text:
# News When the economy collapses

Mentor
cronxeh said:
And France with their meager 60 million population and $35 billion military budget/year is not really a good example. I'd say whatever bad can happen to them will happen 5 fold to the US if we come anywhere near that economic crunching point Certainly it would be worse for us (and the rest of the world), but my point was to get you to look up how near we are to that crunching point in relation to another country like France. Ie: France's debt-to-gdp ratio is 68%. The US's debt-to-gdp ratio is 24%. Yes, that's right - the US's national debt would have to more than double for us to be in the same economic predicament France is in. Art said: The ecomomy began to slip as Clinton's presidency ended because business people in general and foreign investors in particular were worried about the prospect of a republican president given their previous dire record on controlling spending. The same used to happen in Britain when it appeared a labour gov't was in the offing for the same reasons. Yes, I've heard that argument before - extrordinary prescience, that the business community knew 6 months before the election that Gore would lose, yet Gore didn't even know until 3 weeks after the election! The real truth is that Clinton rode the wave of an internet boom he had little to do with. He rode it up and when the bottom fell out (as it was destined to - the market was overvalued), he rode it down part-way, then handed it to Bush to try to pull it out of the death-spiral. In addition, the Clintons were lucky that they were a mediocre President: they attempted, and failed, to pass the largest increase in government spending since Social Security. But since he failed, he gets to claim success! :uhh: Last edited: #### Integral Staff Emeritus Science Advisor Gold Member russ_watters said: Certainly it would be worse for us (and the rest of the world), but my point was to get you to look up how near we are to that crunching point in relation to another country like France. Ie: France's debt-to-gdp ratio is 68%. The US's debt-to-gdp ratio is 24%. Yes, that's right - the US's national debt would have to more than double for us to be in the same economic predicament France is in. Yes, I've heard that argument before - extrordinary prescience, that the business community knew 6 months before the election that Gore would lose, yet Gore didn't even know until 3 weeks after the election! The real truth is that Clinton rode the wave of an internet boom he had little to do with. He rode it up and when the bottom fell out (as it was destined to - the market was overvalued), he rode it down part-way, then handed it to Bush to try to pull it out of the death-spiral. In addition, the Clintons were lucky that they were a mediocre President: they attempted, and failed, to pass the largest increase in government spending since Social Security. But since he failed, he gets to claim success! :uhh: What you say here is pretty much correct. Clinton can not really be given credit for the boom, or blame for the bust. It is an example of how little effect the president actually has on the economy. Unless, of course, they do something completely stupid, like start an unnecessary war. Bush can and should take full blame for the current situation. In a worst case scenario, the Bushes and their good buddy Ronny Raygun will be seen as the beginning of the end of the American civilization. We need to keep in mind that at least 3 generations must pass before the true place in history is know for any event or president. The true effect of Reagan and the Bushes will not be know until the adulthood of our grandchildren. All we can do now is express biased opinions. #### Integral Staff Emeritus Science Advisor Gold Member That last post was not really much on topic of this thread. I apologize. I wonder what level of natural disaster will be required to precipitate the decline and fall of the American economy. Perhaps the possible Richter 9 earthquake in the Pacific Northwest could be the trigger. Just imagine Seattle and San Francisco suddenly below sea level, the entire population of the pacific from San Francisco to The Olympic peninsula homeless. All bridges across the Columbia, gone; the Columbia dams, gone; Portland, gone. Sorry no time to continue, but how could the nation recover from this disaster? #### Art russ_watters said: Yes, I've heard that argument before - extrordinary prescience, that the business community knew 6 months before the election that Gore would lose, yet Gore didn't even know until 3 weeks after the election! Not exactly prescient, just prudent. It's not unusual for people to move out of the market until they see how an event plays out. If you followed the markets even remotely you'd know that. russ_watters said: The real truth is that Clinton rode the wave of an internet boom he had little to do with. He rode it up and when the bottom fell out (as it was destined to - the market was overvalued), he rode it down part-way, then handed it to Bush to try to pull it out of the death-spiral. That's funny I remember you arguing on another thread that the internet boom was only a tiny part of America's economy but now that that no longer fits your arguement it has suddenly become the lynch pin of the economy. At least make an effort to be consistant. BTW isn't it an amazing coincidence that the economy boomed only throughout his term in office. russ_watters said: In addition, the Clintons were lucky that they were a mediocre President: they attempted, and failed, to pass the largest increase in government spending since Social Security. But since he failed, he gets to claim success! :uhh: :rofl: :rofl: :rofl: Sorry but this is just nonsense. Clinton stated from the start he was going to balance the budget, the republicans first tried to stop him by pushing through huge tax decreases for the wealthy which he vetoed (you might remember it closed the gov't down as they tagged them onto the main budget bill). When plan 'A' didn't work they then had the gall to try to make it look like balanced budgets were their idea by trying to push through the balanced budget act. :rofl: They really have no shame. BTW I wonder what happened to that policy after they got their republican president in??????? The current deficit is entirely due to the neocons warmongering and the enormous tax decreases for the wealthy. Bush even boasts about his tax reductions being the largest for over 20 years!! #### Manchot I don't care what anyone's position is on whether Clinton just rode the dot-com bubble or not. (Having said that, I do agree with Integral: the president has very little effect on the economy unless he changes something large, like implementing large tax cuts or going to war.) The truth that cannot be denied is that a) there was a budget surplus when Clinton left office that was going to level our debt, and b) that there is now a sizable budget deficit, mostly due to Bush's tax cuts and war. Instead of tax and spend, we have tax and borrow. Really conservative. Furthermore, our GDP-to-debt ratio in comparison with France's is completely irrelevant. Our economy is magnitudes larger than theirs, and as such, works completely differently. For one thing, the strength of many smaller nations' currencies depends on the strength of our dollar, which means that if confidence is lost in it, it could have a resounding effect throughout the world, multiplying the effect by several fold. At the same time, France has a much lesser effect on the strength of the Euro. #### russ_watters Mentor Integral said: We need to keep in mind that at least 3 generations must pass before the true place in history is know for any event or president. The true effect of Reagan and the Bushes will not be know until the adulthood of our grandchildren. All we can do now is express biased opinions. I've always thought it was 20 years, but then, I'm impatient... That last post was not really much on topic of this thread. I apologize. I wonder what level of natural disaster will be required to precipitate the decline and fall of the American economy. Perhaps the possible Richter 9 earthquake in the Pacific Northwest could be the trigger. Just imagine Seattle and San Francisco suddenly below sea level, the entire population of the pacific from San Francisco to The Olympic peninsula homeless. All bridges across the Columbia, gone; the Columbia dams, gone; Portland, gone. Sorry no time to continue, but how could the nation recover from this disaster? A 9.0 earthquake up there would be rough, but submerge Seattle? I doubt it. I suppose you're talking about a tsunami, but San Francisco is pretty high up for a coastal city - a check off Google Earth shows that half a mile inland, it's 90 feet above sea level. I really think Katrina is about the worst mother nature has to offer for the US. It cost roughly$100 billion. About the only thing worse would be a Cat5 hurricane making landfall in Miami and bisecting Florida.

Art - find that quote, because that doesn't sound like something I'd say. I've always been very consistent in saying that the President has little effect on the economy. Also, do you know what failed policy I was referring to? And uncertainty is one thing, but what the economy did in 2000 was far beyond what uncertainty gives you - especially considering that people had every reason to believe Clintons policies would continue with a Gore win (with the economy riding high, he would have been shoo-in).

#### russ_watters

Mentor
Manchot said:
The truth that cannot be denied is that a) there was a budget surplus when Clinton left office that was going to level our debt,
That is only "true" if you ignore the fact that the economy is cyclical, make unrealistic estimates of it's future gains, and make promises for other Presidents to keep that Clinton wouldn't have kept (that's the best part about promises for when you're already out of office!). In short, such speculation is utterly meaningless.
Furthermore, our GDP-to-debt ratio in comparison with France's is completely irrelevant. Our economy is magnitudes larger than theirs, and as such, works completely differently.
:rofl: :rofl: :rofl: :rofl: It works differently because it's bigger? That's absurd. It works differently because France's government is driving their economy into the ground.
For one thing, the strength of many smaller nations' currencies depends on the strength of our dollar, which means that if confidence is lost in it, it could have a resounding effect throughout the world, multiplying the effect by several fold. [emphasis added]
As you say, that's an effect - that has nothing to do with your assertion that our economy functions differently.
At the same time, France has a much lesser effect on the strength of the Euro.
That's true, but it has the opposite effect that you are implying it has on France's economy: that works to stabilize and strengthen France's economy. So that doesn't support your assertion either: being small should make France's economy more stable and stronger.

#### Archon

russ_watters said:
That is only "true" if you ignore the fact that the economy is cyclical, make unrealistic estimates of it's future gains, and make promises for other Presidents to keep that Clinton wouldn't have kept (that's the best part about promises for when you're already out of office!). In short, such speculation is utterly meaningless.
Ok. Emphasis on "budget surplus" then? Is that better? What do we have now? Please don't tell me you're planning to argue that our current deficits are the result of our cyclical economy. Because to the best of my knowledge, it's also a fact that Bush has cut taxes and drawn us into war. But surely that has nothing to do with it...

And promises like what? Like "I'm not going to drive our country into massive debt?"

:rofl: :rofl: :rofl: :rofl: It works differently because it's bigger?
Umm....yes? Surely you weren't expecting me to say anything different?

That's absurd. It works differently because France's government is driving their economy into the ground.
You could argue that. But on an even more fundamental level, they're different because the U.S. economy is so much larger and more globally important than the French economy.

As you say, that's an effect - that has nothing to do with your assertion that our economy functions differently.
It's an effect that exists because of the fact that our economies function differently.

That's true, but it has the opposite effect that you are implying it has on France's economy: that works to stabilize and strengthen France's economy. So that doesn't support your assertion either: being small should make France's economy more stable and stronger.
Unless I'm mistaken, he's talking about the effect the state of the French economy has on the rest of the world. He's saying that this is small compared to the effect the state of the American economy has on the rest of the world. Naturally, since so much depends on the strength of the dollar, people are unlikely to allow the U.S. to amass a large GDP-to-debt ratio without acting in some way, often with negative effects on the American economy. The high GDP-to-debt ratio in France has a much smaller effect, so people don't care as much. Among other things. But that's one aspect of it.

#### Futobingoro

russ_watters said:
France's debt-to-gdp ratio is 68%.
The US's debt-to-gdp ratio is 24%.
I don't mean to be nitpicky, but the http://www.whitehouse.gov/omb/budget/fy2006/pdf/hist.pdf [Broken] is 65.7% of GDP for 2005. I might be missing some technicality here (public or federal debt), but the gross US debt is 65.7% of GDP.

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#### russ_watters

Mentor
Futobingoro said:
I don't mean to be nitpicky, but the http://www.whitehouse.gov/omb/budget/fy2006/pdf/hist.pdf [Broken] is 65.7% of GDP for 2005. I might be missing some technicality here (public or federal debt), but the gross US debt is 65.7% of GDP.
The technicality is that if you have the money in your own account, it's surplus in another area - so you subtract that from your debt to find your net debt. One column to the right of the one you were looking at..... (actually, my data may have been old - it's 26.5% for 2004).
Archon said:
But on an even more fundamental level, they're different because the U.S. economy is so much larger and more globally important than the French economy.
Could someone at least make an effort to prove that!?!? Just saying it and repeating it doesn't make it true.

Again, (your follow-up points), having a different effect on the world economy does not in any way imply that it functions differently.

Ie, the stat that I listed (debt to gdp) means that France needs to have some of the highest taxes in the developed world to finance it's debt, yet the citizens don't see that money....because it's going to finance the debt. That is a straightforward economic concept that works exactly the same if you are France, the US, or my neighbor Bob Smith who has 3 morgages on his house (caveat: Bob Smith doesn't exist, but the concept is the same for in-debt Americans). There are, of course, differences in how banks treat you if you are an individual or a country, but that doesn't change the fact that the underlying concept that debt reduces your disposable income.

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#### BobG

Homework Helper
Art said:
Some achievements of Bush that spring to mind are;
He declared a war on terror and succeeded in changing a small terrorist network with supporters numbered in the 1000s into a formidable international fighting force with supporters now in the millions and all for the paltry sum of several hundred billion dollars of U.S. tax payers money. Way to go!!
He's managed to undo 40 years of hard work the U.S. spent in building an excellent relationship with europe and all in just a few years. Quite an accomplishment.
He has alienated just about every muslim on the planet.
He's now turning his 'charm' on the Chinese with similar effects.
He has managed to eliminate the Clinton budget supluses and run up a record deficit by handing out trillions of dollars in tax cuts mainly to his already extremely rich friends.
He has increased american debt even further by borrowing every cent used to fight his personal crusade in Iraq for which he lied to the american people to gain their support. (This is in addition to the budget deficit as he hasn't funded this through the budget)
He has introduced torture as a 'repectable' means to glean information.
The list just goes on and on......... of Bush's 'achievements'
Not all of these are good examples.

You're overestimating the number of new terrorists Bush policies have created. The real impact is to move the problem from Chechnya and Afghanistan into Iraq. That's a result that's bad enough in its own right since fighting terrorists in backwater countries is better than fighting terrorists in the heart of the world's oil supply.

The economy is cyclic and has more to do with budget surpluses and deficits than presidential policies. The important thing is what a president does with the money during the surplus periods. Clinton at least had a reasonable plan for putting the money to good use - at least in January 1999:

Clintion '99 State of the Union speech said:
......Third, we must help all Americans, from their first day on the job -- to save, to invest, to create wealth. From its beginning, Americans have supplemented Social Security with private pensions and savings. Yet, today, millions of people retire with little to live on other than Social Security. Americans living longer than ever simply must save more than ever.

Therefore, in addition to saving Social Security and Medicare, I propose a new pension initiative for retirement security in the 21st century. I propose that we use a little over 11 percent of the surplus to establish universal savings accounts -- USA accounts -- to give all Americans the means to save. With these new accounts Americans can invest as they choose and receive funds to match a portion of their savings, with extra help for those least able to save. USA accounts will help all Americans to share in our nation's wealth and to enjoy a more secure retirement. I ask you to support them. (Applause.)

.....Saving Social Security, Medicare, creating USA accounts -- this is the right way to use the surplus. If we do so -- if we do so -- we will still have resources to meet critical needs in education and defense. And I want to point out that this proposal is fiscally sound. Listen to this: If we set aside 60 percent of the surplus for Social Security and 16 percent for Medicare, over the next 15 years, that saving will achieve the lowest level of publicly-held debt since right before World War I, in 1917. (Applause.)
Alas, he was no more successful in the middle of budget surpluses than Bush was six years later in the middle of budget deficits.

I think it is fair to blame Bush for making the deficits worse through an expensive war that has little impact on the overall problem of terrorism and for trashing the American image among Muslims, Europeans, Asians ... well, probably it would be easier to list the few countries whose image of America hasn't been tarnished by Bush.

#### Art

russ_watters said:
The technicality is that if you have the money in your own account, it's surplus in another area - so you subtract that from your debt to find your net debt. One column to the right of the one you were looking at..... (actually, my data may have been old - it's 26.5% for 2004).
Russ the reason why there is a difference between gross debt and nett debt is because the gov't has borrowed from itself. i.e. It has borrowed from monies put away to pay state pensions. This money will have to be repaid just like the rest of the debt or an awful lot of people are in for a nasty surprise when they retire. One of the things Clinton did when he was in power was refund the state pension scheme to safeguard it for the future but that has now been "reborrowed" by Bush.
BTW What did happen to the republicans' balanced budget bill??

#### Art

BobG said:
Not all of these are good examples.
You're overestimating the number of new terrorists Bush policies have created. The real impact is to move the problem from Chechnya and Afghanistan into Iraq. That's a result that's bad enough in its own right since fighting terrorists in backwater countries is better than fighting terrorists in the heart of the world's oil supply.
I didn't mention the number of terrorists I spoke about the number of people who support terrorism as popular support is the key to a terrorist network surviving and I think it is save to say these supporters can now be numbered in the millions.
BobG said:
The economy is cyclic and has more to do with budget surpluses and deficits than presidential policies. The important thing is what a president does with the money during the surplus periods. Clinton at least had a reasonable plan for putting the money to good use - at least in January 1999:
It is the presidents policies that determine whether or not there will be a budget surplus or deficit both directly by weighing income against expenditure and indirectly by creating a favourable environment for business to thrive and a motivation for people to work.
BobG said:
Alas, he was no more successful in the middle of budget surpluses than Bush was six years later in the middle of budget deficits.
I have no idea what you mean by this??
BobG said:
I think it is fair to blame Bush for making the deficits worse through an expensive war that has little impact on the overall problem of terrorism and for trashing the American image among Muslims, Europeans, Asians ... well, probably it would be easier to list the few countries whose image of America hasn't been tarnished by Bush.
The deficit exists because Bush inherited a balanced budget then drastically reduced the state's income through his tax giveaways whilst at the same time increased expenditure massively in other areas such as defence. Deficits don't just happen. They are a result of policy decisions.
He also scared off foreign investment by effectively devaluing the dollar, a tactic that destroys international confidence and is usually seen as an act of desperation in the financial markets.
And again I'd like to emphasise the current deficit doesn't even include the \$100s billions for the Iraq war as that has not been funded through the budget. The money for that has come from congress on separate bills.

#### phcatlantis

cronxeh said:
I wonder who will be the one who survives the collapse of the government and law n order once the Government debt reaches its tipping point?
Why debt? Why not debt-to-GDP?

#### edward

phcatlantis said:
Why debt? Why not debt-to-GDP?
Debt to GDP is no longer the standard indicator that it once was. Much of GDP is based on consumer spending, which previously would have indicated strong factory and industrial growth, however the products being purchased are to a great degee coming from other countries.

Much of the current consumer spending has involved the building boom. And as with all booms it can not maintain itself indefinitely.

Even worse a large portion of consumer spending has been done on credit. Consumer credit debt is at a record high in this country as is the national debt. At the same time, individual savings are dangerously low.

Debt to GDP is fine as a basic indicator, but a much bigger overall picture must be looked at when considering the long term economic outlook. Especially with a large pecentage of the national debt being owed to foreign countries, while the trade deficit with those countries continues to grow.

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#### BobG

Homework Helper
Art said:
BobG said:
The economy is cyclic and has more to do with budget surpluses and deficits than presidential policies. The important thing is what a president does with the money during the surplus periods.
It is the presidents policies that determine whether or not there will be a budget surplus or deficit both directly by weighing income against expenditure and indirectly by creating a favourable environment for business to thrive and a motivation for people to work.
Next year's budget is based on a prediction of next year's economy, which affects how much tax money comes in. Even if the next year's economy could be predicted with better accuracy, having government expenditures fluctuate up and down with the economy wouldn't necessarily be the best policy. In other words, a few years with a budget deficit isn't that bad as long the surplus years are balancing the deficit years. It's better than government policies that lose any semblance in continuity since their funding fluctuates too rapidly.
Art said:
BobG said:
Alas, he was no more successful in the middle of budget surpluses than Bush was six years later in the middle of budget deficits.
I have no idea what you mean by this??
Bush wasn't the first to suggest privatization of Social Security in some form. Because of budget surpluses, Clinton was actually in a better position to accomplish some serious Social Security reforms, but hit the same brick wall Bush did. Even if it makes sense, it's a hard sell, regardless of who's president.
Art said:
The deficit exists because Bush inherited a balanced budget then drastically reduced the state's income through his tax giveaways whilst at the same time increased expenditure massively in other areas such as defence. Deficits don't just happen. They are a result of policy decisions.
With the economy taking a downturn, a reduction in the amount of taxes the government would take in shouldn't have come as a surprise. Without some serious budget cutting, there was going to be a budget deficit. In any event, it's a technicality. Bush pushing through tax cuts and starting an expensive war when budget deficits were already expected is still fiscally irresponsible.

The bottom line is that, without the Iraq war to complicate matters, Bush would rate as about average in his ability to deal with the economy.
Ironically, if the Iraq war weren't complicating matters, Bush probably would have lost re-election. Usually, a president that inherits a good economy is doomed to a single term, regardless of how he handles the economy, since a downturn that brings the economy back into line with long term averages is almost certain to hit at the most inopportune time.

#### X-43D

The key to end poverty for everyone imo is to stop excessive consumerism and to set limits to business monopolies and government power (crony capitalism).

#### cronxeh

Gold Member
X-43D said:
The key to end poverty for everyone imo is to stop excessive consumerism and to set limits to business monopolies and government power (crony capitalism).

In my humble opinion, the key to our survival as a civilization is to conserve our population numbers. In last 50 years our population has increased by 4 BILLION people. People want cars, house(s), spirits, foods, books, and various other commodities nowadays and with this boom of population we are destined to run out of natural resources as well as create a hazardous environment for the future populations.

#### Smasherman

cronxeh said:
In my humble opinion, the key to our survival as a civilization is to conserve our population numbers. In last 50 years our population has increased by 4 BILLION people. People want cars, house(s), spirits, foods, books, and various other commodities nowadays and with this boom of population we are destined to run out of natural resources as well as create a hazardous environment for the future populations.
My knowledge on this subject is sparce, but don't wealthy families tend to have less children? Kind of a brutal cycle if so.

#### cronxeh

Gold Member
Smasherman said:
My knowledge on this subject is sparce, but don't wealthy families tend to have less children? Kind of a brutal cycle if so.
well if you would draw parallel between intelligence and wealth, then I suppose it makes sense

#### TheStatutoryApe

cronxeh said:
well if you would draw parallel between intelligence and wealth, then I suppose it makes sense
Other parallel's you could draw might be; Wealthier people tend to be spoiled and selfcentered don't want to have to deal with children. Wealthier people have better access to birthcontrol and have an easier time with the financial part of getting an abortion. ect..

#### X-43D

Hard work and materialism encourage population growth. Poor people work harder to survive and therefore need more children. Only post-materialism can result in liberty and happiness for all.

#### edward

X-43D said:
Hard work and materialism encourage population growth. Poor people work harder to survive and therefore need more children. Only post-materialism can result in liberty and happiness for all.
Ironically we may have to have an economic collapse in order for a post-materialism era to happen.

#### Phrak

I wonder who will be the one who survives the collapse of the government and law n order once the Government debt reaches its tipping point? With raising national debt to both domestic and foreign entities,
the US Government will start printing more money and the inflation will take place. ...
You are 8 years behind.

After the tech wreck came the housing bubble. Someone who can answer as to why this occured has at least some credibility in predicting the future economy.

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Staff Emeritus


text:
# What are the possible rational roots of x^3-4x^2+x+2=0 and then determine the rational roots?

Apr 2, 2017

"Possible" rational roots: $\pm 1$, $\pm 2$

Rational root: $1$

Irrational roots: $\frac{3}{2} \pm \frac{\sqrt{17}}{2}$

#### Explanation:

Given:

${x}^{3} - 4 {x}^{2} + x + 2 = 0$

By the rational roots theorem, any rational roots of this cubic are expressible in the form $\frac{p}{q}$ for integers $p , q$ with $p$ a divisor of the constant term $2$ and $q$ a divisor of the coefficient $1$ of the leading term.

That means that the only possible rational roots are:

$\pm 1$, $\pm 2$

In addition, notice that the sum of the coefficients is zero. That is:

$1 - 4 + 1 + 2 = 0$

Hence we can tell that $x = 1$ is a zero and $\left(x - 1\right)$ a factor:

$0 = {x}^{3} - 4 {x}^{2} + x + 2$

$\textcolor{w h i t e}{0} = \left(x - 1\right) \left({x}^{2} - 3 x - 2\right)$

The remaining roots can be found using the quadratic formula.

$x = \frac{3 \pm \sqrt{{\left(- 3\right)}^{2} - 4 \left(1\right) \left(- 2\right)}}{2 \cdot 1} = \frac{3}{2} \pm \frac{\sqrt{17}}{2}$

format: plain


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# Flushright or left of a long Table [duplicate]

I have a very long table that cannot be displayed on the page because it is too long and the right hand side of the table has been cutoff.

How can I modify the \begin{center} to something else so that the whole table will flushright or flushright?

\begin{widetext}
\begin{center}
\begin{table} [!h]
\begin{tabular}{|c||c|c|}
\hline
$\omega$ &  $S$  & Induced   \\[0mm]  \hline \hline
\end{tabular}
\caption{ The shorthand of ... }
\label{}
\end{table}
\end{center}
\end{widetext}

## marked as duplicate by Werner, Heiko Oberdiek, Adam Liter, cmhughes, JesseApr 28 '14 at 3:18

You can use adjustbox package and it adjustbox environment. Use max width= <width> so that the table is resized only if it exceeds that width.

\documentclass{article}
\usepackage{showframe}  %% just for demo
\begin{document}
\begin{table} [!h]
\begin{tabular}{|c||c|c|}
\hline
$\omega$ some dummy &  $S$ some dummmy text to make this long & Induced some more dummy   \\[0mm]  \hline \hline
\end{tabular}
\caption{ The shorthand of ... }
\label{}
\end{table}

\end{document}

On the other hand you can use a box.

\documentclass{article}
\usepackage{showframe}  %% just for demo
\begin{document}
\begin{table} [!h]
\makebox[\textwidth][c]{%
\begin{tabular}{|c||c|c|}
\hline
$\omega$ some dummy &  $S$ some dummmy text to make this long & Induced some more dummy   \\[0mm]  \hline \hline
\end{tabular}
}
\caption{ The shorthand of ... }
\label{}
\end{table}

\end{document}

• THis is an awesome answer, thank you! +1 – wonderich Apr 28 '14 at 2:14


text:
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# how are sampling error and standard error related Heltonville, Indiana

Linked 11 Why does the standard deviation not decrease when I do more measurements? 1 Standard Error vs. So how do we calculate sampling error? According to a differing view, a potential example of a sampling error in evolution is genetic drift; a change is a population’s allele frequencies due to chance. more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed

Because n is in the denominator of the standard error formula, the standard error decreases as n increases. Imagine that you did an infinite number of samples from the same population and computed the average for each one. The real value (in this fictitious example) was 3.72 and so we have correctly estimated that value with our sample. « PreviousHomeNext » Copyright �2006, William M.K. If you measure the entire population and calculate a value like a mean or average, we don't refer to this as a statistic, we call it a parameter of the population.

The standard deviation is most often used to refer to the individual observations. Since sampling is typically done to determine the characteristics of a whole population, the difference between the sample and population values is considered a sampling error.[1] Exact measurement of sampling error Even though all three samples came from the same population, you wouldn't expect to get the exact same statistic from each. To do this, we use the standard deviation for our sample and the sample size (in this case N=100) and we come up with a standard error of .025 (just trust

Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statement Mobile view ENN home Field Exchange Nutrition Exchange en-net Our work Resources Facebook Twitter Change language: English Français Log in to Because to construct it we would have to take an infinite number of samples and at least the last time I checked, on this planet infinite is not a number we The standard error is also related to the sample size. Standard Error of Sample Estimates Sadly, the values of population parameters are often unknown, making it impossible to compute the standard deviation of a statistic.

Why did Moody eat the school's sausages? Statistic Standard Error Sample mean, x SEx = s / sqrt( n ) Sample proportion, p SEp = sqrt [ p(1 - p) / n ] Difference between means, x1 - This makes $\hat{\theta}(\mathbf{x})$ a realisation of a random variable which I denote $\hat{\theta}$. If you take a sample that consists of the entire population you actually have no sampling error because you don't have a sample, you have the entire population.

AP Statistics Tutorial Exploring Data ▸ The basics ▾ Variables ▾ Population vs sample ▾ Central tendency ▾ Variability ▾ Position ▸ Charts and graphs ▾ Patterns in data ▾ Dotplots Browse other questions tagged mean standard-deviation standard-error basic-concepts or ask your own question. Because we need to realize that our sample is just one of a potentially infinite number of samples that we could have taken. The standard error of the mean estimates the variability between samples whereas the standard deviation measures the variability within a single sample.

The phrase "the standard error" is a bit ambiguous. If you go up and down (i.e., left and right) one standard unit, you will include approximately 68% of the cases in the distribution (i.e., 68% of the area under the Burns, N & Grove, S.K. (2009). Both SD and SEM are in the same units -- the units of the data.


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# Confusion with the gain concept of an amplifier

Before asking my question I want to briefly write about the basic meaning of the power gain.

Below is a model of an amplifier:

It shows the source input and amplifier output from left to right. In the middle there is the amplifier with the voltage gain A.

For simplicity above, both the source and amplifier output impedance is zero.

So the voltage at the input of the amplifier is exactly v1. And the voltage at the amplifier output is v2 = A*v2.

For the input and output power hence p1 and p2 we can write:

$$p1 = v1^2/Z1$$ $$p2 = v2^2/Z2$$

Now the power gain is defined using 10 based log:

Power gain(dB) = $10*\log(\frac{p2}{p1})$

which can be rewritten as:

$$10\log[\frac{\frac{v2^2}{Z2}}{\frac{v1^2}{Z1}}]$$

So finally we can write it as:

Power gain(dB) = $$20\log(\frac{v2}{v1}) + 10\log\frac{Z1}{Z2}$$

My question:

Now the second part of the last power gain equation confuses me. It is because, according to that equation the gain is impedance dependent. Only if Z1 = Z2 we can omit it.

But when we have an amplifier such made with an opamp or a single transistor why we never talk about Z1 and Z2 when it comes to gain? We say "the gain of the amplifier is such". Imagine an amplifier made with an opamp using negative feedback, I cant see in the gain equation Z1 = Z2. I guess if this model were for an opamp Z1 would be the input impedance and Z2 would be the total load at the output.(?) But seems we do not take them into account when we derive the power gain.(?)

I hope I could express where I'm stuck at clearly.

• We say "the gain of the amplifier is such" but what we really mean is a voltage gain "A = V2/V1"
– G36
Sep 11, 2018 at 15:20
• But some times I see made amplifier modules and their gain is defined as dB in datasheet. In such case if the gain is given as 20dB, does that mean they are actually talking about voltage gain v2/v1=10? Sep 11, 2018 at 15:24
• Yes, the 20dB gain is 10V/V of a voltage gain.
– G36
Sep 11, 2018 at 15:26
• Fun fact. There is no universally accepted definition for gain. There is no universally accepted units of measurement for gain. The gain isn't always dimensionless: it cal be in [V/A] in some cases or [A/V] in some cases. Gain is coefficient which is dependent on nature and application of a given circuit. [source: The Art of Electronics, by memory] Sep 11, 2018 at 15:40
• Z1 can be almost infinite these days. Think about what that does. Sep 11, 2018 at 17:52

But when we have an amplifier such made with an opamp or a single transistor why we never talk about Z1 and Z2 when it comes to gain?

It still needs to be taken into consideration, however in this diagram the sources do not have series impedance, they are infinite voltage sources (ideal).

In the real world however things are different:
Most often amplifiers are designed to have very low input bias current, and very high impedance. With op amps, it is usually uA's to fA's. With uA's you wouldn't need to worry about pulling the supply down unless the source impedance was more than 10KΩ's and most sources have source impedance's in the ohms or lower range (they can source 10mA or more).

I guess if this model were for an opamp Z1 would be the input impedance and Z2 would be the total load at the output.(?)

Z1 would be the input impedance (or for an op amp more commonly listed is the input bias current). This is an ideal amplifier, and it's output is an ideal voltage source. A real op amp also has source resistance/impedance. (it also does not have infinite gain)

The power gain can be near infinite (or infinite) for an ideal voltage source as described above, which isn't unusual if you think about it, an ideal voltage source pulls power out of nowhere and is not limited, so in the example above, you can have insane values for power gain.

As far as the power gain goes for an op amp, it is limited by the op amps open loop gain $A_v$ and the output impedance

Voltage gain: $$A_v = \frac{V_o}{V_i}$$

Power gain: $$\frac{P_{out}}{P_{in}} = \frac{\frac{V_{out}^2}{R_{out}}}{\frac{V_{in}^2}{R_{in}}}=A_v^2\frac{R_{in}}{R_{out}}$$

Power Gain (in dB): $$10\log{(A_v^2\frac{R_{in}}{R_{out}})}$$

But Rin will always be larger than Rout with almost every amplifier in the real world (output impedance is usually under 10Ω and input impedance is usually over 100k) so you will always have power gain usually in the tens of dB.

Here is an example of a power gain for an op amp:
Source: https://e2e.ti.com/blogs_/b/analogwire/archive/2013/07/10/how-to-determine-power-gain-and-voltage-gain-in-rf-systems

• But very very high input impedance makes the power gain huge. Look at the second part of the last equation 10log(Z1/Z2). You say Z1 is huge like 100Meg and Z2 can be kOhms. This makes the power gain huge. Is that correct? Sep 11, 2018 at 16:15
• Sorry, missed that, edited. Yes correct Sep 11, 2018 at 17:22

When talking about simple amplifiers the "gain" parameter is usually the voltage gain, not the power gain.

And, yes, the power gain is very dependent on the load impedance. Without a load no power will be delivered. If the output is shorted, no power will be delivered.


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## anonymous one year ago Two electrically-charged spheres are suspended from insulated threads a certain distance from each other. There is a certain amount of electrostatic force between them. Describe specifically (not just increase or decrease) what happens to this force in each of the scenarios below The charge on one sphere is reduced by half The charge on both spheres is doubled The distance between the spheres is increased by a factor of three The distance between the sphere is decreased to one-fourth The charge of each sphere is doubled and the distance between them is doubled

1. Abhisar

brb

2. Abhisar

There?

3. shamim

Force between two electrically charged objects is F=k(q1*q2/r^2)

4. shamim

Frm my equation If charge of any 1 sphere is reduced to half then force between them will b half of the previous force

5. Michele_Laino

let's suppose that Q_1 is the charge on sphere #1, and Q_2 is the charge on sphere #2, then the electrostatic force has the subsequent magnitude: $F = k\frac{{{Q_1}{Q_2}}}{{{d^2}}}$ where d is the distance between the spheres

6. Michele_Laino

first scenarios: the new charge on sphere #1 is Q_1/2, so we have: ${F_1} = k\frac{{\left( {{Q_1}/2} \right){Q_2}}}{{{d^2}}} = ...$

7. Michele_Laino

scenario*

8. Michele_Laino

second scenario: the new charges are: 2Q_1 and 2Q_2 on the spheres #1 and #2 respectively. The new force is: ${F_2} = k\frac{{{{\left( {2Q} \right)}_1}\left( {2{Q_2}} \right)}}{{{d^2}}} = ...$

9. Michele_Laino

third scenario: the new distance is 3d, so the new force is: ${F_3} = k\frac{{{Q_1}{Q_2}}}{{{{\left( {3d} \right)}^2}}} = ...$

10. Michele_Laino

fourth scenario: the new distance is d/4, so the electrical force is: ${F_4} = k\frac{{{Q_1}{Q_2}}}{{{{\left( {d/4} \right)}^2}}} = ...$

11. Michele_Laino

fifth scenario: the new charges are 2Q_1 and 2Q_2 on the spheres #1 and #2 respectively, furthermore the new distance is 2d, so the electrostatic force is: ${F_5} = k\frac{{{{\left( {2Q} \right)}_1}\left( {2{Q_2}} \right)}}{{{{\left( {2d} \right)}^2}}} = ...$

12. Michele_Laino

hint: in each of five scenarios, you have to express the new electrostatic force, namely F_1, F_2, F_3, F_4, and F_5 in terms of the force F

13. anonymous

ie. F_1 is one-half the magnitude initial electrostatic force, F_2 is four times the magnitude initial electrostatic force, F_3 is one-ninth the magnitude initial electrostatic force, F_4 is sixteen times the magnitude initial electrostatic force and F_5 is equal in magnitude to the initial electrostatic force.


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# What is the future value of $7,890 at the end of 5 periods at 8% compounded interest? What is the pr 1 answer below » What is the future value of$7,890 at the end of 5 periods at 8% compounded interest?

What is the present value of $7,890 due 9 periods hence, discounted at 6%? What is the future value of 16 periodic payments of$7,890 each made at the end of each period and compounded at 10%?

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# Thread: Tricky Line Integral

1. ## Tricky Line Integral

integral P of (x + y)dx + dy, where P is given by g(t) = (t,t^2), 0 <= t <= 1.

2. $\displaystyle x = t \implies dx = dt, y = t^2 \implies dy = 2t\,dt$.

Therefore

$\displaystyle \int_P{[(x + y)\,dx + dy]} = \int_0^1{[(t + t^2)\,dt + 2t\,dt]}$

$\displaystyle = \int_0^1{(3t + t^2)\,dt}$.

Go from here.


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RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Computer Optics: Year: Volume: Issue: Page: Find

 Computer Optics, 2016, Volume 40, Issue 1, Pages 121–129 (Mi co123)

IMAGE PROCESSING, PATTERN RECOGNITION

Multi-sensor systems for monitoring access to restricted areas: capabilities of the intrusion detection video analytical channel

B. N. Epifanceva, A. A. Pyatkovb, S. A. Kopeykina

b LLC “MAGMA-computer”

Abstract: The analysis of research publications dealing with security of critical facilities and their protection from terrorist threats suggests that the multisensor systems proposed in the literature need to be supplemented with a video analytical system to enhance the reliability of detecting the intrusion to the restricted access area and recognition of intruders' intentions. An algorithm for solving the first part of the problem, based on the implementation of the principle of accumulation, is proposed. The issue of recognizing the intruders' intentions is partially addressed. The probabilities of an error of the first and second kind for the proposed intrusion detection algorithm are estimated.

Keywords: restricted access area, intrusion detection, video analytical system, the principle of accumulation, detection rates.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-48-04172_ð_ñèáèðü_à The work is partially funded by the Government of the Omsk Region and RFBR grant No. 15-48-04172/15 of 05.07.2015.

DOI: https://doi.org/10.18287/2412-6179-2016-40-1-121-129

Full text: PDF file (442 kB)
Full text: http://www.computeroptics.smr.ru/.../400116.html
References: PDF file   HTML file

Revised: 22.01.2016

Citation: B. N. Epifancev, A. A. Pyatkov, S. A. Kopeykin, “Multi-sensor systems for monitoring access to restricted areas: capabilities of the intrusion detection video analytical channel”, Computer Optics, 40:1 (2016), 121–129

Citation in format AMSBIB
\Bibitem{EpiPyaKop16} \by B.~N.~Epifancev, A.~A.~Pyatkov, S.~A.~Kopeykin \paper Multi-sensor systems for monitoring access to restricted areas: capabilities of the intrusion detection video analytical channel \jour Computer Optics \yr 2016 \vol 40 \issue 1 \pages 121--129 \mathnet{http://mi.mathnet.ru/co123} \crossref{https://doi.org/10.18287/2412-6179-2016-40-1-121-129}


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## Dot product arithmetic

If you square the magnitude of a vector you get the dot product, correct?

||v||^2 = v . v

Can you also say that

||v|| = sqrt(v . v)?
 PhysOrg.com physics news on PhysOrg.com >> A quantum simulator for magnetic materials>> Atomic-scale investigations solve key puzzle of LED efficiency>> Error sought & found: State-of-the-art measurement technique optimised
 Recognitions: Gold Member Of course, basic algebra.

 Quote by Vorde Of course, basic algebra.
Thanks. I didn't know if some weird cosine rule existed in there

Recognitions:
Gold Member

## Dot product arithmetic

Okay. Just to cement this:

If ##\vec{v} = <a,b>## and ##\vec{w} = <c,d>## then ##\vec{v} \cdot \vec{w} = ac+bd## and ##\vec{v} \cdot \vec{v} = a^2+b^2##

So if ##|| \vec{v} || ^2 = \vec{v} \cdot \vec{v} = a^2+b^2## then ##\sqrt{|| \vec{v} || ^2} = || \vec{v} || = \sqrt{a^2+b^2}##
 Hey cytochrome. If the inner product is valid then all of your statements are true.

 Quote by cytochrome Thanks. I didn't know if some weird cosine rule existed in there
cosine rule cannot bother you here because the angle between "vectors" is zero.


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# Two unrelated equations, $w^2 = -\frac{15}{4} - 2i$ and $z^2 - (3-4i)z + (2-4i) = 0$?

I am to solve the equation $z^2 - (3-4i)z + (2-4i) = 0$, and also have to help me that $w^2 = -\frac{15}{4} - 2i$.

I can ofcourse find $w$, but I fail to see how this helps me in solving $z^2 - (3-4i)z + (2-4i) = 0$. What am I missing? Is there a connection I fail to see?

-
Try completing the square. –  Hans Lundmark Sep 13 '12 at 9:23

$z=\frac{3-4i±\sqrt{(3-4i)^2-4\cdot 1\cdot (2-4i)}}{2}$

Now, $(3-4i)^2-4\cdot 1\cdot (2-4i)=-15-16i=1^2+(4i)^2-2\cdot 1\cdot 4i=(1-4i)^2$

So, $z=\frac{3-4i±(1-4i)}{2}=1$ or $2-4i$

$4w^2=-15-8i=(1)^2+(4i)^2-2\cdot 1\cdot 4i=(1-4i)^2$

$2w=±(1-4i)$

Alternatively, let $-\frac{15}{4}-2i=(a-ib)^2$

So, $-\frac{15}{4}-2i=a^2-b^2-i2ab$

$a^2-b^2=-\frac{15}{4}$ and $2ab=2$

$(a^2+b^2)^2=(a^2-b^2)^2+4a^2b^2=(a^2-b^2)^2+(2ab)^2=\frac{289}{16}$

$a^2+b^2=\frac{17}{4}$

$a^2=\frac{1}{4},b^2=4$

So,$a=±\frac{1}{2},b=±2$ as $a,b$ are of same sign.

The square root of $(3-4i)^2-4\cdot 1\cdot (2-4i)=-15-16i$ can be calculated in the same way.

-


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# pyrk.inp.validation module¶

pyrk.inp.validation.validate_g(valname, val, llim)[source]

Raises errors if the value is less than the lower limit (llim) or if it is of the wrong type

Parameters: valname (string) – the name of the value being validated val (should be a number (or pint.Quantity)) – the value to be validated llim (the same type as val) – the lower limit of acceptable value for val
pyrk.inp.validation.validate_ge(valname, val, llim)[source]

Raises errors if the value is less than the lower limit (llim) or if it is of the wrong type

Parameters: valname (string) – the name of the value being validated val (should be a number (or pint.Quantity)) – the value to be validated llim (the same type as val) – the lower limit of acceptable value for val
pyrk.inp.validation.validate_le(valname, val, ulim)[source]

Raises errors if the value is greater than the upper limit (ulim) or if it is of the wrong type

Parameters: valname (string) – the name of the value being validated val (should be a number (or pint.Quantity)) – the value to be validated ulim (the same type as val) – the upper limit of acceptable value for val
pyrk.inp.validation.validate_not_none(valname, val)[source]

Checks that val is not None

Parameters: valname (string) – the name of the value being validated val (any) – the value to be validated
pyrk.inp.validation.validate_num(valname, val)[source]

Checks that val is a number

Parameters: valname (string) – the name of the value being validated val (should be a number (or pint.Quantity)) – the value to be validated
pyrk.inp.validation.validate_supported(valname, val, supported)[source]

Checks that val is not None

Parameters: valname (string) – the name of the value being validated val (any) – the value to be validated supported (list, tuple, anything with “in” functionality) – a list or tuple of allowed values for val


text:
Applied Particle Physics in Energy Research

# Drone Technologies

An experimental propellerless drone (potentially purely electric) based on conventional familiar physics in unfamiliar combinations (without propellers, without conventional rocketry, without electromagnetic radiation) has been under proprietary research and development since November 2015. If this drone can be demonstrated by proof of concept prototypes (initial versions already built & advanced) then the technology could be designed to lift arbitrary weight warehouse cargo (limited only by extent of onboard power) transporting them internally within warehouses. The proposed warehouse drones thus enable cargo re-organization and re-sorting within warehouses to minimize costs and optimize logistical and other benefits from automation.

Proprietary designs have been built into a series of prototypes which demonstrate need for both i) specific mechanical constraints as well as ii) electronic circuits for constant-speed feedback-loop control. Existing prototypes have revealed these features in designs which have been successively and successfully improved. No-cost access to or capital for further time & materials in relevant machine shops, including mechanical and electrical engineering, to continue prototype constructions for specific warehouse applications is sought.

Initial Capital to be deployed on:
• Mechanical precision machining
• Electrical Engineering
• Prototypes for warehouse drone applications

# Hydrogen From Water Technologies

Thermochemical Water Splitting
Initial Capital to be deployed on:
• Mechanical precision machining
• Electrical, Chemical Engineering
• Initial prototype
• Secondary prototype

## Hydrogen From Fossil Fuels Technologies

Thermochemical Hydrocarbon Splitting
Solid Oxide Fuel Cell (SOFC) CHP systems

## Hydrogen Cartridge Charger Technologies

Hydrogen Cell Phone Chargers

# Accelerator Transmutation of Nuclear Waste (ATW)

Spallation Neutrons from Fission Reactor
Vast majority of literature in ATW focuses on sATW

Fusion Neutrons from Tokamak Reactor
New growing literature in ATW focuses on tATW

Fusion Neutrons from Muon Catalyzed Fusion Target
No literature in ATW focuses on mATW but there should be and this is strongly advocated by General Scientifics
Initial Capital to be deployed on:
• Mechanical precision machining
• Electrical, Chemical, Civil Engineering
• Initial prototype
• Secondary prototype

# General Relativity

MIT-Mt Sunapee NH Time Dilation Measurement

Planetary Time Dilation Calculator

### Special Relativity

Light Speed Normalization

### Muon Physics Applications

Muon $$g-2$$: Anomalous Magnetic Moment of the Muon
Muonoscopy discovers new chambers in the Pyramids

### Utilities

Running CERN ROOT on Windows10

Physics Forums already has a nice thread posting machinery tool it is intended to duplicate here. A thread machinery tool is one where an initial user posts a comment with a title and block of text. Next, other users can comment on the post with each user's contribution appearing in a thread sequence with time stamp and user identification stamp for each. Physics Forums recommends 5 Free Javascript Utilities as a starting point.

General Scientifics is a general physics consulting practice focused on deriving solutions to technical challenges faced in industrial applications by means of applications of mathematical, classical, quantum, relativistic, molecular, atomic, nuclear and/or particle-level physics. General Scientifics makes available PhD experimental physics experience designing, constructing and operating custom vacuum chamber hardware, custom electronics hardware and custom software embedded in particle accelerator and detector equipment including analysis of data from experiments and their comparison with theoretical predictions at particle accelerator laboratories in USA (Bates (MIT), Fermilab (SSC), BNL, Wright (Yale)), Russia (BINP) and Japan (KEK). Pre-doctoral support was provided by the US Dept of Energy, Office of Basic Sciences, Particle Physics Division; Russian Academy of Sciences, Siberian Division and post-doctoral by the Japan Society for the Promotion of Science (JSPS) government agencies.

Recent and current experience includes calculations in:
• Relativity:
• mechanical acceleration (drone lift physics);
• basic hydrogen chemistry for hydrogen applications;
• the mATW variant of ATW for accelerator transmutation of nuclear waste.

### Contact

If you would like analyses and potentially solutions in your application fields please send a message so your technology problems can receive the proper attention.

Website created by General Scientifics IT Dept


text:
12:42 AM
There already is tag. And was recently created. Synonym? Keep them separate? Remove?
1

Let $A$ be a separable subset of a metric space $(M,d)$. Show that $\overline {A}$ is separable. Since $A$ is separable ,$A$ has a countable dense subset say $D$.But $\overline D=A$. I can't make $\overline D=\overline A$. What should I do? Please help me.

2 hours later…
2:44 AM
0

Very often when I see a edited question made by another person, I see that the editor didn't change the wrong tag, for example, it's very typical that questions about high school algebra are tagged with linear algebra or abstract algebra. Other times I see that the editor hasn't added a correct t...


text:
# 6. Consider the equation Σ F-mv2/r for the total force exerted on an object in uniform...

###### Question:

6. Consider the equation Σ F-mv2/r for the total force exerted on an object in uniform circular motion. From one trial to a) Then what should happen to the total force? b) Name the pattern of dependence of the total force on the speed To display this pattern of dependence by comparison to a straight line, a graph can be c) What is the theoretical slope of this graph? d) From one trial to another, suppose that the moving mass m is made two times largs plotted with TF on the y-axis and v2 on the x-axis. Then what should happen to the total force acting on it? e) Name the pattern of dependence of the total force on the mass. 30

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# CDS - Accumulated Default Risk?

Say I issue insurance contracts covering fire damage in personal households. Fires occur with a probability of $x$% in a household (and they obviously occur independently from one another). If the potential repair after a fire costs $P$ \$, then the fair value of such an insurance contract would be$x\%\cdot P\$.

Now with CDS we can't really apply this principle, because if the reference entity defaults, then everybody that holds a CDS on this entity will get paid out, i.e. the risk accumulates.

Am I right in assuming that this is indeed a real issue? If not, what is wrong with my reasoning. But if so, how is this issue circumvented in practice?

Most, but far from all, companies maintain a relatively steady debt load. When a bond matures, they fund its principal payout with a new bond.

Sometimes companies do take on more and more debt, meaning that CDS protection sold during earlier times of small debt loads becomes more valuable (and underpriced, from the point of view of the protection seller). Each increase in debt load is effectively a change in capital structure, which is an important and under-modeled part of quantitative finance.

Homeowners insurance runs some similar risks. A homeowner might significantly increase the value of his or her possessions while remaining under the same policy.

I originally thought you were asking about a key difference between CDS and fire insurance. Namely, there is not much risk of many homes burning all at once (modulo the occasional wildfire in the western USA). Corporate bankruptcy, in contrast, tends to happen in bunches.

The lack of independence is handled with copula models, where the event probabilities are translated to the probability of some quadrant of multidimensional space. Typically this quadrant is evaluated under gaussian or student-t distributions, where correlation is easy to define and compute.

There has been much debate in recent years about whether these models are truly capable of dealing with small quadrants, which correspond to tail events.

• It doesn't seem to me that the question was about correlation of defaults of multiple entities but about multiple CDS on a single entity. – AFK Aug 11 '14 at 22:43
• I agree, and will add a section on capital structure. – Brian B Aug 12 '14 at 10:58

There is no issue here.

In the world of insurance:

• An actuarial value is placed on the insurance by examining the probability of the fire event based on historically similar events and other input data.
• The event probability (per annum) times the expected size of the loss, plus some extra safety premium and costs, determines the annual premium that must be paid.
• The insurance company builds a reserve against this risk to cover itself in the event that a fire occurs.
• There is a distribution around the mean loss so that the losses in one year may be more than the expected loss. However insurance companies know about the law of large numbers which says that as the number of similar independent contracts written becomes large, deviations from the mean will become less likely. So they seek to write more contracts.
• They will also be required by Insurance Solvency requirements to hold an extra cash buffer against a large deviation in the number of fires.

The world of CDS is quite different:

• Every CDS trade has a buyer and a seller so a CDS moves credit risk - it does not create it
• The price of the CDS is determined by a market not by historical data.
• Market participants do pay attention to the historical performance of similar companies, or companies with similar ratings, but this is just one ingredient out of many inputs.
• Market prices respond to market news about the CDS reference entity
• Most market participants can hedge their position easily
• Most of the largest holders of CDS are dealers who are hedged.
• Those who do not hedge are investors who sell protection in order to earn a risk-adjusted return. They should be able to survive a credit event.

If a reference credit does default, then all contracts linked to that reference entity will be triggered. Many of the market players are hedged and so have no net loss. Those who are not hedged will make a gain or loss depending on their position. However, across the market the summed gains and losses will offset exactly.

For many of these reasons, the worlds of insurance and derivatives are quite different.


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# Determine whether the sequence converges or diverges. If it converges, find the limit.$a_n = \frac { \cos^2 n}{2^n}$

## converges

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for this problem will use the Squeezed Sam. So there thing that kind of gets tricky with this. A in term is this co sign squared of N to the coastline squared Event is the troublemaker here, and hopefully it's something that's bounded. And if he knows which is the case here, then you can use the squeeze there. So we bound this thing that's giving us some trouble. This's co sign squared of ends that's going to be bigger than or equal to zero and less than or equal to one. You can't be a negative number because this is co sign squared. Okay, so this is going to be something that's true for all in, and we can divide by two to the end. And it's not going to change these bounds either, because to the end is always going to be some positive. Number two, divided by a positive number, are multiplying by a positive number. We don't have to worry about flipping these things that are, and then we can put these limits on take the limit as n goes to infinity limit as n goes to infinity. No Lamma as n goes to Infinity case is on the far left. This is still zero. Now we have the limit in question here by construction. This was our way in term. And then on the right hand side, we have limited as n goes to infinity of one over to the end. So that is also going to go to zero. So this Lim is trapped between zero and zero for it squeezed between zero and zero. That's where the name the serum comes from. So the only logical conclusion is that limit, as in approaches, infinity of a N is actually equal to zero to our sequence does converge and it converges to zero.

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# Algebraic description of compact smooth manifolds?

Given a compact smooth manifold $M$, it's relatively well known that $C^\infty(M)$ determines $M$ up to diffeomorphism. That is, if $M$ and $N$ are two smooth manifolds and there is an $\mathbb{R}$-algebra isomorphism between $C^\infty(M)$ and $C^\infty(N)$, then $M$ and $N$ are diffeomorphic.

(See, for example, Milnor and Stasheff's "Characteristic Classes" book where an exercise walks one through the proof of this fact).

Thus, in some sense, all the information about the manifold is contained in $C^\infty(M)$.

Further, the tools of logic/set theory/model theory etc. have clearly been applied with greater success to purely algebraic structures than to, say, differential or Riemannian geometry. This is partly do the fact that many interesting algebraic structures can be defined via first-order formulas, whereas in the geometric setting, one often uses (needs?) second-order formulas.

So, my question is two-fold:

First, is there a known characterization of when a given (commutive, unital) $\mathbb{R}$-algebra is isomorphic to $C^\infty(M)$ for some compact smooth manifold $M$? I imagine the answer is either known, very difficult, or both.

Second, has anyone applied the machinery of logic/etc to, say, prove an independence result in differential or Riemannian geometry? What are the references?

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Someone edited this post, and in an effort to see exactly what was changed, I accidentally reverted it to it's current form. For whomever edited it, if you'd like to re-edit, feel free. –  Jason DeVito Nov 13 '09 at 19:09

There is a very cool answer to your question, and it goes by the name well-adapted models for synthetic differential geometry. Andrew Stacey already indicated it in his reply, but maybe I can expand a bit more on this.

Synthetic differential geometry is an axiom system that characterizes those categories whose objects may sensibly be regarded as spaces on which differential calculus makes sense. These categories are called smooth toposes.

A model for this is a particular such category with these properties. A well-adapted model is one which has a full and faithful embedding of the category of smooth manifolds. (This is "well adapted" from the point of view of ordinary differential geoemtry: ordinary differential geometry embeds into these more powerful theories of smooth structures).

The striking insight is that this perspective in particular usefully unifies the ideas of algebraic geometry with that of differential geoemtry to a grander whole.

Indeed, the category of presheaves on the opposite of (finitely generated) commutative rings is a model for the axioms, and of course this is the context in which algebraic geometry takes place.

But we are entitled to take probe categories considerably richer than just that of duals of commutative rings. In particular, we may consider a category of commutative rings that have a notion of being "smooth" the way a ring of smooth functions is "smooth". These are the C-infinity rings or generalized smooth algebras. Every ring of smooth functions on a smooth manifold is an example, but there are more.

The formal dual of these rings are spaces called smooth loci. This is a smooth analog of the notion of affine scheme. (Notice that the notion of "smooth" as used here is that of differential geometry, not quite that of algebraic geometry, which is more like "singularity free". But they are not unrelated).

The main theorem going in the direction of an answer to your question is that the category of manifolds embeds fully and faithfully into that of smooth loci. See at the link smooth loci for the details.

But inside the category SmoothLoci, manifolds are characterized as the formal dual to their smooth rings of functions, so that's one way to answer your question.

There is a grand story developing from this point on, but for the moment this much is maybe sufficient as a reply.

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Excuse me, is it possible to reformulate the conclusions of this theory for the question that Jason DeVito asked? Which properties distinguish $C^\infty(M)$ among other $\Bbb R$-algebras? –  Sergei Akbarov Apr 11 '13 at 12:27
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I would also have recommended this book, you may find the answer in it. Moreover you don't need compactness, you can always reconstruct the manifold from its algebra of smooth functions. –  Michael Bächtold Nov 13 '09 at 21:54

Obligatory n-lab reference: generalised smooth algebra. Lots of links to other pages and to other literature, in particular Moerdijk and Reyes' book Models for Smooth Infinitesimal Analysis.

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I'm no expert in this area and can't directly answer your questions, but it seems to me to be related to Connes' non-commutative approach to Riemannian geometry. He considers so-called "spectral triples" which consist of a (certain type of non-commutative) algebra acting on a Hilbert space together with an unbounded operator. One is supposed to think of this as the generalisation of $C^\infty(M)$, $L^2$-integrable spinors and the Dirac operator. The triple must satisfy certain conditions, inspired by the (commutative) manifold case. When the algebra is in fact commutative, I seem to remember that such spectral triples are precisely those arising from a manifold. (Experts: is this right?!)

I think I read about this in "Elements of non-commutative geometry" by but since this isn't really my area I have no idea if this is the best place to start.

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The algebra of a spectral triple being commutative alone does not guarantee yet that it comes from a Riemannian manifold, I think. But it is true that there is a way to characterize Riemannian manifolds in terms of spectral triples. It should be noted that the notion of spectral triple aims at encoding much more than smooth structure: the notion of spectral triple is really an algebraic characterization of Riemannian and (in its graded version) of Spin-structure, hence of metric information on a smooth space. –  Urs Schreiber Nov 13 '09 at 15:18

In certain cases (such as compact complex manifolds) good old fashioned relational approach is sufficient to apply model theory to geometry. One does not need the algebra of functions. You simply name all the submanifolds of your manifold and work with an uncountable language. In this setting not all models are manifolds but you can still prove theorems about manifolds. Here is a survey paper: http://arxiv.org/abs/math/0702468

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# Consider the following infinite series: \sum_{k=0}^\infty\frac{2^k}{1+\frac{1}{x^{2k}}}

Joan Thompson 2022-01-04 Answered
Consider the following infinite series:
$$\displaystyle{\sum_{{{k}={0}}}^{\infty}}{\frac{{{2}^{{k}}}}{{{1}+{\frac{{{1}}}{{{x}^{{{2}{k}}}}}}}}}$$

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## Expert Answer

Hattie Schaeffer
Answered 2022-01-05 Author has 3091 answers
To me, the sum can be derived informally as
$$\displaystyle{\sum_{{{k}={0}}}^{\infty}}{\frac{{{2}^{{k}}{x}^{{{2}{k}}}}}{{{1}+{x}^{{{2}{k}}}}}}={x}{\sum_{{{k}={0}}}^{\infty}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{\log{{\left({1}+{x}^{{{2}{k}}}\right)}}}\right]}$$
$$\displaystyle={x}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{\log{{\left({\prod_{{{k}={0}}}^{\infty}}{1}+{x}^{{{2}^{{k}}}}\right)}}}\right]}$$
$$\displaystyle={x}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{\log{{\left({\prod_{{{k}={0}}}^{\infty}}{\frac{{{1}-{x}^{{{2}^{{{k}+{1}}}}}}}{{{1}-{x}^{{{2}^{{k}}}}}}}\right)}}}\right]}$$
$$\displaystyle=-{x}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{\log{{\left({1}-{x}\right)}}}\right]}$$
$$\displaystyle={\frac{{{x}}}{{{1}-{x}}}}$$
The key is the summand in the original series is proportional to the derivative of logarithm of something simple. Once we turn the sum of log to a log of product and realize the product is a telescoping one, the rest is just following your nose.
###### Not exactly what you’re looking for?
usumbiix
Answered 2022-01-06 Author has 1667 answers
Let $$\displaystyle{\left|{x}\right|}{ < }{1}$$ and $$\displaystyle{k}\geq{0}$$. Using the factorization $$\displaystyle{1}-{x}^{{{2}^{{{k}+{1}}}}}={\left({1}-{x}^{{{2}^{{k}}}}\right)}{\left({1}+{x}^{{{2}^{{k}}}}\right)}$$, we find
$$\displaystyle{\frac{{{2}^{{k}}{x}^{{{2}^{{k}}}}}}{{{1}+{x}^{{{2}^{{k}}}}}}}={\frac{{{2}^{{k}}{x}^{{{2}^{{k}}}}{\left[{1}+{x}^{{{2}^{{k}}}}-{2}{x}^{{{2}^{{k}}}}\right]}}}{{{\left({1}+{x}^{{{2}^{{k}}}}\right)}{\left({1}-{x}^{{{2}^{{k}}}}\right)}}}}$$
$$\displaystyle={\frac{{{2}^{{k}}{x}^{{{2}^{{k}}}}}}{{{1}-{x}^{{{2}^{{k}}}}}}}-{\frac{{{2}^{{{k}+{1}}}{x}^{{{2}^{{{k}+{1}}}}}}}{{{1}-{x}^{{{2}^{{{k}+{1}}}}}}}}$$
So since $$\displaystyle\lim_{{{k}\to\infty}}{\frac{{{2}^{{k}}{x}^{{{2}^{{k}}}}}}{{{1}-{x}^{{{2}^{{k}}}}}}}={0}$$, the series telescopes to $$\displaystyle{\frac{{{x}}}{{{1}-{x}}}}$$
karton
Answered 2022-01-11 Author has 8659 answers

Let f(x) be the series. Rewriting, we get
$$f(x)=\sum_{k=0}^\infty\frac{2^kx^{2k}}{1+x^{2^k}}$$
Let $$s_n$$ be the n-th partial sum. We then have
$$\frac{x}{1-x}-s_n=\frac{x}{1-x}-\frac{x}{1+x}-...-\frac{2^nx^{2^n}}{1+x^{2^n}}$$
$$=\frac{2^{n+1}x^{2^{2^{n+1}}}}{1+x^{2^{n+1}}}$$
One can show that the above tends to 0 as $$n\to\infty$$ for |x|<1

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