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Friday, December 12, 2014 ... /////

Energy conditions from entanglement-glue duality

One of the great conceptual insights in the research of quantum gravity of the recent 5 years or so was the realization that the geometric connection of two regions of the spacetime – according to a theory that respects the postulates of quantum mechanics and allows the spacetime to be curved, too – is physically equivalent to the entanglement between the degrees of freedom that lived in these previously separated regions of the spacetime geometry.

Folks like Mark Van Raamsdonk deserve to be credited for the original discovery of this broader concept. The Maldacena-Susskind ER-EPR correspondence is a particular, simple, well-defined example of the general concept. It claims that the Einstein-Rosen bridge – more generally, a non-traversable wormhole which is a pair of two black holes whose interiors are connected or identified – is equivalent to two perfectly entangled black holes. A high degree of entanglement is capable of changing the "most useful" spacetime topology used to describe the situation. But the spacetime topology itself isn't a well-defined or unique observable on the Hilbert space – it is emergent and one may only mention that it is "easier" to describe one situation with one topology than with another.

Because this realization is a tight link between the quantum information theory on one side; and spacetime geometry within a gravitating quantum theory on the other side (on both sides, we have a quantum theory: we just "visualize" their Hilbert space[s] in two geometrically distinct ways), we may construct dictionaries between various rules, conditions, and concepts in quantum information theory and in general relativity.

It has been previously asked what is the quantum informational interpretation of the "energy conditions" we know in general relativity. Some weeks ago, I wrote a short text on energy conditions on Quora. You may want to read it now.

They're conditions saying that the energy density (with or without plus minus pressure added, or something like that) can never be negative. If it could be negative, the spacetime would be unstable, superluminal spaceships would be allowed, special relativistic causality would be broken, or another tragedy that is ultimately equivalent would materialize.

Energy conditions are examples of conditions expressed in the general relativistic language. There are conditions in "non-gravitating" or "topology-preserving" quantum information theory, too. Subadditivity, positivity of relative entropy, and so on. It turns out that these seemingly completely different types of inequalities – energy conditions vs subadditivity or positivity of relative entropy – are actually equivalent to each other.

You may go through the geometric (general relativistic) visualization of things like the relative entropy and you will find how the inequalities are translated to the other language. There have been several papers like that in literature. A new one was added today:

Inviolable energy conditions from entanglement inequalities
by Mark Van Raamsdonk and 3 collaborators (Lashkari, Rabideau, and Sabella-Garnier). The paper has some overlap with the December 5th paper by Hiroši Ooguri plus three (Lin, Marcolli, Stoica).

Mark et al. use the AdS/CFT and Ryu-Takayanagi methods and achieve the following main results:

The strong subadditivity of entropy (a quantum information condition that always holds) translates to an averaged null energy condition:$\int_\gamma \dd s\,T_{\mu\nu}u^\mu u^\nu \geq 0$ where $\gamma$ is any spatial geodesic and $u^\mu$ is a null vector generating a light sheet
of $\gamma$ defined such that translation by $u^\mu$ produces an equal change in the spatial scale factor at all points. I think that the experts who have thought about energy conditions sufficiently deeply won't be surprised that it's (a version of) the null condition that seems to be equivalent to one of the most important inequalities in quantum information theory. It's the best energy condition.

The monotonicity of relative entropy gets translated to an upper bound on the minimum scale factor defined in some way.

The positivity of relative entropy gets translated to a weak energy condition $T_{\mu\nu} u^\mu u^\nu \geq 0$ verified by observers moving near the boundary at any speed. Just to be sure that people don't complain that the null energy condition was too privileged. ;-) But this condition is derived only for some static spacetimes constrained by additional conditions.

For other, AdS-like spacetimes, the positivity of relative entropy is translated to a Bekenstein-like inequality for some area, $\Delta A \leq K\cdot M$. You find those things summarized on page 4 of the paper. And one clearly needs to go through the details if they are relevant for her.

But these new dictionaries are cool, anyway. You could say that information – and I mean quantum information – is more fundamental than the spacetime geometry, and geometric properties of the spacetime (and even things like the positivity of energy) may be described as some "rough" macroscopic conditions constraining lots of information and the entanglement between various parts of it. I think that most people would adopt this perspective – the spacetime geometry is less fundamental.

Alternatively, you may say that some spacetime geometry is fundamental in a quantum theory of gravity but we may use these dictionaries to learn the details about the character of the relevant Hilbert spaces and constraints imposed by quantum gravity (on the geometry as well as the "matter" that lives upon it) that couldn't be seen from classical geometry itself.

At any rate, these results may be viewed as another deep step towards the unification of physics, a step that reduces the number of truly independent concepts that our contemporary physical image of the real world is built upon. And that's amazing, especially because just decades ago, people (and even string theorists) could think that the set of "irreducible concepts" needed to describe physics including gravity couldn't have been reduced more than it was. String theory – which is just a more "constructive" or "specific" name for any consistent theory of quantum gravity – is unifying gravity with features of other forces and phenomena in ways that couldn't have been even envisioned decades ago. And string theory forced the physicists to learn some truly clever "tricks of unification" employed by Nature that only became clear after decades of investigation.

My broader point is that the people who don't appreciate the deep insights that the string-theory-based research of quantum gravity has generated are uncultural, wild, primitive animals.

snail feedback (62) :

Hi Lubos ,
you wrote:
“They're conditions saying that the energy density (with or without plus minus pressure added, or something like that) can never be negative. If it could be negative, the spacetime would be unstable.”
If I understand well, as a consequence, for quantum (gravity) attraction we need TWO opposing pushing quantum actions within a tight time slot. I would call that: Dual Push gravity.
See perhaps:

“I am absolutely convinced that the actual power of the Bitcoin Foundation – plus a few exchanges and other companies involved in the Bitcoin fad – over a hypothetical Bitcoin-denominated economy would be vastly greater than the actual power of any real-world central bank over a fiat-currency-based economy. What they can do with the value of the Bitcoin if they want is amazing. They would be a government whose control over others would dwarf totalitarian governments”

That’s an elementary mistake made by people who don’t understand how a decentralized protocol like bitcoin works. Since bitcoin mining always (by protocol) mines the largest chain that represents the most prove of work, the bitcoin foundation can do whatever they like but if the majority of miners doesn’t agree and doesn’t update their software these changes have no effect.

Dear Marcel, the main thing that people like you with zero abilities to think as economists don't get is that the mining and similar technicalities are almost completely irrelevant for the economics of the currency.

Even today, the total number of Bitcoins in circulation increases just by a few percent per year - so this increase has an irrelevant impact on the economics of that currency. Meanwhile, during the same year, the value of one Bitcoin - or, almost equivalently, the combined value of all Bitcoins in the world - changes by something like 60% in a random direction. It is the latter - the value of one Bitcoin - and not the former - the total number of Bitcoins - that matters.

The value of the Bitcoin is driven purely by bureaucratic exercises, like making Microsoft accept it here and there which increases the "impression" that the Bitcoin could be used as a widespread currency; or, on the contrary, closure of some exchanges, banning the Bitcoin payments in one country or another, and so on.

Whoever controls these events controls the dynamics of the price of one Bitcoin which is the only thing that really matters here. And the people who control this thing may of course easily make profit. If I know that my friend in Microsoft will announce tomorrow that they accept Bitcoins, I know that the Bitcoin will jump at least 5% by tomorrow, and if I go long Bitcoin, I can earn tons of money. In the same way, one may profit from the opposite motion. Many of these positive and negative events are easily induced by the people who are big fish in this business.

So that's how it works. You make a completely false statement and when corrected start to talk about other things. I'm done.

Oh yeah, this will be a nice "Betthupfrl" for me to read later to decently end this week :-)

Just to be sure, Betthupfrl is some kind of "good night story" and not an insult or something :-D

The fact that Lubos mentioned exchanges in addition to the Bitcoin foundation should have told you he wasn't talking about software changes.

The Bitcoin foundation has no power over anything. They need to mine or buy their bitcoins just like you and me. What are we talking about?

Marcel, the Bitcoin foundation has assets worth $4.8 billion, larger than the value of all the Bitcoins in the world combined ($4.7 billion).

It's clear that the very fact that the wealth of this "foundation" is nonzero is linked to the idea that the Bitcoin as a concept has a merit or future. The assets of this foundation are clearly bound to be comparable to the total value of the Bitcoins in the world.

The foundation undoubtedly has interest for the Bitcoin fad not to fade away, and perhaps grow again, if possible. Individual members of that foundation surely have extra money stored in the Bitcoins, too. Your suggestion that it is a neutral player that doesn't affect anything is absolutely laughable.

It's much more accurate to say that the Federal Reserve has no assets and no power over anything - because its balance sheet says that whatever it possesses is canceled by its obligations. Moreover, it's regulated by rather transparent laws. We know who is the chairwoman. We can look into it.

We don't even know who is the guy or lady or guys who wrote the Bitcoin paper even though he or she or they own 5% of the economy, a percentage that would rival most kingdoms in feudalism if the total value of the Bitcoin were supposed to grow to the size of the world economy.

All your suggestions that there is something clearer, fairer, or more impartial about the Bitcoin relatively to the U.S. dollar are fairy-tales addressed to hopelessly retarded children.

The Bitcoin foundation has no power over anything. They need to mine or
buy their bitcoins just like you and me. What are we talking about?

https://de.wikipedia.org/wiki/Betthupferl

says that Betthupfrl are chocolate cookies. ;-) At any rate, I will think of Betthpfrl when someone says that the Czech language has almost no vowels. ;-)

Your mother is a great invention, at least to my dick.

Lubos...you owe it too yourself to read up on france's history with fiat currency....it isn't pretty.
There are many examples of fiat currencies gone bad...most economies' experiences with fiat currency do not end well-the temptation to mortgage future resources for consumption now is simply too strong for governments to resist.
the very condition you outline as being disastrous to the stability of a fiat currency exist today in the US. deficit spending, out of control printing and distribution go money...below market interests rates...the very things that you cite as dangers to the health of a fiat currency exist today in some of the largest economies in the world.
just sayin.

"But market = people, so the behavoir is not always rational" You can repeat this fact/definition a 100 times, For some reason Lubos doesn't seem capable of understanding the free market.

I guess the argument that Bitcoins can't be made out of thin air is by itself valid. Nevertheless, reading Lubos it looks that that is irrelevant for it to be decent money to pay for products in real life.

Looking at his explanation I now start to understand that it works a bit like a 'chain letter'. The more people get interested, the more its value, the more a bitcoin is worth (bigger piramid, paying to the top where the coins reside, the piramid growth at the bottom, where the loosers are). So if I would have bitcoins I would wait and see what happens with it's value. As long as the piramid grows, the bitcoin will go up in value. I would not spend it, since there is a chance that the value goes through the roof. Until, of course, the world understands it's concept and then it falls to pieces. It is that moment to be aware of and sell it just before that happens.

Thus, the problem with Bitcoins is with its stability in regard to its purchase power, not whether you can create it from thin air or not. Txs, Lubos, I didn't see it that way until today.

Dear David, they may be correlated in some way but I still think it is very important to view deficit spending and fiat currency as two independent things.

Even if France were using gold, it would have to borrow it and it would have a hard time to repay it. Every time the gold price doubles, the debt would double as well, and so on.

Hard to repay and increases the probability of a collapse but I don't think that this fact would prevent French socialists and not only socialists from reaching over 100% of GDP in debt.

The relative entropy of string theory to our observables is very high still. Too high. The relative entropy of QM or Relativity to our observables is low but of course not zero.

Dear Marcel, people are not always rational, and individual people are mostly irrational, but the markets as wholes are far more rational than you can even imagine.

All the prices of stocks, commodities, or anything else correctly take supply and demand and various potential benefits and risks into account very nicely to produce a price that is much more accurately reflecting the objective underlying reality than any other method to compute an objective underlying reality that you could propose.

Gold is a commodity-like copper, soy beans, oil, etc. It obviously has an intrinsic value- jewelry, industrial uses, etc. The advantages of gold (other than the fact that it looks nice and is easily transferable) is that the amount available is limited and there can never be excess supply, unlike oil and other commodities.
Fiat money (legal tender) is necessary for transactions within a country as no commodity can function as a method of exchange in a modern economy. The problem is that the amount of fiat money created by a government must reflect the true economic conditions within that country. If a government prints too much money (or creates too much debt) then people within that country, and people trading with it, will attempt to transfer their fiat money to possession of a commodity or other real assets.
Gold being the most efficient substitute for fiat money will always have a value which will increase relative to any particular currency, if that currency's true value declines. To avoid this the US now restricts gold and silver trading via COMEX to paper profits, as actual gold or silver is no longer delivered. Please refer to Dr. Jim Willie's interviews at goldenjackass.com.
You are correct, we must use a fiat currency. But using gold or silver as a required means of international exchange will prevent governments from printing too much money or issuing too much debt. As the US has already exceeded reasonable limits, the only solution is a huge devaluation of the dollar which will increase price of gold in dollars. Of course this will only apply to those who have gold located outside the US, as within the country gold will either be sieged or made illegal to transfer.

Exactly, demol. I am happy to see that these writings of mine weren't completely useless. I haven't understood these things "from the childhood" - it may be relatively recently when I could see them - and I think everyone should get these points.

The value of one Bitcoin depends on the popularity of the concept which is partly helped by the mining itself - people who mine some Bitcoins become interested. It's also being helped by bogus transactions, people who move Bitcoins from one wallet of theirs to another, and so on. People are trying.

The more people are involved, the greater the price of the Bitcoin - or the price of all Bitcoins in the world - will be. And that's what matters most, not the change of the number of Bitcoins by a few percent per year.

It *is* a pyramid scheme in this sense.

Any currency that is supposed to be introduced from scratch may be viewed as a pyramid scheme because it's always true that if something claims to be a universal method of payment, it has an important new application; that increases the demand; and the higher the demand it, the more overpriced this object will become.

This holds for the Bitcoins. It also holds for the banknotes. If there weren't conventions and laws that mostly guarantee that people will continue to pay by U.S. dollars, the pictures of founding fathers would be almost worthless. This huge increase of the value of one $100 banknote from$0.1 to $100 or what it is exactly is due to the status of the universal currency. If a new independent unpegged currency were spreading gradually, its price would be increasing in the same way. This increase holds for the Bitcoins or any new electronic currency, for banknotes of existing currencies, but also for gold. If someone wanted to turn gold back to the main payment commodity that is enough to pay everything, he would indeed see a gradual increase of the price of gold from today's already huge value to even higher values, as more and more people believe in the new application of gold (they suddenly need gold even to buy a hamburger, and so on), and therefore the demand for gold increases. Of course, it won't happen - most of the world would have to switch back to gold (to stabilize its price) and the gold price would have to multiply by 100+ to back up the common assets and cash that exists. But it did happen in the past. When countries were switching to currencies based on a metal, of course that it meant that the price of the metal increased. And because of inertia, we still see the elevated price of gold. The gold advocates focus on promoting the shiny metal, its stability etc. It's so great. But its real price, if it were not thought to be a form of currency in the post-gold-standard era, would be much lower than it is today. Most of the gold price is due to hype - the special new demand created by the pseudo-currency status. reader Luboš Motl said... And the number seventy-pants is highly queefing with Victorian demonetized peak oil through the overpopulated Oriental jungles, too, although not thirty-furcoat which would underwear the torture. reader Luboš Motl said... Dear Tavrik, gold is pretty for me as well but I can still see that the statements about "gold's being the most efficient one" are just irrational. If you remove all the subjective opinions about beauty, iridium or any other precious metals - and other things - is just as good or better. I agree that the monetary policy should be wise. What it exactly means is debatable but hyperinflation is not wise. My positive point is that the goal of good monetary policy is to make the value of money in the future as predictable as possible, and changes near the stability are good. You simply can't switch all the international trade to gold. Just the U.S. runs a trade deficit comparable to half a trillion dollars per year. It's 10,000 tons of gold. The New York U.S. Bullion Repository https://en.wikipedia.org/wiki/United_States_Bullion_Depository only has 7,000 tons of gold - and most of it doesn't even belong to the U.S. - so you may see that it would be enough for the U.S. to pay less than a year of its deficit-making trade, even if it consumed the gold it doesn't really possess but controls. And what about the next year? ;-) Of course, the U.S. may want to tell its suppliers: Let's agree that gold is actually 100 times more valuable than it is now, so our gold will be enough to cover almost 100 years of similar trade deficit years. Such an increase of gold price 100 times would be great for everyone who owns more gold than others. The problem is that it would be bad for the rest and they won't allow such a jump in gold price. Why should they? Why should China send its products for a 100 times lower price than yesterday? The gold price is dictated by the markets, not by wishful thinking, and any brutal discontinuity in someone's wealth or his need to pay is considered "unfair" and won't emerge peacefully. reader QsaTheory said... People find entanglement with separation very strange, but I have a simple analogy which works very well. Imagine a line going from one particle to another, then push or pull the line. So then both particles get affected without anything actually being transferred. Moreover, one particle well have some effect and the other the opposite, just like up and down in EPR. Maybe somebody has already came up with such analogy. reader davideisenstadt said... First of lubos, may I write that Im quite flattered that you would take the time to respond to my comment? :-) dude, you are one serious intellect. I only bring up (John Law and the) French experience with fiat currency to point out that the underlying assumptions crucial to a healthy fiat currency that you outline above are, all to often, not met by governments. One of the most important functions of currency and modern credit markets is the inter temporal transfer and focusing of resources. When a government has a vested interest in paying back current debt in the future with worthless currency, it finds itself at war with savers and private sources of capital. Thats what is going on in the US now. Nominal rates of return on private capital are near to zero; real rates of return for dollar denominated securities are often negative. Inflation is understated by government generated statistics here, and the rate charged to rent money (interest, as I know you know) is kept artificially low. So, net borrowers and recipients of transfer payments are beneficiaries and savers are victims. Not a sustainable way to run an economy. Of course in the last hundred years or so, we've only seen a handful or so economies crash and burn from these policies...Im sure it couldn't happen in a country like the US, or Germany, or Argentina, or Zimbabwe, or Venezuela ....no not too great of a chance, eh? reader Gordon said... I loved his first paper and could mostly understand it. For this one---not much hope, I suspect, unless I actually learn more basics. reader Luboš Motl said... Dear John, too much debt is bad. But debt itself is a cornerstone of the economy in the modern sense. To demonize any debt means to throw the baby out with the bath water. There are good reasons why some people and company may build or consume something before they earn for that. If there are reasons to expect that they will be able to pay for it in the future, why not lend them at an interest? The activity and products may then be safely used even today. This insight made the economy much more efficient when people began to use it. One may call any economics based on debt a Ponzi scheme but it is a big and stable Ponzi scheme that may continue for centuries because, indeed, it has swallowed almost all (or at least one-half?) people on Earth and their activity. There's nothing wrong with debt. It's a relationship between two parties that allowed one to consume/purchase/invest something he didn't have at this moment, but probably will have in the future. It allowed the producers of this "early consumed/purchased thing" to sell their product although they may be forced to liquidate it at a much lower or no price otherwise. And it will give the lender the interest (which was unfortunately close to zero, but it's because the borrowing desires are not high). So *everyone* wins. When loans break down, it's bad, but there are tons of loans that are reasonably expected to go OK and they surely make everyone more prosperous. reader Uncle Al said... "Local soccer Mom discovers warp drive using a simple kitchen microwave trick." Will she pass peer review? reader Marcel van Velzen said... Lubos, I don’t think rational/irrational (or objective underlying reality) can be properly defined in a market and it is useless to talk about it. Economics assumes demand “as is”, that much I know of economics (not much, that is true) That the fluctuation caused by a group is usually lower than that of individuals shouldn’t surprise us. reader kashyap vasavada said... Nice informative blog Lubos! I take it that a real particle cannot pass through the neck of a wormhole even within the time allocated by the uncertainty principle. Since there are correlations between the two sides, perhaps a virtual particle (field effects) can pass through.I realize this is very vague, poor statement. But I would like to understand this point little better. reader Luboš Motl said... Dear David, first, there is nothing market-unfriendly about the Federal Reserve's purchase of certain things if it is done at a market price. The Federal Reserve is a central bank and central banks may hold diverse kinds of assets that contribute to their balance sheets. Other banks, like the Czech National Bank, own lots of foreign currencies, of course. But bonds or gold or other "financial" assets are equally good. The Federal Reserve's buying of up to 70% of its treasuries (it's been a law since 2010, I guess, an upper bound) is just a way for its balance sheet to grow. But it doesn't affect the economy. Just try to lend to yourself. It doesn't do anything, does it? You lose as much as you earn. If the balance sheet gets much higher, it's not a problem by itself. It is money that is cancelled against the same money with the opposite sign. The balance sheet is *not* debt in any invariant sense. It is just one term contributing to debt and it's silly to overlook the opposite-sign term that exactly cancels it. I don't know what to do with claims like "CPI is not an index". CPI is the consumer price index. It's calculate according to some methodology. You may prefer to define a different inflation index according to another methodoloy, David Price Index DPI, not to be confused with dots per inch. But if you approach all these issues rationally, you will be able to find an expected relationship between CPI and DPI - DPI may be simply higher by 1 percentage point, for example. So any information about CPI may be translated to some information about DPI, within some uncertainties that are largely unavoidable, anyway. So if there's some CPI and the Federal Reserve offers you treasuries, you may calculate the expected DPI, whatever you need to decide whether the offer is good, and you will demand some nominal interest rate on the bond. That's surely what all the lenders with billions of dollars are trying to do as carefully as they can. Comments that some CPI is misleading are completely irrelevant. The people who have billions won't be easily misled. They know what the actual inflation rate they think to describe the reality as seen from their viewpoint is, anyway. reader Guest said... Hawking's proof of the Area Theorem relies on the Null Energy Condition (NEC). See for example Wald's GR textbook, theorem 12.2.6. What about the other energy conditions, however (weak, strong, dominant, etc.)? Any one of those implies the NEC, but imposes further constraints. If we impose any of those, can we obtain a stronger constraint on black hole entropy than the Area Theorem? reader Luboš Motl said... Dear Tavrik, the U.S. has a large absolute deficit because it's the #1 - well, perhaps now #2, according to some - world's economy. But the relevant quantity saying whether it's extreme or not is the ratio of the trade deficit to the GDP. And this ratio is summarized e.g. in this table https://en.wikipedia.org/wiki/List_of_countries_by_current_account_balance_as_a_percentage_of_GDP U.S. was below 3% in 2013, making it completely average country. Mozambique, Liberia, and other countries have deficits of 40% of GDP. ;-) If you don't divide it by the GDP, see https://en.wikipedia.org/wiki/List_of_sovereign_states_by_current_account_balance you see that the U.K.has 1/5 of the trade deficit of the U.S. - indicating it's about the same per capita, and India, France, Turkey are above$100 billion, too.

The U.S. trade deficit will be shrinking with the fracking, I would bet.

While I am sympathetic to your idea of balanced trade sheets, it has conceptual problems. First, it's hard to even calculate it. The current account contains some entries beyond the trade balance, and it's questionable what is the right way to count all the dividends from companies owned by other companies, and taxes from those, and so on.It may be complicated to even define what condition you exactly demand.

Second, the strict condition you suggest may be highly counterproductive because different countries may have different eras in which it's natural that they borrow, or they repay, or lend to others from the overproduction etc. It's just like the people. At different ages, people are more or less likely to borrow, repay, or be investors. Countries may have their "periods", too. That doesn't mean that I think that most of these imbalances are actually justified. But they *might* have very good reasons.

Ukraine will clearly need to borrow a lot in coming years if it wants to remain a civilized country.

Lubos,
I have no problem with debt. As you point out, it's what makes the system work. The problem is when the controls start to break down and everyone starts gaming the system, to where even the regulators are corrupted and then it's financial Ebola. The system is bleeding out its orifices. Basically what we have now is the financial tail wagging the dog of the whole economy, to the degree the bubble of financial assets is far greater than the economy on which it is based, let alone necessary for the economy to function. What the bankers, in all their short-sighted gamesmanship, lose sight of, is that if they do completely blow up the financial system, they are superfluous and that's when the security and military parts of the system take over and start using convenient bankers as pinatas. They just don't think it can happen to them, but we are pretty high up and its a long ways down.
It's nothing personal, as these sorts of people are my relatives, but even wasps are going to need a plan B.
Everything in moderation. Or else. You don't want to hit the end of Mother Nature's chain at a dead run and we are going there.

Thanks Cynholt,
The whole mortality business has been a little too friendly lately. I'm even a widower now, though that's a long story. So I'm getting a little too objective about where the world is headed. I try avoiding the medical system, but now even my daughter looks like she is getting into it. Public health and political science major. Probably be administrative.
I'm just sticking with the horses and the farm.

Dear John, concerning leverage and derivatives, well, I think it is an unavoidable sign of the modern economy.

They don't imply any collapse. When all the things are arranged more accurately, one may afford more of the "tail wagging the dog" i.e. a higher ratio of the "most inclusive money supply" and the underlying assets.

In the predictions of doom, there is a lot of mysterious conspiracy theories. I think that at the end, it is rather clear who is at risk and who goes bust in certain circumstances. In the 2008 crisis, there was a crash in derivatives, but you could still see what were the bad underlying assets - the bad mortgages. Someone or many banks etc. just made a bet on something whose value wasn't being evaluated well.

But that doesn't mean that everything is evaluated totally badly. And I think that in fact people are getting better in evaluating the assets etc. which allows the higher ratio of derivatives-vs-underlying-assets than what we had in the past.

The economy is changing and it is improving in many ways so certain things that would be considered super dangerous in the past ceased to be dangerous today.

At the end, I think that it's right to blame the 2008 crisis on the government-like interventions into the underlying assets, the bad mortgages, like giving the mortgages to lots of people who just shouldn't be able to afford it in normal circumstances. So I do think that it's the government and its activism to be blamed, not some fancy financial products.

But even if some crisis were started primarily by the markets, you know, crises don't mean the end of the world. It's healthy to see crises sometimes in something. People see the weak parts of the system and they may improve it. Some usually weaker companies may be killed and this destruction has some creative powers, too. So I think it's just wrong to try to build a society which will never see another crisis again or a society where everything is 100% safe. It's impossible and even if it were possible, it is counterproductive.

LM

Except that this line - I suppose you mean a long rigid rod - does represent an action at a distance!

In the real world which respects relativity, rods are never rigid. If you tilt the rod at one side, the change of the angle spreads at most by the speed of light. In reality, it spreads by a much lower speed, the speed of sound.

Entanglement, on the other hand, guarantees the (perfect, in some cases) correlation on both sides in the real world, without an action at a distance that relativity prohibits in the real world, so entanglement definitely *cannot* be explained by a rigid rod!

That's really the whole point of the discussions started by Einstein.

Dear Kashyap, it is vague but it is sort of interesting and potentially well-defined, too.

I think it's right to say that the Hawking radiation is able to circumvent the causal restrictions because it uses the uncertainty principle - the Hawking radiation is tunneling of some sort.

And if the two black holes are entangled, this might imply some marginal ability to get something through the wormhole. But there will be something that prevents you from making this transfer macroscopic because whenever it's macroscopic, classical GR should be restored and it bans the penetration through the throat. ;-)

So I am not sure whether the correlations on the two sides of the throat may ever be interpreted as something's getting through.

When I studied (equilibrium) chemical thermodynamics in the 1950s it seem to me to be a perfect structure. “Thermo” could not be modified or altered in any way; it was complete and perfect. I still think so.
At the end of the fifties I measured heat contents and the heats of formation of a series of gold/nickel alloys and was able to calculate the phase diagram of that alloy system. That was neat!
If spacetime really is not fundamental, perhaps it is not accidental that neither time nor space enter into the equations of thermodynamics.

I don't know - my view is that dominant and strong conditions aren't really true in the real world in general, so I am not too interested in them.

Not sure what is the answer to your question. But you realize that it is a question about more or less semiclassical GR, so it's a question at a simpler, less abstract level than this paper?

There is no entanglement in your question, is there?

Good memories, Gene.

Thermodynamics is a great principled science - Einstein called it this way and packed in the same group with relativity because it is not constructing a theory from building blocks but deriving consequences of postulates/principles.

Statistical physics ultimately gave us a "constructive" derivation of the laws. Thermodynamics was reduced to the k=0 limit of statistical physics, when there are infinitely many degrees of freedom.

But the fun in this GR-related stuff - already started by Bekenstein and Hawking and later by Ted Jacobson, before these stringy guys started to make it geometrically clever and diverse - is that the curved spacetime geometry in GR may be interpreted as thermodynamics. So entropy are areas and temperatures are accelerations near the horizon, in some units, and there are similar maps.

A quantum system that admits gravity is an example of a (quantum) statistical mechanics system with some Hilbert space that doesn't have a clear geometry -much like a few atoms can't be clearly assigned temperature. But when you take the objects' size large and the thermodynamic limit, it's exactly the same limit in which the (preferred) classical geometry becomes clear and well-defined.

But one shouldn't take the classical geometry for granted as a starting point - the new insights pretty much exactly say that it would be as silly as starting to to assign temperature to each single atom and then to fix it (because the temperature is only defined statistically).

The spacetime geometry and even topology is much like the description using temperatures - it isn't quite complete. It's just describing certain statistical properties of the system such as the entropy (or entropy density) and temperature at various points. But one may still say lots of things about the spacetime geometry, its evolution and inequalities it obeys - much like one may derive lots of thermodynamic laws from statistical physics even without paying attention to the detailed chemical/microscopic structure.

you make salient points.
While household debt is down, net household wealth hasn't increased in about twenty years or so. Not only that, but by historical standards, like the time span of the graph you so kindly provided, it is still really high. we are still mortaged to the hilt (almost)
I would also point out that here in the states, a large chunk of accumulated wealth is held by those who dont have billions, or even millions, it is held by those who dont have too many alternatives to dollar denominated securities...these people are getting a rough ride now, unlike the treatment they received in the mid 80's.
As for the CPI, ( I would prefer the DPI...a static, rational index that measures the same thing from month to month)and its value as an indicator of real inflation, it is important to note that it is used by the federal government to adjust its inflation protected bonds, to index cost of living increases and so on, so to the extent that it understates inflation, it is used to screw the little guy, so to speak. these people can estimate the real rates of inflation all too accurately, they just have no power to do anything about it, except to substitute inferior goods over time for those that they used to consume...thats how we get from beef to chicken to spam to twigs and bark.
We will all live to get our social security checks each month...its just that milk will cost $20/gal. Also, here in the states, our tax brackets arent adjusted for inflation in any way, thus govt receipts track prices, while its obligations are slowly eroded by inflation. it is precisely this inter-temporal shell game that governments love to play. As wimpy would say...I will gladly pay you tuesday for a burger today. (in our case the burger will be paid for with worthless currency) The reason long term rates are so low is that the federal government is lending money out at zero percent. now, if you can rent it for nothing, and then sublease it at 4 percent, you've got a good business model... what happens when the feds stop this? 1)Interest rates will increase, and of course so will the costs of rolling over our debt. One of the tricks we here have gotten away with using is to shorten the overall length of the debt instruments we sell... this lowers our rate, but reduces the stability of the costs associated with servicing this debt. a one percent rise in interest rates will double our debt service costs... 2) real estate prices will plummet. Since a huge part of this country's perceived wealth is tied up in real estate...people who see their main asset lose, say 30% of its value in a year or so will not be so keen to keep consuming, not a good thing. BTW the only reason long term mortgages are available here in the States today is that the federal government guarantees them (something like over 90% of them) not that it has the resources to back them up at all...its all smoke and mirrors. Its not a true functioning market place as you or I would understand it. Its very enjoyable to pick your brains regarding monetary policy... thanks for sharing your thoughts with me reader Marcel van Velzen said... Dear demol, you got a point but you are grossly exaggerating one aspect of bitcoin. It is true that bitcoin gives advantage to early adapters but things are not that simple. As long as you store bitcoins you cannot enjoy the things you could have bought with them. Everyday there is a chance you die and then the value of your bitcoins collapse to zero. Also it is possible that the bitcoins go to a high value, you don't sell, and then take forever to return to that value again (bitcoins went from$32 to $2 and it took more than a year to return to$32). You also
have to decide, will I buy stuff with my bitcoins when I need them or
am I going to sell all my bitcoins for fiat at once. There is always a
chance a flaw is found or a better coins wins the race. All these things
are happening right now, that's why the price fluctuates. However, it's
a good thing. It can be observed when long stored bitcoins are being
exchanged and in this way bitcoins are distributed more evenly among the
population. All these things are true for gold also but bitcoin is also
programmable currency with special and useful properties. I sent
bitcoins (money) to my son, 10000 km away and it took less than 2
seconds. I have never donated so much money in my life because it's so
easy. A lot of innovation is taking place at the moment (400 million in
venture capital this year): multisignature, HD-wallets (only 12/24 words
are needed to restore all your money), safe hardware wallets, escrow,
games, lottery, side-chains, decentralized marketplace, secretly paying
secret agents by governments, crowd funding, ....

I don't know where all this hatred towards bitcoin comes from. Why does
Lubos call the inventor of bitcoin a jerk? It is just people trying to
create a better form of money. Maybe they will succeed, maybe they will
partially succeed, maybe they will fail maybe an improved coin will take
over. But most of all: IT IS FUN!

Is there any discussion in HEP that the Hilbert Spaces maybe physically real instead of a mathematical abstraction?

It's rather remarkable to accuse me of attempting to circumvent the laws of logic. I'm not sure what you're misunderstanding about my post. Perhaps you've misunderstood, what "allowed to vary freely" means. It means that if people value gold to a certain extent as jewelry etc. *that* is what determines how many apples, oranges, pigs, chickens, houses, computers, etc. a paper that is a promise to a certain amount of gold can buy. The Gold Standard did *not* artificially raise the price of gold in goods. And it was not imposed by the government. Your belief that it was, and your repeating of the "price fixing" meme popular among anti-Gold commentators, perhaps explains a great deal of your opposition to the idea. Unfortunately it's also just totally wrong.

You'll note, however, that under the Gold Standard, prices were more stable over the long term than they have been under fiat currency, even "responsibly managed" fiat currency-see my comment about the historical CPI above. The reason for this is because if the prices deflate too much, people mine more gold, increasing the money supply, and if they mine too much, mining ceases to be profitable, halting the process of inflation. This is, in fact, not optimal, but this is in fact the reason why prices in gold were stable over the long term under a gold standard, not because of "stickers." Nor did it have anything to do with "intrinsic value" of gold. I can't have argued that intrinsic value of gold was a desirable property, *I don't believe intrinsic value exists*. Because there is *no such thing as intrinsic value*. We had an automatically, self-regulated supply of outside money.

Unfortunately we did not have an automatically self-regulated supply of inside money. But we could have, and Canada did when the US did not. And as a result the financial crises common to the US in the 19th century, were unheard of in Canada contemporaneously. Why? Again: Free Banking.

As best I can understand it, Lubos, this all makes sense. Will all this be taught in high school one day?

So sorry for your loss, John. Loss makes us aware of our own mortality, and makes distance and separation palpable. Contemplating the “big picture” stuff is very suitable for this.

My daughter is also pursuing a career in healthcare. She's not at the top of her class, so she's not on track to become a surgeon or a specialist. But she's ranked high enough in her class and has the necessary people skills to make her a fine primary care physician.

BTW, I know that you are not the only John Merryman out there roaming the planet, but I was wondering if you happen to be a direct descendent of John Merryman, the "gentleman farmer" from Maryland who was indicted for treason during the Civil War. From what recall and according to the link below, he was accused of recruiting and arming Confederate soldiers and of cutting off telegraph communications to Union commanders in Washington. He was quite famous for challenging his indictment on constitutional grounds that the President lacks the authority to suspend habeas corpus without Congressional approval. The Supreme Court ruled in his favor and against President Lincoln. This unfortunately is something that today's Supreme Court lacks the backbone to do against President Obama.

http://www.fjc.gov/history/home.nsf/page/tu_merryman_bio_merryman.html

Lubos,
To quote the tagline of a certain doom based financial website, "On a long enough timeline, the survival rate for everyone drops to zero." One of the problems with effectively evaluating risk is that it allows you to take on more risk. Just as more complexity can mean both more adaptability and more things to go wrong. Having spent my life in the horse racing business, I'm reasonably familiar with the nuances of risk, including its explicit necessity in every aspect of existence. Hence the fear side of the fear/hope dichotomy. What drove the 2008 mortgage bubble was the need to turn investment capital into income streams. Period. Blaming the suckers who took on these loans would be like blaming the cattle for the beef industry. The government was an enabler and handy scapegoat. Basically they all knew exactly what they were doing, but their timeframe was short term. I'm not really blaming the banking industry. They have taken on the job of turning paper assets into more paper assets, ie. compounding interest and no one wants to be the one to say its not sustainable, especially when it becomes a religious belief and not just a happy side effect. So the system runs on bs until it can't anymore. The fact is that there is no happy medium, as that could just as well be a flatline on the universal heart monitor, but what goes up, does come down and when it's mostly foam and bubbles, it usually means the wave is cresting and the next move is down.

Thanks for the reply, actually I was thinking of a "hypothetical" infinitely rigid( like a geometric line). Then as the line move , the points on it move to represent the locality principle, but also you will get the non-local behavior, i.e. the best of worlds.

I interpret the end point of the line as non transverse-able "wormholes" like.

Two questions: How is entaglement in the case of black holes defined? Is there any way to make a parallel to the standard definition(s) of two entangled qubits?

I've seen Penrose eternal black hole diagram used to represent two entangled black holes. But that is quite confusing because a black hole is only the one in the future. The one in the past is a white hole. What am I missing here?

Cynholt,
Thanks. We had been separated for awhile, but were still good friends and raising the daughter. I'm not sure she is still thinking of being a doctor. She is actually looking at law school after college.
Yes, he was an ancestor. Usually referred to as "John Merryman of Hayfields." That farm was still in the family, until the 1980's. I've had people ask me if I was using a pseudonym, because of that. There is a lot of family history in this neck of the woods, having been around here since the 1600's. The family church is called Sherwood and the joke was it was from Robin Hood and his Merry men. I don't know what the truth of that is though.

No disrespect intended, but after observing the species in action over a number of years (during which they have obviously been having lots of fun) the thought occurs to me that:

nature is the playpen of theoretical physicists.

Attempting to get a handle on the first question, I found this one by Brian Swingle:

http://arxiv.org/abs/0905.1317

which, according to Preskill, precedes Van Raamsdonk.

Neat.

Hi Lubos, I have a quick side question. Has the stationary QM/QFT equations been generalized(or investigated) to superstring field theories and/or String Matrix. Any reference is appreciated. Thanks.

Doh. The answer to a second question is that it is not a white hole and a black hole in eternal black hole diagram that are treated as two black holes. The white hole is apparently ignored, leaving only one black hole seen from two separate universes. Then, of course, we have a system consisting of two parts like in the case of two distant qubits.

Sorry, QsaTheory, I don't see it. If points move locally, they move at certain speed. How do you get non-local behavior out of that?

And what does 'infinitely rigid' really mean? Surely, if something is so rigid then nothing moves, yet it somehow moves in your declaration? .

Dear John, my condolences etc.

Everyone is dead in the long run. On one hand, it's true and it's why planning of personal finances etc. shouldn't really go to much more distant future than 30 or at most 50 years.

On the other hand, the families (where relatives inherit) and societies may still plan at a longer time scale than that, although it becomes much more fuzzy. For example, we would like to arrange things so that there is a high probability that the societies won't totally decay in 70 years, either, and I think it's possible.

Mortgages.

To compare the buyers of the mortgages to cattle may be problematic but I accept your comparison. However, I still do blame it on cattle. If there is some problem with the cattle industry, of course that it boils down to mistakes done with (lots of) particular cattle.

It is very easy for someone in any industry to choose a scapegoat in the financial world (note that a scapegoat is technically also cattle LOL) but that's not a fair description of the reality.

It may be true that someone wanted to guarantee a continuous profit out of a fluctuating thing that is only likely to generate profit in the long run. But such a conversion isn't impossible, either. A big enough bank or financial object able to carry the risk may convert the long-term profit opportunities to a continuous source of income.

That's why we have saving accounts. The banks are lending our money for a particular period of time - to individuals or if they buy bonds with a maturity date - but they may add the continuous pre-announced interest. Why it is so? They simply can. They do the calculation how much they earn in average, and if the average profit is positive and the standard deviation is small enough, they just take the risk.

All kinds of products, services, contracts, and risk plus profit opportunities are monetized, and that's what makes the global economy advanced and more efficient than in the past. There's nothing wrong about it and of course that it depends on the assumption that the underlying assets are behaving in the microscopic risk-producing way they are expected to. Of course that if they don't, the problems spread outside.

But they don't spread "too much" because the more complicated financial products may be leveraged but they are also diversified. So for the problems to existentially affect them, the wrongness in the underlying assets has to be widespread and systematic.

Dear QsaTheory, special relativity prohibits all nonlocal propagation of any usable information so everything in the "world of nonlocality" is bad and if you choose the best of this world, you should choose nothing. ;-)

In Newton's physics, totally rigid objects were allowed and could move at any velocity and angular velocity.

In special relativity, totally rigid objects may only break at a constant velocity and constant angular velocity. Any attempt to accelerate them, spin them differently, or bend them (of course) will fail. According to relativity, all objects stretch or shrink or bend a little even if the velocity or angular velocity is changed, and these deformations are moving by the speed much lower than the speed of light.

However, the entanglement is known to guarantee correlations even at spacelike separations so it can't be due to signals or information moving in between the two measurement events. To explain the observed perfect correlations in entanglement, you would need signals sent through totally rigid rods etc. but they are forbidden by relativity.

After I have deepened my intuitive understanding by lapping up the content of this post - and very much enjoying the taste of it - I feel like a somewhat closer-to-become tame (even though I still sometimes behaving with predictable viciousness) wild animal.
};-)

FRACTIONAL RESERVE UNIVERSE

The fools think that debt's still expanding
An infinite misunderstanding
There is no free lunch
Here comes the big crunch
We're in for a very hard landing

The Limerick King

https://www.flickr.com/photos/expd/8210143863/


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# Machinations

Self-healing of Byzantine Faults
May 21, 2012, 7:41 pm
Filed under: Uncategorized | Tags: , ,

“You have to have good researchers. Good researchers, all of them.” – Paraphrase of quote from Das Boot.

Sometimes writing a paper can be a challenging slog: goals move as conjectures prove wrong; facts that seem simple turn out to be slippery to prove; experiments must be rerun as the paper evolves; and notation needs to be redone for the sake or readability.

In such a situation, it’s a blessing to have coauthors who will go the extra mile to keep pushing and pushing the paper to completion.  So I would now really like to thank my students Jeffrey Knockel and George Saad for their hard work on our latest paper: “Self-Healing of Byzantine Faults” (abstract below, full paper here).

Note: I wouldn’t exactly compare the research process with serving on a U-boat mission during the decline of the Third Reich.  For me it’s most similar to mountaineering, whose ethos is captured by the motto of the Seattle mountaineers: “Like fun, only different!”

Paper Abstract:

Recent years have seen significant interest in designing networks that are self-healing in the sense that they can automatically recover from adversarial attack. Previous work shows that it is possible for a network to automatically recover, even when an adversary repeatedly deletes nodes in the network. However, there have not yet been any algorithms that self-heal in the case where an adversary takes over nodes in the network. In this paper, we address this gap.

In particular, we describe a communication network over n nodes that ensures the following properties, even when an adversary controls up to $t \leq (1/4-\epsilon)n$ nodes, for any positive $\epsilon$. First, the network provides point-to-point communication with bandwidth and latency costs that are asymptotically optimal. Second, $O(t (log*n)^2)$ message corruptions occur in expectation, before the adversarially controlled nodes are effectively quarantined so that they cause no more corruptions. We present empirical results showing that our approach may be practical.

Special thanks to Prof. Jared for giving me an opportunity to work on the paper “Self-Healing of Byzantine Faults”. This is an exceptional experience for me in the research field.

format: plain


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## Same as in figure 1 with λ = 450 nm

2013-06-26T00:00:00Z (GMT) by

Figure 2. Same as in figure 1 with λ = 450 nm. (a) The maximum intensity of the pulse is I_{m}=0.2 \times 10^{13} \rm \ \rm \ W\,cm^{-2}. (b) I_{m}=0.5 \times 10^{13} \rm \ \rm \ W\,cm^{-2}.

Abstract

Laser control schemes for selective population inversion between molecular vibrational states have recently been proposed in the context of molecular cooling strategies using the so-called exceptional points (corresponding to a couple of coalescing resonances). All these proposals rest on the predictions of a purely adiabatic Floquet theory. In this work we compare the Floquet model with an exact wavepacket propagation taking into account the accompanying non-adiabatic effects. We search for signatures of a given exceptional point in the wavepacket dynamics and we discuss the role of the non-adiabatic interaction between the resonances blurring the ideal Floquet scheme. Moreover, we derive an optimal laser field to achieve, within acceptable compromise and rationalizing the unavoidable non-adiabatic contamination, the expected population inversions. The molecular system taken as an illustrative example is H_{2}^{+}.


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##### Physicists have long thought that the singularities associated with gravity (like the inside of a black hole) should vanish in a quantum theory of gravity.

It now appears that this may indeed be the case. Researchers in Uruguay and Louisiana have just published a description of a quantum black hole using loop quantum gravity in which the predictions of physics-ending singularities vanish, and are replaced by bridges to another universe.

Singularities, such as the infinitely strong crushing forces at the center of a black hole, in a physical theory are bad. What they tell you is that your description of the universe fails miserably to explain what happens as you approach the singularity. Tricks can sometimes resolve what appears to be singular behavior, but essential singularities are signs of a failure of the physical description itself.

General relativity has been summed up by the late John Wheeler’s phrase: “Spacetime tells matter how to move, matter tells spacetime how to curve.” Relativity is riddled with essential singularities, because gravity is both attractive and nonlinear – curvature in the presence of mass tends to lead to more curvature, eventually leading to trouble.

The result is rather similar to a PA system on the verge of producing a feedback whistle. If you whisper into the microphone (small gravitational fields) the positive feedback isn’t enough to send the PA into oscillation, but talking at a normal volume (larger gravitational fields) produces that horrible howl. Whispering is the comparable to the familiar actions of gravity that keep the planets and stars in their courses. The howl is the process that eventually leads to a singularity as the end result of gravitational collapse.

Let’s follow this analogy a bit further. On a PA system, the volume of the feedback is limited by the power capacity of the amplifier, so it can’t reach truly destructive levels (other than to our eardrums.) However, gravity as described by general relativity doesn’t have such a limit. Since gravity is always attractive, and eventually becomes stronger than all the (known) forces that normally give volume to matter, there is nothing to keep gravitational collapse from proceeding until the curvature of the spacetime tends toward infinity – i.e. a singularity.

Remember that this is the prediction of the classical theory of gravity, general relativity. Classical physical theories contain no fundamental limitation on mass-energy density or on the size of spacetime curvature. While this may be (and probably is) incorrect, we rarely run into a problem caused by this error, so have largely ignored the problem for centuries.

Then along came gravitational collapse and black holes. First proposed by geologist John Mitchell in 1783, a black hole is a region of spacetime from which gravity prevents anything, even light, from escaping.

##### The Latest on: Quantum black hole
• on June 27, 2017 at 5:34 am

"The techniques required to detect extremely faint signals produced by distant black holes were developed over decades," Dr ... It was supported by the ARC Centre of Excellence for Engineered Quantum Systems, which aims to develop next-generation quantum ... […]

• on June 26, 2017 at 11:12 am

What's on the other side? Famed physicist Stephen Hawking theorized that even though nothing can escape a black hole because of its immense pressure, it can secrete radiation. This theory works because of quantum mechanics. To simplify the theory ... […]

• on June 26, 2017 at 4:42 am

the magnetic quantum field has always TWO diffe... Proplyds are not young planets, but young black holes accelerated by plasma tails. According to Quantum FFF Theory, Proplyds should be young molecular ( ejected star spot) black holes accelerated by plasma ... […]

• on June 25, 2017 at 2:58 pm

A calculation of the black hole’s evaporation seemed impossible, unless a full theory of quantum gravity that unites Einstein’s general relativity with the framework of quantum field theory could be found. Adami’s recent paper, published in Physical ... […]

• on June 15, 2017 at 12:38 pm

This question is based on page 71 of Thomas Hartman's notes on Quantum Gravity and Black Holes (http://www.hartmanhep.net/topics2015/gravity-lectures.pdf). ds^{2} = \left(1-\frac{2M}{r}\right)d\tau^{2} + \frac{dr^{2}}{1-\frac{2M}{r}} + r^{2}d\Omega_{2}^{ […]

• on June 14, 2017 at 1:40 pm

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• on June 14, 2017 at 9:52 am

Superadiance - the extraction of energy from a rotating black hole - is also known as the Penrose Mechanism and is a precursor of Hawking Radiation - a quantum version of black-hole superradiance. What's in the Black Hole Lab? Dr Weinfurtner said ... […]

• on June 14, 2017 at 9:01 am

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• on June 14, 2017 at 1:00 am

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# The Sorites Paradox - Some Doubtful Thoughts

It has been asked how, if at all, one might resolve the Sorites paradox. I am not convinced a solution is possible, and in this paper I will explain the responses I have become aware of, and why they fail. In the end, I will conclude that there is no solution to the paradox, but I will offer a few suggestions for a way forward.

The first response might simply be to reject the first premise of the argument. In other words, simply deny that a man with 10 hairs is in fact bald, or that 100 grains of sand is in fact a heap. In essence, this would render vague predicates useless at best, meaningless at worst, since no predicate that allows for a vague border case would be permitted to apply to anything. There is one way in which we might stretch this into plausibility, but I will address the other responses first, before returning to this in the conclusion.

The second response is to set some arbitrary boundary. This means selecting one one among the indefinite number of secondary premises beyond which all others will be false. For example, we might say that thirty-thousand and one grains of sand is the boundary below which we no longer regard a collection of grains to be a heap. At first glance, this approach might seem plausible. After all, we do this frequently in practice: setting the legal drinking age, or the number of credit-hours required to count as a ‘full-time’ student, for example. However, there are two core problems with this. First, from the context of the formal argument, there is no good reason to reject any of the subsequent conditional statements, and there appears to be no means by which we could discover a reason. The implicit modus ponens of the conditional compels us to accept them all. Second, as Wright (Vagueness, 1997) pointed out, vague predicates are inherently coarse by virtue of their intended use. So attempts to impose some sort of specificity would destroy their meaning.

The next approach would be to attempt to define a knowledge gap within some middle range of propositions between the edge false and edge true statements. On one interpretation of the idea, we could use a three-value logic, in evaluating the propositions. At some point, starting with grain one, the proposition ‘this is a heap’ would cease being false, and would instead be valued ‘unknown’ or ‘undefined’. Later, the unknown state would transition to true, once we’ve reached the next threshold. This would make it possible to judge the argument invalid, since any number of its premises were neither true nor false. However, this seems to be attempting to win on a technicality, and it suffers from the same problem as the arbitrary boundary solution, in that we have no real way of determining when the states should change.

The next response might be some form of Edgington’s “degrees” of truth (Vagueness, 1997). But this suffers from it’s own serious flaws. For example, consider the statement ‘it is raining’. That has a ‘degree’ of truth of .5. It’s negation, ‘it is not raining’ will have a degree of truth of .5. The consequence of this, is that the following propositions have exactly the same truth value: ’It is raining, and it is raining’, ’It is raining, and it is not raining’. The same problem exists with our heap of sand. So, again, we’re left with no clear way to determine the truth of the conditionals in the Sorites case.

In the end, there does not seem to be any clear resolution to this paradox. However, I would offer one suggestion. vague predicates, in addition to being inherently coarse, also seem to be describing inherently subjective experiences or judgements. While Sorites arguments seem to want to talk about objective properties of objects. Perhaps there is no solution to the paradox, precisely because “tall” or “heap” or “red” or “bald” is not in fact a property of the object being considered, but a property of the experience the subject has of that object. The paradox is perhaps trying to square subjective interpretation with objective matters of fact. And that’s why it cannot be resolved.

[Imported from exitingthecave.com on 1 December 2021]


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# American Institute of Mathematical Sciences

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doi: 10.3934/cpaa.2021133
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## Global boundedness of radial solutions to a parabolic-elliptic chemotaxis system with flux limitation and nonlinear signal production

 1 College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China 2 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China 3 School of Sciences, Southwest Petroleum University, Chengdu 610500, China 4 College of Mathematics and Statistics, Yili Normal University, Yining 835000, China

* Corresponding author

Received  August 2020 Revised  June 2021 Early access August 2021

Fund Project: This work is supported in part by the NSFC under grants 11771062 and 11971082, the Fundamental Research Funds for the Central Universities under grants 2019CDJCYJ001, 2020CDJQY-Z001, Chongqing Key Laboratory of Analytic Mathematics and Applications, and Scientific Research Program of the Higher Education Institution of XinJiang under grant XJEDU2021Y043

The following degenerate chemotaxis system with flux limitation and nonlinear signal production
 $\begin{equation*} \begin{cases} u_t = \nabla\cdot(\frac{u\nabla u}{\sqrt {u^{2}+|\nabla u|^{2}}})-\chi\nabla\cdot(\frac{u\nabla v}{\sqrt {1+|\nabla v|^{2}}}) \quad &in\quad B_{R}\times(0, +\infty), \\ 0 = \Delta v-\mu (t)+u^{\kappa}, \quad \mu(t): = \frac{1}{|\Omega|}\int_{\Omega}u^{\kappa}(\cdot, t) \quad &in\quad B_{R}\times(0, +\infty) \end{cases} \end{equation*}$
is considered in balls
 $B_R = B_R(0)\subset \mathbb{R}^n$
for
 $n\geq 1$
and
 $R>0$
with no-flux boundary conditions, where
 $\chi>0, \kappa>0$
. We obtained local existence of unique classical solution and extensibility criterion ruling out gradient blow-up, and moreover proved global existence and boundedness of solutions under some conditions for
 $\chi, \kappa$
and
 $\int_{B_R}u_{0}$
.
Citation: Hong Yi, Chunlai Mu, Shuyan Qiu, Lu Xu. Global boundedness of radial solutions to a parabolic-elliptic chemotaxis system with flux limitation and nonlinear signal production. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021133
##### References:

show all references


text:
Bayesian Analysis

Mixtures of $g$-Priors for Analysis of Variance Models with a Diverging Number of Parameters

Min Wang

Abstract

We consider Bayesian approaches for the hypothesis testing problem in the analysis-of-variance (ANOVA) models. With the aid of the singular value decomposition of the centered designed matrix, we reparameterize the ANOVA models with linear constraints for uniqueness into a standard linear regression model without any constraint. We derive the Bayes factors based on mixtures of $g$-priors and study their consistency properties with a growing number of parameters. It is shown that two commonly used hyper-priors on $g$ (the Zellner-Siow prior and the beta-prime prior) yield inconsistent Bayes factors due to the presence of an inconsistency region around the null model. We propose a new class of hyper-priors to avoid this inconsistency problem. Simulation studies on the two-way ANOVA models are conducted to compare the performance of the proposed procedures with that of some existing ones in the literature.

Article information

Source
Bayesian Anal., Volume 12, Number 2 (2017), 511-532.

Dates
First available in Project Euclid: 5 July 2016

https://projecteuclid.org/euclid.ba/1467722664

Digital Object Identifier
doi:10.1214/16-BA1011

Mathematical Reviews number (MathSciNet)
MR3620743

Citation

Wang, Min. Mixtures of $g$ -Priors for Analysis of Variance Models with a Diverging Number of Parameters. Bayesian Anal. 12 (2017), no. 2, 511--532. doi:10.1214/16-BA1011. https://projecteuclid.org/euclid.ba/1467722664

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# On the roots of $2z-\sin(2z)$

Recently I have been working on an approach to numerically find the roots of the equation $$2z-\sin(2z)=0$$

As you can see below, I was able to find all the roots in a specific, bounded domain

So far so good. However, I have some observations, which makes me formulate some statements, of which I am not sure if they are correct.

1. The zeros lie on a fixed curve per quadrant.

I now that if $z$ is a zero, than so will $z^*$ (the conjugate), $-z$ and $-z^*$. This limits my root-finding to one quadrant (speed up of factor 4 hooray!). But, is it true there are no zeros outside this 'curve'?

2. The zeros reach an asymptote

Considering one curve, I get the feeling that, numbering the roots $z_n$ in increasing distance from the origin,

$$\lim_{n\to\infty} Re(z_n)-Re(z_{n-1})=\pi.$$

And maybe, but less convinced

$$\lim_{n\to\infty} Im(z_n)-Im(z_{n-1})=0.$$

Is there proof for either of the above statements? Did I miss any exact, closed form solution to the problem?

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I just confirmed that $Re(z_{100000}-z_{99999})=3.141592654$ (that is six more digits of $\pi$ than I had yesterday :)) – Bernhard Nov 24 '12 at 22:48

This is only a partial answer, it only really answers the first question. It won't fit in a comment though, so I'll put it here (is this the right thing to do?).

Write $z = a+bi$. Note that $\sin(a+bi) = \sin a \cosh b +i\cos a \sinh b$. This means that $\sin z = z$ is equivalent to the system of equations $$a = \sin a \cosh b$$ $$b = \cos a \sinh b.$$ If we divide the first equation by $\cosh b$ and the second by $\sinh b$, and use $\sin^2 a + \cos^2 a = 1$, we get $$\left(\frac{a}{\cosh b}\right)^2 + \left(\frac{b}{\sinh b}\right)^2 = 1.$$ This allows us to write $a^2 = \cosh^2 b - b^2 \coth^2 b$. This curve looks like this: (the image is using the equation that comes from $\sin 2z = 2z$, not $\sin z = z$, so you can compare it to your points; it seems to agree nicely). This answers your first question: the points are confined to a curve and you will not find any outside of it. However, it does not tell us anything about where the solutions lie on the curve.

You can do something similar by taking the system of equations and dividing by $\sin a$ and $\cos a$ and using the identity $\cosh^2 b - \sinh^2 b = 1$, which gives us $b^2 = a^2 \cot^2 a - \cos^2 a$. We can plot this together with the previous curve to get a much less nice-looking picture (I'm only plotting the positive values for $b$ here to get a less cluttered image): .
I'd be a little more skeptical of this plot than the other one, since this function behaves so erratically. This gives us a necessary condition for our roots; they have to lie on the intersections pictured. However, I can't really think of how to find precisely which intersections are roots to the equation.

Of course, if you can prove that the real parts are evenly spaced in the limit, then by considering the derivative of the first curve, it is easy to see that the limit of $\operatorname{Im}(z_n) - \operatorname{Im}(z_{n-1})$ is $0$.

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It is definitely OK to post it as an answer, I shouldn't have asked two questions in one. I asked them in order of importance. I will wait to accept to see if anyone comes up with something on the second question. Thanks for this insight! – Bernhard Nov 24 '12 at 18:18
Just some thoughts. In the limit of large $a$, the intersections of the red and blue curve get closer together. This is probably related to the period of $\cos^2 a$. – Bernhard Nov 24 '12 at 22:52
Are you able to numerically get any information about which intersections give zeros? Do you have any guesses? – Ben Nov 25 '12 at 0:43
Hmm, I have some difficulties plotting my dots and the two functions in a single graph, but it may be that every fourth intersection is a zero. I am not sure about this. – Bernhard Nov 25 '12 at 9:22


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# The apparent angular size of the moon is about 1/2 degree, how many full moons could fit across the apparent size of the Andromeda galaxy?

Feb 3, 2017

About $6$

#### Explanation:

The Andromeda galaxy is about $2.5$ million light years distance from us and has a diameter of approximately $140000$ light years.

So it subtends approximately:

$\frac{1.4 \cdot {10}^{5}}{2.5 \cdot {10}^{6}} = 0.056$ radians

In degrees, that's:

$0.056 \cdot \frac{180}{\pi} \approx {3.2}^{\circ}$

So about $6$ times the angle which the full moon subtends.

That having been said, we only usually observe the bright central area of the Andromeda galaxy by naked eye or small telescope under normal conditions, so it appears much smaller than it actually is.


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MAT240F Algebra I

Classes: T11-1, R12; Room: HS610. (Health Sciences, 155 College Street).

Midterm exam: T11-1 on October 17, 2017 , in EX 100

 Course description

This course is an introduction to Linear Algebra, aimed at students in our specialist programs.

Text Book: Linear Algebra (fourth edition) by Friedberg, Insel and Spence, Prentice Hall. (Required reading.)

Prerequisite: High school level calculus

Corequisite: MAT157Y

Course outline

Course syllabus

The required homework problems for this course will be sent online, via Crowdmark. You will be asked to submit the solutions electronically, via Crowdmark. (More information will be given later.)

Our late policy is as follows: No late assignments will be accepted.

You can certainly discuss homework with classmates, but you have to write up the solutions yourself, in your own words. Otherwise it is considered unauthorized aid or assistance (working too closely with another student on an individual assignment so that the end result is too similar), which is an academic offence under the University's Code of Behaviour on Academic Matters. If you find the solutions in books or on the internet, you must quote your source (and still write it up in your own words!) Otherwise, it may count as plagiarism which again is an academic offense.

Here is a sample latex source file and its pdf output for download, in case you want to type up your solutions in latex and need some hints.

If you like, you can attend Peer Study Team (PST) sessions for this course: Thursday 4-6, UC257. The sessions are being run by Phil Blakey and Mira Kuroyedov, who took this course last year.

The calendar' below will be updated as the course is underway. For example, I will give a few additional problems (not to be handed in), informations on tests and exams, required readings, and additional references.

 Calendar

Week 9: November 13-17, 2017
• Required reading (for this and next week): Section 3.2-3.4 (you can skip p.176-179)
• I have also posted some of my course notes on Blackboard (the notes that I'm lecturing off). They may contain mistakes, use at your own risk.
• Homework #7 is due 11pm on Friday, November 17
• Additional homework (do not hand in).
• Section 3.1, Problems 1
• Section 3.2, Problems 1,2b,2d,2f,4a,5,19,21,22
• Just for fun: Let $X$ be a set with $n\ge 3$ elements, and $S_1,\ldots,S_m$ proper subsets of $X$ (thus, $S_j\not=X$) such that every pair of distinct elements of $X$ is contained in exactly one of the $S_j$. Prove that $m\ge n$. (Hint: introduce a suitable incidence matrix, and consider ranks. This problem is quite challenging!)
Week 10: November 20-24, 2017
• Required reading: Section 4.1, 4.2 . We've spend one lecture on dual spaces'; you may want to review my notes on this (will be posted on Blackboard under course materials).

• Material covered this week: The general theory of linear systems of equations. Reduced row echelon form of a matrix. How to read off a basis of the null space and range of a matrix from its reduced row echelon form. How to determine the solution set of $Ax=b$ from the reduced row echelon form of $(A|b)$. Dual spaces, dual bases, dual maps and their matrices.
• Homework #8 is due 11pm on Monday, November 27
• Additional homework (do not hand in).
• Section 3.3, Problems 1, 2, 4, 10
• Section 3.4, Problems 1, 2 (pick a few)
• Section 4.1, problems 1, 3, 8, 9,
• (1) Prove that if $A,B\in M_{m\times n}(F)$ are in reduced row echelon form, and with $N(L_A)=N(L_B)$, then $A=B$.
• (2) Prove that the reduced row echelon form of a matrix is uniquely determined by the matrix, and does not depend on the order of the row operations used. Hint: Use (1) above.

Week 11: November 27 - December 1, 2017
• Required reading: We will be treating determinants a bit different from the textbook. An online reference along similar lines is Denton-Waldron: p.96-p.138; I will also post my notes.

• Material covered this week: Definition of determinant as a multi-linear form on column vectors, the formula as a sum over pemutations. Properties of determinants: Row and column operations, $\det(A^t)=\det(A)$ , $\det(AB)=\det(A)\det(B)$. The fact that $\det(A)\neq 0$ if and only if $A$ invertible. Computing $\det(A)$ be reducing to upper triangular form.
• Homework #9 is due 11pm on Monday, December 4
• Additional homework (do not hand in).
• Section 4.2, problems 1, 2, 5, 7, 9, 11. Also compute the last few determinants using the method explained on p. 234 (using that the determinant of an upper/lower triangular matrix is the product of diagonal entries).
• Section 4.3, problems 22. (We've encountered the Vandermonde matrix as the change-of-basis matrix from the standard basis of polynomials to the basis given by Lagrange polynomials. Now it's time to calculate its determinant.)
• Section 4.3, problem 28, Section 4.4, problem 1, Section 4.5, problems 3-10

Week 12: December 4-6, 2017
The final exam is on December 18, 2-5 pm. The official source for time, location, etc is the Arts and Sciences webpage. Please note that there are three exam locations; you have to go to the correct room depending on the first letter of your last name. I've posted last year's exam on Blackboard, under course materials. This year's exam will be a somewhat similar format. For further practice, you can use exams from earlier years on Dror Bar-Natan's MAT240 websites.


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## Keywords (5)

Publications
Brezin-Gross-Witten model as “pure gauge” limit of Selberg integrals

# Brezin-Gross-Witten model as “pure gauge” limit of Selberg integrals,10.1007/JHEP03(2011)102,Journal of High Energy Physics,A. Mironov,A. Morozov,Sh.

Brezin-Gross-Witten model as “pure gauge” limit of Selberg integrals
The AGT relation identifies the Nekrasov functions for various $$\mathcal{N} = 2$$ SUSY gauge theories with the 2d conformal blocks, which possess explicit Dotsenko-Fateev matrix model (β-ensemble) representations the latter being polylinear combinations of Selberg integrals. The “pure gauge” limit of these matrix models is, however, a non-trivial multiscaling large-N limit, which requires a separate investigation. We show that in this pure gauge limit the Selberg integrals turn into averages in a Brezin-Gross-Witten (BGW) model. Thus, the Nekrasov function for pure SU(2) theory acquires a form very much reminiscent of the AMM decomposition formula for some model X into a pair of the BGW models. At the same time, X, which still has to be found, is the pure gauge limit of the elliptic Selberg integral. Presumably, it is again a BGW model, only in the Dijkgraaf-Vafa double cut phase.


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# Question on notation of differentials

Is the following notation acceptable, specifically last part of the last line?

$$f(x) = \csc^4(x) = (\csc(x))^4$$ Let $$u =\csc(x) \rightarrow f(x) = u^4$$ $$f'(x) = \frac{du}{dx} \times \frac{df(x)}{du}$$ $$...$$

Edit: Reworded question: Is it OK to mix Leibniz's and Lagrange's notation?

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It shouldn't be $$\frac{df(x)}{du},$$ but rather $$\frac{df(u)}{du}.$$ This is the Chain Rule, so$$f'(x) = f'(u)u'(x) = \frac{d}{du}f(u)\times\frac{d}{dx}u(x).$$ – Arturo Magidin Apr 14 '11 at 20:41
I think the question can be made to be more precise. The function $csc(x)$ here is not the essential part of your question. This may be helpful to clarify your "notation" question. – Jack Apr 14 '11 at 20:47
@Jack True, good point. My question perhaps should have been "is $\frac{df(u)}{du}$ correct notation? I was nervous to mix Leibniz's and Lagrange's notation. But I gather from Arturo's response that this is acceptable? – Danny King Apr 14 '11 at 20:56
You aren't mixing it, because you are not using primes and Leibnitz notation on the same side of the equal sign; even so, it's still okay to mix the two with some care. E.g.,$$\frac{d^2f}{dx^2} = \frac{d}{dx}f'(x)$$is understandable enough to be acceptable. – Arturo Magidin Apr 14 '11 at 21:47


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## G = C2×C4⋊D8order 128 = 27

### Direct product of C2 and C4⋊D8

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C4⋊D8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×C4×D4 — C2×C4⋊D8
 Lower central C1 — C2 — C2×C4 — C2×C4⋊D8
 Upper central C1 — C23 — C2×C42 — C2×C4⋊D8
 Jennings C1 — C2 — C2 — C2×C4 — C2×C4⋊D8

Generators and relations for C2×C4⋊D8
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 780 in 324 conjugacy classes, 116 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C22×C4, C2×D4, C2×D4, C24, D4⋊C4, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C41D4, C41D4, C22×C8, C2×D8, C2×D8, C23×C4, C22×D4, C22×D4, C22×D4, C2×D4⋊C4, C2×C4⋊C8, C4⋊D8, C2×C4×D4, C2×C41D4, C22×D8, C2×C4⋊D8
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C4○D4, C24, C4⋊D4, C2×D8, C8⋊C22, C22×D4, C2×C4○D4, C4⋊D8, C2×C4⋊D4, C22×D8, C2×C8⋊C22, C2×C4⋊D8

Smallest permutation representation of C2×C4⋊D8
On 64 points
Generators in S64
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)
(1 36 47 61)(2 62 48 37)(3 38 41 63)(4 64 42 39)(5 40 43 57)(6 58 44 33)(7 34 45 59)(8 60 46 35)(9 27 18 55)(10 56 19 28)(11 29 20 49)(12 50 21 30)(13 31 22 51)(14 52 23 32)(15 25 24 53)(16 54 17 26)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(18 24)(19 23)(20 22)(25 55)(26 54)(27 53)(28 52)(29 51)(30 50)(31 49)(32 56)(33 64)(34 63)(35 62)(36 61)(37 60)(38 59)(39 58)(40 57)(41 45)(42 44)(46 48)

G:=sub<Sym(64)| (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,33)(32,34)(49,64)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63), (1,36,47,61)(2,62,48,37)(3,38,41,63)(4,64,42,39)(5,40,43,57)(6,58,44,33)(7,34,45,59)(8,60,46,35)(9,27,18,55)(10,56,19,28)(11,29,20,49)(12,50,21,30)(13,31,22,51)(14,52,23,32)(15,25,24,53)(16,54,17,26), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(18,24)(19,23)(20,22)(25,55)(26,54)(27,53)(28,52)(29,51)(30,50)(31,49)(32,56)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,45)(42,44)(46,48)>;

G:=Group( (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,48)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,33)(32,34)(49,64)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63), (1,36,47,61)(2,62,48,37)(3,38,41,63)(4,64,42,39)(5,40,43,57)(6,58,44,33)(7,34,45,59)(8,60,46,35)(9,27,18,55)(10,56,19,28)(11,29,20,49)(12,50,21,30)(13,31,22,51)(14,52,23,32)(15,25,24,53)(16,54,17,26), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(18,24)(19,23)(20,22)(25,55)(26,54)(27,53)(28,52)(29,51)(30,50)(31,49)(32,56)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,45)(42,44)(46,48) );

G=PermutationGroup([[(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,48),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(31,33),(32,34),(49,64),(50,57),(51,58),(52,59),(53,60),(54,61),(55,62),(56,63)], [(1,36,47,61),(2,62,48,37),(3,38,41,63),(4,64,42,39),(5,40,43,57),(6,58,44,33),(7,34,45,59),(8,60,46,35),(9,27,18,55),(10,56,19,28),(11,29,20,49),(12,50,21,30),(13,31,22,51),(14,52,23,32),(15,25,24,53),(16,54,17,26)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(2,8),(3,7),(4,6),(9,15),(10,14),(11,13),(18,24),(19,23),(20,22),(25,55),(26,54),(27,53),(28,52),(29,51),(30,50),(31,49),(32,56),(33,64),(34,63),(35,62),(36,61),(37,60),(38,59),(39,58),(40,57),(41,45),(42,44),(46,48)]])

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A ··· 4H 4I ··· 4N 8A ··· 8H order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 4 4 4 4 8 8 8 8 2 ··· 2 4 ··· 4 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 D8 C4○D4 C8⋊C22 kernel C2×C4⋊D8 C2×D4⋊C4 C2×C4⋊C8 C4⋊D8 C2×C4×D4 C2×C4⋊1D4 C22×D8 C42 C22×C4 C2×D4 C2×C4 C2×C4 C22 # reps 1 2 1 8 1 1 2 2 2 4 8 4 2

Matrix representation of C2×C4⋊D8 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 13 0 0 0 0 13 0
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 6 6 0 0 0 0 14 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 16 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,13,0],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,6,14,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C2×C4⋊D8 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes D_8
% in TeX

G:=Group("C2xC4:D8");
// GroupNames label

G:=SmallGroup(128,1761);
// by ID

G=gap.SmallGroup(128,1761);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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