Making PoW usefull

[VectorChief]

. . .

http://mathworld.wolfram.com/images/equations/RiemannZetaFunction/NumberedEquation3.gif

The problem here is that conventional mathematics uses a flawed (i.e., partially anti-symmetric [i.e., one divided by infinity is equal to zero and one divided by zero is undefined]) numerical system. The Riemann hypothesis should be provable when using Earth’s numerical system with the system’s zero approached from the positive direction (which is of greater magnitude than its positive infinity) in the place of the traditional infinity of the conventional Riemann zeta function.

Code:

ℝ = {0⁻, −∞, …, −1, …, −⅟∞, −0⁻, −0⁺, ⅟∞, …, 1, …, ∞, 0⁺}

That's very insightful. Thanks!

So, if applied to my frequency spectrum example, the 0-frequency flat-line of non-existence at the bottom, which all non-zero frequencies can sample and thus refer to (as we do now), is actually a whole another reality-bubble of existence within itself (with its own 0 and infinity), which cascades this way further and further indefinitely. Thus the idea of becoming non-existent can only be experienced momentarily, as this state immediately brings forward the realization that you're suddenly everything there is, which then cools down towards a particular finite shape, so that the whole process can repeat itself again and again.

. . .

(Red colorization added.)

“Earth’s numerical system” (username18333) “loops around” at both its −0 (under conventional mathematics, zero) and its 0 (under conventional mathematics, undefined).

Yeah, I guess I see it. You can actually fit a whole another numerical sub-system in between the inner zeroes in the middle, in which they will serve as the outer ones (bigger than infinity). Or in other words, one singular zero in our current system suddenly "wakes up" as the whole numerical system in itself. That's what I was referring to in my example, only your zeroes have never gone to sleep .

[username18333]

. . .

Change is the only constant (except for the first three laws). Note, how not only does the 4th law contain paradoxical statement (change == constant), but it also happens to be the only rule with exceptions (of the first three laws), which in itself is paradoxical as it contains its opposite (rule != exception).

The first reflection is achieved by asking the paradoxical question: "is there me out there, which is not me?". This other "me" needs to be different in some regards, but similar in structure in order to constitute another "me" in a wholistic way. In other words, if the original was autonomous, the reflection would need to be autonomous as well. So it's not a perfect copy in a physical sense, but rather a meta-physically wholistic one with fundamental attributes of the original preserved, but different in all other aspects (made in the image of).

. . .

(Red colorization mine.)

Entropy as a Measure of the Multiplicity of a System

The probability of finding a system in a given state depends upon the multiplicity of that state. That is to say, it is proportional to the number of ways you can produce that state. Here a "state" is defined by some measurable property which would allow you to distinguish it from other states. In throwing a pair of dice, that measurable property is the sum of the number of dots facing up. The multiplicity for two dots showing is just one, because there is only one arrangement of the dice which will give that state. The multiplicity for seven dots showing is six, because there are six arrangements of the dice which will show a total of seven dots.

One way to define the quantity "entropy" is to do it in terms of the multiplicity.

Multiplicity = W
Entropy = S = k lnW

where k is Boltzmann's constant. This is Boltzmann's expression for entropy, and in fact S = klnW is carved onto his tombstone! The k is included as part of the historical definition of entropy and gives the units Joule/Kelvin in the SI system of units. The logarithm is used to make the defined entropy of reasonable size. It also gives the right kind of behavior for combining two systems. The entropy of the combined systems will be the sum of their entropies, but the multiplicity will be the product of their multiplicities. The fact that the logarithm of the product of two multiplicities is the sum of their individual logarithms gives the proper kind of combination of entropies. The multiplicity for ordinary collections of matter is inconveniently large, on the order of Avogadro's number, so using the logarithm of the multiplicity as entropy is convenient.

For a system of a large number of particles, like a mole of atoms, the most probable state will be overwhelmingly probable. You can with confidence expect that the system at equilibrium will be found in the state of highest multiplicity since fluctuations from that state will usually be too small to measure. As a large system approaches equilibrium, its multiplicity (entropy) tends to increase. This is a way of stating the second law of thermodynamics.

“Th[ose] other ‘[you][𝗌]’” (VectorChief) would be the unique arrangements (i.e., microstates) of “the dice” (Nave) that satisfy a definition of “[you]” (i.e., a given macrostate).

[VectorChief]

. . .

Change is the only constant (except for the first three laws). Note, how not only does the 4th law contain paradoxical statement (change == constant), but it also happens to be the only rule with exceptions (of the first three laws), which in itself is paradoxical as it contains its opposite (rule != exception).

The first reflection is achieved by asking the paradoxical question: "is there me out there, which is not me?". This other "me" needs to be different in some regards, but similar in structure in order to constitute another "me" in a wholistic way. In other words, if the original was autonomous, the reflection would need to be autonomous as well. So it's not a perfect copy in a physical sense, but rather a meta-physically wholistic one with fundamental attributes of the original preserved, but different in all other aspects (made in the image of).

. . .

(Red colorization mine.)

Entropy as a Measure of the Multiplicity of a System

The probability of finding a system in a given state depends upon the multiplicity of that state. That is to say, it is proportional to the number of ways you can produce that state. Here a "state" is defined by some measurable property which would allow you to distinguish it from other states. In throwing a pair of dice, that measurable property is the sum of the number of dots facing up. The multiplicity for two dots showing is just one, because there is only one arrangement of the dice which will give that state. The multiplicity for seven dots showing is six, because there are six arrangements of the dice which will show a total of seven dots.

http://hyperphysics.phy-astr.gsu.edu/hbase/therm/imgthe/dicemult.gif

One way to define the quantity "entropy" is to do it in terms of the multiplicity.

Multiplicity = W
Entropy = S = k lnW

where k is Boltzmann's constant. This is Boltzmann's expression for entropy, and in fact S = klnW is carved onto his tombstone! The k is included as part of the historical definition of entropy and gives the units Joule/Kelvin in the SI system of units. The logarithm is used to make the defined entropy of reasonable size. It also gives the right kind of behavior for combining two systems. The entropy of the combined systems will be the sum of their entropies, but the multiplicity will be the product of their multiplicities. The fact that the logarithm of the product of two multiplicities is the sum of their individual logarithms gives the proper kind of combination of entropies. The multiplicity for ordinary collections of matter is inconveniently large, on the order of Avogadro's number, so using the logarithm of the multiplicity as entropy is convenient.

For a system of a large number of particles, like a mole of atoms, the most probable state will be overwhelmingly probable. You can with confidence expect that the system at equilibrium will be found in the state of highest multiplicity since fluctuations from that state will usually be too small to measure. As a large system approaches equilibrium, its multiplicity (entropy) tends to increase. This is a way of stating the second law of thermodynamics.

“Th[ose] other ‘[you][𝗌]’” (VectorChief) would be the unique arrangements (i.e., microstates) of “the dice” (Nave) that satisfy a definition of “[you]” (i.e., a given macrostate).

Interesting.

So, for a given macrostate the microstates represent an inner world, while other macrostates represent an outer world. The inner world can also be seen as a multiplicity of choices one could make, which would then translate into the multitude of parallel realities one would simultaneously exist in (according to each outcome of a choice). The fact that complex numbers have "real" and "imaginary" parts might have something to do with the above, while "rational" and "transcendental" would correspond to behavioral patterns. Transcendental numbers are particularly interesting as they are finite in magnitude, but infinite in structure.

I wonder what happens when infinity is introduced into the picture? Would it mean that a singular infinite intelligence is equivalent to an infinite number of finite intelligences and the essence that connects all of the "copies" is the Identity or the presence of "I"? Everybody is "I", and that's the same "I" as in the Infinite, that's the ultimate sameness in our otherwise difference, the sameness that everyone can wake up to. Feel the stillness of "I" in you, for that is the absolute Is-ness of existence shining through. There is no opposite to being "I", as there is no opposite to existence.

These might be relevant to this discussion in general:
https://www.youtube.com/watch?v=5FELdBsixGg
https://www.youtube.com/watch?v=3yFiqdCjNMk
https://www.youtube.com/watch?v=PwSvV3RAH4s

[username18333]

. . .

Interesting.

So, for a given macrostate the microstates represent an inner world, while other macrostates represent an outer world. The inner world can also be seen as a multiplicity of choices one could make, which would then translate into the multitude of parallel realities one would simultaneously exist in (according to each outcome of a choice). The fact that complex numbers have "real" and "imaginary" parts might have something to do with the above, while "rational" and "transcendental" would correspond to behavioral patterns. Transcendental numbers are particularly interesting as they are finite in magnitude, but infinite in structure.

. . .

If a limit of a transcendental number should be converted to Earth’s numerical system, the both should equal each other without exception.

[VectorChief]

http://mathworld.wolfram.com/images/equations/e/NumberedEquation5.gif

If a limit of a transcendental number should be converted to Earth’s numerical system, the both should equal each other without exception.

You got me intrigued with Earth's numerical system. The name sounds just about right and it is hidden under layers upon layers of onion router, exactly like those zeroes loop around the infinity.

I will look into that.
Does it shake the foundations of math so badly that it gets banned if published in the open?

Also what would be the result of -1+1? Which of those inner zeroes?

[username18333]

If a limit of a transcendental number should be converted to Earth’s numerical system, the both should equal each other without exception.

You got me intrigued with Earth's numerical system. The name sounds just about right and it is hidden under layers upon layers of onion router, exactly like those zeroes loop around the infinity.

I will look into that.
Does it shake the foundations of math so badly that it gets banned if published in the open?

Also what would be the result of -1+1? Which of those inner zeroes?

Code:

[( −1 − (−1) = −1 + 1 = −0⁻ ) ∧ ( 1 + (−1) = 1 − 1 = −0⁺ )]  ⇔  [( −0 = −0⁻ ∧ −0⁺ ) ∧ ( −0⁻ < −0⁺ )]

In the first cases, one approaches nought from the negative direction. In the last cases, one approaches nought from the positive direction.

[VectorChief]

Also what would be the result of -1+1? Which of those inner zeroes?

Code:

[( −1 − (−1) = −1 + 1 = −0⁻ ) ∧ ( 1 + (−1) = 1 − 1 = −0⁺ )]  ⇔  [( −0 = −0⁻ ∧ −0⁺ ) ∧ ( −0⁻ < −0⁺ )]

In the first cases, one approaches nought from the positive direction. In the last cases, one approaches nought from the negative direction.

The answer to -1+1 that I held in mind was −0⁻ + −0⁺, which would not be reducible further in Earth's numerical system, but would represent zero under conventional mathematics. This makes addition commutative:

−0⁻ + −0⁺ = −0⁺ + −0⁻ = −0  <-  not on the axis, but rather an imaginary bridge?

When it comes to subtraction, we get the following:

−0⁻ - −0⁺ = −0⁻ + −0⁻ = −0⁻
−0⁺ - −0⁻ = −0⁺ + −0⁺ = −0⁺

Subtracting polarized nought from non-polarized nought would destroy the bridge resulting in polarized nought of opposite sign. Adding it would be equivalent to subtracting with sign inverted. Subtracting polarized nought from itself should reestablish the bridge resulting in non-polarized nought.

This in fact adds a three-state modifier to all non-zero numbers as well, something that conventional mathematics didn't have due to its zero being singular (thus adding/subtracting it had no effect). It should look like this:

1 + −0⁻ = 1⁻
1 + −0⁺ = 1⁺
1 + −0  = 1

Which reminds me of the (past, present, future) modifiers used in conjunction with numbers in the new type of numerology (described here, scroll all the way down to Numerology).

These interdimensional three have no energy of their own. They must have the other numbers to function. This also makes them catalytic. It also places them in a circle with the others instead of a line or a column. Some will understand this, and some will not. If you had the numbers one to nine in a column-going down a page that you were looking at, think of the other three as hovering above the column. This is the best we can do to explain something that's out of your normal 4D conception.

In your example: -1+1 = −0⁻ < −0⁺ = +1-1  =>  -1+1 < +1-1
Does Earth's addition commute? Is strict inequality due to implied modifiers?

PS: Also you might have mixed the terms positive and negative in your description.

[username18333]

Code:

[( 0 ⋅ (−1 ÷ 0) = 0 ⋅ (−0⁻) = −1 ) ∧ ( 0 ⋅ (1 ÷ 0) = 0 ⋅ (−0⁺) = 1 ) ∧ ( −1 < 1 )]  ⇒  ( 0 ⋅ (−0⁻) < 0 ⋅ (−0⁺) )  ⇒  ( −0⁻ < −0⁺ )

Code:

[( (0 ⋅ (−1)) ÷ 0 = 0⁻ ÷ 0 = −1 ) ∧ ( (0 ⋅ 1) ÷ 0 = 0⁺ ÷ 0 = 1 ) ∧ ( −1 < 1 )]  ⇒  ( 0⁻ ÷ 0 < 0⁺ ÷ 0 )  ⇒  ( 0⁻ < 0⁺ )

Direction of approach can be used to determine what might otherwise be indeterminate.

[VectorChief]

Code:

[( 0 ⋅ (−1 ÷ 0) = 0 ⋅ (−0⁻) = −1 ) ∧ ( 0 ⋅ (1 ÷ 0) = 0 ⋅ (−0⁺) = 1 ) ∧ ( −1 < 1 )]  ⇒  ( 0 ⋅ (−0⁻) < 0 ⋅ (−0⁺) )  ⇒  ( −0⁻ < −0⁺ )

Code:

[( (0 ⋅ (−1)) ÷ 0 = 0⁻ ÷ 0 = −1 ) ∧ ( (0 ⋅ 1) ÷ 0 = 0⁺ ÷ 0 = 1 ) ∧ ( −1 < 1 )]  ⇒  ( 0⁻ ÷ 0 < 0⁺ ÷ 0 )  ⇒  ( 0⁻ < 0⁺ )

Direction of approach can be used to determine what might otherwise be indeterminate.

I see. It's getting trickier down the road, as we are probably approaching some existential paradox here.

Singularity (one zero) leads to things being undefined (division by zero).
Duality (two zeroes) necessitates trinity (three zeroes) with neutral balance point between the two, or otherwise addition lacks commutativity (-1+1 != +1-1).

Maybe those "interdimensional" modifiers provide a way out of this, because placing two zeroes (or three for that matter) on a single numerical line would always necessitate something in between: (a+b)/2.

The way I look at Earth's numerical system (according to how you describe it) is more like a pair of gears or maybe even spirals placed next to each other, rather than a single numerical line. Maybe this approach doesn't require addition to be commutative, because the time arrow is always forward and every moment in existence is unique (see my signature) so there is no way to get the same perspective (number) by following two different paths (order of addition)?

I guess the following quotes by Kurt Vonnegut would be appropriate here:

"Everything is Nothing with a twist."

https://s-media-cache-ak0.pinimg.com/736x/0f/24/16/0f2416cafb8738b50ab815863360a0d0.jpg

"So it goes."

[username18333]

. . .

I see. It's getting trickier down the road, as we are probably approaching some existential paradox here.

Singularity (one zero) leads to things being undefined (division by zero).
Duality (two zeroes) necessitates trinity (three zeroes) with neutral balance point between the two, or otherwise addition lacks commutativity (-1+1 != +1-1).

Maybe those "interdimensional" modifiers provide a way out of this, because placing two zeroes (or three for that matter) on a single numerical line would always necessitate something in between: (a+b)/2.

The way I look at Earth's numerical system (according to how you describe it) is more like a pair of gears or maybe even spirals placed next to each other, rather than a single numerical line. Maybe this approach doesn't require addition to be commutative, because the time arrow is always forward and every moment in existence is unique (see my signature) so there is no way to get the same perspective (number) by following two different paths (order of addition)?

. . .

Limits may be taken from below

   (3)

or from above

   (4)

if the two are equal, then "the" limit is said to exist

   (5)

Code:

(−0⁺ + −0⁻) ÷ 2 = (⅟₀ + ⁻⅟₀) ÷ 2 = ⁽¹⁻¹⁾⁄₂₍₀₎ = −0⁺ ÷ 0 = −0⁺

Code:

(1 + 1) ÷ 2 = 2⁻ ÷ 2 = 1⁺

Code:

[( 1 + (−1 − 1) = 1 + −2 = 1 − 2 = −1⁺ ) ∧ ( (−1 − 1) + 1 = −2 + 1 = −1⁻ )]  ⇒  [( 3 + (1 − 2) = (3 + 1) − 2 = 4⁻ − 2 = 2⁺ ) ∧ ( 3 + (−2 + 1) = (3 + (−2)) + 1 = (3 − 2) + 1 = 1⁺ + 1 = 2⁺ )]  ⇒  ( 3 + (1 − 2) = 3 + (−2 + 1) )  ⇒  ( 1 − 2 = −2 + 1 )  ⇒  ( −1⁺ = −1⁻ )

Your “existential paradox” (VectorChief) arises from the notion that hyperreality’s nothing (here, nought) and everything (here, zero) “exist” (Stover) (i.e., are “elements of” the real).

[VectorChief]

Limits may be taken from below

http://mathworld.wolfram.com/images/equations/Limit/NumberedEquation3.gif    (3)

or from above

http://mathworld.wolfram.com/images/equations/Limit/NumberedEquation4.gif    (4)

if the two are equal, then "the" limit is said to exist

http://mathworld.wolfram.com/images/equations/Limit/NumberedEquation5.gif    (5)

So, the existence [of the limit] would be equivalent of saying "as above, so below", which is a realization that "now" is the only time and "here" is the only place [in all of existence].

The significance of this phrase is that it is believed to hold the key to all mysteries. All systems of magic are claimed to function by this formula. "'That which is above is the same as that which is below'...Macrocosmos is the same as microcosmos. The universe is the same as God, God is the same as man, man is the same as the cell, the cell is the same as the atom, the atom is the same as...and so on, ad infinitum."

Code:

... ⇒  ( −0⁻ < −0⁺ )

Code:

...  ⇒  ( 0⁻ < 0⁺ )

This would then be "as within, so without", which means that Earth's "nought" and Earth's "zero" are one and the same, but neither actually exists (thus strict inequality) because it's the hole in The Torus?

Gnosis is the concept... That knowledge of self is knowledge of the principle that originated you.

The quote is one of the seven principles of Hermeticism. Briefly, it tells us that things which appear to be very different have attributes that are actually quite similar. It also tells us that by studying one thing we can learn about something else.

Code:

...  ⇒  ( −1⁺ = −1⁻ )

Your “existential paradox” (VectorChief) arises from the notion that hyperreality’s nothing (here, nought) and everything (here, zero) “exist” (Stover) (i.e., are “elements of” the real).

So, if neither "nothing" nor "everything" exists, then only the "twist" is real. I guess "the Vortex of the Torus" would be an appropriate description of Earth's numerical system then.  

[username18333]

So, you're trying to reinvent limits? Nice, but you're some centuries late in that.

Yes, the singularity of 1/x can be "removed", even on the complex plane if you add an infinity point to make it a sphere, it becomes a simple reflection
Now Riemann's function is a wee bit more difficult

Code:

( ∀𝑥 𝑥 ∈ (−0⁺, 0⁺) )  ⇒  ( −0 ± 𝑥 = {−𝑥, 𝑥} ) ∧ ( 0 ± 𝑥 = {0⁻ + 𝑥, 0⁺ − 𝑥} )

It is not “an infinity point” (coric), for such a point would not accomodate conventional mathematics’ “hyperreal numbers.” Instead, it is an origin—one that has been missed sorely.

Earth’s set of all real numbers is, essentially, a Möbius strip fashioned from a line where one surface extends from 0⁻ to −0⁻, the other from −0⁺ to 0⁺, and all “edges” are retained.

[VectorChief]

So, you're trying to reinvent limits? Nice, but you're some centuries late in that.

Yes, the singularity of 1/x can be "removed", even on the complex plane if you add an infinity point to make it a sphere, it becomes a simple reflection
Now Riemann's function is a wee bit more difficult

Code:

( ∀𝑥 𝑥 ∈ (−0⁺, 0⁺) )  ⇒  ( −0 ± 𝑥 = {−𝑥, 𝑥} ) ∧ ( 0 ± 𝑥 = {0⁻ + 𝑥, 0⁺ − 𝑥} )

It is not “an infinity point” (coric), for such a point would not accomodate conventional mathematics’ “hyperreal numbers.” Instead, it is an origin—one that has been missed sorely.

Earth’s set of all real numbers is, essentially, a Möbius strip fashioned from a line where one surface extends from −0⁻ to  0⁻, the other from −0⁺ to 0⁺, and all “edges” are retained.

An interesting fact is that the Möbius strip and the Torus are topologically related.

Topologically, the Möbius strip can be defined as the square [0, 1] × [0, 1] with its top and bottom sides identified by the relation (x, 0) ~ (1 − x, 1) for 0 ≤ x ≤ 1, as in the diagram on the right.

A less used presentation of the Möbius strip is as the topological quotient of a torus.[7] A torus can be constructed as the square [0, 1] × [0, 1] with the edges identified as (0, y) ~ (1, y) (glue left to right) and (x, 0) ~ (x, 1) (glue bottom to top). If one then also identified (x, y) ~ (y, x), then one obtains the Möbius strip. The diagonal of the square (the points (x, x) where both coordinates agree) becomes the boundary of the Möbius strip, and carries an orbifold structure, which geometrically corresponds to "reflection" – geodesics (straight lines) in the Möbius strip reflect off the edge back into the strip.

The Möbius ladder might help demonstrate this a bit better.

http://upload.wikimedia.org/wikipedia/commons/7/71/Moebius-ladder-16-animated.svg

Which brings us to the Hopf fibration, that's where the fundamental existential "twist" is hidden.

http://upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Hopfkeyrings.jpg/250px-Hopfkeyrings.jpg

A couple more pages and we will finally be getting to the point of making PoW useful!

[username18333]

. . .

PoW is as useful as it is necessary. When PoW seems to be useless the question one should ask is, “Is it [the PoW schema] necessary?” The necessity of PoW, I have found, is directly proportionate to that of its coin. In light of this and the discussion here, it would follow that Bitcoin might not be necessary and that, because of that, its PoW is widely deemed unnecessary and, thus, “useless.”

[iamnotback]

Very interesting. The trouble is there is no way he can compete with someone using obsolete XBT mining equipment for space heating, where the effective cost of electricity becomes negative, regardless of the price of electricity.

Making POW useful requires nothing more a than changing the mindset. There are many situations where the heat produced has more value than the electricity consumed. Ever used electricity to produce heat? If the objective is to use electricity to produce heat, then POW mining of crypto currency becomes simply a way to reduce costs.

Once and for all, you are irrefutably rebutted:

“We will re-use the PoW as a heater…”

First, reusing work is always a smart idea; second, humans do need heat. Third, such an arrangement is highly conducive to Bitcoin’s goal (of achieving a Bittorrent-like resistance to coercive manipulation).

However, what is the effect on the MC and MR? Instead of an expected ($100 BTC) per x hashes, we’d move to ($100 BTC) + ($5 worth of heat) per x hashes. If $100 were previously being spent (at the hardware-efficient frontier), spending would be drawn upward toward $105 (eventually Bitcoin’s difficulty would adjust). Inefficient miners (those who didn’t use their waste heat) would be put out of business, but the total spending would still equal $100 (producing $95 BTC + $5 Heat).

The “Mining Heater” is just another increase in hardware-efficiency, resulting in a higher difficulty and an increase in ( energy_used / block ).

Proof-of-Work as Space Heaters Belies Economics of Specialization

Specialization enables economies-of-scale.

An example of an erroneous posited caveat[4] that proof-of-work mining resources would not become power-law distribution centralized due to the posited high electrical cost of dissipating heat in centralized mining farms coupled with the posited free electricity cost of using the “waste” heat of ASIC mining equipment as space heaters, is (in hindsight) incorrect because:

• Two-phase immersion cooling is 4000 times more efficient at removing heat from high-power density data centers[5], reducing the 30 - 50% electricity overhead to 1%[6].
• Electricity proximate to hydroelectric generation or subsidized electriciy costs approximately 50 - 75% less than the average electricity cost.
• Heating is rarely needed year-round, 24 hours daily, at full output. Not running mining hardware at full output continuously renders its purchase cost depreciation much less economic because the systemic hashrate is always increasing and (because) ASIC efficiency is always increasing[7]. The posited purchase of obsolete mining equipment[8] is incorrect because mining rewards from obsolete hardware are commensurately lower so the depreciation is not proportionally reduced. To make it profitable enough to be worthwhile (to justify the pita of jerry–rigging a space heater for equipment not designed for the purpose) requires running so many 10s or 100s of kWH of relatively much less efficient (i.e. obsolete) hardware generating more heat than can be typically utilized (unless infernos are in sufficient decentralized demand).
  

[4] https://blog.ethereum.org/2014/06/19/mining/
[5] http://www.allied-control.com/immersion-cooling
[6] http://www.allied-control.com/publications/Analysis_of_Large-Scale_Bitcoin_Mining_Operations.pdf#page=9
[7] https://www.reddit.com/r/Bitcoin/comments/335107/i_am_thinking_of_using_a_bitcoin_miner_to_heat_my/

[8]	https://bitcointalk.org/index.php?topic=918758.msg10109255#msg10109255 https://bitcointalk.org/index.php?topic=1527954.msg16816538#msg16816538