text:
Create a cross-section, time series data set using Matrix operations

I have developed the following code, which produces what I aim to obtain.

Clear[n, t, gBar, mat0, mat1];
SeedRandom[15];
n = 5;
t = 7;
gBar = 0.05;
mat0 = Array[RandomInteger[{1, 8}] &, {n,t}]; (* Cross-section (n) over time (t) *)
mat1 = Array[0 &, {n, t - 1}];
mat2 = Array[0 &, {n, n}];
Do[
Do[
If[Log[mat0[[i, j + 1]]/mat0[[i, j]]] > gBar,
mat1[[i, j]] = Log[mat0[[i, j + 1]]/mat0[[i, j]]]], {i, 1, n}
], {j, 1, t - 1}
];

desiredOutput = ReplacePart[mat2, {{2, _} -> 1, {3, _} -> 1}]

First, I create a matrix of data (cross-section over time): called mat0. Then, for each row, I conduct a Log[] operation. If the outcome satisfies If statement, then I place the result of the operation to that row in mat1. Here are the resulting matrices, mat0 and mat1, respectively.

After the mat1 is fully created, then I conduct operations using each column. Find the positions of nonzero elements in the 1st column, for example (mat1[2,1] and mat1[3,1] and then create the matrix of desiredOutput in which 2nd and 3rd rows are all one and the rest of the matrix is zero as follows:

How can I make the above code more efficient and shorter?

EDIT 1

This edit does not change anything in the above formulation, but asks a related extension.

This additional code:

Clear[λ, ψ, mu, μ];
λ = 0.6;
ψ = 2;
mu = λ*(mat1[[All, 1]]/gBar)^ψ  (* the 1st column *)
μ = Table[{i, mu[[i]]}, {i, 1, n}]

gives me what I need. I can repeatedly calculate the code for each column in mat1, but it is not efficient to do so. Instead, for the integrity of my research problem, I like to incorporate this additional piece of code into the code given in the question before EDIT 1 and automate it for each column t in mat1.

Regards.

Using mat1 from Henrik's answer, alternative ways to get mat2 and mu:

ClearAll[m2]
{0, {Dimensions[m][[1]] - 1, 0}}, "Fixed"];
Row[MatrixForm /@ m2[mat1] /@ Range[4]] // TeXForm

$$\tiny\left( \begin{array}{ccccc} 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\left( \begin{array}{ccccc} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ \end{array} \right)\left( \begin{array}{ccccc} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\left( \begin{array}{ccccc} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ \end{array} \right)$$

ClearAll[mu]
mu[m_][k_] := MapIndexed[{#2[[1]], #} &, λ*(m[[All, k]]/gBar)^ψ ] ;
Row[MatrixForm /@ mu[mat1] /@ Range[6]] // TeXForm

$$\tiny\left( \begin{array}{cc} 1 & 0. \\ 2 & 39.4565 \\ 3 & 39.4565 \\ 4 & 0. \\ 5 & 0. \\ \end{array} \right)\left( \begin{array}{cc} 1 & 0. \\ 2 & 0. \\ 3 & 0. \\ 4 & 908.776 \\ 5 & 461.235 \\ \end{array} \right)\left( \begin{array}{cc} 1 & 230.886 \\ 2 & 770.496 \\ 3 & 75.1608 \\ 4 & 0. \\ 5 & 0. \\ \end{array} \right)\left( \begin{array}{cc} 1 & 0. \\ 2 & 0. \\ 3 & 0. \\ 4 & 0. \\ 5 & 115.309 \\ \end{array} \right)\left( \begin{array}{cc} 1 & 0. \\ 2 & 0. \\ 3 & 0. \\ 4 & 0. \\ 5 & 115.309 \\ \end{array} \right)\left( \begin{array}{cc} 1 & 39.4565 \\ 2 & 19.8626 \\ 3 & 0. \\ 4 & 621.67 \\ 5 & 0. \\ \end{array} \right)$$

• Really very neat. It is just what I asked for. Thank you and Henrik. Oct 16 '18 at 23:32
• @Tugrul, my pleasure. Thank you for the accept.
– kglr
Oct 16 '18 at 23:33
• I think something is not right in the generation of 0 and 1 matrices. Your m2[m_] matrix should use columns of mat1 to generate 6 matrices of (n,n) dimension each. Oct 17 '18 at 0:17
• Tugrul, mat1 is 5by6 and m2[mat1][k] generates a 5by6 matrix based on column k of mat1. I just displayed 4 of 6 matrices because 6 wouldn't fit on the page.
– kglr
Oct 17 '18 at 0:20
• Yes, I saw the 6 matrices and I understand why you did it that way. But my question is about the dimension of the 1 and 0 matrices. In the original question, the 1-0 matrix is (n,n) dimension, not (n,t-1). Oct 17 '18 at 0:26
• Log is vectorized and should not be computed more often than needed.

• Clip is also vectorized and can replace If here. This way, we also do not have to initialize the matrix mat1.

• Unitize along with Part can help you to generate the second matrix. Also no initialization needed.

Check this out:

mat1 = With[{A = Log[mat0]},
Clip[Subtract[A[[All, 2 ;;]], A[[All, ;; -2]]], {gBar, ∞}, {0., 0.}]
];

mat2 = ({
ConstantArray[0., n],
ConstantArray[1., n]
})[[Unitize[mat1[[All, 1]]] + 1]];

• Thank you very much. Given mat0 and gBar, I run your code, but it gives me the following message: Symbol called with 0 arguments, 1 argument is expected and  the expression 1+Unitize[Symbol[]] cannot be used as a part specification. Can you tell me what the matrix mymat1 is? Oct 16 '18 at 17:14
• I discovered mymat1. It is my mat1. Your code` runs perfect... Oct 16 '18 at 17:25
• Yeah. Sorry. Fixed it. And, of course, you're welcome. Oct 16 '18 at 17:46
• Do you mind if I ask another related question? This extra question is not worth to formulate as a separate question. Please let me know. Oct 16 '18 at 18:56
• Yeah, okay. If it is a short one... =) Oct 16 '18 at 18:59

format: plain


text:
# Why not just drop an engine on fire?

I am interested in the case of Qantas Flight 32 and similar incidents, where there was a fuel leak after a number two engine explosion due to the disintegration of the turbine disc. It was a very dangerous situation since the fuel leaks were very close to the failed, then burning, engine. And to make the matter worse, fuel dumping system was down due to hydraulic failure.

Although the crew managed to shut it down, the engine took considerable amount of time for fire to be extinguished completely.

I wonder in those kind of situations, could adding eventually a "drop-engine-on-fire" feature make the situation less dangerous?

You may consider the following:

• Modern airplanes could fly with one engine out.

• Dropping the engine must be in water to not cause any harm to the people below.

• Even if the main evidence of the investigation (that is, the burning engine) will be lost and maybe never found, I think that saving lives is a bit more important.

EDIT :
In response to comments, I think that every feature (even if its seems silly) is subject to pro-con studies. For example, Boing 747 had fuse pins that are designed to break cleanly in case of excessive loads/vibrations.

• You may not know that dropping things of this size into sea is not allowed, by international conventions, unless the area is free of ships and a permit has been granted (this is also valid for jettisoning space vehicle elements). – mins Dec 16 '16 at 17:10
• @mins: Doing whatever is necessary during an emergency is virtually always allowed. No Clean Water agency is going to pursue Capt. Sullenberger for throwing trash in the Hudson. – abelenky Dec 16 '16 at 17:44
• At first glance, given the very small number of engine fires, it seems there could well be more damage caused by accidental activation of the engine jettison system than would be prevented in cases where it's needed. – jamesqf Dec 16 '16 at 19:30
• "Usually, the engines are mounted on the wing only at three points." That is true so far as it goes, but it rather misses the reason why the engine was there at all - to apply a force to the aircraft!". Those "three points" are actually quite complicated structures which already contain failsafe devices to prevent "accidental" disconnections. Designing a connector that can transmit something like 50 tons of engine thrust, and also be disconnected remotely even after possible damage from fire, etc, would be far from simple - not to mention testing that it actually worked as intended. – alephzero Dec 16 '16 at 19:59
• @alephzero: Beyond making sure it works as intended, is ensuring that it NEVER accidentally activates. I can think of few things worse than having a perfectly good engine inadvertently detach when it is most needed! – abelenky Dec 16 '16 at 21:18

The current position of regulators like the FAA is that dropping things off of airplanes in flight is a bad thing. Dropping extremely large and heavy things like engines would be extremely bad. This poses an unacceptable risk to people and property on the ground (or even in the water). Most aircraft with engine fires do not kill anyone on board, so why risk killing people on the ground?

The preferred method of dealing with engine fires is similar to that of fire protection for buildings; take steps to prevent them, and provide methods for putting them out should they start. We don't go blowing up buildings that are on fire just because it might be quicker than putting it out.

So instead manufacturers are required to provide many features to prevent fires. Areas around the engines are separated into zones by firewalls and seals, required to prevent spread of fire. Any flammable fluid is required to be contained and drained, and separated from ignition sources.

If a fire does start, the engines have fire suppression systems installed. The system has to be tested to show that it provides the required suppression to a fire, and the aircraft generally carries enough suppressing agent for at least 2 uses. Even if this doesn't work, the isolation features built in should prevent the fire from spreading, and possibly allow it to burn itself out of fuel or oxygen. In most cases an aircraft is close enough to an airport to land before this becomes an issue.

These measures have been shown to be adequate. When was the last time someone died because of an engine fire on an airplane? Fires are extremely rare, and fires that can't be dealt with by the existing measures are even more rare.

There are many more reasons why dropping engines is a bad idea.

Some designs allow an engine to shear away from the wing in the event of a crash, but this is not designed to work in flight. Aircraft that drop items like weapons in flight require extensive design and testing to make sure that the object will drop away from the aircraft properly. An aircraft in an emergency like an engine fire could be making extreme maneuvers that would make this even more difficult.

Objects like weapons that are designed to be dropped regularly still malfunction. How can you make an engine that's been installed for decades separate reliably? If you are relying on this feature for safety, there will be added cost for designing, installing, and maintaining the system. Those costs could be reduced if you don't rely on the feature as much. But then if the current measures are enough, why accept that extra expense?

Investigators generally take interest in engine fires. If you drop the engine, there goes the best evidence for figuring out how the fire started, and what could have prevented or mitigated it. Something large like an engine would be easier to find than smaller parts, but there is still the risk of loss or damage hindering the investigation. If the problem is bad enough that airplanes are dropping engines, it would be better to fix the problem than address the symptoms.

If you constrain pilots to only dropping an engine over water, this complicates things further. There are already rules about where airliners can dump fuel, but dumping fuel is generally a less stressful situation. The pilots chose to stay in the air and dump fuel, so any emergencies are generally less urgent and they have time and attention for where they could dump fuel. If there's an engine fire that requires jettison of an engine, things are going to be more hectic in the cockpit, and it may not be the best time to be considering where this would be allowed, taking things like trajectory into account. And going back to the cost issue, if this would be limited to only areas over water, you couldn't rely on this option to always be available, so is it worth the cost?

Airliners can fly with an engine shut down, but missing an engine is a different matter altogether. The aircraft is required to be tested and certified in many different conditions with an engine shut down. How can we test this with a missing engine? Do we drop an extremely valuable engine every time we test? Do we just make the flight with one engine, which is still in testing itself? Do we not test, and accept that after going to all the trouble of allowing jettison of an engine, the aircraft may not be controllable in some situations?

• +1 for We don't go blowing up buildings that are on fire just because it might be quicker than putting it out. – kevin Dec 16 '16 at 17:43
• I can just imagine it now - please welcome the new Chief of the Fire Dept, Michael bay. – David says Reinstate Monica Dec 16 '16 at 20:32
• In particular the FAA has specific regulations about dropping things (FAR 91.15). I would submit that dropping your engine "creates a hazard to persons or property" (unless we constrain pilots to dropping their engines over open ocean with no vessels around, but as fooot noted telling a pilot "Please carry your burning engine 50 miles offshore before you drop it!" might be asking a bit much. – voretaq7 Dec 16 '16 at 20:38
• Also in regard to testing aircraft with a missing engine, aircraft engineers tend to design for the worst case: Engines have ripped off in flight and the plane has returned to make a safe landing, so at least from that angle it's theoretically possible to have a more orderly detachment executed safely - aside from the whole "giant hunk of metal falling from the sky" problem and violating FAR 91.15 anyway. – voretaq7 Dec 16 '16 at 20:44
• @tuskiomi That is a myth and not based in reality. The reality of it is that the "blue ice" falling from planes in the 70's was because of leaking systems, not intentional discharges. Planes have not purposely ejected lavatory waste since somewhere around WWII. – Ron Beyer Dec 16 '16 at 21:22

The fact that the modern aircraft could fly with one engine out does not mean that the offending engine should be dropped. It is better to have it in the aircraft rather than risk the engine taking off with a chunk of aircraft. After all, in case of El Al Flight 1862,

... engine no.3 and its pylon separated from the aircraft and damaged part of the leading edge of the right wing. The no.3 engine then struck engine no.4, causing this engine and its pylon to depart the wing.

Uncontained engine failures are rare and in case of most engine failures, the fire, if there's any is extinguished quite quickly. Also, in most cases, dropping of the engines is not going to wait until a (suitable) water body is found. In case of failures during takeoffs/landings, like the American Airlines Flight 383 or the British Airways Flight 2276, the pilot will have one more thing to worry about.

Then there's always the possibility of uncommanded detonation and failure to detonate when commanded- this will add further checks to the maintenance schedule. Once the engine has been jettisoned, the fuel and other lines have to closed. All this adds complexity to the system, which may well be not worth the cost.

Finally, the engine weighs quite a lot and dropping it would affect the aircraft's weight and balance. This could be a problem if the aircraft is maneuvering, requiring the pilot to take this sudden change into account.

• The experience of El Al 1862 is probably the best argument against a system that would intentionally drop an engine. – Michael Seifert Dec 16 '16 at 19:29
• @MichaelSeifert indeed. Even had there been no further structural damage the aircraft, minus an engine, would have been far harder to fly than had the engine simply failed and remained attached. It was in the end of course not the engine falling that caused damage on the ground (it fell harmlessly in water) but the aircraft spinning out of control while trying to turn to land and crashing into an apartment block. – jwenting Dec 19 '16 at 8:29
• Can remember that accident and the aftermath. The aircrew telling ATC "we've lost an engine" and them thinking it had failed when in fact it had fallen off was a classic example of misunderstood communications in aviation. Later met one of the firefighters who'd responded to the scene, not nice stories. – jwenting Dec 19 '16 at 8:32
• Of course, the problem with El Al 1862 was that the separation was not by design, but because of a pylon failure. It tore off, with significant lateral forces. That's how the #3 engine took out #4 as well as damaging the wing. An intentional drop would presumably have no lateral forces, so the engine would drop straight down. – MSalters Dec 19 '16 at 11:09
• @MSalters: One would hope that the engine would drop straight down, but airflow around aircraft does funny things. See this video (also linked in fooot's answer) of an object being intentionally dropped. – Michael Seifert Dec 20 '16 at 20:09

A major issue is that it's not really the engine itself which is on fire, in fact they are supposed to be on fire in a sense. In many cases an engine fire really means that you have lost control of the fuel/hydraulic fluid flow to the engine. For example if a turbine blade detaches, penetrates the engine casing and cuts a fuel or hydraulic line.

Indeed in the example in the question the damage was caused by the original engine failure and resulting fragmentation rather than a fire in the engine itself as such.

Even if you drop the engine you've still got fuel spewing out and burning so unless you can stop the fuel leak at source you haven't really solved the underlying problem and if you can cut off the fuel dropping the engine doesn't achieve much.

If you've got a leaking tap cutting the tap off with a hacksaw isn't a solution.

You are then also in a situation where you have created a whole new risk of accidentally dropping the engine. There are also lots of fuel, sensor, hydraulic, compressed air and other connections to a jet engine and unless you can guarantee separating all of these cleanly you run a very real risk of doing even more damage or even worse ending up with an engine half hanging off.

There is also the fact that while aircraft are deigned to fly with one or more engine failures actually dropping and engine is going to have a fairly dramatic effect on the trim of the aircraft just in terms of the missing mass.

• If you've got a leaking tap cutting the tap off with a hacksaw isn't a solution. +1 – FreeMan Dec 18 '16 at 17:58

The first reason I can think of is that an aircraft engine on an intercontinental jetliner weighs about 6000-10,000 lbs depending on the type and simply jettisoning the thing would significantly affect the weight and balance of the craft, making things worse. An object that size simply released from the jet presents other hazards such as tumbling and then striking the airframe upon release, causing catastrophic damage and possibly departure from controlled flight.

• That was my first thought too and made me smile quite contently. – user6035379 Dec 16 '16 at 16:54
• Or, how about where this flaming 10,000 lb hunk of metal lands? – SnakeDoc Dec 16 '16 at 16:58
• It would definitely shift the balance but not enough that it can't be compensated for. There have been many incidents of engines falling off for various reasons. All but a couple have landed safely. The times that it caused a crash it was due to other systems damaged because they didn't separate as they were intended to. – TomMcW Dec 18 '16 at 23:36
• Who isn't fan of Donnie Darko ? :) – Antzi Dec 19 '16 at 2:55

Any time the question is "why don't we have...", the answer is almost certainly in the trade-off between weight, maintenance cost, fuel cost, & utility. If it weighs a lot, needs maintenance, burns fuel, and will only be useful once a decade, then it just is not productive.

Putting an engine release system on a plane will be heavy, meaning fewer passengers, or less fuel or cargo on board. All that extra weight has to be flown around, requiring fuel. The release mechanism would need regular maintenance checks, which is another cost. And it would only be useful very, very infrequently (on most planes, it would never be used).

So it is just impossible to justify adding new equipment for a once-in-a-lifetime scenario, which can be better managed with good maintenance, good training, and good planning.

This same line of reasoning goes for "why don't more airplanes have rocket thrusters?", "why don't planes have anti-missile systems?", "why don't planes have airbags?", "Why don't airplanes have parachutes?" and many other random items.

• I think the real reason is dropping, effectively, a bomb from an airplane over even a slightly dense population is likely to cause as much damage and loss of life as-if everyone on the plane died in a crash. Given it's far more unlikely for an engine fire to crash an airplane, that's not really an acceptable risk. – SnakeDoc Dec 16 '16 at 17:10
• @SnakeDoc: The world is a lot more empty than you may realize. 70% of the earth is water, and upwards of 95% of land has no one on it at all. Without even trying, you could drop an engine at any time, and probably not do any damage at all. – abelenky Dec 16 '16 at 17:20
• @abelenky We like to put airports near population centers (KJFK, KLGA, KLAX, KORD, KATL…), and coastal waters can have quite a few ships (pleasure, merchant, and military) sailing around in them: Even if the detachment process could be made simple and lightweight (explosive bolts and the existing mechanisms that clamp fuel/oil/hudraulic/pneumatic lines to a severed pylon) along the lines of military aircraft with drop-tanks for fuel it would never pass the safety-of-people-on-the-ground test for civil aircraft. – voretaq7 Dec 16 '16 at 20:55
• With a release mechanism, the risk of accidentally dropping an engine is probably similar to the risk of a fire in the engine. – 200_success Dec 17 '16 at 0:32
• Ahh - a voice of reason! +1. – Peter Kämpf Dec 17 '16 at 10:30

I think the "uncontrolled bomb" answer is probably sufficient on its own but there are also technical reasons.

In order to be able to drop the engine you need to build the engine with a rapid release system that will safely, reliably and near-instantly disconnect the engine and all the wires and fuel lines connected to the engine. What's more it needs to be able to do this while it is on fire following an emergency failure that caused the fire. It is not obvious that this is actually possible, but even assuming it is you are adding a new complex system to the engine assembly that is, itself, capable of failure and where failure has potentially disastrous consequences.

Thus its possible, likely even, that adding an "engine eject" module to a plane is likely to increase the overall risk to the plane rather than decreasing it.


text:
To get the best deal on Tutoring, call 1-855-666-7440 (Toll Free)

# Probability Worksheets - Page 14

Probability Worksheets
• Page 14
131.
In a box there are 3 white, 2 yellow, and 1 red colored crayons. Which one lists the possible 4 crayons, when 4 of them are picked at the same time?
 a. 2 white, 1 yellow, and 1 red b. 1 white, 1 yellow, and 2 red c. 2 white, 2 red d. 3 yellow, 1 red

Correct answer : (1)
132.
Jake has a set of plane shapes as shown in the figure. Which of the following is the possible outcome if Jake chooses 6 shapes?

 a. 4 rectangles, 1 square, and 1 circle b. 2 squares, 3 rectangles, 1 triangle, and 1 circle c. 2 squares, 2 rectangles, and 2 circles d. 1 rectangle, 2 squares, 2 triangles, and 1 circle

Correct answer : (4)
133.
Find all the possible combinations of letters A, B, C taken 2 at a time.
 a. (A, A), (A, B), (A, C), (B, A), (B, C), (C, A), (C, B) b. (A, A), (A, B), (A, C), (B, B), (B, A), (B, C), (C, C), (C, A), (C, B) c. (A, A), (B, B), (C, C) d. (A, B), (A, C), (B, A), (B, C), (C, A), (C, B)

Correct answer : (4)
134.
A team of 3 students with only one girl in the team is to be formed. Find the number of possible outcomes if there are 2 boys and 3 girls to choose from.
 a. 6 b. 5 c. 3 d. 2

Correct answer : (3)
135.
Sheela has 3 skirts and 4 blouses. How many different ways can she choose an outfit?
 a. 4 b. 10 c. 7 d. 12

Correct answer : (4)
136.
A die is rolled. Find the probability of rolling a 2 or 4.
 a. $\frac{1}{2}$ b. $\frac{1}{6}$ c. $\frac{2}{3}$ d. $\frac{1}{3}$

Correct answer : (4)
137.
Find the probability that the card drawn from a standard deck of cards is a heart or a diamond.
 a. $\frac{3}{4}$ b. $\frac{1}{4}$ c. $\frac{1}{3}$ d. $\frac{1}{2}$

Correct answer : (4)
138.
Clara has 4 blue marbles, 5 green marbles, and 3 yellow marbles. Find the probability of drawing either a green or an yellow marble.
 a. $\frac{5}{48}$ b. $\frac{3}{5}$ c. $\frac{3}{4}$ d. $\frac{2}{3}$

Correct answer : (4)
139.
On which color is the spinner most likely to land, if it is spun 10 times?

 a. Blue b. Yellow c. Pink

Correct answer : (2)
140.
Dennis has 4 blue, 2 pink, and 6 white pairs of socks. He picked a pair of socks without looking. Predict the color that is most likely to be picked.
 a. pink b. blue c. white


text:
# Math Help - Differentiate Exponents

1. ## Differentiate Exponents

Find first and second derivatives, critical number, inflection points.

f(x) = xe^x
f'(x) = ('x) (e^x) + x(e^x)('x)
f'(x) = e^x+xe^x
f'(x) = e^x(1+x)
Critical number x = -1

f"(x) = e^x(1) + (1+x)(e^x)
f"(x) = e^x +e^x(1+x)
Critical Number x=-1

I hope I am close to the correct answer.

2. Originally Posted by confusedagain
f"(x) = e^x(1) + (1+x)(e^x)
f"(x) = e^x +e^x(1+x)
Critical Number x=-1

I hope I am close to the correct answer.
first of all, we do not find critical points from the second derivative, we use the second derivative to find concavity and possible inflection points. second of all,
$f''(-1) = \frac {1}{e} \neq 0$

3. Hello, confusedagain!

A small error . . .

Find first and second derivatives, critical number, inflection points.
. . $f(x) \:= \:xe^x$

. . $f'(x) \:= \:1\cdot e^x + x\cdot e^x \:=\:e^x(1 + e^x)$

Then: . $e^x(1+x) \:=\:0\quad\Rightarrow\quad x = -1$

. . Critical point: $\left(-1,\:-\frac{1}{e}\right)$

$f''(x) \:= \:e^x + (1+x)e^x \:=\:e^x(1 + 1 + x) \:=\:e^x(2 + x)$

Then: . $e^x(2 + x) \:=\:0\quad\Rightarrow\quad x = -2$

. . Inflection point: . $\left(-2,\:-\frac{2}{e^2}\right)$

4. Hey Soroban

If you prefer, use e^x(2 + x)=0\implies x=-2, this yields $e^x(2 + x)=0\implies x=-2$


text:
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Magazine of Aviation Development       (Followers: 2) Mekanika : Jurnal Teknik Mesin i       (Followers: 1) Microgravity Science and Technology       (Followers: 2) New Space       (Followers: 6) Nonlinear Dynamics       (Followers: 20) npj Microgravity       (Followers: 3) Open Aerospace Engineering Journal       (Followers: 1) Perspectives of Earth and Space Scientists i Population Space and Place       (Followers: 9) Problemy Mechatroniki. Uzbrojenie, lotnictwo, inżynieria bezpieczeństwa / Problems of Mechatronics. Armament, Aviation, Safety Engineering       (Followers: 3) Proceedings of the Human Factors and Ergonomics Society Annual Meeting       (Followers: 16) Proceedings of the Institution of Mechanical Engineers Part G: Journal of Aerospace Engineering       (Followers: 46) Progress in Aerospace Sciences       (Followers: 81) Propulsion and Power Research       (Followers: 68) REACH - Reviews in Human Space Exploration       (Followers: 5) Research & Reviews : Journal of Space Science & Technology       (Followers: 17) RocketSTEM       (Followers: 6) Russian Aeronautics (Iz VUZ)       (Followers: 24) Science and Education : Scientific Publication of BMSTU       (Followers: 1) Space and Polity       (Followers: 4) Space Policy       (Followers: 29) Space Research Today       (Followers: 48) Space Safety Magazine       (Followers: 51) Space Science International       (Followers: 202) Space Science Reviews       (Followers: 97) SpaceNews       (Followers: 825) Spatial Information Research       (Followers: 1) Technical Soaring       (Followers: 1) Transport and Aerospace Engineering       (Followers: 1) Transportmetrica A : Transport Science       (Followers: 9) Unmanned Systems       (Followers: 5) Вісник Національного Авіаційного Університету       (Followers: 2)
Similar Journals
 Space Science ReviewsJournal Prestige (SJR): 3.262 Citation Impact (citeScore): 7Number of Followers: 97      Hybrid journal (It can contain Open Access articles) ISSN (Print) 1572-9672 - ISSN (Online) 0038-6308 Published by Springer-Verlag  [2656 journals]
• Distinguishing Tidal Disruption Events from Impostors
• Abstract: Recent claimed detections of tidal disruption events (TDEs) in multi-wavelength data have opened potential new windows into the evolution and properties of otherwise dormant supermassive black holes (SMBHs) in the centres of galaxies. At present, there are several dozen TDE candidates, which share some properties and differ in others. The range in properties is broad enough to overlap other transient types, such as active galactic nuclei (AGN) and supernovae (SNe), which can make TDE classification ambiguous. A further complication is that “TDE signatures” have not been uniformly observed to similar sensitivities or even targeted across all candidates. This chapter both reviews those events that are unusual relative to other TDEs, including the possibility of TDEs in pre-existing AGN, and summarises those characteristics thought to best distinguish TDEs from continuously accreting AGN, strongly flaring AGN, SNe, and Gamma-Ray Bursts (GRBs), as well as other potential impostors like stellar collisions, “micro-TDEs,” and circumbinary accretion flows. We conclude that multiple observables should be used to classify any one event as a TDE. We also consider the TDE candidate population as a whole, which, for certain host galaxy or SMBH characteristics, is distinguishable statistically from non-TDEs, suggesting that at least some TDE candidates do in fact arise from SMBH-disrupted stars.
PubDate: 2021-05-05

• Correction to: Small-Scale Dynamic Aurora
• Abstract: A Correction to this paper has been published: https://doi.org/10.1007/s11214-021-00821-y
PubDate: 2021-04-26

• The BepiColombo Planetary Magnetometer MPO-MAG: What Can We Learn from the
Hermean Magnetic Field'
• Abstract: The magnetometer instrument MPO-MAG on-board the Mercury Planetary Orbiter (MPO) of the BepiColombo mission en-route to Mercury is introduced, with its instrument design, its calibration and scientific targets. The instrument is comprised of two tri-axial fluxgate magnetometers mounted on a 2.9 m boom and are 0.8 m apart. They monitor the magnetic field with up to 128 Hz in a $$\pm 2048$$  nT range. The MPO will be injected into an initial $$480 \times 1500$$  km polar orbit (2.3 h orbital period). At Mercury, we will map the planetary magnetic field and determine the dynamo generated field and constrain the secular variation. In this paper, we also discuss the effect of the instrument calibration on the ability to improve the knowledge on the internal field. Furthermore, the study of induced magnetic fields and field-aligned currents will help to constrain the interior structure in concert with other geophysical instruments. The orbit is also well-suited to study dynamical phenomena at the Hermean magnetopause and magnetospheric cusps. Together with its sister instrument Mio-MGF on-board the second satellite of the BepiColombo mission, the magnetometers at Mercury will study the reaction of the highly dynamic magnetosphere to changes in the solar wind. In the extreme case, the solar wind might even collapse the entire dayside magnetosphere. During cruise, MPO-MAG will contribute to studies of solar wind turbulence and transient phenomena.
PubDate: 2021-04-21

• Editorial: Topical Collection on Understanding the Diversity of Planetary
Atmospheres
• PubDate: 2021-04-19

• Meteoroids as One of the Sources for Exosphere Formation on Airless Bodies
in the Inner Solar System
• Abstract: This manuscript represents a review on progress made over the past decade concerning our understanding of meteoroid bombardment on airless solar system bodies as one of the sources of the formation of their exospheres. Specifically, observations at Mercury by MESSENGER and at the Moon by LADEE, together with progress made in dynamical models of the meteoroid environment in the inner solar system, offer new tools to explore in detail the physical phenomena involved in this complex relationship. This progress is timely given the expected results during the next decade that will be provided by new missions such as DESTINY+, BepiColombo, the Artemis program or the Lunar Gateway.
PubDate: 2021-04-19

• In Dedication to Adam P. Showman
• PubDate: 2021-04-14

• The Mars Environmental Dynamics Analyzer, MEDA. A Suite of Environmental
Sensors for the Mars 2020 Mission
• Abstract: NASA’s Mars 2020 (M2020) rover mission includes a suite of sensors to monitor current environmental conditions near the surface of Mars and to constrain bulk aerosol properties from changes in atmospheric radiation at the surface. The Mars Environmental Dynamics Analyzer (MEDA) consists of a set of meteorological sensors including wind sensor, a barometer, a relative humidity sensor, a set of 5 thermocouples to measure atmospheric temperature at ∼1.5 m and ∼0.5 m above the surface, a set of thermopiles to characterize the thermal IR brightness temperatures of the surface and the lower atmosphere. MEDA adds a radiation and dust sensor to monitor the optical atmospheric properties that can be used to infer bulk aerosol physical properties such as particle size distribution, non-sphericity, and concentration. The MEDA package and its scientific purpose are described in this document as well as how it responded to the calibration tests and how it helps prepare for the human exploration of Mars. A comparison is also presented to previous environmental monitoring payloads landed on Mars on the Viking, Pathfinder, Phoenix, MSL, and InSight spacecraft.
PubDate: 2021-04-13

• The SuperCam Instrument Suite on the Mars 2020 Rover: Science Objectives
and Mast-Unit Description
• Abstract: On the NASA 2020 rover mission to Jezero crater, the remote determination of the texture, mineralogy and chemistry of rocks is essential to quickly and thoroughly characterize an area and to optimize the selection of samples for return to Earth. As part of the Perseverance payload, SuperCam is a suite of five techniques that provide critical and complementary observations via Laser-Induced Breakdown Spectroscopy (LIBS), Time-Resolved Raman and Luminescence (TRR/L), visible and near-infrared spectroscopy (VISIR), high-resolution color imaging (RMI), and acoustic recording (MIC). SuperCam operates at remote distances, primarily 2–7 m, while providing data at sub-mm to mm scales. We report on SuperCam’s science objectives in the context of the Mars 2020 mission goals and ways the different techniques can address these questions. The instrument is made up of three separate subsystems: the Mast Unit is designed and built in France; the Body Unit is provided by the United States; the calibration target holder is contributed by Spain, and the targets themselves by the entire science team. This publication focuses on the design, development, and tests of the Mast Unit; companion papers describe the other units. The goal of this work is to provide an understanding of the technical choices made, the constraints that were imposed, and ultimately the validated performance of the flight model as it leaves Earth, and it will serve as the foundation for Mars operations and future processing of the data.
PubDate: 2021-04-12

• Correction to: Radiometric Calibration Targets for the Mastcam-Z Camera on
the Mars 2020 Rover Mission
• Abstract: A Correction to this paper has been published: https://doi.org/10.1007/s11214-021-00828-5
PubDate: 2021-04-07

• Earth’s Nitrogen and Carbon Cycles
• Abstract: Understanding the Earth’s geological nitrogen (N) and carbon (C) cycles is fundamental for assessing the distribution of these volatiles between solid Earth (core, mantle and crust), oceans and atmosphere. This Special Communication about the Earth’s N and C cycles contains material that is relevant for researchers who are interested in the Topical Collection on planetary evolution “Reading Terrestrial Planet Evolution in Isotopes and Element Measurements”. Variations in the fluxes of N and C between these major reservoirs through geological time influenced the evolution and determined the unique composition of the Earth’s atmosphere. Here we review several key geological aspects of the N and C cycles of which our understanding has significantly advanced during the last decade through field-based, experimental and theoretical studies. Subduction zones are the most important pathway of both N and C from the Earth’s surface into the deep Earth. A key question in the flux quantification is how much of the volatile elements is stored in the downgoing slab and introduced into the mantle and how much is returned back to the surface and the atmosphere through arc magmatism. For N, the retention of N as $$\text{NH}_{4}^{+}$$ in minerals has a major influence on fluxes between reservoirs. The temperature-dependent stability of $$\text{NH}_{4}^{+}$$ -bearing minerals determines whether N is predominantly retained in the slab to mantle depths (in subduction zones with a low geothermal gradient) or devolatilized (in subduction zones with a high geothermal gradient). Several lines of evidence suggest that the mantle is regassing with respect to N due to a net influx of subducted N over time, but this issue is highly debated and evidence to the contrary also exists. Nevertheless, there is consensus that the majority of the planetary N budget is stored in the Earth’s mantle, with the continental crust also constituting a significant N reservoir. For C, release from the subducting slab occurs through decarbonation reactions, dissolution and formation of carbonatitic liquids, but reprecipitation of C in the slab or the forearc mantle wedge may limit the effectiveness of direct return of C into the atmosphere. Carbon release through regional metamorphism in collision zone orogens also has potentially profound effects on C release into the atmosphere and consensus has emerged that such orogens are sources rather than sinks of atmospheric CO2. On shorter timescales, contact metamorphism through interaction of mantle-derived magmas with C-bearing country rocks, and the resulting release of large quantities of CH4 and/or CO2, has been linked to global warming events.
PubDate: 2021-04-02

• Relative Atomic Solar System Abundances, Mass Fractions, and Atomic Masses
of the Elements and Their Isotopes, Composition of the Solar Photosphere,
and Compositions of the Major Chondritic Meteorite Groups
• Abstract: This brief special communications article gives data for atomic abundances and mass fractions for the elemental and isotopic solar system composition, the atomic masses of the elements and their isotopes, the composition of the solar photosphere, and the compositions of the major chondritic meteorite groups. This additional material is relevant for researchers who are interested in this Topical Collection on planetary evolution.
PubDate: 2021-04-01

• The Diversity of Planetary Atmospheric Chemistry
• Abstract: Atmospheres in our solar system range from oxidizing to reducing, transient to dense, veiled by clouds and hazes to transparent. Observations already suggest that exoplanets exhibit an even more diverse range of atmospheric chemistry and composition. Nevertheless, there are commonalities across the atmospheres of our solar system that provide valuable guidance and lessons for observing and interpreting exoplanetary atmospheres. Lessons gleaned from decades of study of planetary atmospheric chemistry are synthesized and explored to understand their implications for exoplanets.
PubDate: 2021-03-30

• High Resolution Imaging Camera (HiRIC) on China’s First Mars
Exploration Tianwen-1 Mission
• Abstract: The High-Resolution Imaging Camera (HiRIC) is one major payload of China’s first Mars exploration mission, and its main objective is to obtain the detailed observation images of the key areas on the Martian surface. In this paper, the leading group of HiRIC shows a full blueprint of the HiRIC. The HiRIC can achieve a high resolution (0.5 m at an altitude of 265 km) with a wide swath width of 9 km. The HiRIC adopts an Off-Axis Three-Mirror Astigmatic (TMA) optical system with a focal length of 4640 mm, an F-number of 12 and a Field of View (FOV) of 2° × 0.693°. In order to reduce the instrument weight, carbon-based material is widely used in the opto-mechanical structure which is in ultra-lightweight design, thus, a light-weight camera with a total mass of 42 kg is obtained. The Time Delay and Integration (TDI) Charge Coupled Devices (CCDs) and Complementary Metal-Oxide-Semiconductor Transistor (CMOS) detectors are all set on the imaging plane to achieve the push-broom imaging and frame imaging, respectively. And the high Signal-to-Noise Ratio (SNR) >100:1 can achieve in multi observation types for various scientific imaging tasks. After 4-year design and fabricate, the HiRIC has been assembly. The testing results show that the instrument is in good condition, and the Modulation Transfer Function (MTF) can achieve 0.18 at Nyquist frequency. The HiRIC can achieve a well image on China first Mars exploration mission.
PubDate: 2021-03-29

• The Location of Magnetic Reconnection at Earth’s Magnetopause
• Abstract: One of the major questions about magnetic reconnection is how specific solar wind and interplanetary magnetic field conditions influence where reconnection occurs at the Earth’s magnetopause. There are two reconnection scenarios discussed in the literature: a) anti-parallel reconnection and b) component reconnection. Early spacecraft observations were limited to the detection of accelerated ion beams in the magnetopause boundary layer to determine the general direction of the reconnection X-line location with respect to the spacecraft. An improved view of the reconnection location at the magnetopause evolved from ionospheric emissions observed by polar-orbiting imagers. These observations and the observations of accelerated ion beams revealed that both scenarios occur at the magnetopause. Improved methodology using the time-of-flight effect of precipitating ions in the cusp regions and the cutoff velocity of the precipitating and mirroring ion populations was used to pinpoint magnetopause reconnection locations for a wide range of solar wind conditions. The results from these methodologies have been used to construct an empirical reconnection X-line model known as the Maximum Magnetic Shear model. Since this model’s inception, several tests have confirmed its validity and have resulted in modifications to the model for certain solar wind conditions. This review article summarizes the observational evidence for the location of magnetic reconnection at the Earth’s magnetopause, emphasizing the properties and efficacy of the Maximum Magnetic Shear Model.
PubDate: 2021-03-29

• The Process of Stellar Tidal Disruption by Supermassive Black Holes
• Abstract: Tidal disruption events (TDEs) are among the brightest transients in the optical, ultraviolet, and X-ray sky. These flares are set into motion when a star is torn apart by the tidal field of a massive black hole, triggering a chain of events which is – so far – incompletely understood. However, the disruption process has been studied extensively for almost half a century, and unlike the later stages of a TDE, our understanding of the disruption itself is reasonably well converged. In this Chapter, we review both analytical and numerical models for stellar tidal disruption. Starting with relatively simple, order-of-magnitude physics, we review models of increasing sophistication, the semi-analytic “affine formalism,” hydrodynamic simulations of the disruption of polytropic stars, and the most recent hydrodynamic results concerning the disruption of realistic stellar models. Our review surveys the immediate aftermath of disruption in both typical and more unusual TDEs, exploring how the fate of the tidal debris changes if one considers non-main sequence stars, deeply penetrating tidal encounters, binary star systems, and sub-parabolic orbits. The stellar tidal disruption process provides the initial conditions needed to model the formation of accretion flows around quiescent massive black holes, and in some cases may also lead to directly observable emission, for example via shock breakout, gravitational waves or runaway nuclear fusion in deeply plunging TDEs.
PubDate: 2021-03-25

• Current Sheets, Plasmoids and Flux Ropes in the Heliosphere
• Abstract: Our understanding of processes occurring in the heliosphere historically began with reduced dimensionality - one-dimensional (1D) and two-dimensional (2D) sketches and models, which aimed to illustrate views on large-scale structures in the solar wind. However, any reduced dimensionality vision of the heliosphere limits the possible interpretations of in-situ observations. Accounting for non-planar structures, e.g. current sheets, magnetic islands, flux ropes as well as plasma bubbles, is decisive to shed the light on a variety of phenomena, such as particle acceleration and energy dissipation. In part I of this review, we have described in detail the ubiquitous and multi-scale observations of these magnetic structures in the solar wind and their significance for the acceleration of charged particles. Here, in part II, we elucidate existing theoretical paradigms of the structure of the solar wind and the interplanetary magnetic field, with particular attention to the fine structure and stability of current sheets. Differences in 2D and 3D views of processes associated with current sheets, magnetic islands and flux ropes are discussed. We finally review the results of numerical simulations and in-situ observations, pointing out the complex nature of magnetic reconnection and particle acceleration in a strongly turbulent environment.
PubDate: 2021-03-23

• Current Sheets, Plasmoids and Flux Ropes in the Heliosphere
• Abstract: Recent accumulation of a critical mass of observational material from different spacecraft complete with the enhanced abilities of numerical methods have led to a boom of studies revealing the high complexity of processes occurring in the heliosphere. Views on the solar wind filling the interplanetary medium have dramatically developed from the beginning of the space era. A 2-D picture of the freely expanding solar corona and non-interacting solar wind structures described as planar or spherically-symmetric objects has dominated for decades. Meanwhile, the scientific community gradually moved to a modern understanding of the importance of the 3-D nature of heliospheric processes and their studies via MHD/kinetic simulations, as well as observations of large-scale flows and streams both in situ and remotely, in white light and/or via interplanetary scintillations. The new 3-D approach has provided an opportunity to understand the dynamics of heliospheric structures and processes that could not even be imagined before within the 2-D paradigm. In this review, we highlight a piece of the puzzle, showing the evolution of views on processes related to current sheets, plasmoids, blobs and flux ropes of various scales and origins in the heliosphere. The first part of the review focuses on introducing these plasma structures, discussing their key properties, and paying special attention to their observations in different space plasmas.
PubDate: 2021-03-23

• The Navigation and Terrain Cameras on the Tianwen-1 Mars Rover
• Abstract: The Navigation and Terrain Cameras (NaTeCams) are binocular stereo cameras mounted on the Tianwen-1 Mars Rover. The NaTeCams are primarily designed to provide support for the guidance, navigation, and control of the rover, and they can also be used for scientific observations. NaTeCam is a color imaging system using a complementary metal oxide semiconductor (CMOS) active pixel sensor (APS). An overview of the camera design is provided. Preflight calibration and performance are also discussed. Specific details about the camera operation and ground test are presented.
PubDate: 2021-03-17

• Do Intrinsic Magnetic Fields Protect Planetary Atmospheres from Stellar
Winds'
• Abstract: The accumulation of detailed ion flux measurements from long-lived spacecraft orbiting the solar system’s terrestrial planets have enabled recent studies to estimate the rate of solar wind driven atmospheric ion escape from Venus, Earth, and Mars, as well as the influence of solar wind and solar extreme ultraviolet (EUV) ionizing radiation on the atmospheric ion escape rates. Here, we introduce the basic forces and processes of ion escape, review the recent studies of ion escape rates, and provide a general framework for understanding ion escape as a process that can be limited by potential bottlenecks, such as ion supply, solar wind energy transfer, and transport efficiency, effectively determining the state of the ion escape process at each planet. We find that ion escape from Venus and Earth is energy-limited, though exhibit different dependencies on solar wind and EUV, revealing the influence of Earth’s intrinsic magnetic field. In contrast, ion escape from Mars is in a supply-limited state, mainly due to its low gravity, and has likely contributed relatively little to the total loss of the early Martian atmosphere, in comparison to neutral escape processes. Contrary to the current paradigm, the comparisons between the solar system planets show that an intrinsic magnetic dipole field is not required to prevent stellar wind-driven escape of planetary atmospheres, and the presence of one may instead increase the rate of ion escape. Anticipating the atmospheres of the exoplanets that will begin to be characterized over the coming decade, and the need to explain their evolution, we argue that a modern, nuanced, and evidence-based view of the magnetic field’s role in atmospheric escape is required.
PubDate: 2021-03-09
DOI: 10.1007/s11214-021-00791-1

• Slow-Mode Magnetoacoustic Waves in Coronal Loops
• Abstract: Rapidly decaying long-period oscillations often occur in hot coronal loops of active regions associated with small (or micro-) flares. This kind of wave activity was first discovered with the SOHO/SUMER spectrometer from Doppler velocity measurements of hot emission lines, thus also often called “SUMER” oscillations. They were mainly interpreted as global (or fundamental mode) standing slow magnetoacoustic waves. In addition, increasing evidence has suggested that the decaying harmonic type of pulsations detected in light curves of solar and stellar flares are likely caused by standing slow-mode waves. The study of slow magnetoacoustic waves in coronal loops has become a topic of particular interest in connection with coronal seismology. We review recent results from SDO/AIA and Hinode/XRT observations that have detected both standing and reflected intensity oscillations in hot flaring loops showing the physical properties (e.g., oscillation periods, decay times, and triggers) in accord with the SUMER oscillations. We also review recent advances in theory and numerical modeling of slow-mode waves focusing on the wave excitation and damping mechanisms. MHD simulations in 1D, 2D and 3D have been dedicated to understanding the physical conditions for the generation of a reflected propagating or a standing wave by impulsive heating. Various damping mechanisms and their analysis methods are summarized. Calculations based on linear theory suggest that the non-ideal MHD effects such as thermal conduction, compressive viscosity, and optically thin radiation may dominate in damping of slow-mode waves in coronal loops of different physical conditions. Finally, an overview is given of several important seismological applications such as determination of transport coefficients and heating function.
PubDate: 2021-03-08
DOI: 10.1007/s11214-021-00811-0

JournalTOCs
School of Mathematical and Computer Sciences
Heriot-Watt University
Edinburgh, EH14 4AS, UK
Email: journaltocs@hw.ac.uk
Tel: +00 44 (0)131 4513762


text:
# How do we represent the reaction of hydrochloric acid with stoichiometric calcium carbonate? Do we need to consider oxidation states?

May 27, 2017

This is an acid-base reaction, in which no redox transfer occurs........

#### Explanation:

We can write out the stoichiometric equation:

$C a C {O}_{3} \left(s\right) + 2 H C l \left(a q\right) \rightarrow C a C {l}_{2} \left(a q\right) + C {O}_{2} \left(g\right) + {H}_{2} O \left(l\right)$

And the net ionic equation:

$C {O}_{3}^{2 -} + 2 {H}_{3} {O}^{+} \rightarrow C {O}_{2} \left(g\right) \uparrow + 3 {H}_{2} O \left(l\right)$

And we can include oxidation numbers on reactants and products:

$\stackrel{+ I V}{C} {O}_{3}^{2 -} + 2 {H}_{3} {O}^{+} \rightarrow \stackrel{+ I V}{C} {O}_{2} \left(g\right) \uparrow + 3 {H}_{2} O \left(l\right)$

Are these equations balanced with respect to mass and charge? They need to be if we purport to represent chemical reality!

The oxidation numbers of hydrogen and carbon remain at $\stackrel{- I I}{O}$ and $\stackrel{+ I}{H}$ respectively.


text:
## absorption cross-section, $$\sigma$$

https://doi.org/10.1351/goldbook.A00038
Molar Napierian @A00037@ divided by the @A00543@. When integrated against the logarithm of wavenumber (or frequency) it is called the integrated absorption cross-section.
Source:
Green Book, 2nd ed., p. 32 [Terms] [Book]


text:
# How to add vertical line in table between a multicolumn and a column

I hope you can help me out as I've been struggling with this for quite a while.

I'm trying to add a vertical line before and after the multicolumn holding "Capacity [MW]" so the two lines (before "Spain" and after "Portugal") continues to the top - please see picture.

My code is here. Please note, that I use the \usepackage{makecell}.

\begin{table}[htbp]
\caption{Assumed transmission capacity for the Balmorel model.}
\centering
\begin{tabular}{l|c|c|c|c}
\hline
\hline
\textbf{Technology type}                &\textbf{Fuel}      &\textbf{Spain}       &\textbf{Portugal}                &\textbf{Efficiency}\\
\hline
PT $\rightarrow$ ES                    & 2954              & 3500              & 3500                          &5\% \\
ES $\rightarrow$ PT                    & 2085              & 4200              & 4200                          &5\% \\
\hline
FR $\rightarrow$ ES                    & 450               & 2800              & 8000                          &5\% \\
ES $\rightarrow$ FR                    & 325               & 2500              & 8000                          &5\% \\
\hline
\end{tabular}
\label{tab:transcap}
\end{table}

• Hi esdd, Nice. That works! Appreciate your help! – Henryh Dec 6 '16 at 15:42

You can add a | to multicolumn commands (\multicolumn{2}{c|}) like this

\documentclass{article}
\usepackage{makecell}

\begin{document}
\begin{table}[htbp]
\caption{Assumed transmission capacity for the Balmorel model.}
\centering
\begin{tabular}{l|c|c|c|c}
\hline
\hline
\textbf{Technology type}                &\textbf{Fuel}      &\textbf{Spain}       &\textbf{Portugal}                &\textbf{Efficiency}\\
\hline
PT $\rightarrow$ ES                    & 2954              & 3500              & 3500                          &5\% \\
ES $\rightarrow$ PT                    & 2085              & 4200              & 4200                          &5\% \\
\hline
FR $\rightarrow$ ES                    & 450               & 2800              & 8000                          &5\% \\
ES $\rightarrow$ FR                    & 325               & 2500              & 8000                          &5\% \\
\hline
\end{tabular}
\label{tab:transcap}
\end{table}
\end{document}

In other combinations (e.g. \multicolumn{2}{|c|} in central columns) vertical lines seem to be not well aligned.


text:
Welcome! This page has way too much self-indulgent detail, but if you’re really interested in what I’m all about you can probably find it here.

## Contact

You’re always welcome to email me. Seriously!

## Location

I grew up in Michigan. I’ve lived in California for the past several years, first in LA, then in Sacramento, and currently in San Francisco.

## Outdoors

I play tennis three or four times a week, and have for the past year or so. I’ve gotten a lot better in that time!

I’ve enjoyed almost every sport I’ve tried—baseball, basketball, tetherball, football, softball, soccer, table tennis, pickleball, kickball, dodgeball, wall ball, ultimate, and inner tube water polo, to name a few.

I also like a ton of other outdoorsy stuff. Most often this is hiking, but I’ve also done parasailing, whitewater rafting, mountain biking, skiing, rock climbing, kayaking, and more.

## Video Games

I really like making video games. One of my favorites I’ve worked on was a virtual reality survival game in space called Disasteroids.

I have also played more video games than I care to admit. I’ve written reviews for some of them. I tend to like indie platformers and puzzle games.

## Books

Here’s a list of some books. I read almost exclusively nonfiction, because I like learning new things and trying to better understand how the world works. I wish I had more time to read fiction.

My favorite fiction book is Infinite Jest, and my favorite nonfiction book is probably Learn You a Haskell for Great Good! because it set me down a path to a lot more learning about topics I find super interesting. My most treasured book from my childhood is 1000 Play Thinks.

## Music

I enjoy making music. I usually make electronic stuff. I learned the basics of how to play piano in third grade, and my skills have slowly improved since then.

I also like the flip side of that—dancing, music festivals, concerts, or just listening to stuff while I work. I have a playlist cycle four years in the making, aptly named Dance to This, Dance to That, Dance to These, and Dance to Those. I think it’s cool to see how my taste has changed over time.

## Travels

I’ve been to all 48 continental United States. I’ve seen several parts of Canada. I’ve been to the following European and Asian cities: Athens, Barcelona, Beijing, Florence, Hong Kong, Lisbon, London, Madrid, Paris, and Rome. I’d like to see more of the world!

## Programming

My favorite programming language is probably Haskell, but for quick stuff I’ll usually reach for Python since that’s what I’m fastest in.

I’ve been using Vim as my editor for many years. I also actually like Emacs quite a bit, but I got used to Vim first. The editor wars seem silly to me; clearly ed is the winner.

You can read my dotfiles if you’re into that sort of thing. The quick explanation is: Vim, tmux, and trying to remember to automate things.


text:
# Entropic Regularization of Optimal Transport¶

In [1]:
addpath('toolbox_signal')

Out[1]:

## Entropic Regularization of Optimal Transport¶

We consider two input histograms $p,q \in \Si_N$, where we denote the simplex in $\RR^N$ $$\Si_{N} \eqdef \enscond{ p \in (\RR^+)^N }{ \sum_i p_i = 1 }.$$ We consider the following discrete regularized transport $$W_\ga(p,q) \eqdef \umin{\pi \in \Pi(p,q)} \dotp{C}{\pi} - \ga E(\pi).$$ where the polytope of coupling is defined as $$\Pi(p,q) \eqdef \enscond{\pi \in (\RR^+)^{N \times N}}{ \pi \ones = p, \pi^T \ones = q },$$ where $\ones \eqdef (1,\ldots,1)^T \in \RR^N$, and for $\pi \in (\RR^+)^{N \times N}$, we define its entropy as $$E(\pi) \eqdef - \sum_{i,j} \pi_{i,j} ( \log(\pi_{i,j}) - 1).$$

When $\ga=0$ one recovers the classical (discrete) optimal transport. We refer to the monograph Villani for more details about OT. The idea of regularizing transport to allows for faster computation is introduced in Cuturi.

Here the matrix $C \in (\RR^+)^{N \times N}$ defines the ground cost, i.e. $C_{i,j}$ is the cost of moving mass from a bin indexed by $i$ to a bin indexed by $j$.

The regularized transportation problem can be re-written as a projection $$W_\ga(p,q) = \ga \umin{\pi \in \Pi(p,q)} \KLdiv{\pi}{\xi} \qwhereq \xi_{i,j} = e^{ -\frac{C_{i,j}}{\ga} }$$ of $\xi$ according to the Kullback-Leibler divergence. The Kullback-Leibler divergence between $\pi, \xi \in (\RR^+)^P$ is $$\KLdiv{\pi}{\xi} = \sum_{i,j} \pi_{i,j} \pa{ \log\pa{ \frac{\pi_{i,j}}{\xi_{i,j}} } - 1}.$$

This interpretation of regularized transport as a KL projection and its numerical applications are detailed in BenamouEtAl.

Given a convex set $\Cc \subset \RR^N$, the projection according to the Kullback-Leiber divergence is defined as $$\KLproj_\Cc(\xi) = \uargmin{ \pi \in \Cc } \KLdiv{\pi}{\xi}.$$

## Iterative Bregman Projection Algorithm¶

Given affine constraint sets $(\Cc_1,\ldots,\Cc_K)$, we aim at computing $$\KLproj_\Cc(\xi) \qwhereq \Cc = \Cc_1 \cap \ldots \cap \Cc_K.$$

This can be achieved, starting by $\pi_0=\xi$, by iterating $$\forall \ell \geq 0, \quad \pi_{\ell+1} = \KLproj_{\Cc_\ell}(\pi_\ell),$$ where the index of the constraints should be understood modulo $K$, i.e. we set $\Cc_{\ell+K}=\Cc_\ell$.

One can indeed show that $\pi_\ell \rightarrow \KLproj_\Cc(\bar \pi)$. We refer to BauschkeLewis for more details about this algorithm and its extension to compute the projection on the intersection of convex sets (Dikstra algorithm).

## Iterative Projection for Regularized Transport aka Sinkhorn's Algorithm¶

We can re-cast the regularized optimal transport problem within this framework by introducing $$\Cc_1 \eqdef \enscond{\pi \in (\RR^+)^{N \times N} }{\pi \ones = p} \qandq \Cc_2 \eqdef \enscond{\pi \in (\RR^+)^{N \times N} }{\pi^T \ones = q}.$$

The KL projection on $\Cc_1$ sets are easily computed by divisive normalization of rows. Indeed, denoting $\pi = \KLproj_{\Cc_1}(\bar \pi)$, one has $$\forall (i,j), \quad \pi_{i,j} = \frac{ p_i \bar\pi_{i,j} }{ \sum_{s} \bar\pi_{i,s} }$$ and similarely for $\KLproj_{\Cc_1}(\bar \pi)$ by replacing rows by colums.

A fundamental remark is that, if $\bar\pi = \diag(a) \xi \diag(b)$ (a so-called diagonal scaling of the kernel $\xi$), then one has $$\KLproj_{\Cc_1}(\bar \pi) = \diag(\tilde a) \xi \diag(b) \qandq \KLproj_{\Cc_2}(\bar \pi) = \diag(a) \xi \diag(\tilde b)$$ where the new scaling reads $$\tilde a = \frac{p}{\xi(b)} \qandq \tilde b = \frac{q}{\xi^T(a)}$$ where $\frac{\cdot}{\cdot}$ is entry-wise division.

This means that the iterates of Bregman iterative projections, starting with $a_0 \eqdef \ones$ always have the form $\pi_\ell = \diag(a_\ell) \xi \diag(b_\ell)$ and these diagonal scaling weights are updated as follow $$a_{\ell+1} \eqdef \frac{p}{\xi(b_\ell)} \qandq b_{\ell+1} \eqdef \frac{q}{\xi^T(a_{\ell+1})}.$$ This algorithm is in fact the well known Sinkhorn algorithm Sinkhorn.

## Transport Between Point Clouds¶

We first test the method for two input measures that are uniform measures (i.e. constant histograms) supported on two point clouds (that do not necessarily have the same size).

We thus first load two points clouds $x=(x_i)_{i=1}^{N_1}, y=(y_i)_{i=1}^{N_2},$ where $x_i, y_i \in \RR^2$.

Number of points in each cloud.

In [2]:
N = [300,200];

Out[2]:

Dimension of the clouds.

In [3]:
d = 2;

Out[3]:

Point cloud $x$, of $N_1$ points inside a square.

In [4]:
x = rand(2,N(1))-.5;

Out[4]:

Point cloud $y$, of $N_2$ points inside an anulus.

In [5]:
theta = 2*pi*rand(1,N(2));
r = .8 + .2*rand(1,N(2));
y = [cos(theta).*r; sin(theta).*r];

Out[5]:

Shortcut for displaying point clouds.

In [6]:
plotp = @(x,col)plot(x(1,:)', x(2,:)', 'o', 'MarkerSize', 10, 'MarkerEdgeColor', 'k', 'MarkerFaceColor', col, 'LineWidth', 2);

Out[6]:

Display of the two clouds.

In [7]:
clf; hold on;
plotp(x, 'b');
plotp(y, 'r');
axis('off'); axis('equal');

Out[7]:

Cost matrix $C_{i,j} = \norm{x_i-y_j}^2$.

In [8]:
x2 = sum(x.^2,1); y2 = sum(y.^2,1);
C = repmat(y2,N(1),1)+repmat(x2.',1,N(2))-2*x.'*y;

Out[8]:

Target histograms, here uniform histograms.

In [9]:
p = ones(N(1),1)/N(1);
q = ones(N(2),1)/N(2);

Out[9]:

Regularization strength $\ga$.

In [10]:
gamma = .01;

Out[10]:

Gibbs Kernel.

In [11]:
xi = exp(-C/gamma);

Out[11]:

Initialization of $b_0=\ones_{N_2}$ ($a_0$ does not need to be initialized).

In [12]:
b = ones(N(2),1);

Out[12]:

One sinkhorn iterations.

In [13]:
a = p ./ (xi*b);
b = q ./ (xi'*a);

Out[13]:

Exercise 1

Implement Sinkhorn algorithm. Display the evolution of the constraints satisfaction errors $\norm{ \pi \ones - p }, \norm{ \pi^T \ones - q }$ (you need to think about how to compute these residuals from $(a,b)$ alone). isplay the violation of constraint error in log-plot.

In [14]:
exo1()

Out[14]:
In [15]:
%% Insert your code here.

Out[15]:

Compute the final matrix.

In [16]:
Pi = diag(a)*xi*diag(b);

Out[16]:

Display it.

In [17]:
clf;
imageplot(Pi);

Out[17]:

Exercise 2

Display the regularized transport solution for various values of $\gamma$. For a too small value of $\gamma$, what do you observe ?

In [18]:
exo2()

Out[18]:
In [19]:
%% Insert your code here.

Out[19]:

Compute the obtained optimal $\pi$.

In [20]:
Pi = diag(a)*xi*diag(b);

Out[20]:

Keep only the highest entries of the coupling matrix, and use them to draw a map between the two clouds. First we draw "strong" connexions, i.e. linkds $(i,j)$ corresponding to large values of $\pi_{i,j}$. We then draw weaker connexions.

In [21]:
clf;
hold on;
A = sparse( Pi .* (Pi> min(1./N)*.7) ); [i,j,~] = find(A);
h = plot([x(1,i);y(1,j)], [x(2,i);y(2,j)], 'k');
set(h, 'LineWidth', 2); % weaker connections.
A = sparse( Pi .* (Pi> min(1./N)*.3) ); [i,j,~] = find(A);
h = plot([x(1,i);y(1,j)], [x(2,i);y(2,j)], 'k:');
set(h, 'LineWidth', 1);
plotp(x, 'b'); % plot the two point clouds.
plotp(y, 'r');
axis('off'); axis('equal');

Out[21]:

## Transport Between Histograms¶

We now consider a different setup, where the histogram values $p,q$ are not uniform, but the measures are defined on a uniform grid $x_i=y_i=i/N$. They are thue often refered to as "histograms".

Size $N$ of the histograms.

In [22]:
N = 200;

Out[22]:

We use here a 1-D square Euclidean metric.

In [23]:
t = (0:N-1)'/N;

Out[23]:

Define the histogram $p,q$ as translated Gaussians.

In [24]:
Gaussian = @(t0,sigma)exp( -(t-t0).^2/(2*sigma^2) );
normalize = @(p)p/sum(p(:));
sigma = .06;
p = Gaussian(.25,sigma);
q = Gaussian(.8,sigma);

Out[24]:

Add some minimal mass and normalize.

In [25]:
vmin = .02;
p = normalize( p+max(p)*vmin);
q = normalize( q+max(q)*vmin);

Out[25]:

Display the histograms.

In [26]:
clf;
subplot(2,1,1);
bar(t, p, 'k'); axis tight;
subplot(2,1,2);
bar(t, q, 'k'); axis tight;

Out[26]:

Regularization strength $\ga$.

In [27]:
gamma = (.03)^2;

Out[27]:

The Gibbs kernel is a Gaussian convolution, $$\xi_{i,j} = e^{ -(i/N-j/N)^2/\gamma }.$$

In [28]:
[Y,X] = meshgrid(t,t);
xi = exp( -(X-Y).^2 / gamma);

Out[28]:

Initialization of $b_0=\ones_{N}$.

In [29]:
b = ones(N,1);

Out[29]:

One sinkhorn iteration.

In [30]:
a = p ./ (xi*b);
b = q ./ (xi'*a);

Out[30]:

Exercise 3

Implement Sinkhorn algorithm. Display the evolution of the constraints satisfaction errors $\norm{ \pi \ones - p }, \norm{ \pi^T \ones - q }$ (you need to think how to compute it from $(a,b)$. isplay the violation of constraint error in log-plot.

In [31]:
exo3()

Out[31]:
In [32]:
%% Insert your code here.

Out[32]:

Display the coupling. Use a log domain plot to better vizualize it.

In [33]:
Pi = diag(a)*xi*diag(b);
clf;
imageplot(log(Pi+1e-5));

Out[33]:

One can compute an approximation of the transport plan between the two measure by computing the so-called barycentric projection map $$t_i \in [0,1] \longmapsto s_j \eqdef \frac{\sum_{j} \pi_{i,j} t_j }{ \sum_{j} \pi_{i,j} } = \frac{ [a \odot \xi(b \odot t)]_j }{ p_i }.$$ where $\odot$ and $\frac{\cdot}{\cdot}$ are the enry-wise multiplication and division.

This computation can thus be done using only multiplication with the kernel $\xi$.

In [34]:
s = (xi*(b.*t)) .* a ./ p;

Out[34]:

Display the transport map, super-imposed over the coupling.

In [35]:
clf; hold on;
imagesc(t,t,log(Pi+1e-5)); colormap gray(256);
plot(s,t, 'r', 'LineWidth', 3);
axis image; axis off; axis ij;

Out[35]:

## Wasserstein Barycenters¶

Instead of computing transport, we now turn to the problem of computing barycenter of measures. A barycenter $q$ solves $$\umin{q} \sum_{k=1}^K W_\ga(p_k,q)$$ where $\la_k$ are positive weights with $\sum_k \la_k=1$. This follows the definition of barycenters proposed in AguehCarlier.

With of the histograms.

In [36]:
N = 70;

Out[36]:

Load input histograms $(p_k)_{k=1}^K$.

In [37]:
names = {'disk' 'twodisks' 'letter-x' 'letter-z'};
r = [.35 .19 .12*N .12*N];
vmin = .05;
P = [];
for i=1:length(names)
p = normalize( rescale(p)+vmin );
P(:,:,i) = p;
end
K = size(P,3);

Out[37]:

Display the input histograms.

In [38]:
a = mat2cell(P, N,N,ones(K,1));
clf;
imageplot(a, '', 2,2);

Out[38]:

In this specific case, the kernel $\xi$ associated with the squared Euclidean norm is a convolution with a Gaussian filter $$\xi_{i,j} = e^{ -\norm{i/N-j/N}^2/\gamma }$$ where here $(i,j)$ are 2-D indexes.

The multiplication against the kernel $\xi(a_\ell)$ can now be computed efficiently, using fast convolution methods. This crucial points was exploited and generalized in SolomonEtAl to design fast optimal transport algorithm.

Regularization strength $\ga$.

In [39]:
gamma = (.04)^2;

Out[39]:

Define the $\xi$ kernel. We use here the fact that the convolution is separable to implement it using only 1-D convolution, which further speeds up computations.

In [40]:
n = 41; % width of the convolution kernel
t = linspace(-n/(2*N),n/(2*N),n)';
g = exp(-t.^2 / gamma); g2 = g*g';
xi = @(x)conv2(conv2(x, g, 'same')', g, 'same')';

Out[40]:

Display the application of the $\xi$ kernel on one of the input histogram.

In [41]:
clf;
imageplot({P(:,:,1) xi(P(:,:,1))});

Out[41]:

Weights for isobarycenter.

In [42]:
lambda = ones(K,1)/K;

Out[42]:

It is shown in BenamouEtAl that the problem of Barycenter computation boilds down to optimizing over couplings $(\pi_k)_k$, and that this can be achieved using iterative Bregman projection that defines iterates $(\pi_{k,\ell})_k$. These iterates can be written using diagonal scalings $\pi_{k,\ell} \eqdef \diag(a_{k,\ell}) \xi \diag(b_{k,\ell}).$

Initialize the scaling factors.

In [43]:
b = ones(N,N,K); a = b;

Out[43]:

The first step of the Bregman projection method corresponds to the projection on the fixed marginals constraints $\pi_k \ones=p_k$. This is achieved by updating $$\forall k=1,\ldots,K, \quad a_{k,\ell+1} = \frac{p_k}{ \xi( b_{k,\ell} ) }.$$

In [44]:
for k=1:K
a(:,:,k) = P(:,:,k) ./ xi(b(:,:,k));
end

Out[44]:

The second step of the Bregman projection method corresponds to the projection on the equal marginals constraints $\forall k, \pi_k^T \ones=q$. This is achieved by first computing the target barycenter using a geometric means $$\log(q_\ell) \eqdef \sum_k \lambda_k \log( b_{k,\ell} \odot \xi( a_{k,\ell} ) ).$$

In [45]:
q = zeros(N);
for k=1:K
q = q + lambda(k) * log( max(1e-19, b(:,:,k) .* xi(a(:,:,k)) ) );
end
q = exp(q);

Out[45]:

And then one can update the other maginals to be equal to this barycenter at step $\ell$. $$\forall k=1,\ldots,K, \quad b_{k,\ell+1} \eqdef \frac{q_\ell}{ \xi(a_{k,\ell+1}) }.$$

In [46]:
for k=1:K
b(:,:,k) = q ./ xi(a(:,:,k));
end

Out[46]:

Exercise 4

Implement the iterative algorithm to compute the iso-barycenter of the measures. Plot the decay of the error $\sum_k \norm{\pi_k \ones - p_k}$.

In [47]:
exo4()

Out[47]:
In [48]:
%% Insert your code here.

Out[48]:

Display the barycenter.

In [49]:
clf;
imageplot(q);

Out[49]:

Exercise 5

Compute barycenters for varying weights $\la$ corresponding to a bilinear interpolation inside a square.

In [50]:
exo5()

Out[50]:
In [51]:
%% Insert your code here.

Out[51]:

## Bibliography¶

• [Villani] C. Villani, (2009). Optimal transport: old and new, volume 338. Springer Verlag.
• [Cuturi] M. Cuturi, (2013). Sinkhorn distances: Lightspeed computation of optimal transport. In Burges, C. J. C., Bottou, L., Ghahramani, Z., and Weinberger, K. Q., editors, Proc. NIPS, pages 2292-2300.
• [AguehCarlier] M. Agueh, and G Carlier, (2011). Barycenters in the Wasserstein space. SIAM J. on Mathematical Analysis, 43(2):904-924.
• [CuturiDoucet] M. Cuturi and A. Doucet (2014). Fast computation of wasserstein barycenters. In Proc. ICML.
• [BauschkeLewis] H. H. Bauschke and A. S. Lewis. Dykstra's algorithm with Bregman projections: a convergence proof. Optimization, 48(4):409-427, 2000.
• [Sinkhorn] R. Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Statist., 35:876-879, 1964.
• [SolomonEtAl] J. Solomon, F. de Goes, G. Peyr , M. Cuturi, A. Butscher, A. Nguyen, T. Du, and L. Guibas. Convolutional Wasserstein distances: Efficient optimal transportation on geometric domains. Transaction on Graphics, Proc. SIGGRAPH, 2015.
• [BenamouEtAl] J-D. Benamou, G. Carlier, M. Cuturi, L. Nenna, G. Peyr . Iterative Bregman Projections for Regularized Transportation Problems. SIAM Journal on Scientific Computing, 37(2), pp. A1111-A1138, 2015.


text:
American Institute of Mathematical Sciences

November  2014, 34(11): 4671-4688. doi: 10.3934/dcds.2014.34.4671

Supercritical problems in domains with thin toroidal holes

 1 Departamento de Matemática, Pontificia Universidad Catóica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile 2 Dipartimento SBAI, Università di Roma "La Sapienza", via Antonio Scarpa 16, 00161 Roma

Received  September 2013 Revised  December 2013 Published  May 2014

In this paper we study the Lane-Emden-Fowler equation (P)_ \epsilon \quad \left\{ \begin{aligned} &\Delta u+|u|^{q-2}u=0\ &\hbox{in}\ \mathcal D_ \epsilon,\\ & u=0\ &\hbox{on}\ \partial\mathcal D_ \epsilon.\\ \end{aligned}\right. Here $\mathcal D_ \epsilon=\mathcal D\setminus \left\{x\in \mathcal D\ :\ \mathrm{dist}(x,\Gamma_l)\le \epsilon \right\}$, $\mathcal D$ is a smooth bounded domain in $\mathbb{R}^N$, $\Gamma_l$ is an $l-$dimensional closed manifold such that $\Gamma_l\subset\mathcal D$ with $1\le l\le N-3$ and $q={2(N-l)\over N-l-2} .$ We prove that, under some symmetry assumptions, the number of sign changing solutions to $(P)_ \epsilon$ increases as $\epsilon$ goes to zero.
Citation: Seunghyeok Kim, Angela Pistoia. Supercritical problems in domains with thin toroidal holes. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4671-4688. doi: 10.3934/dcds.2014.34.4671
References:

show all references

References:
 [1] Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453 [2] Craig Cowan. Supercritical elliptic problems involving a Cordes like operator. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021037 [3] Bo Tan, Qinglong Zhou. Approximation properties of Lüroth expansions. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2873-2890. doi: 10.3934/dcds.2020389 [4] Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021039 [5] Rafael G. L. D'Oliveira, Marcelo Firer. Minimum dimensional Hamming embeddings. Advances in Mathematics of Communications, 2017, 11 (2) : 359-366. doi: 10.3934/amc.2017029 [6] Braxton Osting, Jérôme Darbon, Stanley Osher. Statistical ranking using the $l^{1}$-norm on graphs. Inverse Problems & Imaging, 2013, 7 (3) : 907-926. doi: 10.3934/ipi.2013.7.907 [7] Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233 [8] Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $\mathbb{R}^{N}$. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006 [9] Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021060 [10] Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447 [11] Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021017 [12] Shuang-Lin Jing, Hai-Feng Huo, Hong Xiang. Modelling the effects of ozone concentration and pulse vaccination on seasonal influenza outbreaks in Gansu Province, China. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021113 [13] Takeshi Saito, Kazuyuki Yagasaki. Chebyshev spectral methods for computing center manifolds. Journal of Computational Dynamics, 2021  doi: 10.3934/jcd.2021008 [14] Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2021, 13 (1) : 55-72. doi: 10.3934/jgm.2020031 [15] Clara Cufí-Cabré, Ernest Fontich. Differentiable invariant manifolds of nilpotent parabolic points. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021053 [16] Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209 [17] Ahmad Mousavi, Zheming Gao, Lanshan Han, Alvin Lim. Quadratic surface support vector machine with L1 norm regularization. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021046 [18] Mingshang Hu, Shige Peng. G-Lévy processes under sublinear expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 1-22. doi: 10.3934/puqr.2021001 [19] Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228 [20] Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341

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text:
# Distribution of the first vertex on a camera subpath in bidirectional path tracing

Suppose we are sampling camera rays of a perspective camera with a finite aperture. The PBRT implementation of the sampling scheme looks like this:

Float PerspectiveCamera::GenerateRay(const CameraSample &sample,
Ray *ray) const {
ProfilePhase prof(Prof::GenerateCameraRay);
// Compute raster and camera sample positions
Point3f pFilm = Point3f(sample.pFilm.x, sample.pFilm.y, 0);
Point3f pCamera = RasterToCamera(pFilm);
*ray = Ray(Point3f(0, 0, 0), Normalize(Vector3f(pCamera)));
// Modify ray for depth of field
// Sample point on lens
Point2f pLens = lensRadius * ConcentricSampleDisk(sample.pLens);

// Compute point on plane of focus
Float ft = focalDistance / ray->d.z;
Point3f pFocus = (*ray)(ft);

// Update ray for effect of lens
ray->o = Point3f(pLens.x, pLens.y, 0);
ray->d = Normalize(pFocus - ray->o);
}
ray->time = Lerp(sample.time, shutterOpen, shutterClose);
ray->medium = medium;
*ray = CameraToWorld(*ray);
return 1;
}

Let $$X_0\in M$$ and $$\Omega_1\in S^2$$ denote the sampled point on the camera and direction, respectively. The next surface point in a path tracer would be $$Y:=X+d_M(X,\Omega)\Omega,$$ where $$d_M(x,\omega):=\inf\left\{t>0:x+t\omega\in M\right\}\;\;\;\text{for }x\in M\text{ and }\omega\in S^2.$$ What is the conditional distribution of $$Y$$ given $$(X,\Omega)$$?

From page 303 in Veach's thesis the conditional distribution of $$Y$$ given $$X$$ should have a density with respect to the projected solid angle kernel $$\sigma_M^\perp(x,B):=\int_B\sigma_{S^2}({\rm d}\omega)|\langle\nu_M(x),\omega\rangle|\;\;\;\text{for }(x,B)\in M\times\mathcal B(S^2).$$

Formally, we assume that the lens is part of our scene geometry $$M$$, which is assumed to be an oriented embedded $$C^1$$-submanifold of $$\mathbb R^3$$, $$\nu_M$$ denotes the outer normal field on $$M$$ and $$\sigma_M$$ denotes the surface measure on $$\mathcal B(M)$$. Moreover, let $$S^2$$ denote the unit $$2$$-sphere, $$S^2(x):=\left\{y\in\mathbb R^3:|x-y|=1\right\}=x+S^2\;\;\;\text{for }x\in\mathbb R^3$$ and $$\omega_{x\to y}:=\frac{y-x}{|y-x|}\in S^2\;\;\;\text{for }x,y\in\mathbb R^3\text{ with }x\ne y.$$ For simplicity, assume that $$d_M(x,\omega):=\inf\left\{t>0:x+t\omega\in M\right\}<\infty\;\;\;\text{for all }x\in M\text{ and }\omega\in S^2\tag1$$

• I noticed that you have been posting a lot in recent days. While I think it is great that you are interested in light transport, the mathematical level you are using to approach your problems is far too high for the majority of this community. Your questions are mostly research-oriented and, as such, I'm not sure how much this forum can help you. It is clear that you are coming from a pure math background like Veach. Realistically though, the intersection of people who fully understand his thesis and also lurk here is probably nil. I hope someone can prove me wrong... – Hubble Jan 17 at 0:27
• @Hubble Thank you for your comment. – 0xbadf00d Jan 17 at 5:05
• Is this what you're looking for? agraphicsguy.wordpress.com/2016/02/04/… – lightxbulb Jan 17 at 21:20


text:
## Variational Autoencoders

Generative models in deep learning become popular since 2014, when GAN (Generative Adversial Nets) was introduced by Ian Goodfellow. Another model that is widely used for generating data is VAE (Variational Autoencoders) and its derivation CVAE (Conditional Variational Autoencoders).

In this post, we will have a brief introduction to VAE and how it works.

## The intuition

Latent variable in Autoencoder model is an indispensable part. It stores the information that is needed to reconstruct the original data, and this is what we can use to generate new things. We could use VAE to generate images, languages and even music!

Let’s first define several notions before we talk about the generative process:

• $X$, which is the data from the dataset and we want to learn from.
• $z$, which is the latent variable stores information.
• $P(X)$, probability distribution of all the data.
• $P(z)$, probability distribution of the latent variables.
• $P(X|z)$, probability distribution of generated data given the latent variable z.

The generative process would be: $P(X) = {\displaystyle \int }P(X|z)P(z)$. The intuition behind this framework is that, if it can reconstruct the training samples, it would be possible for it to produce simialr samples. Since normal distribution $N(0, I)$ can be used to approximate any distribution within the same dimension, we take $P(z)=N(0,I)$ and all we need to do is to maximize the equation above.

## The model

As its name suggests, Autoencoders containt two parts: $Q(z|X)$ which encode the data sample $X$ onto the latent $z$, and $P(X|z)$ which decodes the latent $z$ and try to reconstruct the original data. Hopefully, we want the latent $z$ produced by our encoder net can best produced $X$. However, how can we make sure that our $Q(z|X)$ would be close to the true distribution $Q_{real}(z|X)$. Here is where the KL-divergence comes from.

Here, we won’t work through the entire proof and equations of how it works, but basically it measures how different are two distribution. Therefore, we could use it as our loss function, and we can use grdient descent later to optimize it. If you want to explore about how it works in VAE, here is the thing you should check out.

## The loss function

The VAE objective function is:

$log(P(X)) - D[Q(z|X)\vert\vert P(z|X)] = E[log(P(X|z))] - D[Q(z)\vert \vert P(z)]$

On the left hand side, it is the quantity that we want to maximize. The error term $D[Q(z|X)\vert \vert P(z|X)]$ should be close to $0$ if we want some $z$ that is most likely to reproduce the data. On the right hand side is the loss function that we can optimize through gradient descent. The former one is the reconstruction loss to make the generating image looks like the training data, while the latter one is the KL-divergence between the distribution of latent $z$ and normal distribution $N(0,I)$.

The loss function written in python with Keras framework:

However, we have encountered a problem now: how should we get $z$ from the encoder. Obviously, directly sampling won‘t work here since it does not have gradient! Therefore, we have to use a method called reparametrization trick to make it differentiable.

## CVAE

The idea behind Conditional VAE is the same as vanilla VAE, it just adds a label input in the input. In such a way, one could use the label and learned latent z to generate any thing he wants.

Take MNIST dataset as an example, one could add a one-hot vector to label the input:

the vector above could be the label for number $3$, and when generating images, one could use the vector above the generating images with written number $3$.

## Others

If you want to have a deeper understanding about VAE and have hand-on expirience, here are some resources you can check on:


text:
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# The Elevation of city (A) is 96 feet Below sea Level; The elevation of city (C) is 24 feet below sea level.

+1
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+1015

The Elevation of city (A) is 96 feet Below sea Level; The elevation of city (C) is 24 feet below sea level.

The elevation of city (D) is 1/4 of the elevation of city (C).

Part A: Write an expression to represent the elevation, in feet, of city D.

Part B: Solve the expression and explain why the answer is negative or positive.

Please explain both parts Thankyou so much!

Jul 22, 2019

#3
+19833
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Jul 22, 2019
#4
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(a) Sea level is $$0$$, therefore the elevation of city (A) is $$-96$$, and the elevation of city (C) is $$-24$$. For this part we do not need the elevation of city (A). The elevation of city (D) is $$\frac{1}{4}$$ of city (C). The key word is OF! Of means to multiply. Therefore we have the expression$$-24\cdot\frac{1}{4}$$.

(b) Part b is fairly simple. All you have to do is compute $$-24\cdot\frac{1}{4}=-6$$. As for explaining why it's negative we go back to how a negative times a positive is a negative.

This wasn't a very challenging problem. Are you sure you couldn't have solved it?

-$$\pi$$   KeyLimePi

Jul 22, 2019
#11
+107095
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I would like to suggest that you do not use sentences like

"This wasn't a very challenging problem. "

What is easy for you may well be very difficult for someone else.

Melody  Jul 23, 2019
#12
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You should put EP’s post back. It’s the best post in the past week! I posted a copy on #2

You should put your post back too!  It’s not as funny but it’s true!

Guest Jul 23, 2019
edited by Guest  Jul 25, 2019
#13
+107095
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It was not EPs post. EP has assured me he did not write it.

For the benefit of others,  EP and Nickolas are both being trolled.

Melody  Jul 23, 2019
#14
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Thanks for your help guys yes I was alittle confused with the last post Ep did for this same question I didnt really no what to do at the part were 1/4 of the elvation of city (C)!

I wasint sure for 1/4 part.

Thanks Melody

ElectricPavlov

KeylimePie

Nickolas  Jul 23, 2019
#15
+1015
+1

Thankyou for explaing that KeylimePi

I see

Nickolas  Jul 23, 2019
#16
+107095
+1

Did you see my answer over here Nickolas?


text:
liesymm - Maple Programming Help

Home : Support : Online Help : Mathematics : Differential Equations : Lie Symmetry Method : liesymm : liesymm/getcoeff

liesymm

 getcoeff
 extract the coefficient part of a basis wedge product

 Calling Sequence getcoeff(expr)

Parameters

 expr - term involving one wedge product

Description

 • This routine is part of the liesymm package and is loaded via with(liesymm).
 • Given an expression expr involving a single wedge product w find c such that $\mathrm{expr}=cw$.
 • If more than one wedge product is encountered, an error message is given.

Examples

 > $\mathrm{with}\left(\mathrm{liesymm}\right):$$\mathrm{setup}\left(x,y,z\right):$
 > $\mathrm{getcoeff}\left(ad\left(x\right)\right)$
 ${a}$ (1)
 > $\mathrm{getcoeff}\left(ad\left(x\right)+bd\left(x\right)\right)$
 ${a}{+}{b}$ (2)
 > $\mathrm{map}\left(\mathrm{getcoeff},\left[ad\left(x\right),bd\left(y\right)\right]\right)$
 $\left[{a}{,}{b}\right]$ (3)
 > $\mathrm{getcoeff}\left(\frac{1\left(d\left(x\right)\right)&^\left(d\left(y\right)\right)}{c}\right)$
 $\frac{{1}}{{c}}$ (4)
 > $\mathrm{getcoeff}\left(c\right)$
 ${c}$ (5)