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## semi decidability: there are two languages \$ L_1 \$ and \$ L_2 \$ so that \$ L_1 \$ and \$ L_2 \$ are recursive, but \$ L_1L_2 \$ is not recursive

True or false statements.

There are two languages $$L_1$$ Y $$L_2$$ such that $$L_1$$ Y $$L_2$$ are recurvise but $$L_1L_2$$ It is not recursive.

For me, the statements are true because I can describe a non-recursive one (I understood recursively enumerable) procedure of two recurring languages.

Test:

Suppose that $$L_1$$ Y $$L_2$$ they are recursive languages Then there is a Turing M1 machine that accepts any input in $$L_1$$ and reject any entry that is not in $$L_1$$. Similarly, there is a Turing M2 machine that accepts any input in $$L_2$$ and reject any entry that is not in $$L_2$$. To determine if $$w ∈ L_1L_2$$, we guess in a non-determining way where to divide the chain into $$w_1w_2$$and run M1 in $$w_1$$ and M2 in $$w_2$$. So, if w is in the language, M1 and M2 will eventually accept.

## Does it involve \$ L_1L_2 = L_2L_1 \$ \$ L_1 = L_2?

Leave $$L_1, L_2 subseteq Sigma ^ *$$ be two languages, where $$Sigma$$ It is a finite alphabet.

Make $$L_1L_2 = L_2L_1$$ to imply $$L_1 = L_2$$?

What if $$L_1$$ Y $$L_2$$ Are they regular languages?

Can you give counterexamples?

## demonstrating L1 * ∪ L2 * ⊆ (L1∪L2) *

x∈ L1 * ∪ L2 * ⇔ x∈ L1 * ∨ x ∈L2 * ⇔ x ∈ (L1) * ∨ x∈ (L2) * ⇔ x ∈L1 * ∪ L2 * ⇔ x∈ (L1∪L2) *

Is it enough to try it this way?


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# Algebraic Representation of Vectors - Problem 1 2,200 views

So we know two things about vectors. First of all, a vector has length and direction and that means you can translate it to any other point in the plane and it’s still the same vector. And second, we learnt that the components of a vector drawn from the origin to point p, are exactly the coordinates of point p.

So let’s use that here to find the component form of any vector QR. So I have QR drawn here from one point to another any two points. Let me observe that this horizontal length here would be x2 minus x1. And this vertical length would be y2 minus y1.

Now we translate this vector down to the origin so that its initial point is at the origin. These lengths are still going to be the same this is still going to minus x2 minus x1, y2 minus y1. And that means that the coordinates of this point would be x2 minus x1 and y2 minus y1. And that means these are the components of this vector, but they are the same vector. So the components are going to be x2 minus x1, y2 minus y1. So it’s really easy to come up with the components of a vector from one point to another.

Just literally subtract like components. Make sure you always subtract in the same order though. Let’s take a look at an example. It says determine the components of PQ and express it in the form with these corner brackets.

All right here’s P and q, so PQ, vector PQ, remember it starts at point Q and ends at point q. So you have to subtract from Q to p. So it will be 6 minus 2, 11 minus 9. And that’s . PQ here, 4 minus 4, 8 minus -1, 8 plus 1. It’s going to be . PQ here -8 minus -5, -8 plus 5 is -3. -7 minus -17 is -7, plus 17, 10. And here, 0 minus 6, -6, 5 minus 0, 5. Always remember you got to subtract the same order, so second minus first second minus first.

That’s how you find the components of any vector from one point to another.

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Equilibrium constant

1. Jun 4, 2005

~angel~

Samples of A (2.5 mol) and B (1.0 mol) are placed in a 5.0L container and the following reaction takes place:

2A(g) <--> B(g) + C(g)

At equilibrium, the concentration of A is 0.20M. What is the value of Kc?

Thanks. It's probably really simple but I can't seem to get it. I'm not that strong on this stuff.

Also, apparently the relationship between Kc and Kp is Kc = Kp. Why is that the case?

Thanks again.

2. Jun 4, 2005

Dr.Brain

$K_p=K_c (RT)^{dn}$

where $dn$= No. of moles of products- No. of moles of reactants

Kc = Kp because the number of moles of the gaseous componentson both sides are the same.

For your first query , what is law of mass action?....How is Kc related to the products and reactants?

Last edited by a moderator: Jun 5, 2005
3. Jun 4, 2005

GCT

$$Kc= \frac{[C]}{[A]^{2}}$$

$$Kc= \frac{[C_b+.25mol/5.0L][.25mol/5.0L]}{[.200M]^{2}}$$

$$Kc= \frac{[1.0mol/5.0L+.25mol/5.0L][.25mol/5.0L]}{[.200M]^{2}}$$

solve for $$Kc$$

4. Jun 5, 2005

siddharth

Initially, the concentration of all the species are
[A] = 2.5(mol) / 5.0(L) = c1 -I
= 1.0(mol) / 5.0(L) = c2 -II
[C] = 0

Now, let 'a' be the extent of the reaction

Concentration of all species at equilibirium are

[A]=c1(1-2a) -III
=c2+c1(a)
[C]=c1(a)

Now it is given that the concentration of [A] at equilibirium is 0.2 mol/L.
From III, c1(1-2a)=0.2
Now, the value of c1 is known.Therefore the value of a can be found.
From this the value of Kc=([C])/([A]^2) {The concentration of species at equilibirium}

GCT:
Instead of 0.25, shouldn't it be 0.75? Initially, number of moles of A are 2.5. Finally, the number of moles of A 0.2*5=1 . Therefore the number of moles of A reacted = 2.5-1=1.5 . By molar ratio, the number of moles of B and C formed are 0.75.

5. Jun 5, 2005

~angel~

Ok, I get it. Thanks guys

6. Jun 5, 2005

GCT

I'm not quite sure why you're trying to find the number of moles, when the equilibrium concentration of A is given. Anyways I neglected to use the appropriate initial concentration for A.

$$.25mol/5.0L=.5M,~mol~A~reacted~=~.5M-.20M=.3M$$

you'll need to divide by 2, to find the equilvalent moles of B and C

$$.3M~A~reacted(1~mol~B/2~mol~A)~=~.15M~B,C~formed$$

7. Jun 5, 2005

siddharth

It should be 2.5mol/5L=.5M
Anyway, finding the number of moles is the same thing. If 0.75 moles of B and C are formed, then the concentration of each is 0.75/5= 0.15 M. That's why in your initial post, instead of 0.25/5, it should be 0.75/5=0.15M. (As you wrote in the last post)

Last edited: Jun 5, 2005


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# Discrepancy in finding derivative of implicit equation

I was learning implicit differentiation and came across a discrepancy in finding $$\frac{\mathrm{d}y}{\mathrm{d}x}$$ of $$y=x+\frac{1}{y}$$: when differentiating without simplifying there is no $$x$$ in $$\frac{\mathrm{d}y}{\mathrm{d}x}$$, but after simplifying there is.

When I simplified the equation to $$y^2=xy+1$$ I got the right answer. Why did this happen?

I could only conclude it has to do something with how implicit functions are defined. I could not find anything on the internet.

• What's the discrepancy you are facing? Aug 6 '21 at 7:43
• @Anurag A when differentiating without simplifying you see there will be no x in dy/dx but after simplifying there is Aug 6 '21 at 7:46

By differentiating $$y=x+\frac{1}{y}$$ we find $$y'=1-\frac{y'}{y^2} \tag{1}$$ On the other hand, $$y^2=xy+1$$ gives $$2yy'=y+xy'\tag{2}$$ (1) and (2) are both valid equations.
Infact, by using $$x=y-\frac{1}{y}$$ in (2), we remove $$x$$ from (2) and we get (1): $$2yy'=y+xy'=y+\left(y-\frac{1}{y}\right)y'$$ that is $$yy'=y-\frac{y'}{y}$$ and after dividing by $$y$$ (recall that here $$y\not=0$$), $$y'=1-\frac{y'}{y^2}.$$ By reversing the process we may obtain (2) from (1).


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# (schematic) Using a micro controller to drive Opto-coupler > 555 timer > MOSFET

I'm trying to use a micro-controller to activate an opto-coupler connected to 555 timer which is connected to a MOSFET. The circuit will drive a solenoid which draws 2amps at 12v. I'm new to circuit design but have combined a few schematics I have found to come up with the one below. I think I have some resistor values wrong, and I'm unsure if the 555 can drive the MOSFET directly???

5ms pulse fom micro converted to .5 second pulse by monostable 555

R1 470 ohm 1/4 watt) (R2 2.2k 1/4 watt) (R3 470 ohm 1/4 watt) (R4 45k 1/4 watt) (R5 2.2k 1/4 watt(Micro controller to opto =5v) (555timer, MOSFET, Solenoid =12v)

(MOSFET=1PP096N03L) (555= TLC555) (opto=PC817)

• IPP096N03L
• Solenoid is 130ft. of 30AWG magnet wire. 13.4 Ohms

The solenoid in this circuit is coupled with another solenoid, connected to an identical circuit as this one, except their polarity is reversed. So one circuit magnetizes the core in one direction and the other circuit magnetizes it in the reverse. Since they are coupled, one circuit will create current in the other right? Does that cause any problems, or will it be shunted by the diode (BY229)?

Another thing I was thinking about is having an H bridge connected to the 555 and using the same solenoid, instead of having two. I'm trying to keep the cost down as I need a number of these circuits, and an H bridge design seems to be more expensive from what I can find. Thanks

1. The OPTO's emitter should be grounded, and the junction of its collector and R3 should be connected to 555 TRIGGER.

2. the formula for determining the 555's timeout is t = 1.1RC, so with 45K and 0.1$\mu$F in there, the the output pulse will last for 5 milliseconds, not 500.

3. The 555's output can be connected directly to the MOSFET gate, so
you can get rid of R1, R2, and the 1N4744.

4. Depending on the drive requirements of the MOSFET and how fast you want it to switch, you may have to switch to a bipolar 555.

5. The degree to which the solenoids will interact depends on how tightly they're coupled, and rather than try to to predict how they might interact, it would be better to determine that empirically. That is, try it and measure the result(s)

Please post the data sheets - or links to their locations - for the solenoid, the MOSFET, and the flyback diode.

UPDATE:

Just for fun, here's your circuit redrawn so that it works, and simulated in LTspice:

Here are links to the files you'll need to run the sim. Download the files to the same folder, left-click on the .asc file to bring up the schematic and then left-click on the running man icon to start it.

Enjoy! :)

• Thank you for your response, very helpful. I have added the datasheets. – Mike H. Apr 15 '16 at 4:38
• can you elaborate on your response number 4 please? The MOSFET will only be switched once every few seconds. – Mike H. Apr 15 '16 at 5:23
• @MikeH.: On number 4: If you use a CMOS 555, its output drive current is is about 5 to 10% that of a bipolar 555, so the MOSFET's gate capacitance will be charged/discharged more slowly by the CMOS 555, which will cause the MOSFET to switch more slowly and dissipate more power during its switching transitions than it would if driven by a bipolar 555. At your switching frequency that won't matter much, but the crisper switching of the solenoid's inductance might. – EM Fields Apr 15 '16 at 11:52
• I just downloaded LTspice , looks like it has a learning curve, is it possible for you to attach the file you made so I can play around with it? – Mike H. Apr 18 '16 at 5:53
• @MikeH: It's done. – EM Fields Apr 18 '16 at 6:41


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# How could world civilization lose the knowledge of a global event?

Every 1000 years, the earth, moon, and sun perfectly align to form a solar eclipse that is seen around the planet. During this time, the barrier separating this reality from the demon world break down, allowing demons to enter our world. During this short window, the demons cause death and destruction on a planet wide scale, toppling empires overnight. After 7 1/2 minutes, the eclipse ends and the barrier comes into place, forcing the demons back into their reality.

You would think that an event as wide scale and traumatic as this would be remembered by the world. There would be historical records of it happening by authors or writers, leaving behind articles or making references to it. This is how we have knowledge of ancient civilizations, like the Romans or the Greeks, who were great record keepers and left evidence of themselves for their future generations. However, civilizations that spring up after the eclipse lose knowledge of it. As generations pass, they lose knowledge of this event and no records exist of it happening, keeping it out of history books.

How could this be possible? Why would a world have no memory of an event as wide scale as this?

• I presume this isn't our Earth? Total Solar Eclipses: How Often Do They Occur (and Why)? It is a popular misconception that the phenomenon of a total eclipse of the sun is a rare occurrence. Quite the contrary. Approximately once every 18 months (on average) a total solar eclipse is visible from some place on the Earth’s surface.. space.com/…. – chasly - supports Monica Jul 1 '20 at 11:08
• In addition to the astronomical problem, students of history would notice multiple civilization collapses happening on a regular schedule, and try to find out why – Patricia Shanahan Jul 1 '20 at 11:17
• MiB agent #: "yo ya think our new recruits up to da job?" Agent \$: "this is their problem since the beginning, why should we have da clean up this human shit." – user6760 Jul 1 '20 at 12:36
• @chaslyfromUK It would, however, be basically impossible for a solar eclipse to be visible to the entire planet, given that it's night on half of it. Now if you're on a fictional world that has all of its inhabitable land on one side (say, nothing but ocean on the other side), it might be possible. (Though you might still have some sailors who would miss the eclipse because they were in the middle of the ocean...) – Darrel Hoffman Jul 1 '20 at 20:45
• it could be deliberately banned and erased from memory by a dictatorial government. – njzk2 Jul 1 '20 at 21:07

I suggest reading Isaac Asimov's 1941 novelette "Nightfall". It is very similar to your plot, depending on an eclipse that happens once every 2,000 years causing a global catastrophe. Civilization never lasts more than 2,000 years before collapsing. Each time, the eclipse fades into myth and religious texts before the cycle repeats.

The story follows a group of researchers who have combined religious texts, recent research into gravity, and evidence of previous civilizations having collapsed. They think they understand what goes wrong, and try to preserve knowledge across the collapse that is about to happen.

The story contains some ideas on how knowledge is lost, partly in the immediate reaction and partly through the passage of time.

I hope many who have not already read "Nightfall" will do so before reading the spoilers. Science Fiction Writers of America selected it as the best science fiction short story written before 1965.

Spoiler alert! Pass mouse over the following.

The key problem is that civilization has to start again at a very low level, after each period of chaos. The scientists are trying to preserve knowledge and carry it forward giving the next cycle a head start. There are several groups opposing the effort.

The Cultists have built a religion around interpretation of the Book of Revelations, which apparently attempts to describe the phenomenon, and preserves a memory of stars.

There is a skeptical newspaper reporter who has been writing columns debunking the claim that civilization is about to end.

That prediction has caused an economic and stock market collapse, making ordinary people very angry.

All this is in a cycle in which archeology and astronomy have progressed to an unusual degree so the scientists think they know what to expect. Highly relevant to the question, there is a twist that takes the place of the demons. The planet is in the middle of a large and active giant cluster. With all the suns set or eclipsed, they see tens of thousands of bright stars. The realization of the size and grandeur of the universe hits suddenly in minutes, not as something learned and understood over centuries. The Book of Revelations had tried to warn, but the scientists had thought it was exaggerated.

• It would be better if you put some of those ideas in the answer, so that one could retrieve them here without having to fetch and read the book. You can use use the spoiler tag in case you don't want to ruin the fun of reading the book – L.Dutch - Reinstate Monica Jul 1 '20 at 12:01
• First, I need to learn how to use the spoiler tag. I don't want to spoil one of the best science fiction stories of all time for those who have not already read it. – Patricia Shanahan Jul 1 '20 at 12:19
• I've edited your answer to include a dummy spoiler. This will show you how to do it. – chasly - supports Monica Jul 1 '20 at 12:30
• Another way to show spoilers is to use Caesar Cipher rot(13). This can easily be created and recovered using online resources. See here for a website .v2.cryptii.com/caesar/rot13 -- Instructions: Clear text from the left hand pane and type your plain-text. The code appears on the right. To decode, type the coded version on the left and plain-text appears on the right. This method is used routinely on Puzzling Stack Exchange to avoid giving away answers, but you'll probably have to explain it here. Puzzling is here: puzzling.stackexchange.com – chasly - supports Monica Jul 1 '20 at 12:40
• I have added spoilers, but strongly urge reading the story. I am familiar with Rot(13) from USENET, but did not know it was still being used. The spoiler tag seems simpler and needs less explanation. – Patricia Shanahan Jul 1 '20 at 16:19

The old tales aren't taken seriously because science wasn't and still isn't very advanced in your civilisation.

The Greeks kept records but they made little distinction between actuality, heroic tales and the activities of gods. The current scientists might say, "Oh well, something might have happened but maybe a volcano erupted and spread ash across the skies." Note that science would not be as advanced as ours because if they understood about orbits etc., scientists would be able to predict that an eclipse was going to happen.

There is a big frame challenge here.

Astronomically you are going to have a very hard time justifying this scenario. For the shadow of a moon to completely cover a planet would mean that the moon would have to be bigger than the planet!

In fact you would have to have your people live on the moon, then they could be eclipsed by the earth.

Actually we are eclipsed from seeing the Sun every night, that's why it goes dark.

For anyone with even a little scientific knowledge to believe your world you would have to design a very unusual solar system. I'm not even sure it is possible.

• Isaac Asimov's Nightfall does depend on having a very unusual solar system. – Patricia Shanahan Jul 1 '20 at 11:54
• "The Greeks kept records but they made little distinction between actuality, heroic tales and the activities of gods": I sincerely hope that this is intended as some sort of British humor. Most certainly Greek historians made a very clear distinction between real stuff, legendary stuff, and mythological stuff. We have zero problems reading, for example, Xenophons Anabasis and distinguishing between stuff which actually happened on one hand, and legendary and mythological stories on the other. (What they didn't really do is make a clear note of their sources for supposedly real stuff.) – AlexP Jul 1 '20 at 12:59
• @AlexP They really didn't start doing so until Herodotus, and even then despite trying to stick to just the facts Herodotus included a lot of weird stuff that was folkloric at best (e.g., re-interpreting the myth of Europa as a historical event) or because he was way too trusting of secondary sources he couldn't cross-verify himself (i.e., giant ants that dig gold). What an ancient Greek would consider plausible is very different from what a modern person would. – user2352714 Jul 1 '20 at 17:35
• @user2352714: And please who was keeping historical records before Herodotus? Speaking about the imprecision of historical records before Herodotus is like speaking about the unreliability of steam locomotives before Robert Stephenson. – AlexP Jul 1 '20 at 17:38
• @AlexP The point is Herodotus, who was the first person to care about the difference between folklore/hearsay and verifiable fact, only published Histories halfway through the traditionally defined Greek Classical Era (430 B.C. in a period typically defined as spanning 510-323ish B.C.), and even he published a lot of stuff that modern audiences would consider supernatural or unreliable due to the way he collected evidence. So for half the Classical Era at minimum most people weren't making much of a distinction between history and story. – user2352714 Jul 1 '20 at 23:13

The demons are so traumatic and out of the ordinary that many survivors convince themselves that it didn't happen. They blank out the actual demons, or think they are hallucinations stemming from the trauma. Possibly the magic of the demons adds to the effect.

Discussion of the demons might be regarded as poor taste, or even actually dangerous. "Speak of the devil" is proverbial even in this world, but in that one, superstitions might spring up very quickly.

Furthermore, the tales DO get passed down -- mostly orally because the demons have trashed your ways to pass them on in writing -- and get written off as obvious myths because the demons have clearly impossible powers and traits.

It's believed that something like this has happened, multiple times. In and around Mediterranean around 1177 BC in the span of a few short years the Minoan culture vanished, as did what's now believed to be several cultures further west disappearing entirely, known collectively as the Sea Peoples.

I this excellent video Eric Cline discusses in great detail the who, what, how, and why of it.

A similar complete loss of civilizations happened around 800 AD during the Classic Mayan Collapse when another complex network of civilizations stretching perhaps as far north as trading partners in North America completely disappeared.

In both of these cases, the collapse of civilization was so total that many of the worst-hit cultures are lost completely. No distinct memory of them remains. Even the cultures that "survived" didn't wholly.

## TL;DR What's the Point?

These total collapses of civilization had some things in common:

• A broadly distributed supply chain of essential goods and services
• Takeaway: as shock-tolerant as the system might be, no culture retained to itself all of the knowledge and tools to supply essential needs (food, manufacturing)
• Long term stresses on the supply chain (drought + famine + disease)
• One or a few shocks that might have, by themselves been survivable (earthquakes, eruptions)

• Large-scale migrations of peoples in the worst-hit areas to once less afflicted (or perceived to be less afflicted)
• Governments closing the borders to waves of refugees further stretching their already badly-stressed resources
• Mounting frustration by refugees and governments alike, leading to wars and riots

Eventually:

In a very short amount of time, several cultures were geographically stacked in a very small region "super melting pot". Language and culture evolved and blended into something unrecognizably new in an incredibly short time. Old culture, language, writing, and science all faded away.

## How to Apply to Your Situation

• The long term stress can be your eclipse. It's only 7 $$1 \over 2$$ minutes long, but no one knows when (or if) it will happen again. I imagine people groups worst hit by the eclipse will hear stories of peoples that fared better, and start picking up in large waves to emigrate. Further stress might be some other issues going on at the time: one of more epidemics, famine, drought.
• Short term stresses can be earthquakes, fires, volcanic eruptions. The people might freak thinking the eclipse is about to happen again.
• The global supply chain breaks down. You can't get your corn from Indiana or pharmaceuticals from Asia because the farms and factories have been abandoned. No one is minding the satellites because the monitoring stations in California are empty, and no one is available to fix a simple problem like a popped breaker, keeping telecommute away. Local governments don't know how to grow crops or compound medicines and possibly make things worse by trying because "it's easy" (see the Cultural Revolution)
• Eventually, things stabilize in super-densely packed urban centers containing dozens of cultures. They are ungovernable. The people work out amongst themselves new languages and ways of living with one another. Pragmatism and survival is prioritized over luxuries like education.
• Society becomes wholly illiterate. It will eventually re-invent writing.
• Even the generation born after the eclipse, not knowing anything else, will have trouble believing their parents' stories about how this all came to be. The story will be poorly kept, probably embellished greatly, if retained at all.
• The modern civilization might even mistake the demons for someone like the Sea Peoples, thinking that "oh, these people just demonized the invading culture as non-human" when they were really facing actual demons, and if there was no physical evidence like armor made for a non-human body nobody would know any different until it was too late. Archaeologists rationalize purported supernatural occurences like that all the time. – user2352714 Jul 1 '20 at 23:22
• Given that part of the Rosetta Stone was written in 196 BCE in hieroglyphics, the statement that the written Egyptian language disappeared shortly after the Bronze Age collapse a thousand years before is clearly wrong and obvious after a few seconds of thinking about it. The Egyptians wrote in hieroglyphics well after that point, right into the Roman era with some dating as recently as the 4th Century CE. – Keith Morrison Jul 2 '20 at 15:54
• Hmm. I wonder when society will again rely on extensive trading networks for fundamental resources and what kind of disruption might occur (cough cough) – user253751 Jul 2 '20 at 16:35
• @KeithMorrison removed those parts. – James McLellan Jul 2 '20 at 19:07

According to the Bible, several thousand years ago, the entire planet was under water. Most modern people refuse to believe this because it suggests a supernatural explanation.

Demons are a silly superstition that only scientifically backwards people could possibly believe. We're "enlightened" these days and don't accept such obvious nonsense.

Seriously, just do a little digging into how willing most people these days are to accept "supernatural" occurrences, and you'll have all the answer you need.

For bonus points, you probably have a religious fringe that does believe in what really happened (and gets laughed at by the rest of the "scientific" community). Boy, are they going to be smug when it turns out they were right all along. Assuming, of course, they don't all come down with an acute, demon-induced case of discorporation.

To expand on this a bit... (Please try to keep an open mind and try to avoid falling into politics; there is a relevant reason why I'm trying to explain this, which I will summarize at the end.)

What a lot of people don't understand about "science" is that your interpretation of the evidence is extremely dependent on your fundamental axioms (dogma). Creationist dogma is "the Bible is probably accurate". From this starting point, they are able to build a cohesive explanation of science, including plausible (to them) explanations for "evolution", the fossil record, and radiometric dating. This framework is capable of making useful predictions and even addresses some evidence that is problematic for uniformitarianists ("dark" matter, cosmic temporal horizon, certain inconsistencies in radiometric dating).

Uniformitarianist dogma, on the other hand, can be summed up as "God can't exist". This is not an evidence-based conjecture, it is an axiomatic assertion that uniformitarianists will not allow to be challenged. Because of this philosophy, they reject certain hypotheses that a creationist would take as self-evident, and are able to build their own, completely different, but still plausible (to them) view of the universe. This framework is also capable of making useful predictions and even addresses some evidence that is problematic for creationists (radiometric dating, apparent age of the universe).

Obviously, these can't both be correct, and yet good luck trying to convince either side that they're wrong. For that matter, try going after a real nut (flat earth, moon landing hoax, etc.) and see just how effectively they respond to evidence that is contrary to their belief.

The point, and the relevance, is that humans are really good at interpreting evidence to fit their preexisting dogma. For the purposes of this question, if humans somehow arrive at the dogma that demons can't exist, it doesn't matter what evidence there is to the contrary; they will find a way to explain it that doesn't violate their prior dogma. Their ability to do so is isn't conjecture, it is thoroughly demonstrated in real life. (It will help if there aren't a lot of written records immediately following "the event", but even that is no guarantee. Just look at how many cultures have a flood myth, and ask someone at random if they believe there was a global flood at some point in Earth's history. Now replace "flood" with "demon invasion".)

The secondary point and corollary is that you can take a mountain of inspiration for this from real life. Depending on where you want to take the story, it could easily become a social commentary for oppressed minorities, casting the "demon deniers" as the "scientific" majority and the "demon believers" as a marginalized, ostracized, and even persecuted minority. There is more than enough real world experience to draw on for inspiration.

• People don't believe in the supernatural because there are only anecdotal experiences of supernatural occurrences, and supposed supernatural occurrences have not left behind the supporting evidence that would exist had they really occurred (i.e., they did not actually happen). Neither of those apply to this event, which is witnessed by the entire planet, and leaves a tangible trail of evidence in its wake. I don't see how the whole world would wake up the next day and decide "that didn't happen", rather than concluding "that happened but I don't understand it". – Nuclear Hoagie Jul 1 '20 at 14:43
• For a scientifically-minded people, this event indisputably occurred and does not fit their worldview at all. It will be one of the best-documented and well-studied events in their history for precisely that reason. Science currently does not concern itself with the supernatural because there isn't any credible evidence that needs further study in the first place. When scientists make well-documented, reproducible findings that do not fit current theories, those become fields of intense study, rather than being ignored as "supernatural". – Nuclear Hoagie Jul 1 '20 at 14:50
• Wrong. People don't believe in the supernatural because modern "scientists" are dogmatically opposed to believing in the supernatural. There is a huge pile of evidence that can be interpreted as supporting the Bible if you are willing to accept such an explanation. Modern "science" is not. Conversely, look at any of the folks that deny climate change, or the moon landings, or that the Earth is round. Dogma is not swayed by evidence, no matter how persuasive. A dogmatic attitude that "demons don't exist", for which real life gives precedent, will quite suffice for the OP's purposes. – Matthew Jul 1 '20 at 14:55
• Science changes all the time to accommodate new evidence - see heliocentrism, the photoelectric effect, blackbody radiation, or general relativity. I don't see how you can compare a dogma like Flat Earthism that ignores or distorts evidence to a scientific dogma, which by definition, is based entirely on evidence and observation. The whole idea of science is predicated on the notion that if observation does not fit the theory, it's the theory that's wrong. You can certainly have a dogma of "demons don't exist", but it's not at all a scientific dogma if it ignores widespread evidence of demons. – Nuclear Hoagie Jul 1 '20 at 15:48
• What if, one day here on earth, early in the morning, fairies, goblins, kobolds, trolls and all the other assorted wild things came out to play, and wrecked havoc. Then, again, they disappeared the next night, never to be seen again. For many hundreds of years. Now, what would happen tomorrow? Would CNN downplay it as mass hallucination? After some months, it would simply be forgotten. – CGCampbell Jul 1 '20 at 23:10

Demons have a propensity of attacking more advanced civilizations.

Cities and even standalone buildings like farmsteads and temples would see a thorough destruction, while indigenous people who don't build any permanent structures may not see any demons at all. As a result, every civilization which has reached the level of cities (and literacy) would get wiped out. Its buildings would become ruins suitable only to quarry stone, and its remaining written records would be illegible for anyone from subsequent generations.

But humanity as a whole would survive just fine. New people would occupy old lands and, given time, raise their own cities and develop their own writing. But there would be no accounts of the past event - only vague legends and horror stories to spook naughty children.

If the new civilization has a chance of advancing into industrial age and develop archeology and study extinct languages, it would have a chance of figuring out what exactly had happened. But before that, the new civilization won't even know that there was any kind of global event.

Don't speak the name of the Devil:

Many cultures have prohibitions of speaking of or naming evil beings.It is believed that these beings are called up or their attention drawn to the speaker. Why not apply this on a global scale? After all, when faced with clear proof that demons exist, and another clear prohibition not to speak their names, who wouldn't assume some idiot said Chutulu's or Satan's name and unleashed hell on Earth? Given demons, it might even be true. Perhaps speaking of them causes lesser manifestations of these evil entities. Given these minor demons (possessions, etc) and the motivation of demons to silence anyone speaking of them, historians would be plagued by evil and curses, and the locals would NOT view historians kindly. What if that historian wrote the names down? Burn those books, and maybe the historian for good measure.

This race of humans might be very aggressive and opportunistic. They would constantly be warring amongst themselves for territory and domination, making accurate record keeping difficult as it is near impossible to stay out of the conflicts. This is how knowledge can easily be lost to future generations (the library of Alexandria springs to mind).

When a large scale demon attack happens, the current ruling faction will be most represented throughout the world, and therefore be hit hardest as they have the largest military presence. Other factions could easily rise up and seize this opportunity and overthrow them in their weakened state. If successful they could claim the victory as their own to establish dominance, denying the demons had anything to do with it, or even exist in the first place.

To further their dominance they might actually aggressively purge any scholars recording the demon attacks to prevent the truth coming out undermining their glorious victory. A civilisation as hell-bent (pun intended) on keeping the truth a secret to further their own agenda can be very successful in doing so, if they are willing to kill for it.

The ancient civilization blames it on the gods. As a result, a modern, secular society does not believe in it and believes it to be a myth much like the Biblical flood.


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# Trump and Big Data Analytics

Trump’s victory was many things. I think of it as a failure of big data analytics on many fronts. My personal theory is that there is more to it than meets the eye, but whatever does meet the eye seems to indicate a massive failure.

# Donald Trump – Vote Rigging

I am a conspiracy theory nut. So let me propose a conspiracy theory to explain the shocking Trump victory. It is vote rigging, but not the way you were thinking. Before proceeding, let me say that this is merely for fun. Don’t take it too seriously.

# Recap of Modern Physics

Since the post series on Particles and Interactions became a bit longer than I wanted, I thought I would break it up. Let’s start with a recap of modern physics that will you will need to understand the structure of matter.

# Interpretation of Special Relativity

When we looked at Quantum Mechanics, we talked about its various interpretations. The reason we have such interpretations, I said, QM, erişemediğimiz bir gerçekle ilgileniyor mu?,,en,duyusal ve algısal aygıtlarımız aracılığıyla,,en,Özel Görelilik, hareket halindeki makro nesnelerle ilgilidir,,en,ve böyle şeyleri hayal etmekte sorunumuz yok,,en,Öyleyse neden bir yoruma ihtiyacımız olsun ki,,en,Cevap ince bir cevaptır,,en,Özel görelilik,,en,Işık hızı,,en,Işık hızının sabit olması kulağa basit bir ifade gibi geliyor,,en,Ama daha fazlası var,,en,oldukça fazla,,en,Şimdi bu değişmezliğin gerçekte ne anlama geldiğine bakalım,,en,eğer bir yerde duruyorsan,,en,ve sağından sola doğru giden bir ışık huzmesi var,,en,hızı var,,en,Soldan sağa giden başka bir ışık ışınının da bir hızı vardır.,,en,Şimdiye kadar,,en,çok iyi,,en,Şimdi bir roket gemisindesin diyelim,,en,aşağıdaki şekilde gösterildiği gibi,,en,sağdan sola hareket etmek,,en, through our sensory and perceptual apparatuses. On the other hand, Special Relativity is about macro objects in motion, and we have no problem imagining such things. So why would we need to have an interpretation? The answer is a subtle one.

# Speed of Light

The speed of light being a constant sounds like a simple statement. But there is more to it, quite a bit more. Let’s look at what this constancy really means. At first glance, it says that if you are standing somewhere, and there is a ray of light going from your right to left, it has a speed c. And another ray of light going from left to right also has a speed c. So far, so good. Now let’s say you are in a rocket ship, as shown in the figure below, moving from right to left.

# Özel Görelilik Teorisi,,en,Einstein'ı ve özel göreliliği duyduğumuzda,,en,veya özel görelilik teorisi,,en,gerçek adı kullanmak,,en,ünlüleri düşünüyoruz,,en,E = mc ^ 2,,en,denklem,,en,ve ikiz paradoksu gibi garip şeyler,,en,Bunların hepsi doğru ve önemli olsa da,,en,SR'nin çözmeye çalıştığı sorun tamamen farklıdır,,en,Fizikte temel bir ilkeyi savunma girişimidir,,en,Kuantum Mekaniği - Yorumlar,,en,Kuantum Mekaniği hakkında ne zaman konuşsak,,en,ilk sorulardan biri,,en,Ya kedi,,en," Bu soru,,en,Kuantum Mekaniğinin yorumlarıyla ilgili,,en,Standart yorumlama,,en,ünlü Schrödinger'in kedisine götürür,,en,Schrodinger,,en,Kuantum Alan Teorisi,,en,Kuantum Mekaniği hakkındaki bu yazıda,,en

When we hear about Einstein and the special relativity (or the special theory of relativity, to use the real name), we think of the famous $E = mc^2$ equation, and weird things like the twin paradox. While those things are all true and important, the problem SR tries to solve is a completely different one. It is an attempt to defend a basic principle in physics.

# Quantum Mechanics – Interpretations

Whenever we talk about Quantum Mechanics, one of the first questions would be, “What about the cat?” This question, really, is about the interpretations of Quantum Mechanics. The standard interpretation, the so-called Copenhagen interpretation, leads to the famous Schrodinger’s cat.


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Probability

We know what probability means, but what is its formal definition? Letâ€™s use our intuition to define it. If there is no chance that an event will occur, then its probability of occurring should be 0. On the other extreme, if an event is certain to occur, then its probability of occurring should be 100%, or 1. Hence, our probability should be a number between 0 and 1, inclusive. But what kind of number? Suppose your favorite actor has a 1 in 3 chance of winning the Oscar for best actor. This can be measured by forming the fraction 1/3. Hence, a probability is a fraction where the top is the number of ways an event can occur and the bottom is the total number of possible events:

Example:
Flipping a coin

Whatâ€™s the probability of getting heads when flipping a coin?

There is only one way to get heads in a coin toss. Hence, the top of the probability fraction is 1. There are two possible results: heads or tails. Forming the probability fraction gives 1/2.

Example:
Tossing a die

Whatâ€™s the probability of getting a 3 when tossing a die?

A die (a cube) has six faces, numbered 1 through 6. There is only one way to get a 3. Hence, the top of the fraction is 1. There are 6 possible results: 1, 2, 3, 4, 5, and 6. Forming the probability fraction gives 1/6.

Example:
Drawing a card from a deck

Whatâ€™s the probability of getting a king when drawing a card from a deck of cards?

A deck of cards has four kings, so there are 4 ways to get a king. Hence, the top of the fraction is 4. There are 52 total cards in a deck. Forming the probability fraction gives 4/52, which reduces to 1/13. Hence, there is 1 chance in 13 of getting a king.

Example:
Drawing marbles from a bowl

Whatâ€™s the probability of drawing a blue marble from a bowl containing 4 red marbles, 5 blue marbles, and 5 green marbles?

There are five ways of drawing a blue marble. Hence, the top of the fraction is 5. There are 14 (= 4 + 5 + 5) possible results. Forming the probability fraction gives 5/14.

Example:
Drawing marbles from a bowl (second drawing)

Whatâ€™s the probability of drawing a red marble from the same bowl, given that the first marble drawn was blue and was not placed back in the bowl?

There are four ways of drawing a red marble. Hence, the top of the fraction is 4. Since the blue marble from the first drawing was not replaced, there are only 4 blue marbles remaining. Hence, there are 13 (= 4 + 4 + 5) possible results. Forming the probability fraction gives 4/13.

Consecutive Probabilities

Whatâ€™s the probability of getting heads twice in a row when flipping a coin twice? Previously we calculated the probability for the first flip to be 1/2. Since the second flip is not affected by the first (these are called independent events), its probability is also 1/2. Forming the product yields the probability of two heads in a row: $\frac{1}{2}Ã—$ $\frac{1}{2}$ = $\frac{1}{4}$ .

Whatâ€™s the probability of drawing a blue marble and then a red marble from a bowl containing 4 red marbles, 5 blue marbles, and 5 green marbles? (Assume that the marbles are not replaced after being selected.) As calculated before, there is a 5/14 likelihood of selecting a blue marble first and a 4/13 likelihood of selecting a red marble second. Forming the product yields the probability of a red marble immediately followed by a blue marble: $\frac{5}{14}Ã—$ $\frac{4}{13}$ = $\frac{20}{182}$ = $\frac{10}{91}$ .

These two examples can be generalized into the following rule for calculating consecutive probabilities:

To calculate consecutive probabilities, multiply the individual probabilities.

This rule applies to two, three, or any number of consecutive probabilities.

Either-Or Probabilities

Whatâ€™s the probability of getting either heads or tails when flipping a coin once? Since the only possible outcomes are heads or tails, we expect the probability to be 100%, or 1: $\frac{1}{2}$ + $\frac{1}{2}$ = 1. Note that the events heads and tails are independent. That is, if heads occurs, then tails cannot (and vice versa).

Whatâ€™s the probability of drawing a red marble or a green marble from a bowl containing 4 red marbles, 5 blue marbles, and 5 green marbles? There are 4 red marbles out of 14 total marbles. So the probability of selecting a red marble is 4/14 = 2/7. Similarly, the probability of selecting a green marble is 5/14. So the probability of selecting a red or green marble is $\frac{2}{7}$ + $\frac{5}{14}$ = $\frac{9}{14}$ . Note again that the events are independent. For instance, if a red marble is selected, then neither a blue marble nor a green marble is selected.

These two examples can be generalized into the following rule for calculating either-or probabilities:

To calculate either-or probabilities, add the individual probabilities (only if the events are independent).

The probabilities in the two immediately preceding examples can be calculated more naturally by adding up the events that occur and then dividing by the total number of possible events. For the coin example, we get 2 events (heads or tails) divided by the total number of possible events, 2 (heads and tails): 2/2 = 1. For the marble example, we get 9 (= 4 + 5) ways the event can occur divided by 14 (= 4 + 5 + 5) possible events: 9/14.

If itâ€™s more natural to calculate the either-or probabilities above by adding up the events that occur and then dividing by the total number of possible events, why did we introduce a second way of calculating the probabilities? Because in some cases, you may have to add the individual probabilities. For example, you may be given the individual probabilities of two independent events and be asked for the probability that either could occur. You now know to merely add their individual probabilities.


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# If X1, X2, ...., Xn are iid does that necessarily mean X1^2, X2^2, ..., Xn^2 are iids

• August 17th 2011, 12:42 AM
fallingleaves75
If X1, X2, ...., Xn are iid does that necessarily mean X1^2, X2^2, ..., Xn^2 are iids
If X1, X2, Xn are iids, this means that:

Var(X1+X2+...+Xn) = Var(X1) + Var(X2) +...+Var(Xn), right??

but does : Var(X1^2 + X2^2 + ... + Xn^2) = Var(X1^2) + Var(X2^2)+...+Var(Xn^2)?
• August 17th 2011, 01:57 AM
girdav
Re: If X1, X2, ...., Xn are iid does that necessarily mean X1^2, X2^2, ..., Xn^2 are
If $X_1,\ldots,X_n$ are independent then so are $X_1^2,\ldots,X_n^2$. To see that, take $a_1\leq b_1,\ldots,a_n\leq b_n$ and let $g:x\mapsto x^2$. We have $P\left(\bigcap_{j=1}^nX_j^2\in\left[a_j,b_j\right]\right) =P\left(\bigcap_{j=1}^ng(X_j)\in\left[a_j,b_j\right]\right) =P\left(\bigcap_{j=1}^nX_j\in g^{-1}(\left[a_j,b_j\right])\right),$
and since $X_1,\ldots,X_n$ are independent, we have $P\left(\bigcap_{j=1}^nX_j\in g^{-1}(\left[a_j,b_j\right])\right)=\prod_{j=1}^nP\left(X_j\in g^{-1}(\left[a_j,b_j\right])\right)$ and we are done.
We conclude that your formula is true if $Var(X_1^2)$ exists (but the problem doesn't come from the independence).


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# Which of the following satisfy differential equation $(1- \sin x) ^2 y''=\cos x$
$(a)\;y= \sec x +\tan x \\ (b)\;y= \sec x-\cos x \\ (c)\;y= \sin x- \cos x \\ (d)\;y= \sec x- \cos x$


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## LATEX-L@LISTSERV.UNI-HEIDELBERG.DE

 Options: Use Classic View Use Monospaced Font Show Text Part by Default Show All Mail Headers Topic: [<< First] [< Prev] [Next >] [Last >>]

 Suggestions sought Joseph Wright <[log in to unmask]> Sat, 13 Sep 2008 22:30:11 +0100 text/plain (27 lines) Hello all, I wonder if anyone can make a suggestion on how to solve a problem I'm finding rather tricky with my "keys" efforts. The issue is adding code to an existing macro. There are a number of scenarios, but I control the initial definition, and so I have three possibilities: 1) Definitions with a known argument pattern (one parameter #1). Here, I can "recover" the code out of the macro definition, add to in and re-cast the macro. This works for \def:Npn and \def:Npx 2) \def:Npn macros with unknown "p". Here, I can save the argument pattern to a toks, and the body to a second toks. The macro can then be redefined using the information in the two toks as needed. 3) \def:Npx macros with unknown "p". This is the problem. I can of course save the argument and body *as given*, but can't find a method to get the body toks expanded, nor a reliable way of recovering the data from the macro definition itself. Any ideas for (3) would be welcome. One extra point is that ideally I don't want to using \meaning + \scantokens, etc., as this looses catcode information. I'm hoping for a trick at the initial point of definition based on the fact that I can hold onto the various bits and pieces. -- Joseph Wright


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Is plotMDS (from limma package) a PCA (Principal Component Analysis) or a PCoA (Principal Coordinate Analysis)?
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2.5 years ago
harelarik ▴ 90

A. Is the plotMDS function (from the limma package) based on PCA (Principal Component Analysis) or a PCoA (Principal Coordinate Analysis)?

B. What the values on the plot represent? Is it root-mean-square average or root-mean-square deviation (see below)?

• In the limma paper published in Nucleic Acids Res at 2015 (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4402510/) it is stated that: "Distances between samples on the plot represent ‘leading fold change’, which is defined as the root-mean-square average* of the log-fold-changes for the genes best distinguishing each pair of samples. "

• In the R manual it is stated that: "The distance between each pair of samples (columns) is the root-mean-square deviation (Euclidean distance) for the top ‘top’ genes. Distances on the plot can be interpreted as _leading log2-fold-change_, meaning the typical (root-mean-square) log2-fold-change between the samples for the genes that distinguish those samples."

Thank you,

Arik

limma PCA plotMDS RNA-Seq R • 4.0k views
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This video may be helpful for a basic understanding on this topic.

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Sorry but the link does not work.

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Should work now.

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2.5 years ago
Gordon Smyth ★ 5.1k

A. The plotMDS documentation help(plotMDS) explains in detail that's doing multidimensional scaling (MDS) and PCoA is just a synonym for MDS. PCA is a special case of PCoA so in some cases plotMDS is also doing PCA. If the axis labels says "Principal component" then it is PCA (and PCoA). If the axis label says "leading logFC" then it's just PCoA.

B. Both expressions mean the same thing, it's the root-mean-square of the largest logFCs: sqrt(mean(logFC^2)).

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Thank you very much! A. How can I make the function generate PCA? So the axis label will show "Principal component".

Arik

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gene.selection="common"

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Thank you very much!

As common option will "select the same genes for all comparisons" when would you recommend to use it?

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I recommend you use the plotMDS default settings, unless your dataset is very large (hundreds or thousands of samples) in which case gene.selection="common" is much faster.

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Dear Dr. Smyth,

Thank you very much!

Arik