually, and not go back to the idea of what happens when you don't have a metric again--because in physics, 99% of the time you do still have a metric, and forms are not necessary. You can get by just fine by using those cotangent basis vectors and their linear combinations and wedge products. They obey the same algebra as forms without being as abstract.
What you should understand is that differential forms made the work of doing calculus on manifolds a lot easier, but it's written for the general setting of a manifold that might not have a metric. This results in notation that is, for many physics applications, overly handcuffed. Often, metrical operations get abstracted out by using Hodge duality, for instance. Why? Because asserting a volume form is a weaker condition than asserting a metric.
Differential forms' typical notations can lead to confusion over what's geometrical and what's not. @littleO said (paraphrasing, with slight tweaks) that a differential form integral can be thought of as chopping up a manifold into $k$-vector pieces and then plugging that $k$-vector into into the $k$-form that is being integrated. Let's take that to its logical end: that means, when you integrate a volume form $dV$, the geometry of the manifold isn't coming from $dV$! It's coming from that $k$-vector that is being plugged in.
Differential forms are commonly used for integration in these settings because the metric does not appear in their integrals. That, and the coordinate free manipulations they enable? That's what makes them useful to a physicist. But it also leads to misunderstandings, like thinking that you can no longer integrate vector fields (you absolutely can).
At any rate, for most applications in physics, you can just think of differential forms as vectors (or higher dimensional things, like planes and volumes and so on), just described using that cotangent basis, instead of your regular old tangent basis.
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2019-08-18 13:16:01
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http://www.mathblogging.org/posts/?type=post
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X
# Posts
### May 23, 2015
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I had the chance to attend the Canon Media Awards Night, as a guest of the Science Media Centre (who are one of the sponsors). It was a good year for data journalism. Harkanwal Singh and his team won “Best use of interactive graphics” and “Best multimedia storytelling” for projects based on effective communication of publicly-available data. Perhaps more importantly […]
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Here are the slides of the talk I’m giving on Monday to kick off the Categorical Foundations of Network Theory workshop in Turin: • Network theory. This is a long talk, starting with the reasons I care about this subject, and working into details of one particular project: networks in electrical engineering and control theory. […]
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Even after the Snowden revelations, there remained one big mystery about what the NSA was doing and how. The NSA’s classified 2013 budget request mentioned, as a priority item, “groundbreaking cryptanalytic capabilities to defeat adversarial cryptography and exploit internet traffic.” There was a requested increase, of several hundred million dollars, for “cryptanalytic IT services” and “cryptanalysis and […]
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A correspondent writes (I added the links):Our district is looking at revamping our year map and I would like to suggest a map that has the qualities of Algebra: Themes, Tools, Concepts (ATTC): particularly its ‘integratedness’ and how well it spirals through the topics. I’ve read on your blog about separating topics and lagging homework. But what are some things one might consider when deciding what the next week’s topic would be (or what topics to put in what […]
### May 22, 2015
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(a symbol) We can repudiate completely and which we can abandon without regret because one does not know what this pretended sign signifies nor what sense one ought to attribute to it. Cauchy in 1847 in regard to the square root of negative one..The 143rd day of the year; there are 143 three-digit primes. Also, 1432 is a divisor of 143143. HT to Matt McIrvin who found the pattern for numbers such that n^2 divides n.n (where the dot represents concatenation) and then […]
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We are in the middle of moving and I just for now don’t have access to my compute that has Mathematica. I hadn’t really worried too much about it since we really only use it for 3D printing projects, but guess what . . . I was reading through this book in the library today:…
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This morning, on my way to the airport (and to Montpellier for a seminar), Rock, my favourite taxi-driver, told me of a strange ride he endured the night before, so strange that he had not yet fully got over it! As it happened, he had picked an elderly lady with two large bags in the […]
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via TrigonometryIsMyBitch
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PROPs were developed in topology, along with operads, to describe spaces with lots of operations on them. But now some of us are using them to think about 'signal-flow diagrams' in control theory---an important branch of engineering. I talked...
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Cubistic Singularity by Ivan Doncevic http://im-possible.info/english/art/pencil/ivan-doncevic.html#cubistic-singularity Author - http://webbugt.deviantart.com/
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June is just a couple of days away and the holiday season is not an excuse to forget about math. I know that summer is the time when we forget about school, but for me it is also the time to visit a little more and see how cities around the world embrace the beauty […]
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Unizor - Creative Minds through Art of Mathematics - Math4TeensProblemConstruct a common perpendicular h to two given skew lines a and b.AnalysisAssume a common perpendicular h to skew lines a and b is constructed. Let points of its intersection with these lines be A and B correspondingly. Then h must be an intersection of two planes: plane σ that is perpendicular to line a at point A and plane τ that is perpendicular to line b at point B:σ⊥a; A∈στ⊥b; B∈τh = σ∩τWe don't know […]
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$\begin{array}{c}345 - 158 = 345 - 100 - 50 - 8\\ = 245 - 50 - 8\\ = 245 - 45 - 5 - 8\\ = 200 - 5 - 8\\ = 195 - 8\\ = 187\end{array}$ $\begin{array}{c}345 - 158 = 245 - 58\\ = 195 - 8\\ = 187\end{array}$Why do both look correct on PC but only one on iPhone?
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$\newcommand{\PG}{\text{PG}}$ Denser than a Geometry Hello everyone. In my last post, I discussed an unavoidable minor theorem for large matroids of density greater than $\binom{n+1}{2}$, which as a consequence characterised exactly the minor-closed classes of matroids that grow like the the graphic … Continue reading →
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The Current Population Survey provided reliably comparable data on the number of uninsured Americans--until last year.
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The recent story about the retracted paper on political persuasion reminded me of the last time that a politically loaded survey was discredited because the researcher couldn’t come up with the data. I’m referring to John Lott, the “economist, political commentator, and gun rights advocate” (in the words of Wikipedia) who is perhaps more well […] The post John Lott as possible template for future career of “Bruno” Lacour appeared first on Statistical […]
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This semester had, hands-down, the best set of students I’ve ever had. Every semester I do a post-mortem of what worked and what didn’t and how consistent those things have been over the years. Though, mostly irrelevant to this article, but to give a frame of reference, this semester I taught a once-a-week low-level, introductory […]
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Wow! I was going to write a comment synthesizing the very generous thoughts and ideas in response to my last post, but the meta-comment got to be almost as long as the original post, so... here I am. Several categories of possible actions jump out to me:Representing the incorrect idea: By not putting this idea on the board, I'm subtly signaling that it's wrong, or in some way not worth recording. This occurred to me-- which is why I attempted to record the idea of a unit rate by writing 1 hour […]
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This lesson will extend the concept of selections and, in particular, will talk about the number of ways to select any number of objects from ‘n’ distinct objects. Let’s go back to color mixing! Suppose you have the following three colors with you: And you wish to make as many different colors as possible by […]
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I’ve been on both sides of this conversation. And this discussion can get heated. Do we tell students what we want them to know? Do we let them explore? Do we let the students develop their own understanding? Do we model proper techniques for the students? And this isn’t a particularly new debate. John Dewey […]
Editor's Pick
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Viral Math, Part Deux: Terrance F. Ross:Singapore’s mind-bending logical riddles are so last month. Enter: Vietnam, the latest country to be swept up in what could easily be known as “the viral-math epidemic of 2015.”
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I found this blackboard in the maths common room of Lafayette College, a beautiful old campus university near the small town of Easton (just north of Philadelphia in the US). While the board contains some nice mathematics, I was particularly taken by the psychedelic fractal border on top of the board. I believe this was […]
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KrazyDad is a website choc full of mathematics based puzzles Many of these were featured in my blog post about maths puzzles a few months ago.
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I discovered a picture of me in my student lab – one of the students optimized me for a class project using dominos(!) My second blog post ever was about Bob Bosch’s optimization art – see some of his domino art here. It’s worth revisiting opt art. Bob Bosch wrote about his domino optimization models […]
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Another teacher and I started a math-science journal at my school three years ago. We’ve built it up to the point where it is very student-run, and we teachers truly are advisers. Today we had our launch event for the journal, and it is the current issue is now “live.” I’m so proud of the kids […]
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When I wanted to show you a code snippet in the past, I displayed the code in the text of my blog post.... read more >>
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Welcome to my latest instance of Math for Kids! Today I had the pleasure to make an interactive mathematical presentation at my son’s school to the 7th / 8th grade Math Team, about 30 math-enthusiastic kids (twelve and thirteen years … Continue reading →
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Last Thursday, at the OAME 2015 math conference, I presented a double-session entitled Rethinking Math Class. I am going to try to recap it here, as best I can, with links to everything! So expect this to be a long post...First, we played Quadratic Headbanz, which I have blogged about here. Given that 64 people had signed up for my session I had to make a second set of headbanz. As my original set featured equations I decided to make the second set with graphs which you can get here. As it […]
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2015-05-23 10:13:42
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https://studydaddy.com/question/assignment-2-expected-value-and-consumer-choicesconsumers-choices-are-prey-to-su
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Assignment 2: Expected Value and Consumer ChoicesConsumers’ choices are prey to subtle discrepancies that arise in cognitive accounting. Learning how and when you are prey to these discrepancies is an important step in improving your decision making.As the readings for this module demonstrate, people value gains and losses differently under different scenarios. For example, contestants in a game show might choose a guaranteed $10 prize over a 50 percent chance of winning$20 despite the fact that the expected values are the same.Using the readings for this module, the Argosy University online library resources, and the Internet, address the following:What is mental accounting and how does it impact consumer decision making?How might a company take advantage of consumers’ mental accounting? Give examples.As a marketer, how might you frame certain decisions to benefit from the disparities that arise in one’s cognitive accounting?As a consumer, how would you avoid the pitfalls posed by the inequalities of one’s cognitive accounting?Write a 3–5-page paper in Word format. Apply APA standards to citation of sources. Use the following file naming convention: LastnameFirstInitial_M4_A2.doc.
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2018-12-19 02:24:26
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https://quant.stackexchange.com/questions/42423/how-is-payment-calculated-for-a-mortgage-when-already-missed-one-payment-d30
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# How is payment calculated for a mortgage when already missed one payment (D30)?
How is the interest and principal payment calculated when mortgage has already missed one payment? Are the new payments calculated off the new balance (non-amortized balance) or the original scheduled balance?
For example: 100k mortgage, 360 terms, 5% nominal rate, monthly payments, country: Mortgage
Period Payment Int Princ Remaining Balance
0 100,000.00
1 536.82 416.67 120.15 99,879.85
2 536.82 416.17 120.66 99,759.19
3 536.82 415.66 121.16 99,638.03
4 536.82 415.16 121.66 99,516.37
5 536.82 414.65 122.17 99,394.20
6 536.82 414.14 122.68 99,271.52
Now consider 6th scheduled payment (last in table), and consider that this was missed. Therefore, actual situation in 6th period
Period Payment Int Princ Balance
6 Missed Missed Missed 99,394.20 <- same as 5th period
How is payment for 7th period calculated? Which of the following is most appropriate?
( a ) scheduled payment for period 7 (= 536.82)
+ missed payment for period 6 (= 536.82)
+ late fee
( b ) payment calculated of Balance 99,394.20 over 353 terms (= 538.15)
+ missed payment for period 6 (= 536.82)
+ late fee
( c ) payment calculated of balance 99,394.20 over 353 terms (= 538.15)
+ missed payment for period 6 compounded ( =(1+5/1200)*536.82 )
+ late fee
• What country/jurisdiction does this relate to? – Magic is in the chain Oct 30 '18 at 18:08
• Sorry, I should have added this in description. Let me add it now. Its USA – toing Oct 31 '18 at 0:08
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2019-12-14 07:43:10
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https://baripedia.org/wiki/Consumer_Price_Index_(CPI)
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# Consumer Price Index (CPI)
### From Baripedia
In this chapter we focus on how to measure the cost of living, and its evolution. This will be useful to compare the purchasing power of different incomes at different points in time. For example, a salary of 200 can buy more than a salary of 2000 if the cost of living in the first case is more than 10 times lower than in the second case.
The consumer price index allows us to measure changes in the cost of living. It is a measure of the evolution of the general price level faced by the consumer, and therefore of inflation, i.e. the percentage change in the price level from one period to the next).
# Construction and CPI issues
## Definition and construction of the CPI
The CPI is the measure of the cost of the basket of goods and services purchased by the 'typical consumer'. It tells us the evolution of its cost of living. If the CPI increases, the typical consumer will have to spend more money to consume the same basket of goods and services, and therefore his cost of living will have increased.
Construction of the CPI and the inflation rate :
1. the basket of the typical consumer is defined and fixed by conducting consumption surveys to determine the weight given to each good in the total expenditure ;
2. price surveys are conducted at regular intervals;
3. the value of the basket is calculated at different points in time based on the prices collected;
4. a base year is chosen and the value of the index is calculated in each year by taking the ratio of the cost of the basket to the base year and multiplying it by 100;
5. the percentage inflation rate is given by the annual change in the CPI: ${\displaystyle Inflation={\frac {IPC_{t}-IPC_{(t-1)}}{IPC_{(t-1)}}}\times 100}$.
## Construction of the CPI: example
1. We define the basket of goods purchased by a typical consumer:
Dépense d’un consommateur représentatif par catégorie de bien (pour nous: hamburgers et hot-dogs).
2. We do price surveys every year:
3. The cost of the (fixed) basket is calculated each year :
4. A base year is chosen for the index and the CPI is calculated:
5. The CPI is used to calculate the annual inflation rate :
## Problems with the CPI
The CPI is not a perfect measure of the cost of living. There are several reasons for this:
• substitution bias: price changes will affect the composition of the typical consumer's basket. The index overestimates the increase in the cost of living by not taking into account the ability of consumers to substitute goods.
• introduction of new goods: this gives more choice to consumers who can substitute the consumption of certain goods with new goods, thus reducing their cost of living (downloading films from the internet is not part of the CPI, but the cinema ticket is). The increase in the cost of living is again overestimated.
• Improved quality of goods: for the same price, the consumer can buy goods that give him greater satisfaction or that perform better. The CPI overestimates the increase in the cost of living by ignoring quality.
• It is not a "true" cost-of-living index: it does not take into account health insurance premiums, taxes, social security contributions, etc. It is not a "real" cost-of-living index. It does not take into account health insurance premiums, taxes, social security contributions, etc. (only consumer goods and services are considered).
• Heterogeneity of consumption baskets: young versus old, poor versus rich, etc. The average consumption basket does not really exist → limits of the average basket if the composition of society changes or the prices of the goods consumed by each group do not evolve in the same way → comparisons between difficult people (and even more so between countries!).
## CPI versus GDP deflator
The GDP deflator was given by : GDP deflator = ${\displaystyle {\frac {PIB{\text{nominal}}}{PIB{\text{real}}}}\times 100}$
Differences from the CPI
1. The CPI focuses on the price evolution of goods consumed in the economy, while the GDP deflator focuses on the price evolution of goods produced in the domestic economy: the price of imported goods is included in the former but not in the latter.
2. The CPI compares the evolution of the cost of a basket of goods that is fixed, while the GDP deflator looks at the evolution of the price of commonly produced goods in relation to the price of goods produced the previous year. (Paasche index versus Laspeyres index)
## Indice de Paasche et Laspeyres
The GDP deflator is a "Paasche index":
GDP deflator = ${\displaystyle {\frac {\Sigma _{j}^{t}p_{j}^{t}}{\Sigma _{j}p_{j}^{(base)}q_{j}^{t}}}}$
in the case of hot dogs and hamburgers = ${\displaystyle {\frac {p_{HD}^{t}\times q_{HD}^{t}+p_{H}^{t}\times q_{H}^{t}}{p_{HD}^{base}\times q_{HD}^{t}+p_{H}^{base}\times q_{H}^{t}}}}$
The consumer price index is a "'Laspeyres index'":
${\displaystyle {\frac {\Sigma _{j}p_{j}^{t}q_{j}^{base}}{\Sigma _{j}p_{j}^{base}q_{j}^{base}}}}$
in the case of hot dogs and hamburgers = ${\displaystyle {\frac {p_{HD}^{t}\times q_{HD}^{base}+p_{H}^{t}\times q_{H}^{base}}{p_{HD}^{base}\times q_{HD}^{base}+p_{H}^{base}\times q_{H}^{base}}}}$
## CPI versus GDP deflator
Two other alternative measures of price changes are the Producer Price Index (PPI), which measures changes in the cost of a (fixed) basket of goods and services purchased by producers (this is used to predict changes in the CPI), and the Import Price Index (IPI).
# Correction of macroeconomic variables for inflation
## Inflation correction
To be able to compare the purchasing power of a certain income in different years, the (nominal) value of this income must be corrected by the evolution of the cost of living. Ex: ${\displaystyle salary_{2004}^{2000}=salary^{2000}\times {\frac {CPI_{2004}}{;}}{CPI_{2000}}={\text{purchase power of salary 2000 in 2004}}}$.
Example 1 :
• George Washington's income in 1789 was USD 25,000.
• George Bush's income in 2007 was USD 450'000.
• The consumer price index with base 100 in 1789 is 2000 in 2007.
Which of the two George's has a higher real income (higher purchasing power)? ${\displaystyle {\text{Washington's income in 2007 = Washington's income in 1789}}\times {\text{CPI in 2007}}{\text{CPI in 1789}}=25000\times {\frac {2000}{100}}=500000}$.
Eexample 2 :
• LeBron James' salary in 2003 (his first year in the NBA) is $4 million. • Michael Jordan's salary in 1984 (his first year in the NBA) is$550,000.
• CPI in 2003 with base 100 in 1984 is \$200.
James has a real salary almost 4 times higher than Jordan's in his first year of the NBA.
## The base year
Price indices are generally set arbitrarily at 100 for a reference period: beware of comparisons!
If one index is 174 and another is 130, it is necessary to have the same base year (or index year or "year 100") in order to know which one has moved the fastest.
For example:
${\displaystyle IPC1995=102.6}$ and ${\displaystyle IPC1997=103.9}$ (1993 = 100)
Inflation rate (between 1995 and 1997) = ${\displaystyle {\frac {103.9-102.6}{102.6}}=0.01267=1.267\%}$ Or more simply, by approximation:
Inflation rate (between 1995 and 1997) = ${\displaystyle 103.9-102.6=1.3\%}$
(approximation valid only for small variations, i.e. < 10%)
## Inflation and deflation
It can be seen that inflation is low between January 2010 and December 2009 and that there is deflation between January 2009 and January 2010.
## Importance of measuring the CPI correctly
The CPI is used continuously in the political and economic life of countries and economic policy authorities as well as individuals rely on the observation of the CPI to make their decisions.
By "revising" the CPI downwards (see the biases we saw earlier), the government can show a larger increase in real wages than the actual increase and justify existing economic policies. The monetary policy adopted by the Central Bank is chosen, among other things, on the basis of the evolution of the CPI.
The CPI is used to index certain contracts, such as pensions, wages, electricity price regulation, etc. The CPI is also used for the indexation of certain contracts.
To evaluate the profitability of an investment we need a measure of the evolution of the cost of living:
Real interest rate = nominal interest rate - inflation rate.
# Summary
The CPI shows the cost of a basket of goods and services in a given year compared to the cost of the same basket in a base year. The percentage change in the CPI gives us the rate of inflation.
The CPI is an imperfect measure of the cost of living for four reasons:
• substitution bias;
• the importance of new goods;
• unmeasurable changes in the quality of goods;
• heterogeneity in the consumption baskets of different individuals.
The GDP deflator differs from the CPI in two respects:
• The CPI focuses on a standard consumption basket and the deflator focuses on the goods produced in the economy;
• The CPI uses a fixed basket of goods and services while the GDP deflator adjusts the composition of the basket to reflect the structure of production each year.
Monetary variables measured at different points in time must be adjusted by their purchasing power (CPI) in order to be compared. In order to say something about the evolution of individuals' purchasing power over time, the CPI must be measured correctly.
The CPI is used to set wages, pensions, the price of certain goods under public regulation.
The real interest rate is what determines the decision to invest and is given by the nominal interest rate minus the inflation rate.
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2021-11-30 06:18:03
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https://www.ideals.illinois.edu/handle/2142/22216
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## Files in this item
FilesDescriptionFormat
application/pdf
9210873.pdf (4MB)
(no description provided)PDF
## Description
Title: Thermoregulation by foxes Author(s): Klir, John Jan Doctoral Committee Chair(s): Heath, James E. Department / Program: Molecular and Integrative Physiology Discipline: Molecular and Integrative Physiology Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Biology, Animal Physiology Abstract: The main objective of this study was to develop a model of the thermoregulatory control system which could be used to predict the responses of unrestrained foxes of different species to naturally occurring thermal stress. The studied species included the red fox (Vulpes vulpes), arctic fox (Alopex lagopus), and kit fox (Vulpes macrotis). The model was used to test hypothesis whether species of foxes occupying different habitats do or do not use the same thermoregulatory control system.First, infrared (IR) thermography was used to study the control of surface temperature in undisturbed foxes exposed to naturally occurring thermal stress. The resting metabolic rate (RMR) and evaporative water loss (EWL) in the red and arctic fox was measured as oxygen consumption at various ambient temperatures (T$\sb{\rm a}$) using metabolic chamber. Total heat flow from the animal's surface (Q$\sb{\rm t}$) was calculated using the surface temperature measurements.Second, red foxes were surgically implanted in the POAH with two thermodes to control the temperatures of this region. The temperature of the POAH (T$\sb{\rm poah}$) was monitored with an implanted thermocouple. Deep body temperature (T$\sb{\rm b}$), surface temperature, and metabolic rate (MR) were measured. The animals were exposed to various T$\sb{\rm a}$ in a temperature chamber.The most important thermoregulatory surfaces include the area of the dorsal head, face, nose, pinna, lower legs, and paws in the red and kit fox, and the face, nose, front of the pinna, lower legs, and paws in the arctic fox. Although the thermoregulatory effective surface areas represent only about 30% of the total surface area, the animals can lose more than 70% of the total radiative and convective heat loss through these areas. These surfaces are relatively large in the kit fox, small in the arctic fox, and intermediate in the red fox.MR increased during both heating and cooling of the POAH. Resting T$\sb{\rm poah}$ was lower than T$\sb{\rm b}$ at all temperatures which indicates presence of some form of brain cooling mechanism. The surface temperature responses to POAH heating or cooling indicated that the thermoregulatory vasomotor responses can occur within one minute following POAH stimulation.The data support the hypothesis that species of foxes occupying different habitats use the same central thermoregulatory control system, and that they differ basically only in thermoregulatory effectors such as relative size of the thermoregulatory effective surface areas, insulation, vasomotor control, and evaporative heat loss. Issue Date: 1991 Type: Text Language: English URI: http://hdl.handle.net/2142/22216 Rights Information: Copyright 1991 Klir, John Jan Date Available in IDEALS: 2011-05-07 Identifier in Online Catalog: AAI9210873 OCLC Identifier: (UMI)AAI9210873
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2017-01-18 22:21:03
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http://reference.iucr.org/mediawiki/index.php?title=Difference_Patterson_map&diff=prev&oldid=4821
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# Difference Patterson map
(Difference between revisions)
Revision as of 13:38, 10 November 2017 (view source) (Added German and Spanish translations (U. Mueller))← Older edit Latest revision as of 14:51, 18 June 2019 (view source) (Lang (Fr, It)) Line 1: Line 1: - Differenz-Patterson-Karte (''Ge''). Mapa de Patterson de diferencia (''Sp''). + Carte de différence de Patterson (''Fr''). Differenz-Patterson-Karte (''Ge''). Mappa di differenza di Patterson (''It''). Mapa de Patterson de diferencia (''Sp''). == Definition == == Definition ==
## Latest revision as of 14:51, 18 June 2019
Carte de différence de Patterson (Fr). Differenz-Patterson-Karte (Ge). Mappa di differenza di Patterson (It). Mapa de Patterson de diferencia (Sp).
## Definition
An application of Patterson methods for solution of crystal structures, typically proteins with heavy-atom derivatives, where the Patterson function is calculated using structure-factor coefficients based on the difference between the heavy-atom derivative and the native molecule.
## Discussion
Patterson methods for determining diffraction phases depend on the symmetries of interatomic vectors that show up as peaks in a three-dimensional map of the Patterson function. For small molecules containing a heavy atom, the heavy-atom positions can be determined directly from the Patterson function calculated using measured structure-factor amplitudes. For proteins, there are too few heavy atoms for this approach to be successful. However, if an isomorphous derivative crystal is available (i.e. one whose symmetry and dimensions and contents, with the exception of heavy-atom addition, are minimally changed), a Patterson map of derivative (FPH) minus native (FP) structure factors will be dominated by the vectors between the heavy atoms, and thus allow a solution of the coordinates of the heavy atoms.
A true difference Patterson function, representing the difference between the Patterson of the derivative minus the Patterson of the native protein, should be calculated using as coefficients $|F^2_{PH} - F^2_P|$.
In practice, protein crystallographers normally calculate a modulus difference-squared synthesis, also known as an isomorphous difference Patterson, using coefficients ( | FPH | − | FP | )2.
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2019-12-12 11:32:59
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https://gmatclub.com/forum/m19-184200.html
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# M19-20
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Math Expert
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16 Sep 2014, 01:06
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Difficulty:
45% (medium)
Question Stats:
59% (00:30) correct 41% (00:35) wrong based on 196 sessions
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Which of the following always equals $$\sqrt{9 + x^2 - 6x}$$ ?
A. $$x - 3$$
B. $$3 + x$$
C. $$|3 - x|$$
D. $$|3 + x|$$
E. $$3 - x$$
[Reveal] Spoiler: OA
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16 Sep 2014, 01:06
Expert's post
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Official Solution:
Which of the following always equals $$\sqrt{9 + x^2 - 6x}$$ ?
A. $$x - 3$$
B. $$3 + x$$
C. $$|3 - x|$$
D. $$|3 + x|$$
E. $$3 - x$$
$$\sqrt{9 + x^2 - 6x} = \sqrt{(3 - x)^2} = |3 - x|$$ (by the definition of a square root).
_________________
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WE: Analyst (Accounting)
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21 Nov 2014, 03:18
1
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Hi Bunuel,
I got the roots for the equation as (x-3)^2 (reading the equation as x^2 - 6x +9 - We've flipped the values of a and b in the equation a^2 - 2ab + b^2). Based on my approach |x - 3| would be the correct answer and on yours |3 - x| is the correct answer.
Does my above para make any sense? If it does, and if the question has both of these options, which would be the correct answer? If it doesn't please help me understand my mistake.
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Cheers!!
JA
If you like my post, let me know. Give me a kudos!
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21 Nov 2014, 04:51
3
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joseph0alexander wrote:
Hi Bunuel,
I got the roots for the equation as (x-3)^2 (reading the equation as x^2 - 6x +9 - We've flipped the values of a and b in the equation a^2 - 2ab + b^2). Based on my approach |x - 3| would be the correct answer and on yours |3 - x| is the correct answer.
Does my above para make any sense? If it does, and if the question has both of these options, which would be the correct answer? If it doesn't please help me understand my mistake.
The point is that |x - 3| = |3 - x|. Both indicate the distance between x and 3.
_________________
Kudos [?]: 132827 [3], given: 12378
Current Student
Joined: 04 Jul 2014
Posts: 294
Kudos [?]: 345 [1], given: 413
Location: India
GMAT 1: 640 Q47 V31
GMAT 2: 640 Q44 V34
GMAT 3: 710 Q49 V37
GPA: 3.58
WE: Analyst (Accounting)
### Show Tags
21 Nov 2014, 05:05
1
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Bunuel wrote:
The point is that |x - 3| = |3 - x|. Both indicate the distance between x and 3.
So, I understand that both of our answers are correct and that both these values won't appear as options in a single question.
_________________
Cheers!!
JA
If you like my post, let me know. Give me a kudos!
Kudos [?]: 345 [1], given: 413
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21 Nov 2014, 05:07
1
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Expert's post
joseph0alexander wrote:
Bunuel wrote:
The point is that |x - 3| = |3 - x|. Both indicate the distance between x and 3.
So, I understand that both of our answers are correct and that both these values won't appear as options in a single question.
____________
Yes, that's correct.
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### Show Tags
24 Jul 2015, 07:44
Bunuel wrote:
Official Solution:
Which of the following always equals $$\sqrt{9 + x^2 - 6x}$$ ?
A. $$x - 3$$
B. $$3 + x$$
C. $$|3 - x|$$
D. $$|3 + x|$$
E. $$3 - x$$
$$\sqrt{9 + x^2 - 6x} = \sqrt{(3 - x)^2} = |3 - x|$$ (by the definition of a square root).
i answered choice a : x-3
i took the the expression as x^2-6x+9 ---> (x-3)^2. no my understanding is that on GMAT when some expression is under root sign, only positive root is considered...? is it correct. if it is so, the we need not take mod value...? pls help
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### Show Tags
24 Jul 2015, 07:51
Expert's post
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riyazgilani wrote:
Bunuel wrote:
Official Solution:
Which of the following always equals $$\sqrt{9 + x^2 - 6x}$$ ?
A. $$x - 3$$
B. $$3 + x$$
C. $$|3 - x|$$
D. $$|3 + x|$$
E. $$3 - x$$
$$\sqrt{9 + x^2 - 6x} = \sqrt{(3 - x)^2} = |3 - x|$$ (by the definition of a square root).
i answered choice a : x-3
i took the the expression as x^2-6x+9 ---> (x-3)^2. no my understanding is that on GMAT when some expression is under root sign, only positive root is considered...? is it correct. if it is so, the we need not take mod value...? pls help
Exactly because the square root function cannot give negative result the answer is |3-x|, an absolute value of 3-x, which also cannot be negative.
MUST KNOW: $$\sqrt{x^2}=|x|$$:
The point here is that since square root function can not give negative result then $$\sqrt{some \ expression}\geq{0}$$.
So $$\sqrt{x^2}\geq{0}$$. But what does $$\sqrt{x^2}$$ equal to?
Let's consider following examples:
If $$x=5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=x=positive$$;
If $$x=-5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=-x=positive$$.
So we got that:
$$\sqrt{x^2}=x$$, if $$x\geq{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<0$$.
What function does exactly the same thing? The absolute value function: $$|x|=x$$, if $$x\geq{0}$$ and $$|x|=-x$$, if $$x<0$$. That is why $$\sqrt{x^2}=|x|$$.
I'd advice to go through basics and only after to practice questions:
Theory on Abolute Values: math-absolute-value-modulus-86462.html
Absolute value tips: absolute-value-tips-and-hints-175002.html
DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58
Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html
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10 Dec 2015, 03:55
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. please elaborate the explanation.
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12 Dec 2015, 08:37
Bhavanasg wrote:
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. please elaborate the explanation.
Hope it helps.
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22 Aug 2016, 14:02
I think this is a high-quality question and I agree with explanation.
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12 Sep 2016, 15:23
1
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After solving this question I got the result as |X-3| and I answered it as X-3, And got this wrong.
Thanks for giving the link which clears the basic of Absolute value including the result:
|A-B| = |B-A|
Which is clearly applicable in this question provided options.
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### Show Tags
30 Mar 2017, 20:19
1
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Bunuel wrote:
riyazgilani wrote:
Bunuel wrote:
Official Solution:
Which of the following always equals $$\sqrt{9 + x^2 - 6x}$$ ?
A. $$x - 3$$
B. $$3 + x$$
C. $$|3 - x|$$
D. $$|3 + x|$$
E. $$3 - x$$
$$\sqrt{9 + x^2 - 6x} = \sqrt{(3 - x)^2} = |3 - x|$$ (by the definition of a square root).
i answered choice a : x-3
i took the the expression as x^2-6x+9 ---> (x-3)^2. no my understanding is that on GMAT when some expression is under root sign, only positive root is considered...? is it correct. if it is so, the we need not take mod value...? pls help
Exactly because the square root function cannot give negative result the answer is |3-x|, an absolute value of 3-x, which also cannot be negative.
MUST KNOW: $$\sqrt{x^2}=|x|$$:
The point here is that since square root function can not give negative result then $$\sqrt{some \ expression}\geq{0}$$.
So $$\sqrt{x^2}\geq{0}$$. But what does $$\sqrt{x^2}$$ equal to?
Let's consider following examples:
If $$x=5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=x=positive$$;
If $$x=-5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=-x=positive$$.
So we got that:
$$\sqrt{x^2}=x$$, if $$x\geq{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<0$$.
What function does exactly the same thing? The absolute value function: $$|x|=x$$, if $$x\geq{0}$$ and $$|x|=-x$$, if $$x<0$$. That is why $$\sqrt{x^2}=|x|$$.
I'd advice to go through basics and only after to practice questions:
Hello Bunuel, Vyshak
I thought the equation was as follows:
$$\sqrt{x^2}=x$$, if $$x>{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<={0}$$.
Am i wrong? Please correct me if so.
Thanks
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30 Mar 2017, 22:01
manishtank1988 wrote:
Hello Bunuel, Vyshak
I thought the equation was as follows:
$$\sqrt{x^2}=x$$, if $$x>{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<={0}$$.
Am i wrong? Please correct me if so.
Thanks
You can put = sign in any of the two because $$\sqrt{0^2}=0=-0$$.
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31 Mar 2017, 10:00
Dear Bunuel,
The answer choices CAN"T have the following answers in their list: $$|3 - x|$$ and $$|x - 3|$$ Because the following rule:
$$|3 - x|$$ = $$|x - 3|$$
Am I right?
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31 Mar 2017, 15:07
Bunuel wrote:
manishtank1988 wrote:
Hello Bunuel, Vyshak
I thought the equation was as follows:
$$\sqrt{x^2}=x$$, if $$x>{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<={0}$$.
Am i wrong? Please correct me if so.
Thanks
You can put = sign in any of the two because $$\sqrt{0^2}=0=-0$$.
Understood thanks...
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21 Aug 2017, 05:28
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This question is based on the principle of $$\sqrt{x^2}=|x|$$
Hence its $$|x-3|=|3-x|$$
Straight away C.
Press Kudos id this helps!
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Re: M19-20 [#permalink] 21 Aug 2017, 05:28
Display posts from previous: Sort by
# M19-20
Moderators: Bunuel, chetan2u
Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.
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2017-11-20 21:53:00
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https://math.stackexchange.com/tags/commutative-algebra/new
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# Tag Info
1 vote
Accepted
• 753
### Presentation of Witt vectors of polynomial rings
The main trick I know of for getting your hands on Witt vectors (of $\mathbb F_p$-algebras) is to reduce to the case of perfect $\mathbb F_p$-algebras, where we can appeal to the theorem that $W(-)$ ...
• 7,103
1 vote
Accepted
### Questions about the Cayley-Hamilton theorem for modules
I think I'm a bit late! Either way: Yes, we require $\varphi(M) \subseteq IM$ to guarantee that $p_j \in I^j$ (this is particularly useful for commutative algebra when $I$ is prime). Note that $I = R$...
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### Geometric interpretation of Minimal Prime Ideals
First, note that when one is looking for minimal primes over an ideal $I$, one can work with the radical $\sqrt{I}$ instead: $\sqrt{I}=\bigcap_{P\supset I, \text{ P prime}} P$ (ref), so if $P$ is a ...
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I think I've found an answer to my question. We construct a sequence $(x_i\in M)$ by: $x_0=0$ and $\pi x_i=x_{-1}$. Now define a map $K\rightarrow M$ by $\pi^{-i}a\mapsto a.x_i$, then the kernel is ...
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2023-03-23 11:00:39
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http://sourceforge.net/p/lily4jedit/feature-requests/20/
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## #20 Enhancement: smart autocompletion
open
nobody
None
5
2009-10-20
2009-10-20
No
Currently, the autocompletion of LilyPond command is simple but stupid: for example, when typing
\r
the first suggestions you get are
\ragged-bottom
\ragged-last-bottom
\relative
although \ragged-bottom is used much less often than \relative. See where this is going?
What would be Truly Awesome® (as in Mozilla Firefox >3.0 'Awesome' URL bar) would be to either have the command list in a pre-defined arbitrary (non-alphabetic) order, or, even better, to have an Intelligent Autocompletion Learning Thingy feature, that would automatically put the most used command on top of the suggestions list.
(I suspect the first solution would be a nightmare to maintain, and the second would require significant changes in Sidekick. But still, that would just be Plain Coolness™.)
## Discussion
• Bertalan Fodor
2009-10-21
Actually I don't like the new URL bars, they are often too slow or not relevant.
Anyway, this is a good idea, but I don't know how to define the more often keywords.
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2015-01-26 23:53:39
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http://carlacasciari.it/flexsurvreg.html
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Weibull Analysis: Tableau + R Integration by Monica Willbrand 1. Results Patients A total of 192 patients (119 adults, 73 children) was available for Conditioning. Regression for a Parametric Survival Model Description. 1802, df = 1 AIC = 1432. Gompertz-Cox Regression •Distribution -Gompertz distribution. Survival analysis in health economic evaluation Contains a suite of functions to systematise the workflow involving survival analysis in health economic evaluation. In the previous chapter (survival analysis basics), we described the basic concepts of survival analyses and methods for analyzing and summarizing survival. It only takes a minute to sign up. flexsurvreg(cfwei, t = tgrid, trans = tmat) These can be plotted (Figure 5) to show the fit of the parametric models compared to the non-parametric estimates. The other parameters are ancillary parameters that determine the shape, variance, or higher moments of the distribution. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): flexsurv is an R package for fully-parametric modelling of survival data. 6 Also, for rabbit antithymocyte. For the Weibull (and exponential, log-normal and log-logistic) distribution, flexsurvreg simply acts as a wrapper for survreg: The maximum likelihood estimates are obtained by survreg, checked by flexsurvreg for optimization convergence, and converted to flexsurvreg's preferred parameterization. 2020-04-21T13:54:18Z http://oai. 00791 N = 338, Events: 229, Censored: 109 Total time at risk: 1913. The Cox cumulative hazards were replaced with parametric equivalents and were used as arguments in the mssample function for prediction purposes. Comparing the results from flexsurvreg with survreg, we see that the estimates are identical for all models. 我找到了flexsurv包,它实现了广义的gamma分布. Potapczuk National Aeronautics and Space Administration Glenn Research Center Cleveland, Ohio 44135 Didier Guffond and E. For example, flexsurvreg can be used to create custom models, or use the large range of existing ones, including Royston-Parmar spline models. Cox Proportional Hazard), and Non-Parametric Models. Between 1992 and 2015, 51 410 measurements of age-specific body mass were obtained from 10 854 individual chicks. Option to summary. There are differences between Parametric Models (e. 1802, df = 1 AIC = 1432. model_parameters() for Stan-models (brms, rstanarm) gains a group_level argument to show or hide parameters for group levels of random effects. I am using flexsurvreg from the flexsurv package in order to fit a Gompertz model to survival data. 1 #----- # MLE for log-normal distribution #----- # 240F Complete Data #----- data. Regression for a Parametric Survival Model Description. dweibullPH and related functions give the Weibull distribution in proportional hazards parameterisation, and "weibullPH" is supported as a built-in model for flexsurvreg. flexsurvreg(formula = su_obj ~ 1, data = orca, dist = "exponential") Estimates: est L95% U95% se rate 0. This procedure can handle complex survey sample designs, including designs with stratification, clustering, and unequal weighting. in Cost-effectiveness Analyses: A Comparison. Building a linear model in R R makes building linear models really easy. 0 2020-03-01. note to exams office: when making copies please make single collated exam paper per student. Viewed 1k times 2. Understanding the Cox Regression Models with Time-Change Covariates Mai Zhou University of Kentucky The Cox regression model is a cornerstone of modern survival analysis and is widely used in many other fields as well. Flexible parametric models for time-to-event data, including the Royston-Parmar spline model, generalized gamma and generalized F distributions. dweibullPH and related functions give the Weibull distribution in proportional hazards parameterisation, and "weibullPH" is supported as a built-in model for flexsurvreg. 1 De nitions: The goals of this unit are to introduce notation, discuss ways of probabilisti-cally describing the distribution of a 'survival time' random variable, apply these to several common parametric families, and discuss how observations of survival times can be right. tail=FALSE: Jan 26, 2018: codecov. 6 Also, for rabbit antithymocyte. in Cost-effectiveness Analyses: A Comparison. Anybody can ask a question. Censoring or left-truncation are specified in 'Surv' objects. 예를 들어, 생존이 Weibull 분포를 따른다고 가정하지만 (수학적 위험이 변하기 때문에 지수가 너무 간단합니다. Survival analysis is a mature scientific discipline with a variety of statistical methods and associated computer programs available to the analyst. Sampled mixture model parameters. Distribution Parametrization STATA, SAS and R. Survival Distributions, Hazard Functions, Cumulative Hazards 1. Drosophila melanogaster IIS is propagated by eight Drosophila insulin-like peptides (DILPs), homologs of both mammalian insulin and IGFs, with various spatiotemporal expression patterns and functions. Results Patients A total of 192 patients (119 adults, 73 children) was available for Conditioning. and Mark G. The models are fitted by maximizing the full log-likelihood, and estimates and confidence intervals for any function of the model parameters can be. 분포를 이루기 때문에 많은 분야에서 사용된다. If you want updates on when I'll upload new video go. flexsurvreg). Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. (我知道rbind效率不高,但你可以随时切换到大型数据集的data. 1 #----- # MLE for log-normal distribution #----- # 240F Complete Data #----- data. Description This function returns predictions from a flexsurvreg object. flexsurv-package: flexsurv: Flexible parametric survival and multi-state models: flexsurvreg: Flexible parametric regression for time-to-event data: summary. Of particular interest is the accuracy of the estima-. flexsurvreg to return a tidy data frame. For 50 cells, the Gompertz model may be the better fit 4 out of 5 times. Things like. For example, flexsurvreg can be used to create custom models, or use the large range of existing ones, including Royston-Parmar spline models. Building a linear model in R R makes building linear models really easy. align = "center", warning = FALSE) options(width = 95, show. This site contains. The type of data available, the manner the data were obtained, the mathematical models used to analyze the data, and the integrity of the conclusions can be very confusing for someone not steeped. Survival analysis is a mature scientific discipline with a variety of statistical methods and associated computer programs available to the analyst. 代替オプションは、パッケージflexsurvを使用することです。これは、survivalパッケージにいくつかの追加機能を提供します - パラメトリック回帰関数flexsurvreg()には、あなたが求めるものを行う素敵なプロット方法が含まれています。 上記のように肺を使用する。. Censoring or left-truncation are specified in 'Surv' objects. 我已经使用flexsurvreg来估计威布尔分布的参数，并得到以下输出。我想重建生存函数来估计给定时间t的生存率。 flexsurvreg（式. Furthermore, the residuals were assumed to be independently distributed and. If absent predictions are for the subjects used in the original fit. Not only is the package itself rich in features, but the object created by the Surv () function, which contains failure time and censoring information, is the basic survival analysis data structure in R. Thursday, 28 February. Objects of class flexsurvreg, which can be named. The sampled mixture model parameters are contained in a list containing the following: beta1. Observed outcome variables. , ## dist = "weibull") ## ## Estimates: ## est L95% U95% se ## shape 3. " - David Pearce (exaggerated compliment) "Thank you so much, Dr. Keywords:˜survival. flexsurv: A Platform for Parametric Survival Modeling in R: Abstract: flexsurv is an R package for fully-parametric modeling of survival data. If for some reason you do not have the package survival, you need to install it rst. To find the p-value for your test statistic:. 我找到了flexsurv包,它实现了广义的gamma分布. Predictions. Note that, when used inappropriately, statistical models may give. , credit cards and student loans), see Chen (2015). A lot of functions (and data sets) for survival analysis is in the package survival, so we need to load it rst. flexsurv: Flexible Parametric Survival and Multi-State Models. The aes argument stands for aesthetics. Data Science Stack Exchange is a question and answer site for Data science professionals, Machine Learning specialists, and those interested in learning more about the field. When creating a CTSTM from a flexsurvreg object, the user must simply set the argument point_estimate = FALSE and choose the number of samples of the parameters to draw. A list of class "flexsurvreg" containing information about the fitted model. and Mark G. This is a post about linear models in R, how to interpret lm results, and common rules of thumb to help side-step the most common mistakes. " It could also. Credit risk assessment using survival analysis for progressive right-censored data: a case study in Jordan May 2017 Journal of Internet Banking and Commerce 22(1):1-18. The main model-fitting function, flexsurvreg, uses the familiar syntax of survreg from the standard survival package (Therneau 2016). 2 heemod: Models For Health Economic Evaluation in R Where X is a vector2 giving the probability of being in a given state at the start of the model, and Tt is the product of multiplying t matrices T. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. You're going to have to tell us a little more. This is a hotfix release to correct some of the failing tests and other minor breakages resulting from the new release of ggplot2 3. ggstatsplot is an extension of ggplot2 package for creating graphics with details from statistical tests included in the information-rich plots themselves. Vuori et al. This can be a convenient/faster alternative to summary. Proportional hazards generalized gamma model Crowther and Lambert (2013) discuss using the stgenreg Stata package to construct a pro-portional hazards parameterisation of the three. 7%) reached the age of 24 days, while the other 3614 died at. rm(list=ls()) require(survival) require(flexsurv) require(doParallel) no_cores <- detectCores() - 1. and Mark G. The Cox cumulative hazards were replaced with parametric equivalents and were used as arguments in the mssample function for prediction purposes. Censoring or left-truncation are specified in 'Surv' objects. Any parametric time-to-event distribution may be fitted if the user supplies a probability density or hazard function, and ideally also their cumulative versions. General function to return predictions, either corresponding to the observed data, or to a user-supplied "newdata". If you want to have the color, size etc fixed (i. Je voudrais reconstruire la fonction de survie pour estimer le taux de survie à un instant donné t. If absent predictions are for the subjects used in the original fit. 2020-04-21T13:54:18Z http://oai. (c) Growth analyses Between 1992 and 2015, 51 410 measurements of age-specific body mass were obtained from 10 854 individual chicks. These can be plotted against nonparametric estimates (plot. Code and data are available on my github-repo under file name ‘p180’. Allowed values include "survival" (default) and "cumhaz" (for cumulative hazard). edu Spring, 2001; revised Spring 2005, Summer 2010 We consider brie y the analysis of survival data when one is willing to assume a parametric form for the distribution of survival time. In this example, we fit a Weibull model to. Returns an object of class "flexsurvreg_list". Or copy & paste this link into an email or IM:. The flexsurvreg function was used to fit generalized gamma models. in Cost-effectiveness Analyses: A Comparison. Therefore I conclude that flexsurv is an alternative when fitting with gamma distribution. A survival analysis can be defined as consisting of two parts: the core survial object with a time indicator plus the corresponding event status (used to calculate the baseline hazard). The best answers are voted up and rise to the top. These outcome variables can be observed variables or continuous latent variables. 1802, df = 1 AIC = 1432. Package: flexsurv Type: Package Title: Flexible Parametric Survival and Multi-State Models Version: 1. knowledgable about the basics of survival analysis, 2. I flexsurvreg() or flexsurvspline() function in flexsurv package (fully parametric models) I survreg() function in survival package Specialised software then needed to deduce quantities needed for decision modelling:transition probabilities, expectedtotal time spent in some state over some horizon:::. R help archive by subject. The response must be a survival object as returned by the Surv function, and any covariates are given on the right-hand side. A tool to provide an easy, intuitive and consistent access to information contained in various R models, like model formulas, model terms, information about random effects, data that was used to fit the model or data from response variables. Kaplan-Meier), Semi-Parametric Models (e. overall survival) • Time-to-event data may not be complete for all patients, and so some observations may be censored. flexsurvreg for the required form of the model and the data. 1 The Accelerated Failure Time Model Before talking about parametric regression models for survival data, let us introduce the ac- celerated failure time (AFT) Model. 3, in R, version 3. flexsurvreg to return a tidy data frame. Writing Equation in Slope-Intercept Form (y=mx+b) to Find the Slope and y-Intercept. Potapczuk National Aeronautics and Space Administration Glenn Research Center Cleveland, Ohio 44135 Didier Guffond and E. Survival Distributions, Hazard Functions, Cumulative Hazards 1. 左側截尾現象，又被叫做延時進入 (delayed entry): 由於觀察對象實際進入研究時的年齡各不相同，對所有人的觀察時間，都從出生日開始算起的研究，實施難度極大。此時，應當注意把進入研究之前的生存時間 (進入實驗時的年齡)，考慮進來，因爲這些人至少活到了. The type of data available, the manner the data were obtained, the mathematical models used to analyze the data, and the integrity of the conclusions can be very confusing for someone not steeped. To give users more flexibility in terms of modifying the aesthetic defaults for all geoms included in the ggstatsplot plots (each plot typically has multiple geoms), the package now uses a new form of syntax. This procedure can handle complex survey sample designs, including designs with stratification, clustering, and unequal weighting. The best answers are voted up and rise to the top. align = "center", warning = FALSE) options(width = 95, show. If you want to have the color, size etc fixed (i. packages, flexsurvreg, cmprsk, survival, and rms. Definition: Schoenfeld Residuals Test. Therefore the same model can be fitted more. Or put it another way: as R is a typical "the reference implementation is the specification" programming environment there is no true "de jure" R, only a de facto R. A tool to provide an easy, intuitive and consistent access to information contained in various R models, like model formulas, model terms, information about random effects, data that was used to fit the model or data from response variables. Active 3 years, 7 months ago. 예를 들어, 생존이 Weibull 분포를 따른다고 가정하지만 (수학적 위험이 변하기 때문에 지수가 너무 간단합니다. Exercise for survival analysis Alessio Crippa February 28, 2018 Survival analysis, Exercises ConsidernowtheWhitehallstudy,alargeprospectivecohortof17,260maleBritishCivilServants. 1 De nitions: The goals of this unit are to introduce notation, discuss ways of probabilisti-cally describing the distribution of a ‘survival time’ random variable, apply these to several common parametric families, and discuss how observations of survival times can be right. The Schoenfeld Residuals Test is used to test the independence between residuals and time and hence is used to test the proportional Hazard assumption in Cox Model. The main model-fitting function, flexsurvreg, uses the familiar syntax of survreg from the standard survival package (Therneau 2016). in collaboration with Department of Mathematics & Statistics of Williams College, Williamstown, MA created a reference document describing corresponding parametrization of selected distributions between TreeAge Pro, STATA, SAS and R. parameters_model() now explicitely get arguments to define the digits for decimal places used in output. Se muestran algunos paquetes que se requieren y diferentes calculos, También se muestran algunas gráficas para proponer algún modelo, dependiento del criterio y de la gráfica para así ayudar a seleccionar -empíricamente- un modelo paramétrico. All parametric survival models were run using the procedure 'flexsurvreg' in the package 'flexsurv' in R (v. Interpret Regression Coefficient Estimates - {level-level, log-level, level-log & log-log regression}. 05575 NA NA. lifetime T ∼ Exp(λ) λ > 0,T ≥ 0 pdf f(t) = λexp(−λt), t ≥ 0;. Simulation definition is - the act or process of simulating. the type of survival curves. The Gauss-Markov assumptions* hold (in a lot of situations these assumptions may be relaxed - particularly if you are only interested in an approximation - but for now assume they strictly hold). Common Shape Parameter Likelihood Ratio Test. If absent predictions are for the subjects used in the original fit. R: A language and environment for statistical. flexsurv::flexsurvreg(formula = missing_arg(), data = missing_arg(), weights = missing_arg()) survival Note that model = TRUE is needed to produce quantile predictions when there is a stratification variable and can be overridden in other cases. Should behave similarly to other predict methods in base R and common packages. nb Ben Bolker ; Re: [R] best ordination method for binary variables David L Carlson ; Re: [R] PCA with spearman and kendall correlations David L Carlson ; Re: [R] ARMA and AR in R Rui Barradas ; Re: [R] positioning of R windows Duncan Murdoch. R uses the shape/scale parameterization of the Weibull distribution. flexsurvreg for the required form of the model and the data. M ¨ MTT Agrifood Research Finland, Biotechnology and Food Research, Biometrical Genetics, FIN-31600 Jokioinen, Finland (Received 6 July 2005; accepted 27 January 2006). This can be a convenient/faster alternative to summary. The Weibull distribution with shape parameter a and scale parameter b has density given by. I am using a Gompertz. The flexsurvreg function wa s used to fit. rm(list=ls()) require(survival) require(flexsurv) require(doParallel) no_cores <- detectCores() - 1. Therefore the same model can be fitted more. A mathematical definition of Martingale like residuals for the Accelerated Failure Time model (which is a parametric survival model) can be found in Collett's 2003 book Modelling survival data in medical research. Exponential distribution The exponential distribution is the 'canonical model' for survival analysis. The parameter of primary interest (in flexsurv) is colored in red—it is known as the location parameter and typically governs the mean or location for each distribution. R Development Page Contributed R Packages. flexsurvreg is intended to be easy to extend to handle new distributions. These outcome variables can be observed variables or continuous latent variables. 5 Adjusting Survival Curves From a survival analysis point of view, we want to obtain also estimates for the survival curve. I generate a sequence of 5000 numbers distributed following a Weibull distribution with: c=location=10 (shift from origin), b=scale = 2 and; a=shape = 1; sample<- rweibull(5000, shape=1, scale = 2) + 10. flexsurvreg (formula = Surv (time, all) ~ sex + I ((age-65) / 10) + st3, data = orca2, dist = "weibull") Estimates: data mean est L95 % U95 % se exp (est) L95 % shape NA 0. What to report from a Cox Proportional Hazards Regression analysis? I am currently writing up a paper where I have used CPH regression to test the survival of ants. Curtis Kephart is a International Economics Ph. Age-specific mortality rates were compared by fitting parametric survival models implemented using the flexsurvreg function within the flexsurv package, version 0. 1802, df = 1 AIC = 1432. 5000 simulations were used with the mssample function to sample paths from the multi-state model. In order to successfully install the packages provided on R-Forge, you have to switch to the most recent version of R or. The type of data available, the manner the data were obtained, the mathematical models used to analyze the data, and the integrity of the conclusions can be very confusing for someone not steeped. 1 Introduction to (Univariate) Distribution Fitting. org This document is intended to assist individuals who are 1. x can also be a list of flexsurvreg models, with one component for each per-mitted transition in the multi-state model, as illustrated in msfit. 第一件事是从您提供的汇总表中重新创建"原始"数据. Vuori et al. The flexsurvreg function was used to fit generalized gamma models. insight mainly revolves around two types of functions: Functions that find (the names of) information, starting with find_, and functions that get the. The parameterizations of these distributions in R are shown in the next table. Simulation definition is - the act or process of simulating. object: result of a model fit using the survreg function. The R code implements Collett's approach to Martingale. If θ 1 and θ 2 are the scale and shape parameters, respectively, then one may write α 0(t,θ) = θ 1θ 2tθ 2−1 or θθ 2 1 θ 2t θ 2−1 or θ 1t θ 2−1 or probably several other things. flexsurvreg(cfwei, t = tgrid, trans = tmat) These can be plotted (Figure 5) to show the fit of the parametric models compared to the non-parametric estimates. This can be a convenient/faster alternative to summary. For example, flexsurvreg can be used to create custom models, or use the large range of existing ones, including Royston-Parmar spline models. 1 Overview This tutorial aims to support the interpretation of parametric time-to-event models by explaining how to calculate the hazard ratio, which is a conventional e ect size to evaluate clinical relevance of treatment e ects. packages, flexsurvreg, cmprsk, survival, and rms. I'll give a quick overview of them here, but have a look at the vignette for more examples. Here's the stepwise survival curve we'll be using in this demonstration:. Exercise for survival analysis Alessio Crippa February 28, 2018 Survival analysis, Exercises ConsidernowtheWhitehallstudy,alargeprospectivecohortof17,260maleBritishCivilServants. Cox Proportional Hazard), and Non-Parametric Models. Below is a list of all packages provided by project flexsurv: Flexible survival models. JAGS에서 시변 공변량을 허용하는 생존 모델을 작성하려고합니다. Weibull Analysis: Tableau + R Integration by Monica Willbrand 1. Parametric Survival Models Germ an Rodr guez [email protected] 左側截尾現象，又被叫做延時進入 (delayed entry): 由於觀察對象實際進入研究時的年齡各不相同，對所有人的觀察時間，都從出生日開始算起的研究，實施難度極大。此時，應當注意把進入研究之前的生存時間 (進入實驗時的年齡)，考慮進來，因爲這些人至少活到了. The "flexsurv" package for flexible parametric survival models, including splines, generalized gamma / F, and extensible to user-defined models. When analyzing accelerated life testing data, it is important to assess model assumptions, discover inadequacies in the model, note extreme observations and assess the possibility that the test did not account for important factors. I generate a sequence of 5000 numbers distributed following a Weibull distribution with: c=location=10 (shift from origin), b=scale = 2 and; a=shape = 1; sample<- rweibull(5000, shape=1, scale = 2) + 10. R: A language and environment for statistical. , geom_point would get arguments like point. Survival Distributions, Hazard Functions, Cumulative Hazards 1. Should behave similarly to other predict methods in base R and common packages. Similarly, in the UK, the average total debt per household. interpretation of flexsurvreg output from flexsurv package Dear all, I am fitting a parametric regression model to survival data using the flexsurvreg function from the flexsurv package. The flexsurvreg function was used to fit generalized gamma models. Similarly, P = P 0 Iq,whereP 0 is a 3 × 3 covariance matrix, i. Description. seleccionar -empíricamente- un modelo paramétrico. The type of data available, the manner the data were obtained, the mathematical models used to analyze the data, and the integrity of the conclusions can be very confusing for someone not steeped. Drosophila melanogaster IIS is propagated by eight Drosophila insulin-like peptides (DILPs), homologs of both mammalian insulin and IGFs, with various spatiotemporal expression patterns and functions. As a result, flexsurv now depends on the "quadprog" package. If absent predictions are for the subjects used in the original fit. As an example from ?flexsurv::flexsurvreg: library (flexsurv) data (ovarian) fitg <-flexsurvreg (formula = Surv (futime, fustat) ~ age, data = ovarian, dist = "gengamma") For each new sample, this model can make probabilistic predictions at a number of user-specified time points. The Cox cumulative hazards were replaced with parametric equivalents and were used as arguments in the mssample function for prediction purposes. Insulin and IGF signaling (IIS) is a complex system that controls diverse processes including growth, development, metabolism, stress responses, and aging. Not only is the package itself rich in features, but the object created by the Surv () function, which contains failure time and censoring information, is the basic survival analysis data structure in R. Writing Equation in Slope-Intercept Form (y=mx+b) to Find the Slope and y-Intercept. When there are other covariates, the β is interpreted as the same log hazard ratio while all the other covariates are held the same. 17 trillion is on mortgages and $2. When analyzing accelerated life testing data, it is important to assess model assumptions, discover inadequacies in the model, note extreme observations and assess the possibility that the test did not account for important factors. 0 2020-03-01. The Cox cumu lative. Regression for a Parametric Survival Model Description. A copy of the function call, for use in post-processing. I flexsurvreg() or flexsurvspline() function in flexsurv package (fully parametric models) I survreg() function in survival package Specialised software then needed to deduce quantities needed for decision modelling:transition probabilities, expectedtotal time spent in some state over some horizon:::. 1 Introduction to (Univariate) Distribution Fitting. Designed for processes observed at arbitrary times in continuous time (panel data) but some other observation schemes are. Things like. x <- c(1175, 1175, 1521, 1567, 1617, 1665, 1665, 1713, 1761, 1953). Any para-metric time-to-event distribution may be fitted if the user supplies a probability density or hazard function, and ideally also their cumulative versions. Understanding the '# of subjects at risk' page of survival analysis. The second part of the survival model consists of the covariates. A probability forecast refers to a specific event, such as there is a 25% probability of it raining in the next 24 hours. The survival package is the cornerstone of the entire R survival analysis edifice. Understanding the Cox Regression Models with Time-Change Covariates Mai Zhou University of Kentucky The Cox regression model is a cornerstone of modern survival analysis and is widely used in many other fields as well. So we will first create this “new” dataset for prediction consisting of each possible value of the ECOG score in the data. To define a new distri- bution for use in flexsurvreg, construct a list with the following elements:. Kaplan-Meier), Semi-Parametric Models (e. flexsurvreg() in package flexsurv; flexsurv, R package, Block, Li, Savits, 2003, initial and final behavio Gamma function, a smooth curve that connect the fa Resources on scientific writing; Presentation tips (draft) notes on dbSNP, in progress; Notes, Lai & Xie, 2006, stochastic ageing and depe Q-and-A on network reliability model of. x <- c(1175, 1175, 1521, 1567, 1617, 1665, 1665, 1713, 1761, 1953). 30 October 2019 16. The other parameters are ancillary parameters that determine the shape, variance, or higher moments of the distribution. 4 모수적 방법을 이용한 생존함수의 추정과 비교 공학(시멘트의 양, 유리의 버티는 힘), 경영(고객 수), 교통(소방차 수) 모두 모수적 방법을 이용. J'ai utilisé flexsurvreg pour estimer les paramètres d'une distribution de Weibull et obtenu la sortie suivante. An example of this with one categorical and one continuous covariate on each parameter is below:. Package msm updated to version 1. How to use simulation in a sentence. Reliability Basics: Utilizing Residual Plots in Accelerated Life Testing Data Analysis. 5000 simulations were used with the mssample function to sample paths from the multi-state model. The mixture and non-mixture cure models from flexsurvcure can be also be used and are very appropriate for long-term survival estimation. This procedure can handle complex survey sample designs, including designs with stratification, clustering, and unequal weighting. The main model-fitting function, flexsurvreg, uses the familiar syntax of survreg from the standard survival package (Therneau 2016). Previously, we described the basic methods for analyzing survival data, as well as, the Cox proportional hazards methods to deal with the situation where several factors impact on the survival process. Simulation definition is - the act or process of simulating. Potapczuk National Aeronautics and Space Administration Glenn Research Center Cleveland, Ohio 44135 Didier Guffond and E. Look up your test statistic on the appropriate. Firstly, printing an flexsurvreg object (or its res element) already shows the 95% confidence interval: > expFit Estimates: est L95% U95% se rate 0. Any parametric time-to-event distribution may be fitted if the user supplies a probability density or hazard function, and ideally also their cumulative versions. For example, flexsurvreg can be used to create custom models, or use the large range of existing ones, including Royston-Parmar spline models. Performance scores rate how well the patient can perform usual daily activities. The main functions, in the package, are organized in different categories as follow. anu sem1 end of semester, 2016 supplementary p2 part questions. Interpreting Beta: how to interpret your estimate of your regression coefficients (given a level-level, log-level, level-log, and log-log regression)? Assumptions before we may interpret our results:. The use of Markov models in health economic evaluation have been thoroughly described inBeck and Pauker(1983),Sonnenberg. •Shape Model -Hypercholesterolemia. The aes argument stands for aesthetics. " - Anonymous correpondent (misfiring compliment). The Weibull distribution with shape parameter a and scale parameter b has density given by. Broeren University of Illinois at Urbana-Champaign Urbana, Illinois 61801 Harold E. ）= μ + σw ^ log?(T)=μ+σW w ^ W flexsurvreg()flexsurv ?flexsurvreg w ^ W 我们将考虑三种常见的选择：指数，Weibull和log-logistic模型。 此外，还考虑了使用Royston和Parmar（2002）的样条模型对时间 - 事件数据进行灵活的参数化建模。. For example, if the model is fit using flexsurvreg in the flexsurv package, the output should be returned from res. The flexsurvreg function wa s used to fit. All parametric survival models were run using the procedure 'flexsurvreg' in the package 'flexsurv' in R (v. set_covariates(surv_model, age = 18, prognosis = "Poor"). " - Anonymous correpondent (misfiring compliment). Should behave similarly to other predict methods in base R and common packages. 4 Weibull 分布. Takes a survival model estimated with covariates (from survfit, flexsurvreg, or other supported functions) and sets the covariate values for which survival projections will be used. The goal of broomExtra is to provide helper functions that assist in data analysis workflows involving regression analyses. Let's fit a function of the form f(t) = exp(λt) to a stepwise survival curve (e. Analyses were conducted using the package flexsurvreg in R software (R Development Core Team, Vienna, Austria) 43 R Core Team. If for some reason you do not have the package survival, you need to install it rst. There are differences between Parametric Models (e. Comparing the results from flexsurvreg with survreg, we see that the estimates are identical for all models. Censoring or left-truncation are specified in 'Surv' objects. I am using flexsurvreg from the flexsurv package in order to fit a Gompertz model to survival data. Could return predictions as linear predic. The other parameters are ancillary parameters that determine the shape, variance, or higher moments of the distribution. 4 Weibull 分布. これはR Advent Calendar2019の第1日目の記事です。 はじめに R言語の特徴として 統計解析向けの手法がたくさん実装されている CRANやGitHubに誰でもパッケージを公開できる というものがあるかと思います。他にも tidyverse パッケージ群の登場によってデータハンドリング、可視化周りが強く…. Thus cβ is the log hazard ratio when the covariate value increases by c units. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. But the Cox models with time-change covariates are not easy to understand or visualize. data: the data used to fit survival curves. Credit risk assessment using survival analysis for progressive right-censored data: a case study in Jordan May 2017 Journal of Internet Banking and Commerce 22(1):1-18. The type of data available, the manner the data were obtained, the mathematical models used to analyze the data, and the integrity of the conclusions can be very confusing for someone not steeped. CHAPTER 5 ST 745, Daowen Zhang 5 Modeling Survival Data with Parametric Regression Models 5. As a result, flexsurv now depends on the "quadprog" package. If absent predictions are for the subjects used in the original fit. not vary based on a variable from the dataframe), you need to specify it outside the aes(), like this. Melvyn and I hail from the same part of the world, and I learned as a child that many of the local place. flexsurvreg is intended to be easy to extend to handle new distributions. generalize d gamma model s. flexsurvreg(crwei, t = tgrid, trans = tmat) msfit. flexsurv::flexsurvreg(formula = Surv(starttime, stoptime, status) ~ x1 + x2, data=data, dist = "weibull") 检查软件包提供的选项，这些选项可能适合您的需求。 推荐问答. The hard part is knowing whether the model you've built is worth keeping and, if so, figuring out what to do next. The most likely source of the error message is that the data you are putting into the algorithm are not in the format that the function expects. If absent predictions are for the subjects used in the original fit. 1- if I chose the Weibull distribution, does the output inform the goodness. How to use simulation in a sentence. For the Weibull (and exponential, log-normal and log-logistic) distribution, flexsurvreg simply acts as a wrapper for survreg: The maximum likelihood estimates are obtained by survreg, checked by flexsurvreg for optimization convergence, and converted to flexsurvreg's preferred parameterization. interpretation of flexsurvreg output from flexsurv package Dear all, I am fitting a parametric regression model to survival data using the flexsurvreg function from the flexsurv package. Survival models in hesim can be fit using either flexsurvreg or flexsurvspline from the flexsurv package. Interpreting Beta: how to interpret your estimate of your regression coefficients (given a level-level, log-level, level-log, and log-log regression)? Assumptions before we may interpret our results:. I generate a sequence of 5000 numbers distributed following a Weibull distribution with: c=location=10 (shift from origin), b=scale = 2 and; a=shape = 1; sample<- rweibull(5000, shape=1, scale = 2) + 10. 1 左側截尾數據 left-truncation. flexsurvreg クラスは logLik() や AIC() には対応しているものの、deviance() や anova() には対応していないようである。 以下は phreg の結果から AIC を引き出すためのメソッド関数を私が自作したものなので、コピペして使ったらよい。. Contents • Introduction to survival analysis • Commonly-used extrapolation methods • Extrapolation method selection • Relevant packages in R. The type of data available, the manner the data were obtained, the mathematical models used to analyze the data, and the integrity of the conclusions can be very confusing for someone not steeped. 30 October 2019 17 Adjusted Hazard Ratios. 左側截尾現象，又被叫做延時進入 (delayed entry): 由於觀察對象實際進入研究時的年齡各不相同，對所有人的觀察時間，都從出生日開始算起的研究，實施難度極大。此時，應當注意把進入研究之前的生存時間 (進入實驗時的年齡)，考慮進來，因爲這些人至少活到了. 1 De nitions: The goals of this unit are to introduce notation, discuss ways of probabilisti-cally describing the distribution of a ‘survival time’ random variable, apply these to several common parametric families, and discuss how observations of survival times can be right. A new version of multistateutils has been released onto CRAN containing a few new features. Each cell will have unknown life distribution parameters that, in general, are different. Things like. Designed for processes observed at arbitrary times in continuous time (panel data) but some other observation schemes are. I am using a Gompertz distribution (a 2-parameter distribution) to describe the hazard function and I want to compare two groups. Understanding the '# of subjects at risk' page of survival analysis. Results Patients A total of 192 patients (119 adults, 73 children) was available for Conditioning. The parameter of primary interest (in flexsurv) is colored in red—it is known as the location parameter and typically governs the mean or location for each distribution. •Scale Model -All covariates and interactions as in Cox Model. This is a package in the recommended list, if you downloaded the binary when installing R, most likely it is included with the base package. 5000 simulations were used with the mssample function to sample paths from the multi-state model. ## ----setup, include = FALSE----- library(knitr) library(kfigr) opts_chunk$set(comment = NA, fig. A tool to provide an easy, intuitive and consistent access to information contained in various R models, like model formulas, model terms, information about random effects, data that was used to fit the model or data from response variables. In the previous chapter (survival analysis basics), we described the basic concepts of survival analyses and methods for analyzing and summarizing survival. The Cox cumulative hazards were replaced with parametric equivalents and were used as arguments in the mssample function for prediction purposes. ggplot2 considers the X and Y axis of the plot to be aesthetics as well, along with color, size, shape, fill etc. familiar with vectors, matrices, data frames, lists, plotting, and linear models in R, and 3. Of particular interest is the accuracy of the estima-. ggsurvplot (): Draws survival curves with the 'number at risk' table, the cumulative number of events table and the cumulative number of censored subjects table. I'll give a quick overview of them here, but have a look at the vignette for more examples. Any user-defined parametric model can also be employed by supplying a list with basic information. The mixture and non-mixture cure models from flexsurvcure can be also be used and are very appropriate for long-term survival estimation. newdata: data for prediction. Purpose To report the final results on treatment outcomes of a randomized trial comparing conventional and hypofractionated radiotherapy in high-risk, organ-confined prostate cancer (PCa). Custom distributions. R help archive by subject. flexsurv::flexsurvreg(formula = missing_arg(), data = missing_arg(), weights = missing_arg()) survival Note that model = TRUE is needed to produce quantile predictions when there is a stratification variable and can be overridden in other cases. Hello R users, I'm trying to do simulations for comparing cox and weibull I have come across this problem: Warning messages: 1: In survreg. Reliability Basics: Utilizing Residual Plots in Accelerated Life Testing Data Analysis. scale NA 13. anu sem1 end of semester, 2016 supplementary p2 part questions. Multi-state models for time-to-event data can also be fitted with the same functions. 2 heemod: Models For Health Economic Evaluation in R Where X is a vector2 giving the probability of being in a given state at the start of the model, and Tt is the product of multiplying t matrices T. generalize d gamma model s. This procedure can handle complex survey sample designs, including designs with stratification, clustering, and unequal weighting. 我已经使用flexsurvreg来估计威布尔分布的参数，并得到以下输出。我想重建生存函数来估计给定时间t的生存率。 flexsurvreg（式. Developing relevant economic models with R for health technology assessment Devin Incerti 2 What is a relevant model? > Based on available clinical evidence > Quantifies decision uncertainty > Transparent and reproducible > Reusable and adaptable. The R code implements Collett's approach to Martingale. Survival Analysis in R June 2013 David M Diez OpenIntro openintro. The survminer R package provides functions for facilitating survival analysis and visualization. packages, flexsurvreg, cmprsk, survival, and rms. Hedge funds and Survival analysis by Blanche Nadege Nhogue Wabo Thesis submitted to the Faculty of Graduate and Postdoctoral Studies In partial fulfillment of the requirements For the M. newdata: data for prediction. You're going to have to tell us a little more. The R output gives me the. 2-5 For busulfan, this has led to the introduction of therapeutic drug monitoring aiming at an optimal target exposure, which has been proven superior over fixed dosing in a randomized clinical trial. Within that library, the command survreg fits parametric survival models. Results Patients A total of 192 patients (119 adults, 73 children) was available for Conditioning. Distribution Parametrization STATA, SAS and R. Custom distributions. 3, in R, version 3. 85 trillion in debt, of which $8. lung dataset : measures survival in patients with advanced lung cancer from the North Central Cancer Treatment Group. It is designed to be used with semi-Markov multi-state models of healthcare data, but can be used for any system that can be. Sampled mixture model parameters. If the lines are straight, with slope = 1, an exponential distribution is a possibility. Any user-defined parametric distribution can be fitted, given at least an R function defining the probability density or hazard. Keywords:˜survival. Allowed values include "survival" (default) and "cumhaz" (for cumulative hazard). 000918 Reproducing the the confidence interval manually. flexsurv::flexsurvreg(formula = missing_arg(), data = missing_arg(), weights = missing_arg()) survival Note that model = TRUE is needed to produce quantile predictions when there is a stratification variable and can be overridden in other cases. Important note for package binaries: R-Forge provides these binaries only for the most recent version of R, but not for older versions. The mixture and non-mixture cure models from flexsurvcure can be also be used and are very appropriate for long-term survival estimation. Jackson MRC Biostatistics Unit Abstract flexsurv is an R package for fully-parametric modeling of survival data. 4 Weibull 分布. In the context of an outcome such as death this is known as Cox regression for survival analysis. Therefore the same model can be fitted more. tail=FALSE: Jan 26, 2018: codecov. The Gauss-Markov assumptions* hold (in a lot of situations these assumptions may be relaxed - particularly if you are only interested in an approximation - but for now assume they strictly hold). A copy of the function call, for use in post-processing. JAGS에서 시변 공변량을 허용하는 생존 모델을 작성하려고합니다. parameters_table() and print. If the lines are straight, with slope = 1, an exponential distribution is a possibility. Any user-defined parametric model can also be employed by supplying a list with basic information. (c) Growth analyses Between 1992 and 2015, 51 410 measurements of age-specific body mass were obtained from 10 854 individual chicks. The maximum likelihood method can be used to estimate distribution and acceleration model parameters at the same time: The likelihood equation for a multi-cell acceleration model utilizes the likelihood function for each cell, as described in section 8. Data Science Stack Exchange is a question and answer site for Data science professionals, Machine Learning specialists, and those interested in learning more about the field. Hedge funds and Survival analysis by Blanche Nadege Nhogue Wabo Thesis submitted to the Faculty of Graduate and Postdoctoral Studies In partial fulfillment of the requirements For the M. Active 3 years, 7 months ago. org This document is intended to assist individuals who are 1. Package MGLM updated to version 0. Weibull Analysis: Tableau + R Integration by Monica Willbrand 1. In this way it does not aim to supplement the modelling strategies found in mstate, msm, or flexsurv, but rather provide tools for subsequent analysis. seleccionar -empíricamente- un modelo paramétrico. The values tabulated are the number of subjects at. The Cox cumulative hazards were replaced with parametric equivalents and were used as arguments in the mssample function for prediction purposes. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. 1 Overview This tutorial aims to support the interpretation of parametric time-to-event models by explaining how to calculate the hazard ratio, which is a conventional e ect size to evaluate clinical relevance of treatment e ects. Custom distributions. newdata: data for prediction. Could return predictions as linear predic. The second part of the survival model consists of the covariates. To define a new distri-bution for use in flexsurvreg, construct a list with the following elements: name: A string naming the distribution. ggsurvplot (): Draws survival curves with the 'number at risk' table, the cumulative number of events table and the cumulative number of censored subjects table. Comparison of hazard rate estimation in R Yolanda Hagar and Vanja Dukic Abstract We give an overview of eight di erent software packages and functions available in R for semi- or non-parametric estimation of the hazard rate for right-censored survival data. The parameter of primary interest (in flexsurv) is colored in red—it is known as the location parameter and typically governs the mean or location for each distribution. Gain insight into your models! When fitting any statistical model, there are many useful pieces of information that are simultaneously calculated and stored beyond coefficient estimates and general model fit statistics. An example of this with one categorical and one continuous covariate on each parameter is below:. org This document is intended to assist individuals who are 1. R makes it easy to fit a linear model to your data. A formula expression in conventional R linear modelling syntax. R help archive by subject. Several built-in parametric distributions are available. insight mainly revolves around two types of functions: Functions that find (the names of) information, starting with find_, and functions that get the. , ## dist = "weibull") ## ## Estimates: ## est L95% U95% se ## shape 3. The second part of the survival model consists of the covariates. If this is called "dist", for example, then there must be a. Insulin and IGF signaling (IIS) is a complex system that controls diverse processes including growth, development, metabolism, stress responses, and aging. 我已经使用flexsurvreg来估计威布尔分布的参数，并得到以下输出。我想重建生存函数来估计给定时间t的生存率。 flexsurvreg（式. There are 3 examples. Data may be right-censored, and/or left-censored, and/or left-truncated. the type of survival curves. a Kaplan Meier curve). There are still 8 subjects at risk at the beginning of day 46, and this is shown on the table. R Development Page Contributed R Packages. # # TITLE: Conditional Probability Curves of Event Time Distributions # AUTHOR: Takahiro Hasegawa # ORIGINAL DATE: June 8, 2016 # MODIFIED DATE: # REFERENCE: Uno H, Hasegawa T, Cronin AM, Hassett MJ. The response must be a survival object as returned by the Surv function, and any covariates are given on the right-hand side. In this way it does not aim to supplement the modelling strategies found in mstate, msm, or flexsurv, but rather provide tools for subsequent analysis. Data Science Stack Exchange is a question and answer site for Data science professionals, Machine Learning specialists, and those interested in learning more about the field. For example, to model the RF → DM transition, patients who experienced LR or death prior to DM were treated as being censored at the time of the earlier competing event. Multi-state models for time-to-event data can also be fitted with the same functions. This code is quite time consuming, so please be patient. Credit risk assessment using survival analysis for progressive right-censored data: a case study in Jordan May 2017 Journal of Internet Banking and Commerce 22(1):1-18. BREAKING CHANGES. The main model-fitting function, flexsurvreg, uses the familiar syntax of survreg from the standardsurvivalpackage (Therneau 2016). Comparing the results from flexsurvreg with survreg, we see that the estimates are identical for all models. fit(X, Y, weights, offset, init = init, controlvals = control, : Ran out of iterations and did not converge what i did is fallowing. Reliability Basics: Utilizing Residual Plots in Accelerated Life Testing Data Analysis. flexsurvreg) to assess goodness-of-fit. note to exams office: when making copies please make single collated exam paper per student. These are location-scale models for an arbitrary transform of the time variable; the most common cases use a log transformation, leading to accelerated failure time models. , geom_point would get arguments like point. 0 dated 2018-08-30. flexsurvreg is intended to be easy to extend to handle new distributions. x can also be a list of flexsurvreg models, with one component for each per-mitted transition in the multi-state model, as illustrated in msfit. 08 trillion on consumer credit (e. Any parametric time-to-event distribution may be fitted if the user supplies a probability density or hazard function, and ideally also their cumulative versions. Data Science Stack Exchange is a question and answer site for Data science professionals, Machine Learning specialists, and those interested in learning more about the field. flexsurvreg to return a tidy data frame. A copy of the function call, for use in post-processing. anu sem1 end of semester, 2016 supplementary p2 part questions. In a typical exploratory data analysis workflow, data visualization and statistical modeling are two different phases: visualization informs modeling, and modeling in its turn can suggest a. dweibullPH and related functions give the Weibull distribution in proportional hazards parameterisation, and "weibullPH" is supported as a built-in model for flexsurvreg. a Kaplan Meier curve). flexsurvreg for the required form of the model and the data. The parameterizations of these distributions in R are shown in the next table. Questions Data inspection. Besides the basics of using SPSS, you learn to describe your data, test the most frequently encountered hypotheses, and examine relationships among variables. lung dataset : measures survival in patients with advanced lung cancer from the North Central Cancer Treatment Group. flexsurv::flexsurvreg(formula = missing_arg(), data = missing_arg(), weights = missing_arg()) survival Note that model = TRUE is needed to produce quantile predictions when there is a stratification variable and can be overridden in other cases. A probability forecast refers to a specific event, such as there is a 25% probability of it raining in the next 24 hours. 00791 N = 338, Events: 229, Censored: 109 Total time at risk: 1913. Observed outcome variables. x can also be a list of flexsurvreg models, with one component for each per-mitted transition in the multi-state model, as illustrated in msfit. Data may be right-censored, and/or left-censored, and/or left-truncated. The aes argument stands for aesthetics. A short course on Survival Analysis applied to the Financial Industry 3. 1 De nitions: The goals of this unit are to introduce notation, discuss ways of probabilisti-cally describing the distribution of a ‘survival time’ random variable, apply these to several common parametric families, and discuss how observations of survival times can be right. rm(list=ls()) require(survival) require(flexsurv) require(doParallel) no_cores <- detectCores() - 1. These are location-scale models for an arbitrary transform of the time variable; the most common cases use a log transformation, leading to accelerated failure time models. Insulin and IGF signaling (IIS) is a complex system that controls diverse processes including growth, development, metabolism, stress responses, and aging. Ecology and Epidemiology Comparison of the Gompertz and Logistic Equations to Describe Plant Disease Progress R. (This article was first published on R - Win-Vector Blog, and kindly contributed to R-bloggers) "R is its packages", so to know R we should know its popular packages (). 4 Weibull 分布. Missov1,2 Adam Lenart3 Laszlo Nemeth1 Vladimir Canudas-Romo3 James W. To define a new distri- bution for use in flexsurvreg, construct a list with the following elements:. A list of class "flexsurvreg" containing information about the fitted model. For example, in August 2015, the consumers in USA own$11. The Cox cumulative hazards were replaced with parametric equivalents and were used as arguments in the mssample function for prediction purposes. MAJOR CHANGES. 2 heemod: Models For Health Economic Evaluation in R Where X is a vector2 giving the probability of being in a given state at the start of the model, and Tt is the product of multiplying t matrices T. x <- c(1175, 1175, 1521, 1567, 1617, 1665, 1665, 1713, 1761, 1953). In flexsurv, input data for prediction can be specified by using the newdata argument in summary. degree in Mathematics and Satistics Department of mathematics Faculty of Science University of Ottawa ⃝c Blanche Nadege Nhogue Wabo, Ottawa, Canada, 2013. " - Anonymous correpondent (misfiring compliment). Package 'flexsurv' flexsurvreg fits parametric models for time-to-event (survival) data. In order to assess the assumption of a common shape parameter among the data obtained at various stress levels, the likelihood ratio (LR) test can be utilized. Standard survival distri-butions are built in, including the three and four. fit(X, Y, weights, offset, init = init, controlvals = control, : Ran out of iterations and did not converge what i did is fallowing. The mixture and non-mixture cure models from flexsurvcure can be also be used and are very appropriate for long-term survival estimation. edu Spring, 2001; revised Spring 2005, Summer 2010 We consider brie y the analysis of survival data when one is willing to assume a parametric form for the distribution of survival time. hesim currently supports parametric (exponential, Weibull, Gompertz, gamma, log-logistic, lognormal, and generalized gamma), splines, and fractional polynomial survival models (see params_surv). packages, flexsurvreg, cmprsk, survival, and rms. This can be a convenient/faster alternative to summary. flexsurvreg クラスは logLik() や AIC() には対応しているものの、deviance() や anova() には対応していないようである。 以下は phreg の結果から AIC を引き出すためのメソッド関数を私が自作したものなので、コピペして使ったらよい。. The "flexsurv" package for flexible parametric survival models, including splines, generalized gamma / F, and extensible to user-defined models. But the Cox models with time-change covariates are not easy to understand or visualize. Credit risk assessment using survival analysis for progressive right-censored data: a case study in Jordan May 2017 Journal of Internet Banking and Commerce 22(1):1-18. Title: Multivariate Response Generalized Linear Models Description: Provides functions that (1) fit multivariate discrete distributions, (2) generate random numbers from multivariate discrete distributions, and (3) run regression and penalized regression on the multivariate categorical response data. Takes a survival model estimated with covariates (from survfit, flexsurvreg, or other supported functions) and sets the covariate values for which survival projections will be used. List defining the survival distribution used. with $\text{coefficient}$ being the covariate coefficient returned in the flexsurvreg output; $\text{covariate}$ being the covariate value for which I intend to construct the function and $\mu$ the mean value of the covariate over the fitting sample. The best answers are voted up and rise to the top. In flexsurv, input data for prediction can be specified by using the newdata argument in summary. Cox regression (or proportional hazards regression) is method for investigating the effect of several variables upon the time a specified event takes to happen. note to exams office: when making copies please make single collated exam paper per student. loglogistic distributionは（survregとは対照的に）は組み込まれていませんが、簡単に焼き立てることができます（flexsurvregの例を参照）。 私はそれをあまりテストしていませんが、 flexsurv は survival の良い代替手段のようです。. Use simulated Gompertz random number to test flexsurv Gompertz and Weibull fitting results Summary: I experimented the sample size. Understanding the Cox Regression Models with Time-Change Covariates Mai Zhou University of Kentucky The Cox regression model is a cornerstone of modern survival analysis and is widely used in many other fields as well. All parametric survival models were run using the procedure 'flexsurvreg' in the package 'flexsurv' in R (v. Note that, when used inappropriately, statistical models may give. Predictions. Gompertz-Cox Regression •Distribution -Gompertz distribution. 5 Adjusting Survival Curves From a survival analysis point of view, we want to obtain also estimates for the survival curve. For the Weibull (and exponential, log-normal and log-logistic) distribution, flexsurvreg simply acts as a wrapper for survreg: The maximum likelihood estimates are obtained by survreg, checked by flexsurvreg for optimization convergence, and converted to flexsurvreg's preferred parameterization. Option to summary. tail=FALSE: Jan 26, 2018: codecov. 1 左側截尾數據 left-truncation. degree in Mathematics and Satistics Department of mathematics Faculty of Science University of Ottawa ⃝c Blanche Nadege Nhogue Wabo, Ottawa, Canada, 2013. Sign up to join this community. Or copy & paste this link into an email or IM:. R: A language and environment for statistical. Estimating the Baseline Function using flexsurvreg package. Important note for package binaries: R-Forge provides these binaries only for the most recent version of R, but not for older versions. org This document is intended to assist individuals who are 1. You're going to have to tell us a little more. Within that library, the command survreg fits parametric survival models. Hedge funds and Survival analysis by Blanche Nadege Nhogue Wabo Thesis submitted to the Faculty of Graduate and Postdoctoral Studies In partial fulfillment of the requirements For the M. In a typical exploratory data analysis workflow, data visualization and statistical modeling are two different phases: visualization informs modeling, and modeling in its turn can suggest a. Or copy & paste this link into an email or IM:. 7 dated 2016-02-17. 4 Weibull 分布. x can also be a list of flexsurvreg models, with one component for each per-mitted transition in the multi-state model, as illustrated in msfit. This test applies to any distribution with a shape parameter. Viewed 1k times 2. 1 De nitions: The goals of this unit are to introduce notation, discuss ways of probabilisti-cally describing the distribution of a ‘survival time’ random variable, apply these to several common parametric families, and discuss how observations of survival times can be right. in Cost-effectiveness Analyses: A Comparison. Censoring or left-truncation are specified in 'Surv' objects. md: Changed badges to point to chjackson/flexsurv-dev instead of jrdnmdhl… Apr 10, 2020: TODO: Bug fix for qllogis with lower. Of particular interest is the accuracy of the estima-. Insulin and IGF signaling (IIS) is a complex system that controls diverse processes including growth, development, metabolism, stress responses, and aging. The "flexsurv" package for flexible parametric survival models, including splines, generalized gamma / F, and extensible to user-defined models. loglogistic distributionは（survregとは対照的に）は組み込まれていませんが、簡単に焼き立てることができます（flexsurvregの例を参照）。 私はそれをあまりテストしていませんが、 flexsurv は survival の良い代替手段のようです。. The models are fitted by maximizing the full log-likelihood, and estimates and confidence intervals for any function of the model parameters can be. But the Cox models with time-change covariates are not easy to understand or visualize. When creating a CTSTM from a flexsurvreg object, the user must simply set the argument point_estimate = FALSE and choose the number of samples of the parameters to draw. 1 Notation. Anybody can ask a question. 5000 simulations were used with the mssample function to sample paths from the multi-state model. 0 Date: 2016-05-10. In this example, we fit a Weibull model to. Thus cβ is the log hazard ratio when the covariate value increases by c units. Hello R users, I'm trying to do simulations for comparing cox and weibull I have come across this problem: Warning messages: 1: In survreg. This is a package in the recommended list, if you downloaded the binary when installing R, most likely it is included with the base package. The routine for generating initial values in flexsurvspline has been improved. 30 October 2019 16. A lot of functions (and data sets) for survival analysis is in the package survival, so we need to load it rst. ## ----setup, include = FALSE----- library(knitr) library(kfigr) opts_chunk\$set(comment = NA, fig. Regression for a Parametric Survival Model Description. The parameterizations of these distributions in R are shown in the next table. Custom distributions. Package MGLM updated to version 0. Analyses were conducted using the package flexsurvreg in R software (R Development Core Team, Vienna, Austria) 43 R Core Team. The Weibull distribution with shape parameter a and scale parameter b has density given by.
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https://bobsegarini.wordpress.com/tag/pete-ham/
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## Frank Gutch Jr: The Best of 2012, Vinylly— The Shoes!, and Notes…..
Posted in Opinion with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on November 28, 2012 by segarini
Whaaa-a-a-at, you say? 2012 ain’t over yet? You’re right. It ain’t. But if the Mayans have it right, it will soon be all over so if you don’t mind the indulgence I’ve decided to do what I do every year— post my list of albums which have floated to the top a month early. I do it for a couple of reasons. One, I hate for my list to get mixed in with the rest of those end-of-the-year lists which swarm late December and early January. The timing is all too predictable and if there is anything I don’t like, it’s predictability (which is why I don’t go gaga every time Keith Richards adds another day to his fossilized remains or Mick Jagger farts).
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2023-03-26 16:13:17
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https://www.cheenta.com/two-similar-triangles-isi-b-math-b-stat-entrance-2018-subjective-solution-of-problem-no-2/
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Select Page
##### Source of the problem
I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2018. Subjective Problem no. 2.
5.5 out of 10
##### Suggested Book
‘Challenge and Thrill of Pre-College Mathematics’ by V,Krishnamurthy, C.R.Pranesachar, ect.
Do you really need a hint? Try it first!
$PQ$ and $RS$ are two chords of the circle $C$ , intersecting at the point $O$. See figure: click here.
Given $PO=3$ cm $SO=4$ cm $[\triangle POR]= 7 cm^2$.
From the triangles $POS$ and $QOS$ we have, $\angle POR=\angle SOQ$ [Opposite angles] $\angle SRP=\angle SQO$ [Angle on the same semi-circle $STP$] $\angle QSO= \angle OPR$ [Angle on the same semi-circle $ST’P$] Therefore the $\triangle POR$ and $\triangle SOQ$ are similar triangles .
$\frac{[\triangle POR]}{OP^2}=\frac{[\triangle SOQ]}{SO^2}.$ $\Rightarrow [\triangle SOQ]=\frac{SO^2}{PO^2}\cdot [\triangle POR]$=$\frac{4^2}{3^2}\cdot 7=12\frac{4}{9}$.(Ans.)
# I.S.I. & C.M.I. Entrance Program
Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.
The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.
## Testing of Hypothesis| ISI MStat 2016 PSB Problem 9
This is a problem from the ISI MStat Entrance Examination,2016 making us realize the beautiful connection between exponential and geometric distribution and a smooth application of Central Limit Theorem.
## ISI MStat PSB 2006 Problem 8 | Bernoullian Beauty
This is a very simple and regular sample problem from ISI MStat PSB 2009 Problem 8. It It is based on testing the nature of the mean of Exponential distribution. Give it a Try it !
## How to roll a Dice by tossing a Coin ? Cheenta Statistics Department
How can you roll a dice by tossing a coin? Can you use your probability knowledge? Use your conditioning skills.
## ISI MStat PSB 2009 Problem 8 | How big is the Mean?
This is a very simple and regular sample problem from ISI MStat PSB 2009 Problem 8. It It is based on testing the nature of the mean of Exponential distribution. Give it a Try it !
## ISI MStat PSB 2009 Problem 4 | Polarized to Normal
This is a very beautiful sample problem from ISI MStat PSB 2009 Problem 4. It is based on the idea of Polar Transformations, but need a good deal of observation o realize that. Give it a Try it !
## ISI MStat PSB 2008 Problem 7 | Finding the Distribution of a Random Variable
This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 7 based on finding the distribution of a random variable. Let’s give it a try !!
## ISI MStat PSB 2008 Problem 2 | Definite integral as the limit of the Riemann sum
This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 2 based on definite integral as the limit of the Riemann sum . Let’s give it a try !!
## ISI MStat PSB 2008 Problem 3 | Functional equation
This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 3 based on Functional equation . Let’s give it a try !!
## ISI MStat PSB 2009 Problem 6 | abNormal MLE of Normal
This is a very beautiful sample problem from ISI MStat PSB 2009 Problem 6. It is based on the idea of Restricted Maximum Likelihood Estimators, and Mean Squared Errors. Give it a Try it !
## ISI MStat PSB 2009 Problem 3 | Gamma is not abNormal
This is a very simple but beautiful sample problem from ISI MStat PSB 2009 Problem 3. It is based on recognizing density function and then using CLT. Try it !
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2020-09-19 15:56:32
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https://datascience.stackexchange.com/questions/24592/do-convolution-layers-in-a-cnn-treat-the-previous-layer-outputs-as-channels
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# Do Convolution Layers in a CNN Treat the Previous Layer Outputs as Channels?
Lets say you have a max pooling layer that gives 10 downsampled feature maps. Do you stack those feature maps, treat them as channels and convolve that 'single image' of depth 10 with a 3d kernel of depth 10? That is how I have generally thought about it. Is that correct?
This visualization confused me: http://scs.ryerson.ca/~aharley/vis/conv/flat.html
On the second convolution layer in the above visualization most of the feature maps only connect to 3 or 4 of the previous layers maps. Can anyone help me understand this better?
Related side question: If our input is a color image our first convolution kernel will be 3D. This means we learn different weights for each color channel (I assume we aren't learning a single 2D kernel that is duplicated on each channel, correct)?
• It seems like this is some non-standard architecture; it is convolving several 3d kernels of different sizes with the first downsampled layer. Is there a description of this architecture somewhere? Nov 11, 2017 at 1:50
• Is what I describe the typical case then? Here is a link to the paper. Though I am mostly just wanting to understand what typically occurs. Nov 11, 2017 at 1:55
• That paper is kind of weird; it describes two different networks, neither of which seems to pertain to the network being displayed. Nov 11, 2017 at 2:03
Yes. The usual convention in a CNN is that each kernel is always the same depth as the input, so you can also think of this as a "stack" of 2D kernels that are associated with the input channels and summed to make one output channel - because under the convention that $N_{in\_channels} = N_{kernel\_depth}$ this is mathematically the same. Expressing as a 3D convolution allows for simpler notation and code.
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2022-08-18 01:46:37
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http://kdsp.chicweek.it/net-ionic-equations-extension-questions-pogil-answers.html
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COMPLETE TEXT BOOK SOLUTION WITH ANSWERS INSTANT DOWNLOAD SAMPLE QUESTIONS Chemistry Ninth Edition Steven S. The titrant was 0. Academic year. You could not by yourself going later than book buildup or library or borrowing from your friends to right to use them. Description: With the Half-Life Laboratory, students gain a better understanding of radioactive dating and half-lives. srcl2 aq | srcl2 aq | srcl2 aqueous | srcl2 aq +naoh aq | srcl2 aq +h2so4 aq | srcl2 aq +na2co3 aq | srcl2 aq + crso4 aq | srcl2 aq + 2agno3 aq | srcl2 aq and l. As understood, achievement does not recommend that you have astonishing points. Mejías Arias, Rosario Martínez Herrero, Julio Serna Galán,. How does one "take up the residue in the minimum amount of water and decolourize with carbon"?. Why? The ions need to be able to move in order to conduct the current. We then write the solubility product expression for this reaction. A net ionic equation also gives information on how different substances react. Formation of Ionic Compounds 1. Net ionic equation: The net ionic equation is an extension of the ionic equation. combustion or. Home Answers Neuron Function Pogil Answers. The reason I ask is because apparently the molecular equation $$\ce{H2SO4 + 2KOH -> 2H2O + K2SO4}$$ has the net-ionic form $$\ce{H+ + OH- -> H2O}$$ It would be helpful if you could describe in general how I should write the dissociation of the strong acids in the net ionic equation. Question Answer; 11. pogil activities for high school chemistry worksheet answers generated on lbartman. Monday, October 29. • Pay attention! Not seen in class but is on test; it’s a simple extension. So if you wanna go from a complete ionic equation to a net ionic equation, which really deals with the things that aren't spectators, well you just get rid of the spectator ions. Remaining is the net ionic equation, containing only the ions that changed in some way, showing the active pieces in the reaction. Naming Ionic Pounds POGIL Answer Key. Each question is worth 5 points. ap_POGIL_Bond Energy answers. LAST UPDATE: Saturday, December 13, 2008 08:23 AM CLASS SCHEDULE AND ASSIGNMENTS. Rainy_violet's Shop. How to complete a POGIL This video explains and demonstrates how to complete a POGIL, process oriented, guided inquiry lesson. compared to F. cooked egg; burning a candle. If G, otherwise known as Gibbs free energy, is less than 0 then the reaction will be there is now a net loss of energy, and buy in your last. Chemistry Pogil Calorimetry Answer Key plus it is not directly done, you could endure even more nearly this life, approximately the world. I wanted to get the lab to a better place so that students aren’t overwhelmed with doing a lot, but rather that they learn a lot and can apply what they’ve learned outside of lab. What element was oxidized in the. Nuclear induction. As understood, achievement does not recommend that you have astonishing points. In this reaction oxygen is acting as the oxidizing agent. Roseville, CA 95747-7100 Toll Free 800-772-8700. 2 (g) A 2Fe. COMPLETE TEXT BOOK SOLUTION WITH ANSWERS INSTANT DOWNLOAD SAMPLE QUESTIONS Chemistry Ninth Edition Steven S. and calcium hydroxide calcium iodide CaI2 magnesium hydroxide and hydrosulfuric acid. 12 Past Management, Surface Soil Nitrogen Properties, and Net Mineralization Rate'of Mineralizable N for Various Soils. Net Ionic Equations - SCHOOLinSITES. The instructor will answer questions from the leader only. Protein Structure POGIL Activities. Balancing Equations POGIL Answer Key. Using Model 2, answer the questions below about this long-term storage. Get solubility pogil answer key chemistry PDF file for free from our online library PDF File: solubility pogil answer key chemistry. You could not by yourself going later than book buildup or library or borrowing from your friends to right to use them. Question 1: If the charges are equal on each side of the membrane, will the net flow of potassium ions (and thus the net current) be into the cell, out of the cell, or balanced in both directions? Please write down your answer, and check your answer here. Calculate the answers to the appropriate number of significant figures. Net Ionic Equation, Balancing Equations, Stoichiometry, Redox Reaction, Classification of Reactions | High School Lab: Inquiry Redox Investigation. Make sure you have the method down. Wednesday, Feb 19. A net ionic equation leaves the spectator ions out. Suppose you wish to express the average rate of the following reac-tion during the time period beginning at time t 1 and ending at time t 2. Extension Questions. Write chemical formulas under the names of the substances in the word equation. Unit #4 Review: Quiz on Thursday Friday 2/24. One end of the horizontal spring is held on a fixed vertical axle, and the other end is attached to a puck of mass m that can mow without friction over a horizontal surface. This takes up the bulk of first year chemistry classes and will actually be divided over several units: Chemical Reactions (this unit) Stoichiometry (next unit) Reaction Energy (later units) Reaction Rates and Equilibria (later units) This unit, which is the first in that list,…. NOTES: Type of rxns summary. polyatomic ions pogil answer key. Identify the main type of bonding and. Discuss with your group members how your answers to Questions 16 and 18 could be used to calculate the enthalpy change (AH) for the reaction in Model 3, and then do the calculation. I mean, sure, there are numbers and formulas and stuff,…. This is just one of the solutions for you to be successful. Using Model 2, answer the questions below about this long-term storage. • Pay attention! Not seen in class but is on test; it’s a simple extension. The vapor pressure of water is well known, and dependent only on the temperature of the liquid water sample. Compare the net ionic equation in Model 2 to the other two equations. Consider the conductivity data shown in Model 1 and the ionization data in Question 3. Chem 115 POGIL Worksheet - Week 5 - Answers Limiting Reagents, Solubility, and Solution Reactions Key Questions & Exercises 1. Displaying top 8 worksheets found for - Redox Pogil. • A net ionic equation an equation that only shows the ions that undergo changes during a chemical reaction. So, you just need to create a set of algebraic equations expressing the number of atoms of each element involved in the reaction and solve it. In addition the NH4 cation does not form a precipitate. Get Started. Pre lab 7 solubility and net ionic equations ap chemistry mrs spencer chapter 4 practice test solved complete and balance each of the following equatio quiz worksheet chemical equations on the ap chemistry Pre Lab 7 Solubility And Net Ionic Equations Ap Chemistry Mrs Spencer Chapter 4 Practice Test Solved Complete And Balance Each Of The Following Equatio…. 8 Packet, Ch. Includes topics, chapters in the text, problem sets, items to memorize, and labs. Carbohydrate has been done for you. Life cannot exist in a completely closed system (no energy or matter comes into or out of the system). Global Economic Trends Guided Answers advanced engineering mathematics zill 4th solution manual, annotations to finnegans wake roland mchugh, ford fiesta workshop manual download, investment science or solution, daihatsu dm954dt engine, 1976 fxe harley davidson super glide manual,. 0 M HCl to this buffer until a total of 101 mL of 1. 175 Longwood Road South Hamilton, ON L8P 0A1: Phone: 844-200-1455: Fax: Email: [email protected] Acids And Bases Pogil Answer Key. Ions Pogil Answer Key Chemistry | checked. nano3 cacl2 | cacl2 + nano3 | cacl2 and nano3 balanced net ionic equation | cacl2 and nano3 reaction | nano3 calit2 | nano3 + cacl2. 0001 700000 350. Notes: Chemical Equations and Types of Reactions 4. Net Ionic Equation, Balancing Equations, Stoichiometry, Redox Reaction, Classification of Reactions | High School Lab: Inquiry Redox Investigation. concept are Pogil answer key polyatomic ions, Pogil lesson plan, Chem 115 pogil work 06, Pogil chemistry activities, Chem 116 pogil work, This activity has been password protected to prevent, Isotopes, Net ionic equation work answers Pogil Ions Worksheets. Some of the worksheets for this concept are Chapter 20 work redox, Chem 116 pogil work, Chem 115 pogil work, Work 25, Chapter 4 work 3 molecular net ionic equations, Academic resource center, Net ionic equation work answers, Coslet ap chemistry. Water temperature (°C) Vapor pressure (mmHg) 20 17. Net Ionic Equation: Everything that participates in the reaction gets recorded in the net ionic equation. Na 2 CO 3 (aq) + FeCl 2 (aq) FeCO 3. Calculate the energy needed to break one mole of P–Cl bonds in Reaction B. Based on your answer to Question I l, write balanced chemical equations for HNO and HNO reacting with water as they are mixed into an aqueous solution. Behavioral interviews really are a new style of interviewing. ANSWER KEY. O P O P OH O NH. The complete ionic equation shows all of the ions that dissociate and the compounds that stay together (solids, liquids, & gases). *** Balance chemical equations. Monday, Feb 17. Drawing atoms Worksheet Answer Key Awesome Bohr Model Worksheet Answers Tecnologialinstante. The ions in an ionic solid. Reviewing the Model: 9. uozaovavl ou aq ppzom '(uonrqos UI urn1souBE111 -10 uopnps 1. Molecular: AgNO3 (aq) + KCl (aq) AgCl (s) + KNO3 (aq) Total Ionic: Ag + (aq) + NO. ----- Pollution is contamination of soil, water and air as a result of Human activities renewable resource. pdf NOTES: Chpt 9 Notes (Stoich). This is an definitely easy means to specifically get lead. Keeping in mind that the sum of the charges in an ionic compound must equal zero, use the chemical formulas in Model 3 to answer the following questions: Identify the charge on the copper cations in copper(I) oxide and copper(II) oxide, respectively. The net ionic equation models just the formation of the precipitate from its ions and removes the spectator ions Essential Question How is the phenomenon of precipitate formation explained on the atomic level? Activity Objectives 1. POGIL Batteries Answer key for classwork pgs 13 to 17 Molarity, Solution Stoichiometry and Dilution Problem This example shows three different types of ways a solution stoichiometry question can be asked, using molarity, [eBooks] Pogil Answer Key To Chemistry Activit pogil-answer-key-to-chemistry-activit 1/5 PDF Drive - Search and download PDF. Na 2 CO 3 (aq) + FeCl 2 (aq) FeCO 3. Net Ionic Equations Sample Questions. Net Ionic Equations Sample Questions. Life cannot exist in a completely closed system (no energy or matter comes into or out of the system). Suppose you wish to express the average rate of the following reac-tion during the time period beginning at time t 1 and ending at time t 2. How did he determine this charge based on his experimentation? Model 3: Rutherford In 1911, the model of the atom changed once again after Ernest Rutherford discovered the nucleus. Read Free Pogil Acids Bases Answer Key Pogil Acids Bases Answer Key Page 1/7. Crayfish dissection Biology Junction from Crayfish Dissection Worksheet Answers, source:biologyjunction. (Spectator ions are omitted from net ionic equations. Mg + ZnCl2 = MgCl + Zn - Chemical Equation Balancer. txt) or read online for free. Finish Net Ionic Equations. 2 Mg + ZnCl 2 → 2 MgCl + Zn. Calculate the answers to the appropriate number of significant figures. 3 g/mol 8) UF6 352 g/mol 9) SO2 64. HW: Solubility Worksheet. Consider the equations in Model 2. It is described by Hooke’s law with spring constant 4. 9, Write ionic and net ionic equations for the remaining reactions in. Make sure your net ionic equation is properly balanced. active vibration control bearing. 201 Textbook. Peterson’s Master AP Chemistry was designed to be as user-friendly as it is complete. Do the P–Cl bonds in different molecules require the same amount of energy to break? Read This! The energies you calculated in Questions 5c and 6a above are called bond energies. Answer questions: Measuring the rate of reaction. compared to F. Finally, the AI reviews the problem on the board and shares the correct answer to ensure understanding. Give the name and the formula of the ionic compound produced by neutralization reactions between the following acids and bases: Acid and Base reactants Name of ionic compound Formula nitric acid and sodium hydroxide sodium nitrate NaNO3 hydroiodic acid. org Created Date: 10/5/2017 9:35:34 AM Mole Ratios Pogil Extension Answers We provide you this proper as with ease as simple. University of Technology Sydney. net ionic equation for the reaction between Ca(NO3)2 and Na2SO4?. The charge was given incorrectly in another answer. Some of the worksheets displayed are Chapter 20 work redox, Chem 116 pogil work, Chem 115 pogil work, Work 25, Chapter 4 work 3 molecular net ionic equations, Academic resource center, Net ionic equation work answers, Coslet ap chemistry. After reading Lesson 7. (Spectator ions are omitted from net ionic equations. Naming Ionic Pounds POGIL Answer Key. ----- Pollution is contamination of soil, water and air as a result of Human activities renewable resource. If you set sights on to download and install the. to review your particular question. In this reaction oxygen is acting as the oxidizing agent. The barium content of the salt can be determined by gravimetric methods. LAST UPDATE: Saturday, December 13, 2008 08:23 AM CLASS SCHEDULE AND ASSIGNMENTS. If you're anything like me (and pray that you aren't), one of your favorite things in the whole world is to name chemical compounds. You get rid of that. Showing top 8 worksheets in the category - Pogil. All of the answers can be found online for almost every single POGIL Currently, there is a significant amount of discussion on teaching list serves about the frustration of people posting answer keys online and students checking the internet instead of doing the work. I mean, sure, there are numbers and formulas and stuff,…. pdf---ANSWERS. We can find the net ionic equation using the following steps:. PRACTICE PROBLEMS ON NET IONIC EQUATIONS page 1 of 3 Show the complete ionic and net ionic forms of the following equations. Right on top. Half-Life : Paper, M&M’s, Pennies, or Puzzle Pieces. Chemistry Pogil Answer Key Gas Variables Chemistry Pogil Answer Key Gas Getting the books Chemistry Pogil Answer Key Gas Variables now is not type of inspiring means. These behavioral interviews are becoming added and further regular nowadays. The right answer. Key Content Questions: 2. Neutral amino acids O NH 2 H 3 C OH O NH 2 HO OH O O HO OH NH 2 O NH 2 H 2 N OH O. doc View Download 45k: v. Chemistry worksheet naming ionic pounds printable worksheet names of ionic pounds key printable worksheets naming binary ionic compounds worksheets worksheets samples. Explain why this is true based on the Second Law of Thermodynamics. 2 Energy changes in a system can be described as endothermic and exothermic processes such as the. Similarly, chemists classify chemical equations according to their. Showing top 8 worksheets in the category - Pogil. Ethanol is one example of alternative fuels for powering our cars and trucks. The solubility lab is where students develop the concept of soluble and insoluble and where they learn to write net ionic equations. My answer was Fe^1+ + OH^1- = FeOH. Why? The ions need to be able to move in order to conduct the current. It provides a worksheet full of examples and practice. carbon sink. It provides a worksheet full of examples and practice. What element was oxidized in the. To identify the products formed in these reactions and summarize the chemical changes in terms of balanced chemical equations and net ionic equations. DA: 51 PA: 60 MOZ Rank: 50. 2650 g) was dissolved in water (200 mL) and excess sulfuric acid added. Precipitate, Balancing Equations, Solubility Rules | High School Animation: Net Ionic Equations Animation. Write the total-ionic and net-ionic equations for the above reaction. (a) Fe(C 2H 3O 2) 3 + Ca(OH) 2 o (b) KI + Pb(NO 3) 2 o (c) MgSO 4 + AgNO 3 o POGIL 2005, 2006 5/6 Authored by Dr. combustion or. Author: KONICA MINOLTA bizhub PRO 1050 Created Date: 5/10/2017 1:15:39 PM. 9, Write ionic and net ionic equations for the remaining reactions in. Explain why it is valid to remove this species from the equation. Predicting Products PRactice ((Reference Sheet)) 4. Molecular and Ionic Equations –A net ionic equation is a chemical equation from which the spectator ions have been removed. Explain why this is true based on the Second Law of Thermodynamics. Naming Ionic Compounds Answer Key Pdf. These three situations are the results of the battles that took place in Question 4. Solomon-Bloembergen Equations In 1955, Solomon, building on the work of Bloembergen, Purcell, and Pound, developed a model using correlation times and spectral densities to explain and predict the T1 and T2 values of pure liquids and other simple substances. Consider the equations in Model 2. Similarly, chemists classify chemical equations according to their. 4Fe(s) + 3O 2 (g) → 2Fe 2 O 3 (s) a. Using the chlorine family of polyatomic ions as a model, predict the name of the BrO 4 1– ion. One end of the horizontal spring is held on a fixed vertical axle, and the other end is attached to a puck of mass m that can mow without friction over a horizontal surface. Use one of the methods in Model 3 that gave the correct answer for average atomic mass to calculate the average atomic mass for oxygen. start HW: 1. I just sit and name compounds all day long, happy in the knowledge that one day the world will need a compound naming guru to save our species. NaCl is a strong electrolyte when dissolved in water, but pure solid NaCl does not conduct electricity. You may find it useful to draw Lewis structures to find your answer. POGIL Batteries Answer key for classwork pgs 13 to 17 Molarity, Solution Stoichiometry and Dilution Problem This example shows three different types of ways a solution stoichiometry question can be asked, using molarity, [eBooks] Pogil Answer Key To Chemistry Activit pogil-answer-key-to-chemistry-activit 1/5 PDF Drive - Search and download PDF. write complete ionic and net ionic equations for chemical reactions in aqueous solutions from Model 1 to support your answer. •Recorder - In charge of summarizing the main points of. 4 POGIL™ Activities for High School Chemistry 15. Precipitate, Balancing Equations, Solubility Rules | High School Animation: Net Ionic Equations Animation. In this lab, students perform a simple redox reaction using an iron nail and copper(II) chloride solution. the conductivity of the solution and the strength of the electrolyte (acid strength). Naming Ionic Compounds Worksheet One Give the name of the following ionic compounds: 1) Na 2CO 3 _____ 2) NaOH _____. * Give your answers to the hundredths place (two places to the right of the decimal point). [email protected] 4 ™ Activities for AP* Chemistry POGIL 9. Molecular Equation Zn (s) + 2HCl (aq) → ZnCl 2 (aq) + H 2 (g) Ionic Equation Net Ionic Equation b. University. Model 2 – Hydrolysis of ATP. How might this affect the final empirical formula? 17. Global Economic Trends Guided Answers advanced engineering mathematics zill 4th solution manual, annotations to finnegans wake roland mchugh, ford fiesta workshop manual download, investment science or solution, daihatsu dm954dt engine, 1976 fxe harley davidson super glide manual,. Next: FeSO4+K3PO4. Net Ionic Equations POGIL. Ionic charges are not yet supported and will be ignored. Created Date: 10/5/2017 9:36:42 AM. worth 10 points each. CHAPTER3 PASSIVE MEMBRANE PROPERTIES, THE ACTION POTENTIAL, AND ELECTRICAL SIGNALING BY NEURONS SYNAPTIC TRANSMISSION NEUROCHEMICAL TRANSMISSION THE MAINTENANCE OF NERVE CELL FUNCTION CHAPTER OUTLINE 1. Naming Ionic Compounds Worksheet One Give the name of the following ionic compounds: 1) Na 2CO 3 _____ 2) NaOH _____. The O*NET Interest Profiler helps you decide what kinds of careers you might want to explore. The learning objectives incorporate the knowledge and skills needed for students to answer the conceptual question. Pogil Isotopes Worksheet. He is committed to traditional approaches to knowledge and understanding, taught via, and in, digital environments. Can you answer the question without calculations? (Hint: What is the critical quantity here that s proportional to volume?. The barium content of the salt can be determined by gravimetric methods. Common Ionic Compounds List Naming Ionic Compounds Worksheet Answer Key Chemical Names and Formulas Worksheet Biology Organic Molecules Worksheet Review Chemistry Stoichiometry Worksheet Answer Key. At this point, the pH of the solution will be very acidic. Na3P04(aq) + + + +3 +. Extension: find the mass of precipitate formed 8. Net Ionic Drill ANSWERS. Use the data in Model 1 to answer the following questions. Finally, the AI reviews the problem on the board and shares the correct answer to ensure understanding. DA: 14 PA: 59 MOZ Rank: 46. You could not by yourself going later than book buildup or library or borrowing from your friends to right to use them. Nucleic acid— 2. In ionic bonding, one atom pulls strongly on the electron, while the pull of the other atom is very weak. Review Slides; Review Videos Topic 4. uozaovavl ou aq ppzom '(uonrqos UI urn1souBE111 -10 uopnps 1. This takes up the bulk of first year chemistry classes and will actually be divided over several units: Chemical Reactions (this unit) Stoichiometry (next unit) Reaction Energy (later units) Reaction Rates and Equilibria (later units) This unit, which is the first in that list,…. double replacement 2. 9, Write ionic and net ionic equations for the remaining reactions in. Example: The net ionic equation for the reaction that results from mixing 1 M HCl and 1 M NaOH is: H+(aq) + OH-(aq) --> H2O(l) The Cl- and Na+ ions do not react and are not listed in the net ionic equation. In this animation, students will witness a precipitate reaction on the particulate level to understand why a net ionic equation represents what happens in these reaction types. naming ionic compounds worksheet pogil naming ionic compounds worksheet answers pogil naming ionic compounds worksheet pogil activities for high school chemistry answers. The charge was given incorrectly in another answer. think of where the subtraction sign is in the equation. I wanted to get the lab to a better place so that students aren't overwhelmed with doing a lot, but rather that they learn a lot and can apply what they've learned outside of lab. worth 10 points each. During each session we will try do the following: 1. This is an area that has captured the attention of scientists and public alike because of its vast and complex potentials. Monday, Feb 17. concept are Pogil answer key polyatomic ions, Pogil lesson plan, Chem 115 pogil work 06, Pogil chemistry activities, Chem 116 pogil work, This activity has been password protected to prevent, Isotopes, Net ionic equation work answers Pogil Ions Worksheets. Ethanol is one example of alternative fuels for powering our cars and trucks. Often in the foundation exams the equations are given, or the variables are named and you have to write the equation using them; or you may need to choose an equation from the Physics equation sheet given in the exam. Example 1: Write each sentence as an algebraic equation. Reaction Information. Experts like you can vote on posts, so the most helpful answers are easy to find. Henry Jakubowski. Notice that in writing the net ionic equation, the positively-charged silver cation was written first on the reactant side, followed by the negatively-charged chloride anion. Chemistry Pogil Calorimetry Answer Key plus it is not directly done, you could endure even more nearly this life, approximately the world. Global Economic Trends Guided Answers advanced engineering mathematics zill 4th solution manual, annotations to finnegans wake roland mchugh, ford fiesta workshop manual download, investment science or solution, daihatsu dm954dt engine, 1976 fxe harley davidson super glide manual,. Q1/ For the reaction of aluminum sulfate with barium chloride , write the following a / blances chemical equation ? b / ionic equation ? c / spectator ions ? d / net ionec equation ? Q2 / a neutralization reaction between an acid and base is a common method of preparing useful salts , give net ionic equation showing how (NH4 )2 ,HPO4 could be prepared ? Q3 / write an acceptable value for each. Solubility Pogil Answers Solubility Pogil Answers Yeah, reviewing a books Solubility Pogil Answers could build up your close contacts listings. Written by Aaron Keller. Extension Questions Ethanol is one example of alternative fuels for powering our cars and trucks. cations ____ 2. dissolved in water liquid solid gas. Na 2 CO 3 (aq) + FeCl 2 (aq) FeCO 3. Lab: Formation of a Precipitate. Question 1: If the charges are equal on each side of the membrane, will the net flow of potassium ions (and thus the net current) be into the cell, out of the cell, or balanced in both directions? Please write down your answer, and check your answer here. Write the net ionic equation for this reaction. The outer shell of the carbon atom contains four electrons. An aqueous solution is a solution in which the solvent is water. On this page you can read or download pogil net ionic equations answers in PDF format. POGIL Chemical Reactions Answers Net Ionic Equation Worksheet and Answers This chemistry video tutorial focuses on how to write net ionic equations. redox or single replacement 3. POGIL Batteries Answer key for classwork pgs 13 to 17 Molarity, Solution Stoichiometry and Dilution Problem This example shows three different types of ways a solution stoichiometry question can be asked, using molarity, [eBooks] Pogil Answer Key To Chemistry Activit pogil-answer-key-to-chemistry-activit 1/5 PDF Drive - Search and download PDF. diphosphorus pentoxide + water → phosphoric acid a. Displaying top 8 worksheets found for - Redox Pogil. Behavioral interviews really are a new style of interviewing. Example 1: Write each sentence as an algebraic equation. Explain why this is true based on the Second Law of Thermodynamics. (Coefficients equal to one (1) do not need to be shown in your answers). 9/7 Friday: Homework: Read Chapters 1-3 in textbook. Reaction rates cannot be calculated from balanced equations. Many teachers, myself included, experience this frustration. Extension Questions. To balance a chemical equation, can we change the formula of either reactants or products? Question 15. uozaovavl ou aq ppzom '(uonrqos UI urn1souBE111 -10 uopnps 1. Stephen Prilliman. Draw the Lewis structure of ozone, O b. Roseville, CA 95747-7100 Toll Free 800-772-8700. 8 m/s² or 32. Feb 18, 2016 - This Pin was discovered by Jeff. Ionic bonds with answers worksheets lesson worksheets ionic compounds worksheet homeschooldressage naming ionic compounds practice worksheet answer key along naming. Kindle file format ionic reaction lab worksheet answers net ionic equation worksheet answers net ionic equation worksheet answers hudson k12 oh net ionic equation. Explain your answers in Question 10 in terms of the analogies developed earlier in the activity (i. The lattice energy (attractive force) of an ionic solid can be approximated using the Coulombic force equation shown below. Pogil Isotopes Worksheet. This can be linked to neuron function pogil answers. Smith’s best attempt to answer the questions in item 4, are presented below. Showing top 8 worksheets in the category - Pogil. Discussion: Ask students the following questions: What is energy? (Possible answers: The ability to do work or cause change and the capacity for vigorous activity. Net Ionic Equations POGIL. com show printable version !!! hide the show to save images bellow, right click on shown image then save as. pdf: File Size: 675 kb: Download File. Identify reactions that produce precipitates. Unit 3 Module C: Chemical Quantities Extension Questions 19. Start studying Gibbs free energy (bozeman video and pogil). Read This!. If a water molecule’s mass is 18. Extension Questions 1. It provides a worksheet full of examples and practice. Next: FeSO4+K3PO4. doc Author: Parents Created Date: 11/13/2013 8:37:37 PM. My answer was Tl2^1+ + Cl2^1- = Tl2Cl2 Next: (NH4)2CO3+Ca(ClO4)2. It was counted wrong. (a) Fe(C 2H 3O 2) 3 + Ca(OH) 2 o (b) KI + Pb(NO 3) 2 o (c) MgSO 4 + AgNO 3 o POGIL 2005, 2006 5/6 Authored by Dr. I mean, sure, there are numbers and formulas and stuff,…. Significant Figures Worksheet 1. Define solubility: The maximum amount of solute that dissolves in a fixed quantity of a particular solvent at a specified temperature. PS 4 - Acids-Bases. Every classification listed has a question about it on this quiz. Write these questions in your notebook then answer: Ch. 6 POGIL™ Activities for High School Chemistry 20. dissolved in water liquid solid gas. 273, Q #,2,4, 5,7 ;. Displaying top 8 worksheets found for - Redox Pogil. Write your response in the space provided following each question. Questions 1-3 are long free-response questions that require about 23 minutes each to answer and are 4. Extension Questions Ethanol is one example of alternative fuels for powering our cars and trucks. This type of reaction is called a precipitation reaction , and the solid produced in the reaction is known as the precipitate. The net ionic equation is the chemical equation that shows only those elements, compounds, and ions that are directly involved in the chemical reaction. The naming of ionic compounds that contain polyatomic ions follows the same rules as the naming for other ionic compounds: simply combine the name of the cation and the name of the anion. Proudly powered by WeeblyWeebly. The key to being able to write net ionic equations is the ability to recognize monoatomic and polyatomic ions, and the solubility rules. Neutral amino acids O NH 2 H 3 C OH O NH 2 HO OH O O HO OH NH 2 O NH 2 H 2 N OH O. Finish Net Ionic Equations. My answer was Fe^1+ + OH^1- = FeOH. Showing top 8 worksheets in the category - Pogil. In double displacement (replacement) reactions, two. Net Ionic Equations POGIL. The net ionic equation models just the formation of the precipitate from its ions and removes the spectator ions Essential Question How is the phenomenon of precipitate formation explained on the atomic level? Activity Objectives 1. pdf Continue Working on Net Ionic Eq Wksht: 5 EC: Complete - Net Ionic Equations. the conductivity of the solution and the strength of the electrolyte (acid strength). 3 : Jan 18, 2019, 10:58 AM: Raymond Thomas: ĉ: AP Test V Review Wks 2016-17 with answers. By definition, a common ion is an ion that enters the solution from two different sources. Similarly, chemists classify chemical equations according to their. The module presents chemical bonding on a sliding scale from pure covalent to pure ionic, depending on differences in the electronegativity of the bonding atoms. Forces and Motion: Basics - PhET Interactive Simulations. 4Fe(s) + 3O 2 (g) → 2Fe 2 O 3 (s) a. Author: KONICA MINOLTA bizhub PRO 1050 Created Date: 5/10/2017 1:15:39 PM. This activity focused on molecular (covalent. It provides a worksheet full of examples and practice. 2 POGIL™ Activities for High School Biology 1. It provides a worksheet full of examples and practice problems along with the answers to the questions and how to get them. Half-Life : Paper, M&M’s, Pennies, or Puzzle Pieces. This section focuses on the effect of common ions on solubility product equilibria. We pay for Chemistry Pogil Calorimetry Answer Key and numerous ebook collections from fictions to scientific research. Ball does not reach the top of the ditch. The second is the use and design of distinctive classroom materials. Give the name and the formula of the ionic compound produced by neutralization reactions between the following acids and bases: Acid and Base reactants Name of ionic compound Formula nitric acid and sodium hydroxide sodium nitrate NaNO3 hydroiodic acid. 9, Write ionic and net ionic equations for the remaining reactions in. 2 (g) A 2Fe. Stephen Prilliman. In Model 2, does HA represent a weak acid or a strong acid? What evidence found in the model supports your answer? 14. Net Ionic Equation, Balancing Equations, Stoichiometry, Redox Reaction, Classification of Reactions | High School Lab: Inquiry Redox Investigation. Redox Pogil. Support your answer with evidence from Model 2. Use MathJax to format equations. Created Date: 8/27/2015 6:57:30 AM. The bond energy. Predicting Products PRactice ((Reference Sheet)) 4. pdf---ANWERS. You could not by yourself going later than book buildup or library or borrowing from your friends to right to use them. Based on the data in Model 1 and the table in Question 3, describe the relationship between: a. AP Chem-Chap 7 & 8 Name Answer Key POGIL Activity Group Members POGIL: Photoelectron Spectroscopy What is the relationship between the ionization energy of an electron and the net attractive force that holds an electron in an atom? Consider the equation you wrote in Question 12. Ozone, O , is not a linear molecule. A certain barium halide exists as the hydrated salt BaX2 - 2H2O, where X is the halogen. •Recorder - In charge of summarizing the main points of. In double displacement (replacement) reactions, two. A student writes the following incorrect chemical equation for a single replacement reaction be- tween lithium bromide and fluorine. Some of the worksheets displayed are Chapter 20 work redox, Chem 116 pogil work, Chem 115 pogil work, Work 25, Chapter 4 work 3 molecular net ionic equations, Academic resource center, Net ionic equation work answers, Coslet ap chemistry. The planet Earth is not a closed system. To identify the products formed in these reactions and summarize the chemical changes in terms of balanced chemical equations and net ionic equations. Identify the spectator ion or ions in each reaction. Roseville, CA 95747-7100 Toll Free 800-772-8700. 30 J of energy to the soccer ball in the ditch. Thursday 9/7. Each experiment will focus on at least one process skill such as teamwork, oral and written communication, management, information processing, critical thinking, problem solving, assessment, experimental design. Use your understanding of lattice energy and Coulombic attraction to answer the following. One equation can't be solved for two unknowns the Ag + and Br-ion concentrations. Reviewing the Model: 9. Sometimes a teacher finds it necessary to ask questions about PE diagrams that involve actual Potential Energy values. The net ionic equation lists only those ions which are not common on both sides of the reaction: Pb+2 (aq) + 21- (aq) ->Pb12 (s) The spectator ions that are present in the solution but play no direct role in the reaction, are omitted in the net ionic equation. Please give as much additional information as possible. double replacement 2. 2 Mg + ZnCl 2 → 2 MgCl + Zn. Model the formation of a precipitate using ionic and net ionic. com/ebsis/ocpnvx. composition or decomposition 4. A potential energy diagram plots the change in potential energy that occurs during a chemical reaction. 6 POGIL™ Activities for High School Chemistry Extension Questions 15. Molecular Equation Ionic Equation Net Ionic Equation b. ANSWER KEY. Keeping in mind that the sum of the charges in an ionic compound must equal zero, use the chemical formulas in Model 3 to answer the following questions: Identify the charge on the copper cations in copper(I) oxide and copper(II) oxide, respectively. Gas variables worksheet answers answer to pogil ap biology pogil activities for ap biology answer key free energy. org) by the end of the day on Thursday, 4/17 for credit. Clinic Management System Data Flow Diagram. You can predict whether a precipitate will form using a list of solubility rules such as those found in the table below. Tuesday, Feb 18. php on line 143 Deprecated: Function create_function() is deprecated in. Write the net ionic equation for this reaction. 6 ™ Activities for High School Biology POGIL Extension Questions 21. Global Economic Trends Guided Answers advanced engineering mathematics zill 4th solution manual, annotations to finnegans wake roland mchugh, ford fiesta workshop manual download, investment science or solution, daihatsu dm954dt engine, 1976 fxe harley davidson super glide manual,. Created Date: 10/5/2017 9:36:42 AM. For example, sodium chloride is soluble. Extension Questions. 2 Energy changes in a system can be described as endothermic and exothermic processes such as the. Often in the foundation exams the equations are given, or the variables are named and you have to write the equation using them; or you may need to choose an equation from the Physics equation sheet given in the exam. In a net ionic equation for a double-replacement reaction with a precipitate, the ions that form the precipitate are the reactants and the precipitate is the product. docx from CHEM chem 101 at University of Maryland, Baltimore. On the actual exam you will have 10 minutes (calculator-free) to write 3 net-ionic equations and answer a question for each. POGIL, SPUR+, and SPIRAL. PhET sims are based on extensive education research and engage students through an intuitive, game-like environment where students learn through exploration and discovery. I wanted to get the lab to a better place so that students aren’t overwhelmed with doing a lot, but rather that they learn a lot and can apply what they’ve learned outside of lab. *** Write ionic and net ionic equations from molecular equations using phase symbols. Highlights. The planet Earth is not a closed system. Polyatomic Ions Worksheet Answers Pogil is an addition to his classic Physics book called "The New Physics". What is the net ionic equation for potassium sulfite and hydrobromic acid? Join Yahoo Answers and get 100 points today. Crayfish dissection Biology Junction from Crayfish Dissection Worksheet Answers, source:biologyjunction. Balancing Equations: Answers to Practice Problems 1. Gas variables worksheet answers answer to pogil ap biology pogil activities for ap biology answer key free energy. We meet the expense of you this proper as with ease as simple way to acquire those all. Each question is worth 5 points. Many important chemical reactions take place in. Identify each of the variables in the equation. 175 Longwood Road South Hamilton, ON L8P 0A1: Phone: 844-200-1455: Fax: Email: [email protected] Acids And Bases Pogil Answer Key. University. The ions in an ionic solid. Ethanol can be produced in different ways, but most often by microorganisms acting on plant materials such as corn. Neutral amino acids O NH 2 H 3 C OH O NH 2 HO OH O O HO OH NH 2 O NH 2 H 2 N OH O. Question 1: 0:36 Question 2: 2:45 Question 3: 4:47 Acids and Bases (Grade 12) How to Write Complete Ionic Equations and Net Ionic Equations This video covers, how to predict products, how to balance a chemical equation, how to identify the solubility of a compound, how Buffer Solution, pH. Net Ionic Equation Worksheet Answers Write balanced molecular, ionic, and net ionic equations (NIE) for each of the following reactions. But they will always be in the Rationale and Answers. Title: Microsoft Word - extranetionicpractice. How might this affect the final empirical formula? 17. Molecular and Ionic Equations –A net ionic equation is a chemical equation from which the spectator ions have been removed. crawl_filtered. Considering your answer to Question 16, write a mathematical equation to show how the total pressure inside the bottle might be calculated using partial gas pressures. hclo4 aq | hclo4 aq | hclo4 aq h2o l | hclo4 aq +h20 l | hclo4 aq +h2o l | hclo4 aq +koh aq | hclo4 aq k2co3 aq | hclo4 aq +k2co3 aq | hclo4 aq +naoh aq | hclo4. Forces and Motion: Basics - PhET Interactive Simulations. Home About. Carbon bonds: All organic compounds contain carbon. Calculate the energy of the X-ray photon used in the PES experiment described. 2007 - Y ESTRUCTURA. A student writes the following incorrect chemical equation for a double replacement. PDF cell cycle pogil extension questions answer key - Bing answers pdf / cellular respiration an overview pogil answers / cellular respiration pogil. One equation can't be solved for two unknowns the Ag + and Br-ion concentrations. NaCl + H2O ( Na+ + Cl(Example 1: Sodium nitrate reacts with potassium acetate in an aqueous solution. Use pages for 222-223. [email protected] UNIT 3: Chemical Reactions Chapter Exam Instructions. What would happen to the size of the central vacuole if a plant Using the equations above, explain the relationship between mitochondria and chloroplasts. Thursday 9/7. The naming of ionic compounds that contain polyatomic ions follows the same rules as the naming for other ionic compounds: simply combine the name of the cation and the name of the anion. 4Fe(s) + 3O 2 (g) → 2Fe 2 O 3 (s) a. If all species are spectator ions, please indicate that no reaction takes place. org) by the end of the day on Thursday, 4/17 for credit. 00 mL) and concentration of acid (0. 9/6 Thursday: Safety quiz; Begin Unit 1; Steel Wool Lab Homework: Read entire lab and complete pre-lab questions. Questions 4–7 are short free-response questions that require about 9 minutes each to answer and are worth 4 points each. 0 g of H 2? Explain how you determined your answer. 2013)to first identify patterns of reasoning and explanations within and across ten different autonomous cooperative. List four materials that contain this stored. During a POGIL session we will work in teams to learn how to communicate effectively, listen well, and think critically to develop each other's understanding of Chemistry. The AI would circulate around the room to answer any questions while students first attempt the problem. Find the empirical formula and molecular formula for this compound. Novasantoso Printable Blank Worksheet Template. 1 because students are explaining how the unique structural features of the neuron allow it to detect information that will be transmitted to other cells, including other neurons. The net ionic equation shows only the chemical species that are involved in a reaction, while the complete ionic equation also includes spectator ions. Extension Questions Model 2 — The Meaning of K HA(aq) H 0(1) [H O'I[A-I K [HA] H30+(aq) A-(aq) 13. Next: FeCl2 + TlNO3. Based on the data in Model 1 and the table in Question 3, describe the relationship between: a. pdf: 6 NOTES: Stoich Map. POGIL Chemical Reactions Answers Net Ionic Equation Worksheet and Answers This chemistry video tutorial focuses on how to write net ionic equations. Terms also used include: exothermic, endothermic, exergonic, endergonic, spontaneous, 2nd Law of Thermodynamics, and coupled processes. Write ionic and net ionic equations for the remaining reactions in Model 1. crawl_filtered. Balancing Equations Worksheet and Key 1. On this page you can read or download net ionic equations pogil answer key in PDF format. When AgNO 3 is added to a saturated solution of AgCl, it is often described as a source of a common ion, the Ag + ion. Displaying top 8 worksheets found for - Redox Pogil. Similarly, chemists classify chemical equations according to their. The following molecular equation represents the reaction that occurs when aqueous solutions of silver nitrate and chromium(III) iodide are combined. Write your response in the space provided following each question. Deprecated: Function create_function() is deprecated in /www/wwwroot/dm. It includes several features to make your preparation easier. In this animation, students will witness a precipitate reaction on the particulate level to understand why a net ionic equation represents what happens in these reaction types. The activity series shown in Model 2 to the right is a ranking, similar to the one you did in Question 7, of several metals. The barium content of the salt can be determined by gravimetric methods. Buffers 5 Extension Questions 17. 6 ™ Activities for AP* Chemistry POGIL Extension Questions 16. Founded in 2002 by Nobel Laureate Carl Wieman, the PhET Interactive Simulations project at the University of Colorado Boulder creates free interactive math and science simulations. For each of the scenarios in Question 3 where the ball successfully leaves the ditch, determine thekinetic energy the ball will have when it reaches the top of the ditch. Worksheets are Bonding basics, Ionic bonding work 1, Trom po no, Bondingbasics2008, , Chemistry name ws 1 ionic bonding key date block, Ionic and covalent compounds name key, Net ionic equation work answers. pdf NOTES: Chpt 9 Notes (Stoich). It provides a worksheet full of examples and. 201 Textbook. My answer was Fe^1+ + OH^1- = FeOH. Molecular Equation Zn (s) + 2HCl (aq) → ZnCl 2 (aq) + H 2 (g) Ionic Equation Net Ionic Equation b. 16: LAB- Types of Chemical Reactions Review Work (*UNIT 2 TEST- Key terms/concepts to Know) Textbook questions- p. Dissolved ions. Assume all reactions. For example, sodium chloride is soluble. Behavioral interviews really are a new style of interviewing. Thus, all strong acid has been removed and for small amounts of H+, the relative amounts of HNO 2 and NO 2 - remain fairly constant. My answer was Pb^2+ +SO4^2- = PbSO4. This is just one of the solutions for you to be successful. Questions 1-3 are long free-response questions that require about 23 minutes each to answer and are 4. Precipitate, Balancing Equations, Solubility Rules | High School Animation: Net Ionic Equations Animation. Finacial Accounting Chapter 10 Ansewrs answer key for experimental variables pogil, vw crossfox 2006 manual, owners manual for lexus 2013 450h, 2001 bmw 325ci. What chemical species is missing in the net ionic equation? 0 b, Explain why it is valid to remove this species from the equation. ANSWER KEY. 4 ™ Activities for AP* Chemistry POGIL 9. The Action Potential, Synaptic Transmission, and Maintenance of Nerve Function Cynthia J. uonenba UI saouv)sqns JO. PRACTICE PROBLEMS ON NET IONIC EQUATIONS page 1 of 3 Show the complete ionic and net ionic forms of the following equations. 2650 g) was dissolved in water (200 mL) and excess sulfuric acid added. Na3P04(aq) + + + +3 +. ions pogil extension questions answers net ionic pogil answers polyatomic ions pogil answer key polyatomic ions pogil worksheet answers ions pogil packet answers. I mean, sure, there are numbers and formulas and stuff,…. All of the answers can be found online for almost every single POGIL Currently, there is a significant amount of discussion on teaching list serves about the frustration of people posting answer keys online and students checking the internet instead of doing the work. Balancing Equations Worksheet and Key 1. Net Ionic Equations POGIL. Remaining is the net ionic equation, containing only the ions that changed in some way, showing the active pieces in the reaction. We then write the solubility product expression for this reaction. The following diagrams represent a hypothetical reaction X Y. 2 POGIL ™ Activities for AP* Biology. 2 and answer Key Concept questions for each section. A net ionic equation also gives information on how different substances react. For each of the following questions or statements, select the most appropriate response and click its letter: Start Congratulations - you have completed Quiz #2-1 PRACTICE: Types of Chemical Reactions. Cd2+ (aq) + 4Cl1– 2(aq) ⇌ CdCl 4 – (aq) a. uonenba UI saouv)sqns JO. Newton's second law is used to convert between weight (force) and mass:. Include the phases in the formula equation, and circle the spectator ions in the complete ionic equation. Identify the main type of bonding and. ANSWER KEY. This is an area that has captured the attention of scientists and public alike because of its vast and complex potentials. The reason I ask is because apparently the molecular equation $$\ce{H2SO4 + 2KOH -> 2H2O + K2SO4}$$ has the net-ionic form $$\ce{H+ + OH- -> H2O}$$ It would be helpful if you could describe in general how I should write the dissociation of the strong acids in the net ionic equation. Notes: Chemical Equations and Types of Reactions 4. A survey of previous A. In this animation, students will witness a precipitate reaction on the particulate level to understand why a net ionic equation represents what happens in these reaction types. Precipitate, Balancing Equations, Solubility Rules | High School Animation: Net Ionic Equations Animation. 4Fe(s) + 3O. Some of the worksheets for this concept are Chapter 20 work redox, Chem 116 pogil work, Chem 115 pogil work, Work 25, Chapter 4 work 3 molecular net ionic equations, Academic resource center, Net ionic equation work answers, Coslet ap chemistry. COMPLETE TEXT BOOK SOLUTION WITH ANSWERS INSTANT DOWNLOAD SAMPLE QUESTIONS Chemistry Ninth Edition Steven S. Solubility Pogil Answers Solubility Pogil Answers Yeah, reviewing a books Solubility Pogil Answers could build up your close contacts listings. Refer to Model 1 as a guide, but think about how a weak acid would be different from a strong acid. pdf FREE PDF DOWNLOAD NOW!!! Source #2: polyatomic ions pogil answer key. Molecular, Complete Ionic, and Net Ionic Equations How to write a net ionic equation (double replacement)? Basic lesson on molecular equations, complete ionic equations, and net ionic equations. Deprecated: Function create_function() is deprecated in /www/wwwroot/dm. Zinc and Lead (II) nitrate react to form Zinc Nitrate and Lead. Redox Pogil. org) by the end of the day on Thursday, 4/17 for credit. Thursday, Feb 20. Net Ionic Equation: Everything that participates in the reaction gets recorded in the net ionic equation. Stephen Prilliman. 50 mol/L H 2SO 4 to completely neutralize each other? Recall that in an acid/base neutralization an acid reacts with a base to produce water and a neutral ionic compound (called a salat). See last week’s answer sheet. I wanted to get the lab to a better place so that students aren’t overwhelmed with doing a lot, but rather that they learn a lot and can apply what they’ve learned outside of lab. COMPLETE TEXT BOOK SOLUTION WITH ANSWERS INSTANT DOWNLOAD SAMPLE QUESTIONS Chemistry Ninth Edition Steven S. an exam after instruction in which there is an answer key with the 'correct' answer. Solubility Pogil Answers Solubility Pogil Answers Yeah, reviewing a books Solubility Pogil Answers could build up your close contacts listings. Bond Energy Pogil. On this page you can read or download pogil net ionic equations answers in PDF format. Key Questions. Phys Rev 1946; 70:460-474,1946. You get rid of that, and then you see what is left over.
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http://scitation.aip.org/content/aip/journal/chaos/23/3/10.1063/1.4812722?showFTTab=true&containerItemId=content/aip/journal/chaos
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• journal/journal.article
• aip/chaos
• /content/aip/journal/chaos/23/3/10.1063/1.4812722
• chaos.aip.org
1887
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Spatially dependent parameter estimation and nonlinear data assimilation by autosynchronization of a system of partial differential equations
USD
10.1063/1.4812722
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Affiliations:
1 Department of Mathematics, Clarkson University, Potsdam, New York 13669, USA
Chaos 23, 033101 (2013)
/content/aip/journal/chaos/23/3/10.1063/1.4812722
http://aip.metastore.ingenta.com/content/aip/journal/chaos/23/3/10.1063/1.4812722
View: Figures
## Figures
FIG. 1.
Three sets of spatially dependent parameters used in simulations. Figures are described by Eq. , with on the left and on the right. Below, with the same ordering, are the parameters described by Eq. . Finally, the swirly parameters are shown in Figures .
FIG. 2.
Autosynchronization of species in Eqs. . Each figure shows drive (top) and response (bottom) pairs. (, 0) and in (a), (, 1000) and in (c), and (, 4788) and in (e). (, 0) and in (b), (, 1000) and in (d), and (, 4788) and in (f). Model parameters are and .
FIG. 3.
Autosynchronization of response parameters in Eqs. . Each figure shows drive (top) and response (bottom) pairs. and in (a), and in (c), and and in (e). and in (b), and in (d), and and in (f).
FIG. 4.
Autosynchronization of species in Eqs. . Each figure shows drive (top) and response (bottom) pairs. (, 0) and in (a), (, 1000) and in (c), and (, 10 660) and in (e). (, 0) and in (b), (, 1000) and in (d), and (, 10 660) and in (f). Model parameters are and .
FIG. 5.
Autosynchronization of response parameters in Eqs. . Each figure shows drive (top) and response (bottom) pairs. and in (a), and in (c), and and in (e). and in (b), and in (d), and and in (f).
FIG. 6.
Globally averaged relative synchronization error between drive and response PDE components and parameters on a log scale. Figures correspond to parameters built by Eq. and simulation displayed in Figures , respectively. Figures show globally averaged relative synchronization error for species and parameters built by Eq. , corresponding to simulations in Figures , respectively.
FIG. 7.
Autosynchronization of species in Eqs. . Each figure shows drive (top) and response (bottom) pairs. (, 0) and in (a), (, 1000) and in (c), and (, 9360) and in (e). (, 0) and in (b), (, 1000) and in (d), and (, 9360) and in (f). Model parameters are those shown in Figures .
FIG. 8.
Autosynchronization of parameters in Eqs. . Each figure shows drive (top) and response (bottom) pairs. and in (a), and in (c), and and in (e). and in (b), and in (d), and and in (f). Model parameters are those shown in Figures .
FIG. 9.
Globally averaged relative synchronization error between drive and response PDE components and parameters on a log scale, estimating perhaps more realistic spiral parameters. Figures (a) and (b) correspond to parameters shown in Figures and simulation displayed in Figures , respectively.
FIG. 10.
Locally averaged patches over which drive system is sampled shown in black. Sampled on subset of 3 × 3 grid points with a distance of 3grid points between patches.
FIG. 11.
Comparison of three different sampling schemes. Shown are relative synchronization errors between drive and response systems for sampling over 3 × 3 grid points (blue) with a distance of 3 grid points between subsequent patches, 2 × 2 grid points (red) with a distance of 2 grid points between subsequent patches, and 1 × 1 grid points (black) with a distance of 1 grid points between subsequent patches. Phytoplankton synchronization errors on left and zooplankton synchronization errors shown on right.
FIG. 12.
Autosynchronization results shown at t = 2000. Both species and both parameters shown compared with drive species and true parameters. Effect of adding diffusion to parameter equations is clearly visible in estimated parameters.
FIG. 13.
Globally averaged relative synchronization errors shown for species and parameters. Local sampling destroys stability of the identical synchronization manifold, however, spatial characteristics of parameters are still observed.
/content/aip/journal/chaos/23/3/10.1063/1.4812722
2013-07-08
2014-04-21
Article
content/aip/journal/chaos
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2014-04-21 05:17:32
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https://www.physicsforums.com/threads/is-there-an-instantaneous-angular-acceleration-for-a-conical-pendulm.602793/
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# Is there an instantaneous angular acceleration for a conical pendulm?
1. May 3, 2012
### jason12345
For a conical pendulum, there is an instantaneous centripetal acceleration. Does this mean there is an instantaneous angular acceleration of the pendulum towards the center?
2. May 3, 2012
### olivermsun
Can you define what your angle and center refer to?
3. May 3, 2012
### jason12345
The angle is between the string and the axis of symmetry the pendulum rotates around.
4. May 3, 2012
### olivermsun
I see, you're talking about a pendulum which swings about the center axis in a cone.
Your angle, as defined, rotates with the pendulum string and remains constant, so I would say "no."
5. May 3, 2012
### jason12345
Thanks for your reply, although I disagree with it :) I could also argue that the radius of the circular motion is constant and so there isn't an acceleration towards the centre - but there is: v^2/r
6. May 3, 2012
### olivermsun
There is an acceleration (which happens to be toward the center) because the radius vector is not constant. Only the radius magnitude is constant.
As far as I can tell, the angular velocity is constant if defined around the axis of symmetry.
7. May 3, 2012
### jason12345
I think you mean velocity where you state radius.
I agree that angular velocity is constant.
8. May 3, 2012
### olivermsun
You're right. Change in radius vector per time (velocity) changes.
Last edited: May 3, 2012
9. May 3, 2012
### greentlc
How does the radius vector not change? Doesn't it's magnitude stay the same, however the direction is changing?
10. May 4, 2012
### olivermsun
The radius does change (dr/dt is nonzero), so that there is a velocity, but he was talking about whether or not there is an acceleration. There is, since d^2/dt^2 = dv/dt is nonzero. A changing radius vector isn't enough to imply an acceleration, although it is enough that the magnitude stays the same while the direction is changing (as you say).
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2018-03-24 06:44:09
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http://clay6.com/qa/52232/which-of-the-following-is-not-a-consequence-of-lanthanide-contraction-
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Browse Questions
# Which of the following is NOT a consequence of Lanthanide contraction?
(1) The decrease in ionic radii of the lanthanides from 103 pm (La$^{3+}$) to 86 pm (Lu$^{3+}$) in the lanthanide series.
(2) The Basic nature increases from Ce(OH)$_3$ to Lu(OH)$_3$
(3) The covalent nature in M(OH)$_3$ will decrease from Ce(OH)$_3$ to Lu(OH)$_3$
(4) The E$^{\circ}$ value increases slightly due to Lanthanide contraction
$\begin {array} {1 1}Both (2) and (3) \\ Only (2) \\ Both (1) and (2) \\ (1), (2) and (4) \end {array}$
Basic nature $\propto \large\frac{1}{\text{Atomic Number}}$ and Covalent Nature $\propto \large\frac{1}{\text{size of cation}}$
$\Rightarrow$ The Basic nature decreases and the covalent nature increases.
edited Aug 6, 2014
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2016-12-08 20:08:39
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https://codegolf.stackexchange.com/questions/85970/generate-pascals-braid
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# Generate Pascal's Braid
This is Pascal's Braid:
1 4 15 56 209 780 2911 10864 40545 151316 564719
1 3 11 41 153 571 2131 7953 29681 110771 413403 1542841
1 4 15 56 209 780 2911 10864 40545 151316 564719
I totally made that up. Blaise Pascal didn't have a braid as far as I can tell, and if he did it was probably made of hair instead of numbers.
It's defined like this:
1. The first column has a single 1 in the middle.
2. The second column has a 1 at the top and at the bottom.
3. Now we alternate between putting a number in the middle or two copies of a number at the top and bottom.
4. If the number goes on the top or the bottom, it will be the sum of the two adjacent numbers (e.g. 56 = 15 + 41). If you tilt your head a little, this is like a step in Pascal's triangle.
5. If the number goes in the middle, it will be the sum of all three adjacent numbers (e.g. 41 = 15 + 11 + 15).
Your task will be to print (some part of) this braid.
## Input
You should write a program or function, which receives a single integer n, giving the index of the last column to be output.
You may choose whether the first column (printing only a single 1 on the middle line) corresponds to n = 0 or n = 1. This has to be a consistent choice across all possible inputs.
## Output
Output Pascal's Braid up to the nth column. The whitespace has to match exactly the example layout above, except that you may pad the shorter line(s) to the length of the longer line(s) with spaces and you may optionally output a single trailing linefeed.
In other words, every column should be exactly as wide as the number (or pair of equal numbers) in that column, numbers in successive columns should not overlap and there should be no spaces between columns.
You may either print the result to STDOUT (or the closest alternative), or if you write a function you may return either a string with the same contents or a list of three strings (one for each line).
## Further Details
You may assume that n won't be less than the index of the first column (so not less than 0 or 1 depending on your indexing). You may also assume that the last number in the braid is less than 256 or the largest number representable by your language's native integer type, whichever is greater. So if your native integer type can only store bytes, you can assume that the largest n is 9 or 10 (depending on whether you use 0- or 1-based n) and if it can store signed 32-bit integers, n will be at most 33 or 34.
Standard rules apply. The shortest code wins.
## OEIS
Here are a few relevant OEIS links. Of course, these contain spoilers for different ways to generate the numbers in the braid:
## Test Cases
These test cases use 1-base indexing. Each test case is four lines, with the first being the input and the remaining three being the output.
1
1
---
2
1
1
1
---
3
1
1 3
1
---
5
1 4
1 3 11
1 4
---
10
1 4 15 56 209
1 3 11 41 153
1 4 15 56 209
---
15
1 4 15 56 209 780 2911
1 3 11 41 153 571 2131 7953
1 4 15 56 209 780 2911
---
24
1 4 15 56 209 780 2911 10864 40545 151316 564719 2107560
1 3 11 41 153 571 2131 7953 29681 110771 413403 1542841
1 4 15 56 209 780 2911 10864 40545 151316 564719 2107560
• The format seems like a bit chameleon to me. – Leaky Nun Jul 20 '16 at 15:49
• @LeakyNun I tried this challenge while it was in the sandbox, and I spent about half as many bytes on calculating the braid as printing it. This seems like an excellent balance to me for an ascii-art challenge. – FryAmTheEggman Jul 20 '16 at 15:52
• @LeakyNun I was hoping that both the sequence generation and the ASCII art are important components of the challenge, because most languages will probably be better at one of those two, so I figured it would be interesting to mix them up. And it introduces an additional component where it's not obvious whether it's better to generate top/bottom and middle separately or to generate the entire thing and then separate out the bisections. – Martin Ender Jul 20 '16 at 15:52
• – Luis Mendo Jul 20 '16 at 18:56
• Nobody has written a solution in Pascal yet. This makes me sad. – DynamiteReed Jul 21 '16 at 17:52
# Jelly, 3130 29 bytes
Q;S⁹o_
3ḶḂç@⁸СIµa"Ṿ€o⁶z⁶Zµ€Z
This is a monadic link; it accepts a 0-based column index as argument and returns a list of strings.
Try it online!
### How it works
Q;S⁹o_ Helper link.
Arguments: [k, 0, k] and [0, m, 0] (any order)
Q Unique; deduplicate the left argument.
; Concatenate the result with the right argument.
S Take the sum of the resulting array.
⁹o Logical OR with the right argument; replaces zeroes in the
right argument with the sum.
_ Subtract; take the difference with the right argument to
remove its values.
This maps [k, 0, k], [0, m, 0] to [0, k + m, 0] and
[0, m, 0], [k, 0, k] to [m + 2k, 0, m + 2k].
3ḶḂç@⁸СIµa"Ṿ€o⁶z⁶Zµ€Z Monadic link. Argument: A (array of column indices)
3Ḷ Yield [0, 1, 2].
Ḃ Bit; yield [0, 1, 0].
I Increments of n; yield [].
С Apply...
ç@ the helper link with swapped arguments...
⁸ n times, updating the left argument with the return
value, and the right argument with the previous value
of the left one. Collect all intermediate values of
the left argument in an array.
µ µ€ Map the chain in between over the intermediate values.
Ṿ€ Uneval each; turn all integers into strings.
a" Vectorized logical AND; replace non-zero integers with
their string representation.
o⁶ Logical OR with space; replace zeroes with spaces.
z⁶ Zip with fill value space; transpose the resulting 2D
array after inserting spaces to make it rectangular.
Z Zip; transpose the result to restore the original shape.
Z Zip; transpose the resulting 3D array.
# Pyth, 44 bytes
The number generation took 20 bytes, and the formatting took 24 bytes.
jsMC+Led.e.<bkC,J<s.u+B+hNyeNeNQ,1 1Qm*;ldJ
Try it online!
jsMC+Led.e.<bkC,J<s.u+B+hNyeNeNQ,1 1Qm*;ldJ input as Q
.u Q,1 1 repeat Q times, starting with [1,1],
collecting all intermediate results,
current value as N:
(this will generate
more than enough terms)
+hNyeN temp <- N[0] + 2*N[-1]
+B eN temp <- [temp+N[-1], temp]
now, we would have generated [[1, 1], [3, 4], [11, 15], [41, 56], ...]
jsMC+Led.e.<bkC,J<s Qm*;ldJ
s flatten
< Q first Q items
J store in J
m dJ for each item in J:
convert to string
l length
*; repeat " " that many times
jsMC+Led.e.<bkC,
C, transpose, yielding:
[[1, ' '], [1, ' '], [3, ' '], [4, ' '], [11, ' '], ...]
(each element with as many spaces as its length.)
.e for each sub-array (index as k, sub-array as b):
.<bk rotate b as many times as k
[[1, ' '], [' ', 1], [3, ' '], [' ', 4], [11, ' '], ...]
jsMC+Led
+Led add to each sub-array on the left, the end of each sub-array
C transpose
sM sum of each sub-array (reduced concatenation)
j join by new-lines
• That is the largest Pyth program I've ever seen. – imallett Jul 20 '16 at 21:45
# Python 2, 120 bytes
a=1,1,3,4
n=input()
y=0
exec"y+=1;t='';x=0;%sprint t;"%(n*"a+=a[-2]*4-a[-4],;v=a[x];t+=[v,len(v)*' '][x+y&1];x+=1;")*3
Try it on Ideone.
# MATL, 38 bytes
1ti:"yy@oQ*+]vG:)!"@Vt~oX@o?w]&v]&hZ}y
Try it online!
Computing an array with the (unique) numbers takes the first 17 bytes. Formatting takes the remaining 21 bytes.
## Explanation
### Part 1: generate the numbers
This generates an array with the numbers from the first and second rows in increasing order: [1; 1; 3; 4; 11; 15; ...]. It starts with 1, 1. Each new number is iteratively obtained from the preceding two. Of those, the second is multiplied by 1 or 2 depending on the iteration index, and then summed to the first to produce the new number.
The number of iterations is equal to the input n. This means that n+2 numbers are generated. Once generated, the array needs to be trimmed so only the first n entries are kept.
1t % Push 1 twice
i: % Take input n. Generage array [1 2 ... n]
" % For each
yy % Duplicate the two most recent numbers
@o % Parity of the iteration index (0 or 1)
Q % Add 1: gives 1 for even iteration index, 2 for odd
*+ % Multiply this 1 or 2 by the most recent number in the sequence, and add
% to the second most recent. This produces a new number in the sequence
] % End for each
v % Concatenate all numbers in a vertical array
G:) % Keep only the first n entries
### Part 2: format the output
For each number in the obtained array, this generates two strings: string representation of the number, and a string of the same length consisting of character 0 repeated (character 0 is displayed as a space in MATL). For even iterations, these two strings are swapped.
The two strings are then concatenated vertically. So n 2D char arrays are produced as follows (using · to represent character 0):
·
1
1
·
·
3
4
·
··
11
15
··
These arrays are then concatenated horizontally to produce
·1·4··15
1·3·11··
Finally, this 2D char array is split into its two rows, and the first is duplicated onto the top of the stack. The three strings are displayed in order, each on a different line, producing the desired output
! % Transpose into a horizontal array [1 1 3 4 11 15 ...]
" % For each
@V % Push current number and convert to string
t~o % Duplicate, negate, convert to double: string of the same length consisting
% of character 0 repeated
X@o % Parity of the iteration index (1 or 0)
? % If index is odd
w % Swap
] % End if
&v % Concatenate the two strings vertically. Gives a 2D char array representing
% a "numeric column" of the output (actually several columns of characters)
] % End for
&h % Concatenate all 2D char arrays horizontally. Gives a 2D char array with the
% top two rows of the output
Z} % Split this array into its two rows
y % Push a copy of the first row. Implicitly display
# Haskell, 101 bytes
a=1:1:t
t=3:4:zipWith((-).(4*))t a
g(i,x)=min(cycle" 9"!!i)<$>show x f n=[zip[y..y+n]a>>=g|y<-[0..2]] Defines a function f :: Int → [String]. • Michael Klein reminded me I didn’t need to call unlines on the result, saving 7 bytes. Thanks! • I saved a byte by replacing " 9"!!mod i 2 with cycle" 9"!!i. • Three more bytes by writing two corecursive lists instead of using drop. • My girlfriend pointed out I can save two more bytes by starting my answers at 0 instead of 1. # C, 183177 176 bytes #define F for(i=0;i<c;i++) int i,c,a[35],t[9];p(r){F printf("%*s",sprintf(t,"%d",a[i]),r-i&1?t:" ");putchar(10);}b(n){c=n;F a[i]=i<2?1:a[i-2]+a[i-1]*(i&1?1:2);p(0);p(1);p(0);} ## Explanation C is never going to win any prizes for brevity against a higher level language, but the exercise is interesting and good practice. The macro F shaves off six bytes at the cost of readability. Variables are declared globally to avoid multiple declarations. I needed a character buffer for sprintf, but since K&R is loose with type checking, sprintf and printf can interpret t[9] as a pointer to a 36-byte buffer. This saves a separate declaration. #define F for(i=0;i<c;i++) int i,c,a[35],t[9]; Pretty printing function, where r is the row number. Sprintf formats the number and computes the column width. To save space we just call this three times, one for each row of output; the expression r-i&1 filters what gets printed. p(r) { F printf("%*s", sprintf(t, "%d", a[i]), r-i&1 ? t : " "); putchar(10); } Entry point function, argument is number of columns. Computes array a of column values a[], then calls printing function p once for each row of output. b(n) { c=n; F a[i] = i<2 ? 1 : a[i-2] + a[i-1]*(i&1 ? 1 : 2); p(0); p(1); p(0); } Sample invocation (not included in answer and byte count): main(c,v) char**v; { b(atoi(v[1])); } ## Updated Incorporated the inline sprintf suggestion from tomsmeding. That reduced the count from 183 to 177 characters. This also allows removing the the braces around the printf(sprintf()) block since it's only one statement now, but that only saved one character because it still needs a space as a delimiter. So down to 176. • Can't you inline the definition of w where it's used? You seem to only use it once. – tomsmeding Jul 21 '16 at 5:12 • You can't use itoa instead of sprintf? – Giacomo Garabello Jul 21 '16 at 13:08 • I considered itoa, but it doesn't exist on my system, and I am using the return value of sprintf to set the field width. – maharvey67 Jul 22 '16 at 19:55 ## PowerShell v2+, 133 bytes param($n)$a=1,1;1..$n|%{$a+=$a[$_-1]+$a[$_]*($_%2+1)};$a[0..$n]|%{$z=" "*$l+$_;if($i++%2){$x+=$z}else{$y+=$z}$l="$_".Length};$x;$y;$x 44 bytes to calculate the values, 70 bytes to formulate the ASCII Takes input $n as the zero-indexed column. Sets the start of our sequence array $a=1,1. We then loop up to $n with 1..$n|%{...} to construct the array. Each iteration, we concatenate on the sum of (two elements ago) + (the previous element)*(whether we're odd or even index). This will generate $a=1,1,3,4,11... up to $n+2. So, we need to slice $a to only take the first 0..$n elements, and pipe those through another loop |%{...}. Each iteration we set helper $z equal to a number of spaces plus the current element as a string. Then, we're splitting out whether that gets concatenated onto $x (the top and bottom rows) or $y (the middle row) by a simple odd-even if/else. Then, we calculate the number of spaces for $l by taking the current number, stringifying it, and taking its .Length. Finally, we put $x, $y, and $x again on the pipeline, and output is implicit. Since the default .ToString() separator for an array when printing to STDOUT is a newline, we get that for free.
### Example
PS C:\Tools\Scripts\golfing> .\pascal-braid.ps1 27
1 4 15 56 209 780 2911 10864 40545 151316 564719 2107560 7865521 29354524
1 3 11 41 153 571 2131 7953 29681 110771 413403 1542841 5757961 21489003
1 4 15 56 209 780 2911 10864 40545 151316 564719 2107560 7865521 29354524
## PHP 265 bytes
<?php $i=$argv[1];$i=$i?$i:1;$a=[[],[]];$s=['',''];$p='';for($j=0;$j<$i;$j++){$y=($j+1)%2;$x=floor($j/2);$v=$x?$y?2*$a[0][$x-1]+$a[1][$x-1]:$a[0][$x-1]+$a[1][$x]:1;$s[$y].=$p.$v;$a[$y][$x]=$v;$p=str_pad('',strlen($v),' ');}printf("%s\n%s\n%s\n",$s[0],$s[1],$s[0]);
Un-golfed:
$a = [[],[]];$s = ['',''];
$p = '';$i=$argv[1];$i=$i?$i:1;
for($j=0;$j<$i;$j++) {
$y = ($j+1) % 2;
$x = floor($j/2);
if( $x == 0 ) {$v = 1;
} else {
if( $y ) {$v = 2 * $a[0][$x-1] + $a[1][$x-1];
} else {
$v =$a[0][$x-1] +$a[1][$x]; } }$s[$y] .=$p . $v;$a[$y][$x] = $v;$p = str_pad('', strlen($v), ' '); } printf("%s\n%s\n%s\n",$s[0], $s[1],$s[0]);
## Python 278 bytes
import sys,math;a=[[],[]];s=['',''];p='';i=int(sys.argv[1]);i=1 if i<1 else i;j=0
while j<i:y=(j+1)%2;x=int(math.floor(j/2));v=(2*a[0][x-1]+a[1][x-1] if y else a[0][x-1]+a[1][x]) if x else 1;s[y]+=p+str(v);a[y].append(v);p=' '*len(str(v));j+=1
print ("%s\n"*3)%(s[0],s[1],s[0])
# Ruby, 120 bytes
Returns a multiline string.
Try it online!
->n{a=[1,1];(n-2).times{|i|a<<(2-i%2)*a[-1]+a[-2]}
z=->c{a.map{|e|c+=1;c%2>0?' '*e.to_s.size: e}*''}
[s=z[0],z[1],s]*$/} ## Matlab, 223 characters, 226 bytes function[]=p(n) r=[1;1];e={(' 1 ')',('1 1')'} for i=3:n;r(i)=sum((mod(i,2)+1)*r(i-1)+r(i-2));s=num2str(r(i));b=blanks(floor(log10(r(i)))+1);if mod(i,2);e{i}=[b;s;b];else e{i}=[s;b;s];end;end reshape(sprintf('%s',e{:}),3,[]) Ungolfed and commented: function[]=p(n) r=[1;1]; % start with first two e={(' 1 ')',('1 1')'} % initialize string output as columns of blank, 1, blank and 1, blank, 1. for i=3:n; % for n=3 and up! r(i)=sum((mod(i,2)+1)*r(i-1)+r(i-2)); % get the next number by 1 if even, 2 if odd times previous plus two steps back s=num2str(r(i)); % define that number as a string b=blanks(floor(log10(r(i)))+1); % get a number of space characters for that number of digits if mod(i,2); % for odds e{i}=[b;s;b]; % spaces, number, spaces else % for evens e{i}=[s;b;s]; % number, spaces, number end; end reshape(sprintf('%s',e{:}),3,[]) % print the cell array of strings and reshape it so it's 3 lines high # PHP, 135124123 120 bytes <?while($i<$argv[1]){${s.$x=!$x}.=${v.$x}=$a=$i++<2?:$v1+$v+$x*$v;${s.!$x}.=str_repeat(' ',strlen($a));}echo"$s
$s1$s";
taking advantage of implicit typecasts and variable variables
a third of the code (37 bytes) goes into the spaces, 64 bytes altogether used for output
breakdown
$i=0;$x=false; $v=$v1=1; $s=$s1=''; // unnecessary variable initializations
for($i=0;$i<$argv[1];$i++) // $i is column number -1 {$x=!$x; //$x = current row: true (1) for inner, false (empty string or 0) for outer
// calculate value
$a=$i<2? // first or second column: value 1
:$v1+(1+$x)*$v // inner-val + (inner row: 1+1=2, outer row: 1+0=1)*outer-val ;${s.$x}.=${v.$x}=$a; // replace target value, append to current row
${s.!$x}.=str_repeat(' ',strlen($a)); // append spaces to other row } // output echo "$s\n$s1\n$s";
## Batch, 250 bytes
@echo off
set s=
set d=
set/ai=n=0,j=m=1
:l
set/ai+=1,j^^=3,l=m+n*j,m=n,n=l
set t=%s%%l%
for /l %%j in (0,1,9)do call set l=%%l:%%j= %%
set s=%d%%l%
set d=%t%
if not %i%==%1 goto l
if %j%==1 echo %d%
echo %s%
echo %d%
if %j%==2 echo %s%
Since the first and third lines are the same, we just have to build two strings. Here d represents the string that ends with the last entry and s represents the string that ends with spaces; the last four lines ensure that they are printed in the appropriate order. i is just the loop counter (it's slightly cheaper than counting down from %1). j is the toggle between doubling the previous number before adding it to the current number to get the next number. m and n contain those numbers. l, as well as being used as a temporary to calculate the next number, also gets its digits replaced with spaces to pad out s; s and d are exchanged each time via the intermediate variable t.
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2019-07-19 16:56:01
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4722103178501129, "perplexity": 2169.012577666169}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526324.57/warc/CC-MAIN-20190719161034-20190719183034-00234.warc.gz"}
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https://scipost.org/submissions/1712.08012v1/
|
# Engineering Gaussian states of light from a planar microcavity
### Submission summary
As Contributors: Mathias Van Regemortel Arxiv Link: http://arxiv.org/abs/1712.08012v1 Date submitted: 2017-12-22 Submitted by: Van Regemortel, Mathias Submitted to: SciPost Physics Domain(s): Theoretical Subject area: Quantum Physics
### Abstract
Quantum fluids of light in a nonlinear planar microcavity can exhibit antibunched photon statistics at short distances due to repulsive polariton interactions. We show that, despite the weakness of the nonlinearity, the antibunching signal can be amplified orders of magnitude with an appropriate free-space optics scheme to select and interfere output modes. Our results are understood from the unconventional photon blockade perspective by analyzing the approximate Gaussian output state of the microcavity. In a second part, we illustrate how the temporal and spatial profile of the density-density correlation function of a fluid of light can be reconstructed with free-space optics. Also here the nontrivial (anti)bunching signal can be amplified significantly by shaping the light emitted by the microcavity.
###### Current status:
Has been resubmitted
### Submission & Refereeing History
Resubmission 1712.08012v3 (29 May 2018)
Resubmission 1712.08012v2 (16 April 2018)
Submission 1712.08012v1 (22 December 2017)
## Invited Reports on this Submission
Toggle invited reports view
### Strengths
1- The writing is impeccable.
2- The mathematical derivations are clear and well detailed.
3- The scheme to manipulate and observe non-classical photon statistic in this system is very interesting and clever.
### Weaknesses
1- A more complete analysis of the scheme robustness in realistic setups is missing.
2- The motivation for engineering antibunched output Gaussian fields and why doing it with this particular setup is not stated clearly enough.
### Report
This manuscript describes how to obtain antibunched output optical field from a driven nonlinear planar micro cavity. The source of anti bunching is the weak nonlinear interaction between intracavity polaritons which effectively acts as a non-degenerate parametric amplifier (NDPA) for opposite k-modes. An interference scheme of the output field and the attenuation of the k=0 driven mode allows to manipulate and measure the level of anti-bunching of a finite-k single mode output mode. The same setup also allows to reconstruct the spatial profil of the intensity correlation of the intra-cavity optical field.
I personally enjoyed reading the manuscript as it is clearly written and great attention as been devoted to the mathematical details. The scheme is also an interesting and clear example of the unconventional photon blockade'' as it emphasizes on the two main ingredients responsible of the phenomenon, i.e. squeezing and displacement of the optical field, in a totally different setup than the two-cavity system where it was initially introduced. The independent control of the squeezing and the field displacement via the detuning of the drive and the spatial attenuation of the k=0 mode, respectively, is in my opinion a very clever idea. The freedom offered by the spatial extend of the system (in contrast to zero-dimension cavity NDPA) adds good value to this system.
That being said, I still have two major concerns about this work. The first one goes in the same direction of the first referee (thanks to the open peer review). As the antibunching is very sensitive to any imperfections, as capture by the effective thermal population of the resulting gaussian output optical field, I think it would be important to have a short section that roughly estimates the effects of the main noise sources in such nonlinear planar micro cavities. The goal is to have an idea of the robustness of these nonclassical signatures in state-of-the-art setups.
The second concern is more about the motivation of this project itself. As much as I think the idea is clever and interesting, I personally always had some doubts on the relevance of this unconventional photon blockade''. In the case of standard photon blockade, anti bunching is a consequence of having a single photon state. Single photon sources and non-Gaussian states are very useful for all sort of quantum information processing schemes. However, the antibunching resulting form the unconventional photon blockade'' only means that the probability of having two photons is decreased, but does not ensure single-photon light field as higher-excitation states still contribute. The consequence is that the output field is still Gaussian, which takes away a lot of its utility for quantum information processing. This project is still interesting and fits very well in the body of literature concerning this phenomenon, but I would like to see a better description of the motivation to chase this nonclassical statistic in this context.
In the same line of thoughts, what makes this system more attractive than the two-cavity setup initially used in the literature or even than a DPA; the simplest system where you can observe UPB? In the conclusion, it is written: ... the planar microcavity geometry differs from the usual two-cavity geometry typically considered in this literature. I agree about this statement, but it should be clearly stated why. I also think that it is an important point (especially compared to a DPA) that should be emphasized right from the start in the introduction.
On a slightly more technical level, I think the condition to reach the bistable regime introduced in section 2.2 (from Eq. 8) is incomplete. While having a large detuning (\delta > \sqrt(3)\gamma/2) is necessary, one should also have a condition on the drive strength F>??. Unless I'm missing something, no matter the detuning, if my drive is infinitively small, I should not reach the bistable regime.
On a lighter note, there are small additional corrections:
1. After Eq. 13, the operator Chi is not defined (in the main text).
2. The phases phi_+/- introduced in Eq. 27 should be defined right after the Equation.
3. The first sentence of section 4.3 should go: ... analyze the Delta > 0 case.
Despite my concerns and comments, I think this work is of great quality and very interesting. Once the points noted above are addressed, I strongly support its publication.
### Requested changes
1- Add a section on imperfection sources in actual experimental setups and a rough estimation of their effects on the antibunching.
2- Describe the motivation for engineering Gaussian output fields that exhibit antibunching.
3- Emphasize more the differences and strength of this particular system compare to the DPA and the two-cavity setup.
3- State the condition on the drive strength to reach the bistable regime.
4- Address the small corrections noted in the report.
• validity: high
• significance: good
• originality: high
• clarity: top
• formatting: excellent
• grammar: perfect
### Author Mathias Van Regemortel on 2018-04-16
(in reply to Report 2 on 2018-02-06)
1) We have added an extra appendix (C) where we discuss the primary sources of noise expected to influence the scheme, with appropriate references in the main text. In particular, we analyze the impact of a disorder potential and pure polariton dephasing and refer to numbers found in literature where possible. We also suggest that the building up of a thermal polariton population can be circumvented by using a pulsed excitation scheme. We would like to thank the referee for drawing our attention to these points.
2-3) The referee is perfectly right when he says that the resulting output field is still Gaussian and therefore no single photon (Fock) state. However, we give some more motivation in the introduction to explain that even this strong nonclassical feature can be useful. Moreover, we emphasize that our scheme has a larger flexibility compared to previous two-cavity models, as the squeezing and interference is now spatially separated. This was a very useful comment and addressing it has certainly improved our manuscript.
4) We mean that there is bistable behavior in function of the drive strength F, so that automatically implies that it only occurs for a range of values F. When the drive strength F is increased, at some point there is a sudden jump in polariton density.
5) We have corrected the small mistakes.
### Strengths
1. The work, through a rigorous theoretical investigation, proposes an experimental scheme by which nonclassical states of polaritons in semiconductor microcavities could be demonstrated. An experimental proof that microcavity polaritons can host nonclassical states has been eluding experimental efforts for over 25 years. This proposal may indicate a promising route towards this goal.
2. The work is written with great clarity and impeccable style.
### Weaknesses
1. Some of the most common effects preventing nonclassical states of polaritons (such as e.g. pure dephasing and additional thermal population) are not discussed in this work. The real feasibility of the experiment remains therefore questionable.
2. Citations to highly relevant works are lacking.
### Report
This work presents a theoretical analysis of a spontaneous four-wave mixing process - otherwise understood as quantum fluctuations of a coherently driven polariton condensate - showing that under an appropriate collection and interference scheme for the emitted light, subpoissonian statistics could be observed even in presence of a very weak polariton nonlinearity. The phenomenon is linked to the Unconventional Photon Blockade, already described in the literature, and the analysis is carried out in terms of optimal squeezing.
While the theoretical investigation is technically correct in my opinion, and very accurate, I still think that the feasibility of the proposed experimental scheme is not thoroughly assessed by the Authors. Microcavity polaritons always present with some extrinsic effects that are almost unavoidable. Disorder is one, and in the present work it is quickly dismissed by a couple of sentences on page 14, while on page 11 it is clearly stated that the proposed phenomenon relies on the k-> -k symmetry of the system. I am wondering if an analysis in terms of small k-broadening of the emission could be carried out without too much hassle, knowing the typical k-broadening of good quality microcavities. In any case, I think that the disorder issue should be discussed in a more insightful way.
My main concern however, is not disorder but rather phenomena like pure dephasing and additional thermal (i.e. incoherent) occupation. In a planar microcavity, polaritons are prone to all sorts of scattering mechanisms, and it is known that a combination of polariton-phonon and polariton-polariton interaction can lead to incoherent occupation of lower lying modes even if the main driving field is resonant with the lowest lying mode (not to speak of the cases with positive detuning considered by the Authors). It appears from Figure 3 that the ideal condition g2(0)=8*sqrt(n_th) is not realized for sizeable values of n_th, for which instead g2(0) seems to grow much more than what expected by this ideal condition. Given the scattering with acoustic phonons at finite temperature, and the phonon induced relaxation, one always expects a very small incoherent background occupation. This occupation is hardly going to be lower than one polariton per mode, which seems much higher than the value n_th<<0.1 which seems to be required from Figure 3. Can the Authors estimate an additional incoherent occupation at finite temperature? Can they refer to previous experiments where it has been reasonably shown that under similar conditions a very small incoherent polariton occupation is produced? The same criticism holds for pure dephasing. In the original UPB proposal, it is shown that the pure dephasing rate must be much smaller than the nonlinear energy per photon, in order for UPB to survive. Acoustic-phonon scattering rates are known as a function of temperature for polaritons, and they should easily lead to an estimate of the pure dephasing rate? Is this low enough at typical temperatures?
Another important point is the fact that few relevant citations are missing from the bibliography. First - but this is not the Authors' fault as the preprint appeared after they submitted theirs - there is now a clear cut experimental demonstration of UPB in circuit QED. See arXiv:1801.04227. Second, a review article on UPB was published recently in PRA 96, 053810 (2017). It is important in my opinion to cite this paper in particular, as it already introduces the idea of optimal squeezing through interference of the output with the driving field, which is therefore not fully original. Finally, there is at least one published experiment where phenomena strongly related to optimal squeezing and UPB have been investigated for microcavity polaritons. This work (and the related theoretical proposal) should also in my opinion be cited as highly relevant to the present work.
As a minor point, figures are not ordered in the same way as they are cited in the text, which is a bit confusing.
### Requested changes
1. Carefully assess the feasibility of the proposed experiment when in presence of both disorder-induced k-broadening, pure dephasing, and extrinsic additional incoherent occupation, using realistic parameters for microcavity polaritons, to the best of the Authors knowledge.
2. Complete the bibliography with the citations described in the main report.
3. Fix the order of the figures.
• validity: ok
• significance: good
• originality: good
• clarity: high
• formatting: excellent
• grammar: excellent
### Author Mathias Van Regemortel on 2018-04-16
(in reply to Report 1 on 2018-01-29)
1) We have added an extra appendix (C) where we discuss the primary sources of noise expected to influence the scheme, with appropriate references in the main text. In particular, we analyze the impact of a disorder potential and pure polariton dephasing and refer to numbers found in literature where possible. We also suggest that the building up of a thermal polariton population can be circumvented by using a pulsed excitation scheme. We would like to thank the referee for drawing our attention to these points.
2) The citations were added to the main text.
3) The order of the figures has been changed.
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2018-06-24 10:37:53
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6754898428916931, "perplexity": 1125.0734453841897}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267866932.69/warc/CC-MAIN-20180624102433-20180624122433-00090.warc.gz"}
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https://support.bioconductor.org/p/9146625/
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tximport: not all(file.exists(files)) are true
1
0
Entering edit mode
• 0
@1e548362
Last seen 14 days ago
Hong Kong
I used the standard protocol of Cold Spring Harb, and get some .genes.results after most procedures. when I want to import RSEM data, I found that the Rstudio software warned all(file.exists(files)) are true, so how can I solve the problem.
dir <- "/Users/Documents/eQTL_analysis traning"
library(tximport)
library(DESeq2)
files <- file.path (dir, samples$Sample, paste0(samples$Sample, ".genes.results"))
names(files) <- samples\$Sample
txi <- tximport (files, type = "rsem", txIn = FALSE, txOut = FALSE)
tximport • 68 views
0
Entering edit mode
@mikelove
Last seen 1 hour ago
United States
Check files to see what files are listed and then figure out which (or all) are not on your machine at the location you specified.
file.exists(files)
R will not guess the right location for you. You have to specify the correct location, relative to your current working directory or using absolute paths.
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2022-10-05 17:29:46
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.34872373938560486, "perplexity": 10905.852238953947}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337663.75/warc/CC-MAIN-20221005172112-20221005202112-00762.warc.gz"}
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https://physics.stackexchange.com/questions/527459/supercharge-in-mathcaln-1-supersymmetric-quantum-mechanics-and-noethers-th
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# Supercharge in $\mathcal{N}=1$ supersymmetric quantum mechanics and Noether's theorem
Consider the $$0+1$$ dimensional Lagrangian
$$L=\frac{1}{2}\dot{X}^2(t)+i \psi(t) \dot{\psi}(t).\tag{1.24}$$
Essentially this the Lagrangian of a particle moving in one dimension, $$X$$, with an additional degree of freedom $$\psi$$. This can be thought of as a Lagrangian for a spinning particle moving in one dimension.
Define the supersymmetry transformations (and think of $$\delta$$ as a fermionic operator on the fields) as
$$\delta X=2i \epsilon \psi\tag{1.28a}$$ and $$\delta \psi=- \epsilon \dot{X}.\tag{1.28b}$$
Noting that $$\psi$$ and $$\delta$$ anticommute, $$X$$ and $$\delta$$ commute, and also that $$\delta$$ is a linear operator, we can easily see that
$$\delta L = i \epsilon \frac{d}{dt}(\psi \dot{X}).\tag{1.29}$$
Thus, the action is invariant since the Lagrangian changes only by a total derivative, under the above transformation. The conserved 'current' (in fact in one dimension it is the conserved charge) gives, by Noether's theorem,
$$\epsilon Q=\frac{\partial L}{ \partial \dot{X}} \delta X+\frac{\partial L}{ \partial \dot{\psi}} \delta \psi-i \epsilon \psi \dot{X}=2i\epsilon \dot{X} \psi-i \epsilon \dot{X} \psi-i \epsilon \psi \dot{X}=0!\tag{1}$$
So the charge turns out to be trivial. However, in these notes, in equation (1.30) it is claimed that the supercharge is, in fact,
$$Q=\psi \dot{X}.\tag{1.30}$$
What am I missing?
The second term in OP's formula (1) for the Noether charge has a sign mistake. The second term should be $$\delta\psi^{\mu} \frac{\partial_L L}{ \partial \dot{\psi}^{\mu}} ~=~(-\epsilon \dot{X}^{\mu})(-i\psi_{\mu}) ~=~i \epsilon \dot{X}^{\mu} \psi_{\mu} ~=~(i\psi_{\mu})(-\epsilon \dot{X}^{\mu}) ~=~\frac{\partial_R L}{ \partial \dot{\psi}^{\mu}}\delta\psi^{\mu} ,$$ depending on where we use a left (right) derivative, i.e. the derivative acts from left (right), respectively. As a result the Noether charge becomes non-zero: $$Q~=~2i\psi_{\mu} \dot{X}^{\mu}.\tag{1.30'}$$ The overall factor $$2i$$ has to do with a strange normalization.
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2022-06-28 13:35:30
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http://drorbn.net/index.php?title=1617-257/homework_13_assignment_solutions
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# 1617-257/homework 13 assignment solutions
## Doing
Solve all the problems in sections 25-26, but submit only your solutions of problems 4 and 8 in section 25 and problems 5 and 6 in section 26. In addition, solve the following problem, though submit only your solutions of parts d and e:
Problem A. Let $\alpha\colon\{(u,v)\in{\mathbb R}^2\colon u^2+v^2\leq 1\}\to{\mathbb R}^3$ be given by $\alpha(u,v)=\left(u-v,\,u+v,\,2(u^2+v^2)\right)$. Let $M$ be the image of $\alpha$.
a. Describe $M$.
b. Show that $M$ is a manifold.
c. Find the boundary $\partial M$ of $M$.
d. Find the volume $V(M)$ of $M$.
e. Find $\int_MzdV$ (where $z$ denotes the third coordinate of ${\mathbb R}^3$).
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2017-03-29 13:16:52
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http://openstudy.com/updates/559d4d44e4b0564dd2d49ff3
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• anonymous
suppose a triangle has sides a,b,and c, and let beta be the angle opposite the length a.what must be true.
Mathematics
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2017-03-27 18:58:19
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https://scicomp.stackexchange.com/questions/3420/recommendations-for-a-lightweight-no-install-c-or-c-based-dense-linear-algebra/3429
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# Recommendations for a lightweight/no-install C or C++ based dense linear algebra solver
Most of my programming is one-off research codes in C for my own use. I have never distributed any code to other than close collaborators. I have developed an algorithm that I am publishing in a scientific journal. I want to provide the source code and perhaps executable code in the online supplement to the article. A colleague requested that I make a generalization to the algorithm which required me to write in C++ (ack!) and which requires that I solve small dense linear systems. If I succeed in getting a user base for the algorithm it will be partly because the entry bar to using it is low (like on the floor). Potential users won't install libraries, etc. in order to use the code. I want the code to be fully stand alone and unencumbered by any license at all. I might simply write my own solver by taking something out of Golub and van Loan but I'd rather use a vanilla solver that someone else has already written if there are any out there. Suggestions appreciated. Thanks!
• possible duplicate of Recommendations for a usable, fast C++ matrix library? – GertVdE Oct 4 '12 at 15:24
• Dear jep, welcome to the forum. Your question is very similar to the one here: scicomp.stackexchange.com/questions/351/… – GertVdE Oct 4 '12 at 15:24
• Library solvers tend to be complex and big for the sake of robustness, efficiency, and generality. If your problems are very small and reasonably well conditioned, I would suggest you to write your own mini-implementation. – Stefano M Oct 4 '12 at 21:52
• @GertVdE, thanks for the quick response on this question. I'm uncomfortable linking to the "Recommendations..." question because both the question and the top answer are too general to provide any help in situations like these. If you'd like to discuss this further, I suggest we take it to the scicomp chat room. – Aron Ahmadia Oct 4 '12 at 23:29
• @AronAhmadia: I think the only way to start settling some of these debates is to start implementing a computational science programming chrestomathy that is both language and library dependent. If the code is clear, and configuration issues can be taken care of (using a shell script, Chef, or Puppet), then debates about performance can be taken care of (or made concrete) by just running the code and timing it on a reference machine. Debates about clarity can be resolved (or at least, made more concrete) by looking at the code. Otherwise, we'll keep having the same arguments. – Geoff Oxberry Oct 5 '12 at 1:46
I would suggest to exactly duplicate the Lapack interface to the function that you need, most probably you just need dgesv. That way people that have Lapack installed can simply link to it and it will just work. For people that don't have Lapack installed, you provide your own simple implementation of this function, or possibly implement it using Eigen or FLENS as others suggested.
In the Fortran land, the Lapack library is such a standard, that most people simply use it and that's it, instead of providing their own implementations.
• +1 Add to it the fact that most Linux distributions (at least Debian based) have binary packages in the repository and all vendor supplied math libs (MKL, SunPerf, ACML, ESSL etc.) carry it. You should always use standard libs as much as possible though if you're on Windows/Mac you might be better off with something C based as installing a free Fortran compiler (gfortran) on them is some amount of work or so I have heard. – stali Oct 5 '12 at 12:59
• I have used lapack many times but I am not currently in fortran land. I expect that the statistical distribution of platforms my user base run on would be similar to that of the world at large meaning mainly windows, a smaller percentage of macs and an even smaller percentage of *nix. My experience with windows is minimal and I prefer to keep it that way. This is the reason I want a stand alone C++ code. I figure I'll have to provide some of my users with help getting the code to compile and run. I need to minimize the work required to do that. – jep Oct 5 '12 at 13:41
• If your user base is Windows/Macs then you're better of with a simple C based (perhaps even your own) implementation. A package that is difficult to install or depends on 5 other libs, specially when there is no first class binary package repository (like Debian) available, will turn your users off for a long time. Remember most Windows/Mac users are used to one click install. Ease of use triumphs everything else. – stali Oct 5 '12 at 14:11
A very early mistake that many people make when getting started in scientific computing is assuming that you need to write all of your code in the same language. I think this is due largely to historical reasons, when it wasn't clear how to make compiled programs communicate with each other across even versions of the same compiler. That said, in this case, if you are going to be using C++ anyway, there are several very good C++ header-only template libraries that might fit your needs.
• I was hoping for a single file. I've done scientific programming for quite a while. I have mixed languages like C and fortran quite a bit but for this project I really just want one file containing all my source code. I suppose I could put a C solver in the C++ code which wouldn't be a big deal. Mainly I want to keep the code as simple as possible. LU with pivoting should be adequate. I'll look at Eigen. Thanks! – jep Oct 5 '12 at 0:57
• @jep, you could also try cherry-picking the routines you need from CLAPACK if you really don't care at all about performance. – Aron Ahmadia Oct 5 '12 at 1:12
• There are good reasons for writing all the dependent code in the same language, in particular, in HPC environments, you have weird compiler/linking issues and 32/64-bit interface issues. For example, how do I know the width of an integer for built-in libraries? How do I know for sure what compiler was used for a built-in library, and can I link against it with this other compiler? Having everything in one language simplifies many of these issues. And yes, there should be documentation provided by the cluster maintainers, but most of the time there isn't. – Victor Liu Oct 5 '12 at 22:57
• @VictorLiu - The issues you are referring to are more tightly coupled to implementations than languages. Comment space is a poor place to get into a serious discussion, but I am happy to engage you in chat or elsewhere if you'd like me to expand my thoughts on this. – Aron Ahmadia Oct 6 '12 at 3:17
If you want a reliable solver for systems of linear equations I would recommend FLENS. It contains an exact re-implementation of LAPACK (it even reproduces the same roundoff errors as LAPACK if a single-threaded BLAS implementation is used). This is true for all FLENS-LAPACK functions (together with the utility functions about 100 routines).
FLENS is under a BSD License and therefore allows to be incorporated into proprietary products.
FLENS is header only and if you only need a subset of FLENS I can give you a stripped-down version containing only those functions you need. FLENS comes with its own reference BLAS implementation. But optionally your users can link against optimized BLAS libraries like ATLAS, OpenBLAS or GotoBALS. For large matrices this gives a performance gain of about 40% compared to Eigen.
And yes, Eigen also uses the LAPACK test suite to check their results. They do this for 3 functions (Lu, Cholesky and Eigenvalues/-vectors of a symmetric matrix). However, their computation of eigenvalues/-vectors of a non-symmetric matrix would fail the LAPACK test suite.
Disclaimer: Yes, FLENS is my baby! That means I coded about 95% of it and every line of code was worth it.
• Michael - Please consider this a friendly warning that you need to follow the rule in the faq regarding disclosing affiliation. – Aron Ahmadia Oct 5 '12 at 0:30
• Sure, but you also could re-phrase your posts from 'I would strongly recommend that you consider Eigen' to something like 'there is for example Eigen'. In this case I delete my remarks about Eigen (although they are all proven to be true) including this one. – Michael Lehn Oct 5 '12 at 0:39
• Your remarks about Eigen are not at issue here (although they seem off-topic to me). You are a primary developer of FLENS, if you are going to recommend it in an answer here, you must disclose your affiliation as developer of the project. – Aron Ahmadia Oct 5 '12 at 0:43
• Ah, ok then. I thought was was implicitly clear by '... I can give you ...'. Is the disclosure in this form ok? – Michael Lehn Oct 5 '12 at 0:47
• I just want to say thanks for doing this; I had similar plans to re-implement a large part of Lapack in C++. However, it seems that for most of the advanced (eigenvalue) routines, you simply defer to calling into Lapack, so it's a bit of false advertising to say that you re-implement the whole thing. On the other hand, I have actually ported the ZGEEV source to C++ in RNP, albeit some parts are still in 1-based indexing from auto-conversion. – Victor Liu Oct 5 '12 at 1:33
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2019-12-15 08:33:14
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https://ctftime.org/writeup/28282
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Tags: crypto
Rating:
# A primed hash candidate
## Statement
>After the rather embarrassing first attempt at securing our login, our student intern has drastically improved our security by adding more parameters. Good luck getting in now!
We are then given access to a remote server running the following python code :
python
ERROR = "Wrong password with hash "
PASSWD = 91918419847262345220747548257014204909656105967816548490107654667943676632784144361466466654437911844
secret1 = "REDACTED"
secret2 = "REDACTED"
secret3 = int("xxx")
def hash(data):
out = 0
data = [ord(x) ^ ord(y) for x,y in zip(data,secret1*len(data))]
data.extend([ord(c) for c in secret2])
for c in data:
out *= secret3
out += c
return out
try:
while True:
if hash(data) == PASSWD:
print(SUCCESS)
break
else:
print(ERROR + str(hash(data)))
except EOFError:
pass
## Analysis
We see that the server asks us for a password. It then hashes it using a custom function and compares it to the PASSWD value. We then need to find an input producing the given hash. As the server is kind enough to give us the hash of inputs we feed him, this will be useful for our attack.
The hashing function works in three steps:
1. Our input is xored with the word secret1.
2. secret2 is appended to the previous xored string.
3. The final string is then mapped to an integer using a pseudo base-conversion.
### Finding secret3
The first thing to remark is that an null string is a valid input:
The given hash is then f(secret2) where f is the pseudo-conversion function.
For any given input m its hash is then hash(m) = f(m xor secret1) * secret3^len(secret2) + f(secret2). We then know that secret3 divides gcd(hash(1)-f(secret2), hash(0)-f(secret2)). We compute the gcd using sage for example :
python
sage: hash0 = 19005887928914280732260134378748151614599045204546
sage: hash1 = 18783496307853128677280688327194704466734557942945
sage: fsecret2 = 102600138716356059007219996705144046117627968461
sage: gcd01 = gcd(hash0-fsecret2, hash1-fsecret2)
sage: factor(gcd01)
233^20
We then deduce that secret3 = 233.
### Finding secret2
This step is actually not necessary since we already know fsecret2, but we can stll invert the base-conversion function that encrypted secret2 by converting the integer in base 10 to an integer base 233, and displaying each 'digit' as an ascii character.
python
chars = []
fsecret2 = 102600138716356059007219996705144046117627968461
while fsecret2 > 0:
modulo = fsecret2%233
chars.append(chr(modulo))
fsecret2 = fsecret2//233
print(b"".join(chars[::-1]))
Running the code gives us secret2 = 'ks(3n*cl3p%3925(*4*2'
### Finding secret1
To find secret1, we can process similarly as we did in the last step. First, we need to hash a very long input, to make sure that all characters of secret1 are indeed xored at one point.
We can then take the hash of 20*'0' = '00000000000000000000', which should be long enough. We find that
hash(20*'0') = 18126456734850052517766482160657835416461226894114798664396414018388402487161697110017734000706
Then, we can substract the known part from secret2 and secret3 to stay with the interesting stuff.
We end with the following code :
python
chars = []
bighash = 18126456734850052517766482160657835416461226894114798664396414018388402487161697110017734000706
interestingpart = (bighash - fsecret2) // (233**20)
while interestingpart > 0:
modulo = interestingpart%233
chars.append(chr(modulo ^ ord('0')))
interestingpart = interestingpart//233
print("".join(chars[::-1]))
Running the code gives us el3PH4nT$el3PH4nT$el, we then deduce that secret1 = 'el3PH4nT\$'
Now that we know all the secrets, we are able to compute back the password from the hash PASSWD. The only difficulty is that we don't know the length of the original password, and so we can't use the same trick as before without modifying it, as we don't know which character of secret1 to xor from.
Nevertheless, we are only interested in the length of the password modulus the length of secret1, so we can bruteforce this part.
python
for l in range(len(secret1)):
interestingpart = (PASSWD-fsecret2)//233**20
index = l
chars = []
while interestingpart>0:
modulo = interestingpart%233
chars.append(chr(modulo ^ ord(secret1[index%9])))
index=(index+8)%9
interestingpart = interestingpart//233
if(hash("".join(chars[::-1])) == PASSWD):
print("".join(chars[::-1]))
This gives us a collision for the given hash: GZZ9t3W3Ar34un44m8PLXX6.
We can get the flag now:
console
|
2022-05-21 16:33:08
|
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|
http://www.contrib.andrew.cmu.edu/~ryanod/?p=1061
|
# §6.4: Applications in learning and testing
In this section we describe some applications of our study of pseudorandomness.
We begin with a notorious open problem from learning theory, that of learning juntas. Let $\mathcal{C} = \{f : {\mathbb F}_2^n \to {\mathbb F}_2 \mid f \text{ is a } k\text{-junta}\}$; we will always assume that $k \leq O(\log n)$. In the query access model, it is quite easy to learn $\mathcal{C}$ exactly (i.e., with error $0$) in $\mathrm{poly}(n)$ time (Exercise 3.36(a)). However in the model of random examples, it’s not obvious how to learn $\mathcal{C}$ more efficiently than in the $n^{k} \cdot \mathrm{poly}(n)$ time required by the Low-Degree Algorithm (see Theorem 3.36). Unfortunately, this is superpolynomial as soon as $k > \omega(1)$. The state of affairs is the same in the case of depth-$k$ decision trees (a superclass of $\mathcal{C}$), and is similar in the case of $\mathrm{poly}(n)$-size DNFs and CNFs. Thus if we wish to learn, say, $\mathrm{poly}(n)$-size decision trees or DNFs from random examples only, a necessary prerequisite is doing the same for $O(\log n)$-juntas.
Whether or not $\omega(1)$-juntas can be learned from random examples in polynomial time is a longstanding open problem. Here we will show a modest improvement on the $n^{k}$-time algorithm:
Theorem 36 For $k \leq O(\log n)$, the class $\mathcal{C} = \{f : {\mathbb F}_2^n \to {\mathbb F}_2 \mid f \text{ is a } k\text{-junta}\}$ can be exactly learned from random examples in time $n^{(3/4)k} \cdot \mathrm{poly}(n)$.
(The $3/4$ in this theorem can in fact be replaced by $\omega/(\omega + 1)$, where $\omega$ is any number such that $n \times n$ matrices can be multiplied in time $O(n^{\omega})$.)
The first observation we will use to prove Theorem 36 is that to learn $k$-juntas, it suffices to be able to identify a single coordinate which is relevant. The proof of this is fairly simple and is left for the exercises:
Lemma 37 Theorem 36 follows from the existence of a learning algorithm which, given random examples from a nonconstant $k$-junta $f : {\mathbb F}_2^n \to {\mathbb F}_2$, finds at least one relevant coordinate for $f$ (with probability at least $1-\delta$) in time $n^{(3/4)k} \cdot \mathrm{poly}(n) \cdot \log(1/\delta)$.
Assume then that we have random example access to a (nonconstant) $k$-junta $f : {\mathbb F}_2^n \to {\mathbb F}_2$. As in the Low-Degree Algorithm we will estimate the Fourier coefficients $\widehat{f}(S)$ for all $1 \leq |S| \leq d$, where $d \leq k$ is a parameter to be chosen later. Using Proposition 3.30 we can ensure that all estimates are accurate to within $(1/3)2^{-k}$, except with probability most $\delta/2$, in time $n^d \cdot \mathrm{poly}(n) \cdot \log(1/\delta)$. (Recall that $2^k \leq \mathrm{poly}(n)$.) Since $f$ is a $k$-junta, all of its Fourier coefficients are either $0$ or at least $2^{-k}$ in magnitude; hence we can exactly identify the sets $S$ for which $\widehat{f}(S) \neq 0$. For any such $S$, all of the coordinates $i \in S$ are relevant for $f$. So unless $\widehat{f}(S) = 0$ for all $1 \leq |S| \leq d$, we can find a relevant coordinate for $f$ in time $n^{d} \cdot \mathrm{poly}(n) \cdot \log(1/\delta)$ (except with probability at most $\delta/2$).
To complete the proof of Theorem 36 it remains to handle the case that $\widehat{f}(S) = 0$ for all $1 \leq |S| \leq d$; i.e., $f$ is $d$th-order correlation immune. In this case, by Siegenthaler’s Theorem we know that $\deg_{{\mathbb F}_2}(f) \leq k-d$. (Note that $d < k$ since $f$ is not constant.) But there is a learning algorithm running in time in time $O(n)^{3\ell} \cdot \log(1/\delta)$ which exactly learns any ${\mathbb F}_2$-polynomial of degree at most $\ell$ (except with probability at most $\delta/2$). Roughly speaking, the algorithm draws $O(n)^\ell$ random examples and then solves an ${\mathbb F}_2$-linear system to determine the coefficients of the unknown polynomial; see the exercises for details. Thus in time $n^{3(k-d)} \cdot \mathrm{poly}(n) \cdot \log(1/\delta)$ this algorithm will exactly determine $f$, and in particular find a relevant coordinate.
By choosing $d = \left\lceil \frac34 k \right\rceil$ we balance the running time of the two algorithms. Regardless of whether $f$ is $d$th-order correlation immune, at least one of the two algorithms will find a relevant coordinate for $f$ (except with probability at most $\delta/2 + \delta/2 = \delta$) in time $n^{(3/4)k} \cdot \mathrm{poly}(n) \cdot \log(1/\delta)$. This completes the proof of Theorem 36.
Our next application of pseudorandomness involves using $\epsilon$-biased distributions to give a deterministic version of the Goldreich–Levin Algorithm (and hence the Kushilevitz–Mansour learning algorithm) for functions $f$ with small $\hat{\lVert} f \hat{\rVert}_1$. We begin with a basic lemma showing that you can get a good estimate for the mean of such functions using an $\epsilon$-biased distribution:
Lemma 38 If $f : \{-1,1\}^n \to {\mathbb R}$ and $\varphi : \{-1,1\}^n \to {\mathbb R}$ is an $\epsilon$-biased density, then $\left|\mathop{\bf E}_{{\boldsymbol{x}} \sim \varphi}[f({\boldsymbol{x}})] – \mathop{\bf E}[f]\right| \leq \hat{\lVert} f \hat{\rVert}_1 \epsilon.$
This lemma follows from Proposition 13(1), but we provide a separate proof:
Proof: By Plancherel, $\mathop{\bf E}_{{\boldsymbol{x}} \sim \varphi}[f({\boldsymbol{x}})] = \langle \varphi, f\rangle = \widehat{f}(\emptyset) + \sum_{S \neq \emptyset} \widehat{\varphi}(S) \widehat{f}(S),$ and the difference of this from $\mathop{\bf E}[f] = \widehat{f}(\emptyset)$ is, in absolute value, at most $\sum_{S \neq \emptyset} |\widehat{\varphi}(S)| \cdot |\widehat{f}(S)| \leq \epsilon \cdot \sum_{S \neq \emptyset} |\widehat{f}(S)| \leq \hat{\lVert} f \hat{\rVert}_1 \epsilon. \qquad \Box$
Since $\hat{\lVert} f^2 \hat{\rVert}_1 \leq \hat{\lVert} f \hat{\rVert}_1^2$ (exercise), we also have the following immediate corollary:
Corollary 39 If $f : \{-1,1\}^n \to {\mathbb R}$ and $\varphi : \{-1,1\}^n \to {\mathbb R}$ is an $\epsilon$-biased density, then $\left|\mathop{\bf E}_{{\boldsymbol{x}} \sim \varphi}[f({\boldsymbol{x}})^2] – \mathop{\bf E}[f^2]\right| \leq \hat{\lVert} f \hat{\rVert}_1^2 \epsilon.$
We can use the first lemma to get a deterministic version of Proposition 3.30, the learning algorithm which estimates a specified Fourier coefficient.
Proposition 40 There is a deterministic algorithm which, given query access to a function $f : \{-1,1\}^n \to {\mathbb R}$ as well as $U \subseteq [n]$, $0 < \epsilon \leq 1/2$, and $s \geq 1$, outputs an estimate $\widetilde{f}(U)$ for $\widehat{f}(U)$ satisfying $|\widetilde{f}(U) - \widehat{f}(U)| \leq \epsilon,$ provided $\hat{\lVert} f \hat{\rVert}_1 \leq s$. The running time is $\mathrm{poly}(n, s, 1/\epsilon)$.
Proof: It suffices to handle the case $U = \emptyset$ because for general $U$, the algorithm can simulate query access to $f \cdot \chi_U$ with $\mathrm{poly}(n)$ overhead, and $\widehat{f \cdot \chi_U}(\emptyset) = \widehat{f}(U)$. The algorithm will use Theorem 30 to construct an $(\epsilon/s)$-biased density $\varphi$ which is uniform over a (multi-)set of cardinality $O(n^2 s^2/\epsilon^2)$. By enumerating over this set and using queries to $f$, it can deterministically output the estimate $\widetilde{f}(\emptyset) = \mathop{\bf E}_{{\boldsymbol{x}} \sim \varphi}[f({\boldsymbol{x}})]$ in time $\mathrm{poly}(n, s, 1/\epsilon)$. The error bound now follows from Lemma 38. $\Box$
The other key ingredient needed for the Goldreich–Levin Algorithm was Proposition 3.40, which let us estimate $$\label{eqn:gl-key2} \mathbf{W}^{S\mid\overline{J}}[f] = \sum_{T \subseteq \overline{J}} \widehat{f}(S \cup T)^2 = \mathop{\bf E}_{\boldsymbol{z} \sim \{-1,1\}^{\overline{J}}}[\widehat{f_{J \mid \boldsymbol{z}}}(S)^2]$$ for any $S \subseteq J \subseteq [n]$. Observe that for any $z \in \{-1,1\}^{\overline{J}}$ we can use Proposition 40 to deterministically estimate $\widehat{f_{J \mid z}}(S)$ to accuracy $\pm \epsilon$. The reason is that we can simulate query access to the restricted function $\widehat{f_{J \mid z}}$, the $(\epsilon/s)$-biased density $\varphi$ remains $(\epsilon/s)$-biased on $\{-1,1\}^{J}$, and most importantly $\hat{\lVert} f_{J \mid z} \hat{\rVert}_1 \leq \hat{\lVert} f \hat{\rVert}_1 \leq s$ by Exercise 3.6. It is not much more difficult to deterministically estimate \eqref{eqn:gl-key2}
Proposition 41 There is a deterministic algorithm which, given query access to a function $f : \{-1,1\}^n \to \{-1,1\}$ as well as $S \subseteq J \subseteq [n]$, $0 < \epsilon \leq 1/2$, and $s \geq 1$, outputs an estimate $\beta$ for $\mathbf{W}^{S\mid\overline{J}}[f]$ which satisfies $|\mathbf{W}^{S\mid\overline{J}}[f] – \beta| \leq \epsilon,$ provided $\hat{\lVert} f \hat{\rVert}_1 \leq s$. The running time is $\mathrm{poly}(n, s, 1/\epsilon)$.
Proof: Recall the notation $\mathrm{F}_{S \mid \overline{J}} f$ from Definition 3.20; by \eqref{eqn:gl-key2}, the algorithm’s task is to estimate $\mathop{\bf E}_{\boldsymbol{z} \sim \{-1,1\}^{\overline{J}}}[(\mathrm{F}_{S \mid \overline{J}} f)^2(\boldsymbol{z})]$. If $\varphi : \{-1,1\}^{\overline{J}} \to {\mathbb R}^{\geq 0}$ is an $\tfrac{\epsilon}{4s^2}$-biased density, Corollary 39 tells us that $$\label{eqn:gl-det-est} \Bigl| \mathop{\bf E}_{\boldsymbol{z} \sim \varphi}[(\mathrm{F}_{S \mid \overline{J}} f)^2(\boldsymbol{z})] – \mathop{\bf E}_{\boldsymbol{z} \sim \{-1,1\}^{\overline{J}}}[(\mathrm{F}_{S \mid \overline{J}} f)^2(\boldsymbol{z})] \Bigr| \leq \hat{\lVert} \mathrm{F}_{S \mid \overline{J}} f \hat{\rVert}_1^2 \cdot \tfrac{\epsilon}{4s^2}\leq \hat{\lVert} f \hat{\rVert}_1^2 \cdot \tfrac{\epsilon}{4s^2} \leq \tfrac{\epsilon}{4},$$ where the second inequality is immediate from Proposition 3.21. We now show the algorithm can approximately compute $\mathop{\bf E}_{\boldsymbol{z} \sim \varphi}[(\mathrm{F}_{S \mid \overline{J}} f)^2(\boldsymbol{z})]$. For each $z \in \{-1,1\}^{\overline{J}}$, the algorithm can use $\varphi$ to deterministically estimate $(\mathrm{F}_{S \mid \overline{J}} f)(z) = \widehat{f_{J \mid z}}(S)$ to within $\pm s \cdot \tfrac{\epsilon}{4s^2} \leq \tfrac{\epsilon}{4}$ in $\mathrm{poly}(n,s, 1/\epsilon)$ time, just as was described in the text following \eqref{eqn:gl-key2}. Since $|\widehat{f_{J \mid z}}(S)| \leq 1$, the square of this estimate is within, say, $\tfrac{3\epsilon}{4}$ of $(\mathrm{F}_{S \mid \overline{J}} f)^2(z)$. Hence by enumerating over the support of $\varphi$, the algorithm can in deterministic $\mathrm{poly}(n,s, 1/\epsilon)$ time estimate $\mathop{\bf E}_{\boldsymbol{z} \sim \varphi}[(\mathrm{F}_{S \mid \overline{J}} f)^2(\boldsymbol{z})]$ to within $\pm \tfrac{3\epsilon}{4}$, which by \eqref{eqn:gl-det-est} gives an estimate to within $\pm \epsilon$ of the desired quantity $\mathop{\bf E}_{\boldsymbol{z} \sim \{-1,1\}^{\overline{J}}}[(\mathrm{F}_{S \mid \overline{J}} f)^2(\boldsymbol{z})]$. $\Box$
Propositions 40 and 41 are the only two ingredients needed for a derandomization of the Goldreich–Levin Algorithm. We can therefore state a derandomized version of its corollary Theorem 3.38 on learning functions with small Fourier $1$-norm:
Theorem 42 Let $\mathcal{C} = \{f : \{-1,1\}^n \to \{-1,1\} \mid \hat{\lVert} f \hat{\rVert}_1 \leq s\}$. Then $\mathcal{C}$ is deterministically learnable from queries with error $\epsilon$ in time $\mathrm{poly}(n, s, 1/\epsilon)$.
Since any $f : \{-1,1\}^n \to \{-1,1\}$ with $\mathrm{sparsity}(\widehat{f}) \leq s$ also has $\hat{\lVert} f \hat{\rVert}_1 \leq s$, we may also deduce from Exercise 3.36(c):
Theorem 43 Let $\mathcal{C} = \{f : \{-1,1\}^n \to \{-1,1\} \mid \mathrm{sparsity}(\widehat{f}) \leq 2^{O(k)}\}$. Then $\mathcal{C}$ is deterministically learnable exactly ($0$ error) from queries in time $\mathrm{poly}(n, 2^k)$.
Example functions which fall into the concept classes of these theorems are decision trees of size at most $s$, and decision trees of depth at most $k$, respectively.
We conclude this section by discussing a derandomized version of the Blum–Luby–Rubinfeld linearity test from Chapter 1.5:
Derandomized BLR Test Given query access to $f : {\mathbb F}_2^n \to {\mathbb F}_2$:
1. Choose ${\boldsymbol{x}} \sim {\mathbb F}_2^n$ and $\boldsymbol{y} \sim \varphi$, where $\varphi$ is an $\epsilon$-biased density.
2. Query $f$ at ${\boldsymbol{x}}$, $\boldsymbol{y}$, and ${\boldsymbol{x}} + \boldsymbol{y}$.
3. “Accept” if $f({\boldsymbol{x}}) + f(\boldsymbol{y}) = f({\boldsymbol{x}} + \boldsymbol{y})$.
Whereas the original BLR Test required exactly $2n$ independent random bits, the above derandomized version needs only $n + O(\log(n/\epsilon))$. This is very close to minimum possible: a test using only, say, $.99n$ random bits would only be able to inspect a $2^{-.01 n}$ fraction of $f$’s values.
If $f$ is ${\mathbb F}_2$-linear then it is still accepted by the Derandomized BLR Test with probability $1$. As for the approximate converse, we’ll have to make a slight concession: we’ll show that any function accepted with probability close to $1$ must be close to an affine function — i.e., satisfy $\deg_{{\mathbb F}_2}(f) \leq 1$. This concession is necessary: the function $f : {\mathbb F}_2^n \to {\mathbb F}_2$ might be $1$ everywhere except on the (tiny) support of $\varphi$. In that case the acceptance criterion $f({\boldsymbol{x}}) + f(\boldsymbol{y}) = f({\boldsymbol{x}} + \boldsymbol{y})$ will almost always be $1 + 0 = 1$; yet $f$ is very far from every linear function. It is, however, very close to the affine function $1$.
Theorem 44 Suppose the Derandomized BLR Test accepts $f : {\mathbb F}_2^n \to {\mathbb F}_2$ with probability $\tfrac{1}{2} + \tfrac{1}{2} \theta$. Then $f$ has correlation at least $\sqrt{\theta^2 – \epsilon}$ with some affine $g : {\mathbb F}_2^n \to {\mathbb F}_2$; i.e., $\mathrm{dist}(f,g) \leq \tfrac{1}{2} – \tfrac{1}{2} \sqrt{\theta^2 – \epsilon}$.
Remark 45 The bound in this theorem works well both when $\theta$ is close to $0$ and when $\theta$ is close to $1$. E.g., for $\theta = 1-2\delta$ we get that if $f$ is accepted with probability $1-\delta$ then $f$ is nearly $\delta$-close to an affine function, provided $\epsilon \ll \delta$.
Proof: As in the analysis of the BLR Test (Theorem 1.31) we encode $f$’s outputs by $\pm 1 \in {\mathbb R}$. Using the first few lines of that analysis we see that our hypothesis is equivalent to $\theta \leq \mathop{\bf E}_{{\substack{{\boldsymbol{x}} \sim {\mathbb F}_2^n \\ \boldsymbol{y} \sim \varphi}}}[f({\boldsymbol{x}})f(\boldsymbol{y})f({\boldsymbol{x}}+\boldsymbol{y})] = \mathop{\bf E}_{\boldsymbol{y} \sim \varphi}[f(\boldsymbol{y}) \cdot (f * f)(\boldsymbol{y})].$ By Cauchy–Schwarz, $\mathop{\bf E}_{\boldsymbol{y} \sim \varphi}[f(\boldsymbol{y}) \cdot (f * f)(\boldsymbol{y})] \leq \sqrt{\mathop{\bf E}_{\boldsymbol{y} \sim \varphi}[f(\boldsymbol{y})^2]}\sqrt{\mathop{\bf E}_{\boldsymbol{y} \sim \varphi}[(f * f)^2(\boldsymbol{y})]} = \sqrt{\mathop{\bf E}_{\boldsymbol{y} \sim \varphi}[(f * f)^2(\boldsymbol{y})]},$ and hence $\theta^2 \leq \mathop{\bf E}_{\boldsymbol{y} \sim \varphi}[(f * f)^2(\boldsymbol{y})] \leq \mathop{\bf E}[(f * f)^2] + \hat{\lVert} f * f \hat{\rVert}_1 \epsilon = \sum_{\gamma \in {\widehat{{\mathbb F}_2^n}}} \widehat{f}(\gamma)^4 + \epsilon,$ where the inequality is Corollary 39 and we used $\widehat{f * f}(\gamma) = \widehat{f}(\gamma)^2$. The conclusion of the proof is as in the original analysis (cf. Proposition 7, Exercise 1.28): $\theta^2 - \epsilon \leq \sum_{\gamma \in {\widehat{{\mathbb F}_2^n}}} \widehat{f}(\gamma)^4 \leq \max_{\gamma \in {\widehat{{\mathbb F}_2^n}}} \{\widehat{f}(\gamma)^2\} \cdot \sum_{\gamma \in {\widehat{{\mathbb F}_2^n}}} \widehat{f}(\gamma)^2 = \max_{\gamma \in {\widehat{{\mathbb F}_2^n}}} \{\widehat{f}(\gamma)^2\},$ and hence there exists $\gamma^*$ such that $|\widehat{f}(\gamma^*)| \geq \sqrt{\theta^2-\epsilon}$. $\Box$
### 8 comments to §6.4: Applications in learning and testing
• xiwu
Hi Ryan,
I think in the proof of Lemma 38, it shall be \widehat{\varphi}(S),
instead of \varphi(S), the Fourier coefficient of the density function at S
records the bias at S. Do I miss anything?
Best.
• Nope, you’re right. Thanks!
• Noam Lifshitz
Minor typo:
There is a missing “]” after “By cauchy schewarz,” in the last proof of this section.
By the way, congratulations for the book!
• Thanks! Sharp eyes. Missed by the book’s copyeditor, in fact
• Noam Lifshitz
Why is $\mathbf E_{\boldsymbol{y} \sim \varphi}[f(\boldsymbol{y})^2]$ equal to $1$?
• Noam Lifshitz
Sorry it is trivial.
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2017-05-23 20:39:42
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http://mathhelpforum.com/statistics/122826-probability-problem.html
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1. ## Probability Problem
A box contains 12 red and 13 white marbles. If 8 marbles are chosen at random (without replacement) determine the probability that 6 are red. Write your answer as a fraction reduced to lowest terms.
2. Originally Posted by sweeetcaroline
A box contains 12 red and 13 white marbles. If 8 marbles are chosen at random (without replacement) determine the probability that 6 are red. Write your answer as a fraction reduced to lowest terms.
${ {12\choose 6}{13\choose 2}\over {25\choose 8}}$
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2016-10-21 22:14:59
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https://gamedev.stackexchange.com/questions/75313/calculating-per-vertex-normal-in-geometry-shader
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# Calculating Per Vertex Normal in Geometry Shader
I am able to calculate normals per face in my Geometry Shader but i want to calculate per vertex normal for smooth shading. My Geometry shader is
#version 430 core
layout ( triangles ) in;
layout ( triangle_strip, max_vertices = 3 ) out;
out vec3 normal_out;
uniform mat4 projectionMatrix;
uniform mat4 viewMatrix;
uniform mat4 modelTranslationMatrix;
uniform mat4 modelRotationXMatrix;
uniform mat4 modelRotationYMatrix;
uniform mat4 modelRotationZMatrix;
uniform mat4 modelScaleMatrix;
void main(void)
{
// Please ignore my modelMatrix and NormalMatrix calculation here
mat4 modelMatrix = modelTranslationMatrix * modelScaleMatrix * modelRotationXMatrix * modelRotationYMatrix * modelRotationZMatrix;
mat4 modelViewMatrix = viewMatrix * modelMatrix;
mat4 mvp = projectionMatrix * modelViewMatrix;
vec3 A = gl_in[2].gl_Position.xyz - gl_in[0].gl_Position.xyz;
vec3 B = gl_in[1].gl_Position.xyz - gl_in[0].gl_Position.xyz;
mat4 normalMatrix = transpose(inverse(modelViewMatrix));
normal_out = mat3(normalMatrix) * normalize(cross(A,B));
gl_Position = mvp * gl_in[0].gl_Position;
EmitVertex();
gl_Position = mvp * gl_in[1].gl_Position;
EmitVertex();
gl_Position = mvp * gl_in[2].gl_Position;
EmitVertex();
EndPrimitive();
}
Since i don't have access to adjacent faces here, i cannot calculate per vertex normals.
How can i calculate per vertex normals in my Geometry Shader?
## 1 Answer
You can't do this if you want smooth shading.
Calculating per-vertex normals involves calculating per-triangle normals, then averaging those normals (optionally giving each a weight) for each triangle that shares a given vertex.
So you need additional information that you just don't have: which triangles share each vertex.
As you will see from a diagram of triangles-with-adjacency (below), even that isn't sufficient to calculate per-vertex normals. Take the vertex numbered "2" in the image below; it should be obvious that there could be any arbitrary number of triangles that also share this vertex but which aren't in the adjacent set.
You'll have to calculate them on the CPU and add an additional input attribute, I'm afraid.
• how can i calculate it on cpu as many of the primitive will be generated in tessellation... – bhawesh May 20 '14 at 6:20
• @bhawesh Well, you can easily fix this problem if you deform your normals when you evaluate your patches. There is not much point trying to do this with a geometry shader if you are using tessellation shaders. – Andon M. Coleman May 23 '14 at 6:40
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2019-11-15 20:46:52
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