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# Global Economics 101 - A Crash Course

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Global Economics 101 - A Crash Course [#permalink]  22 Jan 2008, 08:37
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I've seen a few posts on this site with people asking how to explain economics to an engineer, social sciences major, anyone without a business degree. Then my girlfriend (an engineer), after listening to all this hoopla about a recession, asked me: What are economics, how does the stock market work, what exactly is a recession?

With all the business guru's on this site - this is quite the gamble to post - but what the heck!

A couple warning's first:
This may not be 100% accurate
I use the "S" word a few times. You know the one. If the mod's don't like this, I can pull them.
It is supposed to be both entertaining and educational, enjoy!
It's LOOOOOOONG! Print it out and enjoy!

Ahhh... A little light conversation at 8:30am. Very nice... Do you have any questions about how a bike works? Or how to make chocolate chip pancakes? Or shit, why the sky is blue - even that is an easier question to answer!!!

I'll give it a shot - let's see if I stayed awake in those Econ classes!

Warning: This is a long read, and may not be 100% accurate, but if it was worth my time to write, it's worth your time to read!

The broad definition of a recession is 6 months of consecutive decline in the common market indicators - the most common being the GDP. The GDP or Gross Domestic Product is the value of all the shit manufactured or produced in this country, be it both products and services of American OR foreign companies (i.e. A Honda made in Ohio and sold in Nigeria effects the GDP and GNP). This is not to be confused with the GNP - the Gross National Product, the aggregate of all the shit manufactured by only American companies on U.S. soil AND abroad. (I.e. GM manufacturing a Buick in China, or Google setting up marketing offices in Iceland affects the GNP but not the GDP). Essentially, GDP is concerned with WHERE it was made, GNP is concerned with WHO made it. Back to this 6 month decline thing... Free market capitalism says that we should continually be buying more shit and more services perpetually. If you have too much shit to fit in your house - then you buy a bigger McMansion, further from work (thus resulting in the need to buy more shit, such as Gas, car parts, etc.). This is a theory which Bill Mckibbem (Deep Economy) argues. He states that more is not necessarily better, and that we should debit (that is charge) the GDP and GNP the environmental detriment an enterprise is causing (i.e. Charge $10 back to the GDP for each ton of carbon emissions created). Just some food for thought. Now, in 2008, we have had some big buzz word problems. Sub Prime Mortgages. "Prime" is a set "good rate" which people with "good credit" can borrow at. Sub Prime is a higher interest rate (paid by the individual, not received by the individual - so BAD!). The original idea was that Sub Prime Mortgages will give people money who otherwise couldn't have had access to it... The problem is, nobody stopped to think that perhaps it is not good to lend people money who can't afford to pay it back! "Predatory Lenders" (i.e. Rock Financial), are known to "sell" people on larger mortgages then they can afford. When people can't pay back the mortgage, the bank which backed the mortgage takes back the house. The problem is, Banks are not in business of being land lords - they don't want these homes! So, when a whole shit-load of people default on their mortgage, the banks are hurt, which drives down their stock price. Banks also handout these nifty little things called credit cards! When you load up a credit card, and can't pay it off, the bank is liable - it's BEAUTIFUL! The problem is, the bank not only doesn't want your McMansion, it doesn't want all the bull shit you stuffed in it. So the stock price tanks even further. Wait... what exactly is a stock anyway? Glad you asked... Stock prices are really a measure of consumer confidence (which, ironically, is another one of those "Market Indicators" I originally mentioned) which tied directly to a companies performance (and potential!). When people feel good about a company, they buy stock. When I say people - I really mean savvy investors. When savvy investors discover that Google is not only the most advanced word in Dubya's vocabulary, but also this insane computer company that is going to take over the world, they say, "shit, I wants to own me some of that!" Then the most basic economic principle of supply and demand kicks in. Since everyone wants to get them "some of that" - the price goes up! Any commodity is a good example of supply and demand - see oil for a perfect example... Another way a stock price can change is if the company issues more stock, or buy's stock back - this results in the same rules of supply and demand. Now, when those same Savvy Investors are holding onto all these bank stocks and mortgage house (lenders) stocks, and see that banks are having to cover for all these people who bought 3,000 sq. ft. homes, 90 minutes from their$30,000 a year jobs - they say, shit, this is gonna get ugly! SELL SELL SELL! Buy me some of them secure government bonds! So, the stock market goes down, and people buy more bonds...

Maybe you're asking why Uncle Sam doesn't step in to prevent all of this from happening... Actually, you probably weren't, but you should have been! First of all, in free market capitalism, the markets are self-regulating, meaning that it's not the government's role to step in. But in our capitalism, Uncle Sam still plays a role. It's actually not Uncle Sam - it's Uncle Ben. Ben Bernanke (the Federal Reserve Chairman) is the poor sap in charge of deciding what to do with the Federal Funds rate and how much money to print. Banks have to keep a certain amount of cold hard cash in a vault (a percentage - say 10%, of all the money they are holding for people). If banks had to actually keep 100% of the money given to them as cash, the system wouldn't work, because the Bank wouldn't be able to invest and make money! Back to the Federal Funds Rate... When banks need more money, Uncle Sam tells them to borrow from each other. If one bank has more than the 10% cash needed, they can lend to banks with less than the 10% cash needed. The amount of interest charged by the lending bank is loosely set by the Federal Funds Rate. If Uncle Ben wants to make it easier for banks to get money, he lowers the rate. Sometimes banks can say, "Screw that other bank! I want to borrow from you Uncle Sam!". This can be done, but is costs more money (higher interest rate) - this is called the "discount rate". I know, it's a pretty deceiving thing to call borrowing money at a higher interest rate, that Uncle Sam can be pretty sneaky! I mentioned that Ben can also print more money - a way to infuse money into the economy. My brother always said, I don't get it, if we have a deficit and need more money, why don't we just print more? Well, it's not that cut and dry. If we print more money, there is more supply, which means the cost (or value goes down). The global economy used to rely on the "gold standard" - this meant that economies needed to have hard, tangible, gold in a reserve to back all of the printed money. This is no longer the case. Economies and the value of currencies are now essentially based on the goodwill of the global economy. That is, the trust and faith that a government can support its currency through bonds and taxes.

So what is a bond? Aside from a shrewd English secret agent, they are also a way of raising money. Corporations can raise money in two ways - Stocks (selling actual "shares" of your company), and Bonds (Borrowing money, with the promise to pay it back with interest). Corporate bonds are ranked according to how likely it is that a company will be able to pay you back. If a bond rating agency doesn't think a company will be able to pay back the bond you bought, they say, "Man, that bond is Junk" and label it a junk bond. Junk bonds are "cheaper" to buy, so there is more potential reward to go along with your additional risk. Bonds are commonly issued by governments to raise money (since they can't sell "shares"). Bonds can be local (city bonds) all the way up to Federal Bonds (Uncle Sam). You can usually be pretty confident that Uncle Sam can pay you back. All he has to do is print more money -> which then results in the devaluation of our currency, which is a whole other ball of wax!

OK, uh, what was your original question? Oh yeah, recessions. Recessions are very finicky things. First of all they are very loosely defined. As I mentioned, the generally accepted definition is "negative growth" (what?), for 2 quarters. However, the NBER (National Bureau of Economic Research), is the U.S. governmental organization that actually defines when we are in a recession. They look at more than just GDP numbers, they also take into consideration: employment, corporate profits, consumer confidence, investment, etc. For the sake of sanity, just think about all of this as the GDP - since they are all complimentary (meaning when one goes up, the others usually do to - or in this case, when one goes down...).

Now that you are more confused then when we started - I'm going to throw another buzz word into the mix. The deficit - well, it's really "the deficits" (plural). When Ross Perot said, "Let's talk about the deficit", he was referring to the budget deficit. This will help answer your question about how we're paying hundreds of billions (some estimate the true cost to be trillions) of dollars on this war we're in. Well, it's easy - Uncle Sam says, "Just put it on the card". When the Government spends more money than it makes, we have a budget deficit. Since the world economy relies on America's gluttonous consumption of, well, everything, the deficit does not cause our currency to crash (but it does "hurt" it a little - see "goodwill"). OK, that's the budget deficit (the opposite is budget surplus - when the gov't makes MORE than it spends), but what's the other deficit? Good question! The other commonly discussed deficit is the trade deficit. The trade deficit is the net exports of a country. Uh, ok, that tells me a lot... Let me break it down. The net exports are simply all the shit we export minus all the shit we import. In the US, we have ran a trade deficit since 1991. This means we have imported more than we have exported. When you consider all the outsourcing of manufacturing to lower cost countries, and the amount of cheap crap we bring in, it's not hard to imagine how we got to this trade deficit.

So that's global economics in a nutshell... It sure is confusing. To bring it all together and bring us back to where we started, I'll leave you with this automotive question which puts the global economy in perspective. "Domestic Content" refers to the % of parts and labor which are from the U.S.

What is more patriotic, what supports the US economy more?

Buying a Ford Fusion made with a 30% domestic (US) content and assembled in Mexico
or
Buying a Toyota Camry made with 75% domestic (US) content and assembled in Georgetown, KY, USA

It's all a matter of perspective!
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Re: Global Economics 101 - A Crash Course [#permalink]  24 Jan 2008, 04:57
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Re: Global Economics 101 - A Crash Course [#permalink]  24 Jan 2008, 07:04
nice post. +5.
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Re: Global Economics 101 - A Crash Course [#permalink]  24 Jan 2008, 13:03
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Praetorian wrote:
nice post. +5.

Thanks Praet!
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Re: Global Economics 101 - A Crash Course [#permalink]  25 Jan 2008, 14:09

+1 from me. but thats like printing money and devalueing the Kudos currency. but what tha heck
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Re: Global Economics 101 - A Crash Course [#permalink]  26 Jan 2008, 13:49
I am an MA in Economics and still can't understand what the E word means

Don't worry about kudos inflation - inflation can be a good thing, especially to those who get the new money first.
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Re: Global Economics 101 - A Crash Course [#permalink]  06 Feb 2008, 16:51
a very good post with full of good "shit" +1
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Re: Global Economics 101 - A Crash Course [#permalink]  09 Apr 2008, 10:12
I like your approach of treating the readers like newbies. Works really well!

Can you explain how inflation works?
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Re: Global Economics 101 - A Crash Course [#permalink]  10 Apr 2008, 00:13
zhenmaster wrote:

Can you explain how inflation works?

Inflation = increasing amounts of fiat paper money chasing a finite amount of goods and services. when there is a lot of money available in the system, it has to go somewhere, and the demand for goods and services often created from the increasing amount of money causes the prices of those goods and services to rise.
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Re: Global Economics 101 - A Crash Course [#permalink]  10 Apr 2008, 06:33
Great post jwiner. Kudos!
Re: Global Economics 101 - A Crash Course   [#permalink] 10 Apr 2008, 06:33
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# Global Economics 101 - A Crash Course

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# Tommi's profile - activity

 2019-09-27 17:57:53 -0600 asked a question Conjugate multiplication of square root Is there a simple way to simplify a formula using conjugate multiplication of the square roots? For example, when I perform var('a,b,d') exp = 1/(a+b*sqrt(d)) exp.full_simplify() I would like to get (b*sqrt(d) - a)/(b^2*d - a^2) but what I actually get is just the form that I started with. Even if I specify the assumptions assume(d,'real') assume(d>0) the conjugate multiplication does not happen automatically. I would like to be able to tell Sage, that I want the conjugate multiplication. In some cases that are relevant to me, the conjugate multiplication would simplify my expressions significantly.


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# How does light carry heat

Discussion in 'Physics & Math' started by Magical Realist, Mar 3, 2017.

1. ### DaveC426913Valued Senior Member

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No.

Einstein won the Nobel prize for the photoelectric effect - the ability for light to give a physical kick to matter.

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3. ### Magical RealistValued Senior Member

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Force = mass x acceleration?

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5. ### DaveC426913Valued Senior Member

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While that is one formula, that is not an exhaustive definition of force.

Here is another, relating frequency of a photon to its maximum kinetic energy:
Kmax = h(ƒ − ƒ0)

Last edited: Mar 4, 2017
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7. ### Q-reeusValued Senior Member

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You've had some answers already on that one. I had assumed that at an early stage in any radar techie course, a chart showing the full EM frequency/wavelength spectrum - ELF radio up to gamma rays - would have been mandatory. Can't recall that happening? No intro to Maxwell's equations?

8. ### exchemistValued Senior Member

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I wonder if it may be helpful here to mention that an electric charge can give a "kick' to another charge, or a magnet can give a "kick" to a steel ball, without being in contact with it, by virtue of the field it gives rise to. Which creates an "F", for F=ma, without physical contact.

A photon gives a "kick" to an electron by means of its oscillating electric field (usually, i.e. in electric dipole processes).

9. ### Magical RealistValued Senior Member

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Yes..we went over that chart especially as it pertained to HF, LF, and ELF transmissions. But there was no physics level math or delving into Maxwell's equations. We were just techies, not engineers. Component level trouble shooting, soldering, signal tracing on schematics, test equipment usage, and circuit board swapping was about the whole of it. Then it was liberty call!

Last edited: Mar 4, 2017
10. ### DaveC426913Valued Senior Member

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I asked the Mods to delete some off-topic posts.

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11. ### rpennerFully WiredStaff Member

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There's been some mixing of different physical theories here.

Newtonian Mechanics v. Special Relativity
$\vec{F} = m \vec{a} = m \frac{d\vec{v}}{dt}$ is not a law of nature, it is Newton's definition of a force. Only when there is no net force will bodies move without acceleration.
But non-zero acceleration is tricky in relativity because it depends on one's notion of time in a complicated fashion. Attempts to continue to use this equation in special relativity leads to weird claims that the inertial mass of the body has a speed-dependent mass if you apply the force parallel to the motion and a different speed-dependent mass if you apply the force perpendicular to the motion, which is nonsensical since relativity is the statement that there is no such thing as absolute motion.
So proper treatments of Special Relativity recognize that Newton was really just talking about conservation of momentum (p) and redefine $\vec{F} = \frac{d\vec{p}}{dt}$ which works one way in Newtonian Mechanics and in a different way (compatible with Newton at low speeds) in Special Relativity where the frame-dependent value of momentum for a free body is expressed as:
$\vec{p}^2 = \frac{E^2}{c^2} - m^2 c^2, \; \vec{p} = \frac{ E \vec{v}}{c^2}$
Using simple algebra, if $m = 0$ this is saying: $m = \frac{E}{c^2} \sqrt{1 - \frac{\vec{v}^2}{c^2}} = 0, \; | \vec{v} | = c, \; \vec{p} = \frac{E}{c} \frac{\vec{v}}{ | \vec{v} | } = \frac{E}{c} \hat{v}$
But if $m > 0$, then it follows $m = \frac{E}{c^2} \sqrt{1 - \frac{\vec{v}^2}{c^2}} > 0, \; | \vec{v} | < c, \; \vec{p} = \frac{E}{c^2} \vec{v} = \sqrt{ \frac{E^2}{c^2} - m^2 c^2 } \hat{v} = \left( 1 - \frac{ \vec{v}^2 }{ c^2} \right)^{- \frac{1}{2}} m \vec{v}$
So in Special Relativity, massless bodies traveling at the speed of light can carry non-zero energy and momentum.

Maxwell's equations v. Quantum Electrodynamics
Maxwell in 1865 described the behavior common to experiments with electricity and magnetism and discovered that disturbances propagated changes in forces at a speed faster than any known terrestrial phenomena save one, light. Since then radio confirmed these predictions. QED is a modern synthesis of Maxwell's equations, and Quantum Field Theory (which includes a synthesis of Special Relativity, Quantum Mechanics).
QED deals with details much better when atoms (with quantum effects) and high-frequency light (where per-particle energy is significant).
While Maxwell's equations would say an electromagnetic wave causes electrons to jiggle and jiggling electrons cause electromagnetic waves to be generated, QED describes charged particles like electrons absorbing and emitting photons when this would conserve energy, momentum and angular momentum. This is why somethings are transparent to light: because there is no way photons of the specific energies associated with visible light can be absorbed by a electron bound to an atom without a violation of conservation laws.

Asking classical questions about individual photons is almost always wrong-headed because of the many ways quantum mechanics and quantum field theory enter into the discussion.

QED's origins start with Planck's black body law and Einstein's explanation of the the photoelectric effect. The first says hot things emit light and can be understood as the intersection of QED and Statistical Mechanics.

Classical Thermodynamics v. Statistical Mechanics
So now we want to learn how light makes things warm.

Heat is a way energy presents itself. Thermodynamic equilibrium means every degree of freedom a system has to hold energy has the statistical expectation of holding the same amount of energy (equipartition). So when a beam of light falls on a non-transparent object, some of that energy is absorbed initially as kinetic energy. But physical objects are complicated collections of atoms, so this energy can eventually show up in a chemical reaction, an internal state change of a molecule, a vibration of a molecule, an acoustic vibration, a net separation of atoms in a body, etc. The statical expectation of the measure of any one way the body has of holding energy just got a little higher. That's an increase in heat.

Planck's black body law says the most untransparent objects there are, those that are "black" to every frequency of light, don't just absorb light, but emit it too. (This makes sense because even Maxwell's equations have said that the EM field-electrically charged body relationship is not a one-way one.) So when the details of equipartition are worked out, the object is trying to reach thermodynamic equilibrium with the surrounding electromagnetic field. Human bodies glow in the infrared, electric stove elements and toaster wires glow red, the plasma of the photosphere of the sun is white hot, and some stars are brighter in UV than visible light.

Light from hot things (the Sun, fire, tungsten filament lights, etc) has a broad, smooth spectrum because every frequency gets a chance for equipartition. In contrast, the fluorescent compounds in a light of the same name, low pressure sodium or mercury lights, chemiluminescent reactions, LEDs and lasers use quantum effects of atoms and molecules and bulk materials to emit light which is concentrated strongly at certain frequencies. So hot things glow, but not all things that glow are hot.

Microwave ovens (915 MHz/2.45GHz) and Radar ( 10 MHz - 110GHz) can specifically cause water molecules to torque around in their hydrogen bond networks and transfer motion to nearby molecules. That's why microwaves don't work well on fatty or very dry foods (not enough water/polar molecules) or large solid things (like roast of smallest dimension of 6 cm or more) (microwaves absorbed in outer layers leaving center cold). But this is a bulk electrostatic effect, not related to the free rotation of water molecules. If microwave ovens were tuned to excite the natural rotation frequencies of water, they would not penetrate as far as they do into normal foods.
http://www.schoolphysics.co.uk/age16-19/Wave properties/Wave properties/text/Microwave_ovens/index.html
https://en.wikipedia.org/wiki/Electromagnetic_absorption_by_water

IR and Light can cause intramolecular bonds to vibrate.
Photons carry energy, momentum and angular momentum. Heat is just a name for the incoherent energy of a system which has reached thermal equilibrium. Light in a featureless sealed chamber with the walls all the same temperature is at thermal equilibrium and so we can speak of its temperature. Light with a predominant direction is not at thermal equilibrium, so heat in not a great description for such energy in motion.

Light travels 150 million km from the 5700 K photosphere of the sun through the Earth's atmosphere and a pane of glass to warm your arm held in a sun beam.
The photons are the same frequency as when they left (ignoring GR effects) so have not lost any energy. There are fewer of them than left in your direction because the atmosphere and glass are more nearly transparent to visible light than UV or low frequency IR. The number of photons per square meter of skin is much lower than it would be closer to the sun, because a sphere of radius 150 million km has a much larger surface than one of radius 700 thousand km. So the individual photons have not lost energy, but there's not enough of them to heat you up as hot as the sun.

Last edited: Mar 4, 2017
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12. ### James RJust this guy, you know?Staff Member

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30,368
Magical Realist:

Photons carry "electromagnetic energy", if you like. That is, a photon has the potential to create electromagnetic effects on other things, so in that sense it has energy.

Sure. That's what a light intensity meter does.

For example, light from the sun carries about 1340 Watts of power per square meter.

No. An error a lot of people make is to think that energy is like a substance. Energy isn't a substance. You can't have a bottle of pure energy. Energy is always associated with some kind of system.

A photon is the carrier of the electromagnetic field. You can think of a photon as a particle, which is kind of substantial, unlike energy. Nevertheless, there is energy (a number) associated with that particle.

Roughly speaking, its colour, if we're talking about visible light. Of course, we can also have invisible photons in things like x-rays or radio waves.

"Brightness" has a number of different definitions, but most likely you're thinking of something that physicists would usually call "intensity" or "irradiance". The intensity is the amount of radiant energy incident per second on one square metre of area. I gave you an example of the intensity of sunlight above.

There are two ways to increase the intensity, therefore: one is to transfer more energy per unit time, and the other is to pack more energy into each unit area.

In terms of photons, the more photons you get landing on your retina (a fixed area) per second, the "brighter" the light will look (because it has higher intensity).

Individual photons (of a specific colour) have a fixed amount of energy. You can't change the "brightness" of an individual photon of a particular colour. However, you can change the number of those photons that is emitted from a source per second. This is what the dimmer knob on your light switch does.

No. "Heat" has a specific technical definition, which is this: heat is a transfer of energy from a hotter body to a colder body due to a temperature difference between them. "Heat" is therefore not restricted to being a specific "type" of energy. As I said earlier, heat can be transferred by convection, conduction or radiation. Photons, being radiation, can act as one mechanism for transferring heat, but there are others.

It is important to realise that there's no such thing as a "hot photon" or a "cold photon". Photons don't have a temperature - not individually, anyway. Photons are electromagnetic excitations that can transfer energy from one place to another.

Technically, the frequency of a photon determines its energy. Specifically, the energy of a photon is $E=hf$, where $f$ is the frequency and $h$ is Planck's constant.

However, we often talk in terms of wavelengths because frequency and wavelength are related. For photons in free space, $f=c/\lambda$, where $\lambda$ is the wavelength and $c$ is the speed of light. So, in a sense, we can speak about frequency or wavelength interchangeably when we talk about the energy, bearing in mind that high frequency means short wavelength, and vice versa.
I attempted to explain that in my first reply to you.

If you still don't understand, please ask more directed questions rather than simply complaining that things haven't been explained to your satisfaction.

It depends on the details of whether and how those photons interact with the thing they are (potentially) heating. But, in general, yes, probably.

I explained that in my first reply to you, did I not?

The important feature is that when an atom/molecule absorbs a photon, the photon's energy is transferred to the atom/molecule, which then appears in some other form. For example, the kinetic energy of the atom might increase on absorbing a photon. Or, a molecule might vibrate more, or rotate at a higher rate.
Photons carry momentum even though they are massless. A force requires only a transfer of momentum.

That's a special case, in effect. More generally, force = rate of change (or transfer) of momentum.

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# PDN Vignette

## Introduction

A Personalized Disease Network (PDN) is a connected graph that represents the disease evolution of an individual patient. We use diagnoses, medical interventions, and their dates to build a PDN. Each node in the PDN represents an event (e.g. Heart failure) and each node is connected to another by a directed edge if the dates fulfill a certain criterion. PDN were introduced by Javier Cabrera and proved to be effective for improving goodness of fit of survival models, dominating individual comorbidity variables. (Cabrera et. al. 2016)

## Installation

Like many other R packages, the simplest way to obtain glmnet is to install it directly from CRAN. Type the following command in R console:

install.packages("PDN", repos = "https://cran.us.r-project.org")

Users may change the repos options depending on their locations and preferences. Other options such as the directories where to install the packages can be altered in the command. For more details, see help(install.packages). Here the R package has been downloaded and installed to the default directories. Alternatively, users can download the package source at “https://cran.r-project.org/package=PDN” and type Unix commands to install it to the desired location.

## Quick Overview

The purpose of this section is to give users a general sense of the package, including the components, what they do and some basic usage. We will briefly go over the main functions, see the basic operations and have a look at the outputs. Users may have a better idea after this section what functions are available, which one to choose, or at least where to seek help. More details are given in later sections.

First, we load the glmnet package:

library(PDN)

The package contains five functions: buildnetworks, datecut, draw.PDN, draw.PDN.circle and draw.PDN.cluster, and three sample data sets: comorbidity_data, demographic_data and survival_data.

You can visit the help page of the package using the following command:

help(package = "PDN")

For comorbidity_data, its columns represent different diagnoses, its rows represent different patients, and the entries represent the time difference between the respective event and January 1st 1970. NA means the patient does not have the respective diagnosis.

summary(comorbidity_data)
##       AFIB             HF              MR            CMTHY
##  Min.   :13100   Min.   :12796   Min.   :12925   Min.   :12876
##  1st Qu.:15882   1st Qu.:15018   1st Qu.:14979   1st Qu.:15915
##  Median :16954   Median :16546   Median :17043   Median :17238
##  Mean   :16729   Mean   :16300   Mean   :16467   Mean   :16770
##  3rd Qu.:17854   3rd Qu.:17739   3rd Qu.:17740   3rd Qu.:18168
##  Max.   :19097   Max.   :19097   Max.   :19191   Max.   :19097
##  NA's   :63                      NA's   :57      NA's   :42
##       CHD             HTN              DM             NEO
##  Min.   :12794   Min.   :12794   Min.   :12794   Min.   :13625
##  1st Qu.:13151   1st Qu.:13262   1st Qu.:13590   1st Qu.:14946
##  Median :13602   Median :13866   Median :14474   Median :17517
##  Mean   :14209   Mean   :14319   Mean   :14953   Mean   :16801
##  3rd Qu.:14867   3rd Qu.:15266   3rd Qu.:15998   3rd Qu.:18526
##  Max.   :18837   Max.   :18528   Max.   :18263   Max.   :18900
##  NA's   :3       NA's   :1       NA's   :43      NA's   :71
##       COPD           RENAL
##  Min.   :12829   Min.   :13166
##  1st Qu.:13944   1st Qu.:16932
##  Median :16062   Median :17821
##  Mean   :15916   Mean   :17407
##  3rd Qu.:17955   3rd Qu.:18405
##  Max.   :19130   Max.   :19233
##  NA's   :30      NA's   :31

For demographic_data, it contains the demographic information of each patients It has five columns which are sex, race, hispan, dshyr and prime.

summary(demographic_data)
##  SEX         RACE        HISPAN       DSHYR          PRIME
##  F:40   1      :66   0      :87   Min.   :1995   Min.   : 11.00
##  M:60   21063  :14   9      :10   1st Qu.:2001   1st Qu.: 11.00
##         2      :13   2      : 2   Median :2005   Median : 11.00
##         0      : 2   5      : 1   Mean   :2004   Mean   : 41.84
##         9      : 2   1      : 0   3rd Qu.:2008   3rd Qu.: 32.00
##         21311  : 1   3      : 0   Max.   :2012   Max.   :303.00
##         (Other): 2   (Other): 0

For survival_data, it contains information of survival days from admission to death, as well as censor information regarding whether the patient is dead or not.

summary(survival_data)
##     surdays         event
##  Min.   :  69   Min.   :0.00
##  1st Qu.:1300   1st Qu.:0.00
##  Median :2466   Median :0.00
##  Mean   :2817   Mean   :0.43
##  3rd Qu.:4213   3rd Qu.:1.00
##  Max.   :6562   Max.   :1.00

## Datecuts

In order to create a PDN based on comorbidity_data, we need to find the criterion for each pair of diagnoses that define the connection of the network.

Generally, it is important to choose an appropriate cutoff point such that it captures information of the interaction between the two diagnoses.

For CMTHY and HTN, the distribution of the time difference is shown below:

We used Cox PH regression to find the best criterion for cutoff point following the rule proposed by (Cabrera et. al. 2016).

k1 = datecut(comorbidity_data,survival_data[,1],survival_data[,2])
k1[1:20]
##    AFIB to HF    HF to AFIB    AFIB to MR    MR to AFIB AFIB to CMTHY
##          6000          6000          1000          1000           900
## CMTHY to AFIB   AFIB to CHD   CHD to AFIB   AFIB to HTN   HTN to AFIB
##           900          5500          5500          5500          5500
##    AFIB to DM    DM to AFIB   AFIB to NEO   NEO to AFIB  AFIB to COPD
##          1400          1400          4700          4700          3300
##  COPD to AFIB AFIB to RENAL RENAL to AFIB      HF to MR      MR to HF
##          3300          1900          1900          5300          5300

## Build Networks

By using function buildnetworks, we past comorbidity_data and criterion information we get from the datecut into the function, then we will get the adjacency matrix for the network shown below:

a = buildnetworks(comorbidity_data,k1)
a[1:10,1:10]
##       AFIB to HF HF to AFIB AFIB to MR MR to AFIB AFIB to CMTHY
##  [1,]          0          0          0          0             0
##  [2,]          0          0          0          0             0
##  [3,]          0          1          0          0             0
##  [4,]          0          0          0          0             0
##  [5,]          0          0          0          0             0
##  [6,]          0          0          0          0             0
##  [7,]          0          0          0          0             0
##  [8,]          0          0          0          0             0
##  [9,]          0          0          0          0             0
## [10,]          0          1          0          0             1
##       CMTHY to AFIB AFIB to CHD CHD to AFIB AFIB to HTN HTN to AFIB
##  [1,]             0           0           0           0           0
##  [2,]             0           0           0           0           0
##  [3,]             0           0           1           0           1
##  [4,]             0           0           0           0           0
##  [5,]             0           0           0           0           0
##  [6,]             0           0           0           0           0
##  [7,]             0           0           0           0           0
##  [8,]             0           0           0           0           0
##  [9,]             0           0           0           0           0
## [10,]             0           1           1           0           1

## Visualize Networks

In our case, the nodes have chronological order, and we opt to incorporate this order in the network visualization for a more informative and clean result. Below left is the example PDN from function draw.PDN.circle.

PDN for individual patient:

Here is the example using the functioin draw.PDN, which utilize network and ggplot2 for better visualization:

By looping each patient through draw.PDN.circle, we obtain the visualizations for the set of patients: Here is an example of using draw.PDN.cluster to visulaize network of cluster, we use different colors to represent weighted connection of each nodes.

## Loading required package: survival

## Analyze Networks

In the below example using PDN network information to perform regression, we combine comorbidity information and the adjacency matrix from buildnetworks as our design matrix which is shown below:

x = cbind(!is.na(comorbidity_data),a)
x[1:10,1:10]
##    AFIB HF MR CMTHY CHD HTN DM NEO COPD RENAL
## 1     0  1  1     1   1   1  1   0    1     1
## 2     0  1  1     0   1   1  0   0    1     1
## 3     1  1  0     0   1   1  1   0    1     1
## 4     0  1  1     0   1   1  1   1    1     1
## 5     0  1  0     1   1   1  1   1    1     1
## 6     0  1  0     0   1   1  1   0    0     0
## 7     0  1  0     1   1   1  1   0    1     1
## 8     0  1  0     0   1   1  0   1    1     0
## 9     0  1  0     0   1   1  0   1    1     1
## 10    1  1  0     1   1   1  1   0    0     1

We then use the glmnet package to perform k-fold cross-validation for the Cox model. We extract two optimal lambdas, that is lambda.min: the ?? at which the minimal MSE is achieved, and lambda.1se: the largest ?? at which the MSE is within one standard error of the minimal MSE. We can check the active covariates in our model and see their coefficients based on the lambda we have chosen.

require(survival)
require(glmnet)
## Loading required package: glmnet
## Loading required package: Matrix
## Loading required package: foreach
## Loaded glmnet 2.0-10
y=Surv(survival_data[,1],survival_data[,2])
glm1 = cv.glmnet(x,y,family="cox",alpha=0.8)
b = glmnet(x,y,family="cox",alpha=0.8,lambda=c(glm1$lambda.min,glm1$lambda.1se))$beta b[1:20,] ## 20 x 2 sparse Matrix of class "dgCMatrix" ## s0 s1 ## AFIB . . ## HF . . ## MR . . ## CMTHY . . ## CHD . . ## HTN . . ## DM . -0.52194109 ## NEO . . ## COPD . . ## RENAL 0.42446 0.54860645 ## AFIB to HF . . ## HF to AFIB . . ## AFIB to MR . . ## MR to AFIB . . ## AFIB to CMTHY . . ## CMTHY to AFIB . . ## AFIB to CHD . -0.03440171 ## CHD to AFIB . . ## AFIB to HTN . . ## HTN to AFIB . . Through stepwise model selection based on the AIC criterion, we can see that the network variables dominate individual comorbidity variables. We conclude that the network information is significant for improving the model compared to just comorbidity information. isel = b[,2]!=0 bcox = coxph(y~.,data=data.frame(x[,isel])) step1 = step(bcox) step1$anova


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# Evaluating integral

## Homework Statement

If ##y(x-y)^2=x##, then find
$$\int \frac{dx}{x-3y}$$

## The Attempt at a Solution

I tried expanding the given equation. That gives me a cubic in y. I found no way to solve the cubic equation. I plugged the equation in Wolfram Alpha to see if it gives any nice solution but nope, no luck there too. Now I think that this is some sort of trick question but I don't really know where to begin.

Any help is appreciated. Thanks!

## Answers and Replies

Related Calculus and Beyond Homework Help News on Phys.org
ehild
Homework Helper
What about changing to the variable y/x=t? Just a guess...

ehild

What about changing to the variable y/x=t? Just a guess...

ehild
I should have really attempted the problem properly before posting it here. I just woke up and gave it a try. I substituted x-y=t. The integral changes to dx/(3t-2x) and the equation changes to ##(x-t)t^2=x##. I differentiated the equation wrt x and after some simplification, I got
$$\frac{dx}{3t-2x}=\frac{dt}{t^2-1}$$

Now its a very easy integral to solve. Thanks for the effort ehild.

I should have really attempted the problem properly before posting it here. I just woke up and gave it a try. I substituted x-y=t. The integral changes to dx/(3t-2x) and the equation changes to ##(x-t)t^2=x##. I differentiated the equation wrt x and after some simplification, I got
$$\frac{dx}{3t-2x}=\frac{dt}{t^2-1}$$

Now its a very easy integral to solve. Thanks for the effort ehild.
I think that's wrong. Sides, once you got t, then what? How about checking it with the integral:

$$\int_1^2 \frac{1}{x-3y}dx,\quad y(x-y)^2=x$$

While we're at it, what is the geometric interpretation of that integral anyway?

I think that's wrong. Sides, once you got t, then what?
I don't see what's wrong here. I differentiated both the sides of the equation wrt x. Did I do some mistake while simplifying the expression?

Just realised that the final equation after differentiating I get is wrong. It is actually
$$\frac{dx}{3t-2x}=\frac{t \, dt}{t^2-1}$$

Just realised that the final equation after differentiating I get is wrong. It is actually
$$\frac{dx}{3t-2x}=\frac{t \, dt}{t^2-1}$$
To check it, I solved the definite integral above and obtained:

$$\int_1^2 \frac{1}{x-3y} dx=\int_a^b \frac{t}{t^2-1}dt$$

where a=1-lowroot and b=2-highroot with the roots being the real roots of the equations $y(1-y)^2=1$ and $y(2-y)^2=2$

which agrees numerically with direct substitution of y(x) from the solution of $y(x-y)^2=x$.

Always a good idea to check them numerically if we're trying to intercept an earth-bound meteor.

Given ::##y\cdot(x-y)^2 = x##

Now Diff. both side w. r . to ##x## , We Get

##\displaystyle y\cdot 2(x-y)\left\{1-\frac{dy}{dx}\right\}+(x-y)^2 \cdot \frac{dy}{dx} = 1##

##\displaystyle 2y\cdot(x-y)\left\{1-\frac{dy}{dx}\right\}+\frac{dy}{dx}\cdot (x-y)^2-(x-y)^2 = 1-(x-y)^2##

##\displaystyle \left\{1-\frac{dy}{dx}\right\}\left(2xy-y^2-x^2+2xy-y^2\right) = 1-(x-y)^2##

##\displaystyle \left\{1-\frac{dy}{dx}\right\}\left(-x^2+4xy-3y^2\right) = 1-(x-y)^2##

##\displaystyle \left\{\frac{dy}{dx}-1 \right\}\left(x^2-4xy+3y^2\right) = 1 - (x-y)^2##

##\displaystyle \left\{\frac{dy}{dx}-1 \right\}.\left(x-y\right).\left(x-3y\right) = 1-(x-y)^2##

##\displaystyle \left(dy - dx\right).(x-y).(x-3y) = \left\{1-(x-y)^2\right\}dx##

##\displaystyle \frac{dx}{(x-3y)} = \frac{\left(dy - dx\right)\cdot(x-y)}{1-(x-y)^2}##

Now Integrate both side , We Get

##\displaystyle \int\frac{dx}{(x-3y)} = \int \frac{\left(dx - dy\right)\cdot(x-y)}{(x-y)^2 -1}##

Let ##(x-y) = t\Leftrightarrow d(x-y) = dt\Leftrightarrow \left(dx - dy\right) = dt##

So ##\displaystyle \int\frac{dx}{(x-3y)} = \int \frac{t}{t^2-1}dt##

So again Let ##\displaystyle t^2 - 1 = u\Leftrightarrow tdt = \frac{1}{2}du##

So ##\displaystyle \int\frac{dx}{(x-3y)} = \frac{1}{2}\cdot \int\frac{1}{u}du = \frac{1}{2}\cdot \ln \left|u\right|+C##

So ##\displaystyle \int\frac{dx}{(x-3y)} = \frac{1}{2}\cdot \ln \left|(x-y)^2 - 1\right|+C##


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## Hermite polynomials – the Rodrigues formula

Required math: calculus

Required physics: Schrödinger equation

Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.17(a).

There are several theorems concerning Hermite polynomials, which show up in the solution of the Schrödinger equation for the harmonic oscillator.

First, we’ll look at the Rodrigues formula (which is a different formula from the Rodrigues formula for Legendre polynomials).

Suppose we start with ${u=e^{-x^{2}}}$ and take its derivative. We have

 $\displaystyle u'$ $\displaystyle =$ $\displaystyle -2xe^{-x^{2}}$ $\displaystyle u'+2xu$ $\displaystyle =$ $\displaystyle 0$

We can now take the derivative of the second equation ${n+1}$ times and use Leibniz’s formula for the ${n}$th derivative of a product, which is

$\displaystyle (fg)^{(n)}=\sum_{k=0}^{n}\left(\begin{array}{c} n\\ k \end{array}\right)f^{(k)}g^{(n-k)}$

We get

$\displaystyle \left(xu\right)^{(n+1)}=\sum_{k=0}^{n+1}\binom{n+1}{k}x^{(k)}u^{(n+1-k)}$

Since any derivative of ${x}$ higher than the first gives zero, we have

$\displaystyle \left(xu\right)^{(n+1)}=xu^{(n+1)}+\left(n+1\right)u^{(n)}$

Applying this to the original equation, we get

$\displaystyle u^{(n+2)}+2xu^{(n+1)}+2\left(n+1\right)u^{(n)}=0$

Defining yet another variable ${v\equiv\left(-1\right)^{n}u^{(n)}}$ we get (the factor of ${\left(-1\right)^{n}}$ is inserted to make things come out right at the other end):

$\displaystyle v''+2xv+2\left(n+1\right)v=0 \ \ \ \ \ (1)$

Finally, defining ${y\equiv e^{x^{2}}v}$, we have

 $\displaystyle v$ $\displaystyle =$ $\displaystyle e^{-x^{2}}y$ $\displaystyle v'$ $\displaystyle =$ $\displaystyle e^{-x^{2}}\left[y'-2xy\right]$ $\displaystyle v''$ $\displaystyle =$ $\displaystyle -2xe^{-x^{2}}\left[y'-2xy\right]+e^{-x^{2}}\left[y''-2y-2xy'\right]$ $\displaystyle$ $\displaystyle =$ $\displaystyle e^{-x^{2}}\left[y''-4xy'+\left(4x^{2}-2\right)y\right]$

Substituting this into 1 we get, after dividing out the common factor of ${e^{-x^{2}}}$:

 $\displaystyle y''-4xy'+\left(4x^{2}-2\right)y+2x\left(y'-2xy\right)+2\left(n+1\right)y$ $\displaystyle =$ $\displaystyle 0$ $\displaystyle y''-2xy+2ny$ $\displaystyle =$ $\displaystyle 0$

This last equation is the same as that obtained from the Schrödinger equation earlier (with different variable names):

 $\displaystyle \frac{d^{2}f}{d\xi^{2}}-2\xi\frac{df}{d\xi}+\left(\epsilon-1\right)\xi$ $\displaystyle =$ $\displaystyle 0$ $\displaystyle \epsilon$ $\displaystyle =$ $\displaystyle \frac{2E}{\hbar\omega}$ $\displaystyle \xi$ $\displaystyle =$ $\displaystyle \sqrt{\frac{m\omega}{\hbar}}x$

We can see by comparing the two forms of the equation that a solution to the latter is

 $\displaystyle f$ $\displaystyle =$ $\displaystyle y$ $\displaystyle$ $\displaystyle =$ $\displaystyle e^{\xi^{2}}v$ $\displaystyle$ $\displaystyle =$ $\displaystyle \left(-1\right)^{n}e^{\xi^{2}}u^{(n)}$ $\displaystyle$ $\displaystyle =$ $\displaystyle \left(-1\right)^{n}e^{\xi^{2}}\frac{d^{n}}{d\xi^{n}}e^{-\xi^{2}}$

Since this is a solution it must be a multiple of the Hermite polynomial. To see that it is actually the Hermite polynomial itself, consider the derivative term. Each derivative of ${e^{-\xi^{2}}}$ will have a term multiplying the previous derivative by ${-2\xi}$, so the term with the highest power of ${\xi}$ in the ${n}$th derivative will be ${\left(-2\xi\right)^{n}=\left(-1\right)^{n}2^{n}\xi^{n}e^{-\xi^{2}}}$. We now see why the factor of ${\left(-1\right)^{n}}$ was introduced earlier: by the usual convention, the coefficient of the highest power of a Hermite polynomial is ${2^{n}}$, which is what we obtain from the formula above. Thus the Rodrigues formula for Hermite polynomials is

$\displaystyle H_{n}(x)=\left(-1\right)^{n}e^{x^{2}}\frac{d^{n}}{dx^{n}}e^{-x^{2}}$

We can apply this formula directly to get the first few polynomials. We get

 $\displaystyle H_{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle H_{1}$ $\displaystyle =$ $\displaystyle 2x$ $\displaystyle H_{2}$ $\displaystyle =$ $\displaystyle e^{x^{2}}\frac{d}{dx}\left(-2xe^{-x^{2}}\right)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 4x^{2}-2$ $\displaystyle H_{3}$ $\displaystyle =$ $\displaystyle -e^{x^{2}}\frac{d}{dx}\left(-2e^{-x^{2}}+4x^{2}e^{-x^{2}}\right)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 8x^{3}-12x$ $\displaystyle H_{4}$ $\displaystyle =$ $\displaystyle e^{x^{2}}\frac{d}{dx}\left(4xe^{-x^{2}}+8xe^{-x^{2}}-8x^{3}e^{-x^{2}}\right)$ $\displaystyle$ $\displaystyle =$ $\displaystyle e^{x^{2}}\frac{d}{dx}\left(12xe^{-x^{2}}-8x^{3}e^{-x^{2}}\right)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 16x^{4}-48x^{2}+12$

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The following proposition generalizes the previous proposition, which proves to be a special case of the following one with $\gcd(c,m)=1.$

## Proposition: Cancellation of Congruences with General Factor

Let the $a,b,c$ be integers, and $m > 1$ be a positive integer and let $d=\gcd(c,m)$ be the greatest common divisor of $c$ and $m$. Then, from the equaility of the congruences $$(ac)(m)\equiv (bc)(m)$$ it follows that $$a\left(\frac md\right)\equiv b\left(\frac md\right).$$

### Example

$3\cdot 2(8)\equiv 7\cdot 2(8)\Longrightarrow 3(4)\equiv 7(4).$

| | | | | created: 2019-04-13 17:21:46 | modified: 2019-06-22 15:20:11 | by: bookofproofs | references: [1272], [8152]

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# Which is the stronger acid in each of the following pair H_3PO_4 or H_3AsO_4?

Jul 31, 2015

${\text{H"_3"PO}}_{4}$ is the stronger acid.

#### Explanation:

For oxyacids with the same number of $\text{OH}$ groups and the same number of $\text{O}$ atoms, acid strength increases with increasing electronegativity of the central atom.

Since $\text{P}$ is more electronegative than $\text{As}$, we predict that ${\text{H"_3"PO}}_{4}$ should be a stronger acid than $\text{H"_3"AsO"_4}$.

This is confirmed by their ${K}_{\text{a}}$ values .

$\text{H"_3"PO"_4}$ has K_a = 7.5 × 10^-3, while${\text{H"_3"AsO}}_{4}$ has K_a = 5.0 × 10^-3.


text:
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Assuming ideal diodes, the voltage at $V_0$ for circuit in the below figure is _______ (Express the voltage in terms of x, y and z)

edited | 121 views


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Article Contents
Article Contents

# Optimal control for stochastic heat equation with memory

• In this paper, we investigate the existence and uniqueness of solutions for a class of evolutionary integral equations perturbed by a noise arising in the theory of heat conduction. As a motivation of our results, we study an optimal control problem when the control enters the system together with the noise.
Mathematics Subject Classification: Primary: 45D05, 93E20; Secondary: 60H30.

 Citation:

•  [1] S. Bonaccorsi, F. Confortola and E. Mastrogiacomo, Optimal control for stochastic Volterra equations with completely monotone kernels, SIAM J. Control Optim. 50 (2012), 748-789.doi: 10.1137/100782875. [2] S. Bonaccorsi and W. Desch, Volterra equations perturbed by noise, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 557-594.doi: 10.1007/s00030-012-0167-0. [3] S. Bonaccorsi, G. Da Prato and L. Tubaro, Asymptotic behavior of a class of nonlinear heat conduction problems with memory effects, SIAM J. Math. Anal. 44 (2012), 1562-1587.doi: 10.1137/110841795. [4] T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B 9 (2008), 525-539.doi: 10.3934/dcdsb.2008.9.525. [5] T. Caraballo, I. D. Chueshov, P. Marín-Rubio and J. Real, Existence and asymptotic behavior for stochastic heat equations with multiplicative noise in materials with memory, Discrete Contin. Dyn. Syst. 18 (2007), 253-270.doi: 10.3934/dcds.2007.18.253. [6] Ph. Clément and G. Da Prato, White noise perturbation of the heat equation in materials with memory, Dynam. Systems Appl. 6 (1997), 441-460. [7] Ph. Clément and G. Da Prato and J. Prüss, White noise perturbation of the equations of linear parabolic viscoelasticity, Rend. Istit. Mat. Univ. Trieste 29 (1997), 207-220 (1998). [8] Ph. Clément and J. A. Nohel, Abstract linear and nonlinear Volterra equations preserving positivity, SIAM J. Math. Anal. 10 (1979), 365-388.doi: 10.1137/0510035. [9] Ph. Clément and J. Prüss, Completely positive measures and Feller semigroups, Math. Ann. 287 (1990), 73-105.doi: 10.1007/BF01446879. [10] M. Conti, E. M. Marchini and V. Pata, Semilinear wave equations of viscoelasticity in the minimal state framework, Discrete Contin. Dyn. Syst. 27 (2010), 1535-1552.doi: 10.3934/dcds.2010.27.1535. [11] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. [12] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. [13] M. Fuhrman and G. Tessitore, Nonlinear Kolmogorov equations in infinite dimensional spaces: The backward stochastic differential equations approach and applications to optimal control, Ann. Probab., 30 (2002), 1397-1465.doi: 10.1214/aop/1029867132. [14] M. Grasselli and V. Pata, Upper semicontinuous attractor for a hyperbolic phase-field model with memory, Indiana Univ. Math. J., 50 (2001), 1281-1308.doi: 10.1512/iumj.2001.50.2122. [15] M. Grasselli and V. Pata, A reaction-diffusion equation with memory, Discrete Contin. Dyn. Syst., 15 (2006), 1079-1088.doi: 10.3934/dcds.2006.15.1079. [16] M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126.doi: 10.1007/BF00281373. [17] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170 Springer-Verlag, New York-Berlin, 1971. xi+396 pp. [18] A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224.doi: 10.1137/0521066. [19] R. K. Miller, Linear Volterra integrodifferential equations as semigroups, Funkcial. Ekvac., 17 (1974), 39-55. [20] S. Monniaux and J. Prüss, A theorem of the Dore-Venni type for noncommuting operators, Trans. Amer. Math. Soc. 349 (1997), 4787-4814.doi: 10.1090/S0002-9947-97-01997-1. [21] J. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204. [22] J. Prüs, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87. Birkhüser Verlag, Basel, 1993.doi: 10.1007/978-3-0348-8570-6. [23] R. B. Sowers, Multidimensional reaction-diffusion equations with white noise boundary perturbations, Ann. Probab., 22 (1994), 2071-2121.doi: 10.1214/aop/1176988495. [24] F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Translated from the 2005 German original by Jorgen Sprekels. Graduate Studies in Mathematics, 112. American Mathematical Society, Providence, RI, 2010. xvi+399 pp. [25] I. I. Vrabie, $C_0$-semigroups and Applications, North-Holland Mathematics Studies, 191. North-Holland Publishing Co., Amsterdam, 2003, xii+373 pp. [26] J. Yong and X. Y. Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations. Applications of Mathematics, (New York), 43. Springer-Verlag, New York, 1999.


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# About the implication of the timestep in simulation

Hi all,

I am using Honeybee to perform the e+ simulation. I originally performed the simulations on annual basis but now I need to extract the hourly electricity demand and hence I need to rerun the simulations on hourly basis. I found that the simulation time increases a lot and the result file size becomes enormous, especially the .eso and .sql file. May I ask

1) I thought that simulations, no matter which timestep is chosen, are ran on a hourly basis, so the simulation time shouldn't be that different. Am I totally wrong? 2) What is the contributor to such a mega size of .eso and .sql file? 3) Is there any solution if I just want to extract the hourly electricity demand and PV generation?

Thanks a lot.

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EnergyPlus simulations are run at timesteps no larger than an hour. HVAC equipment can have smaller timesteps, but generally, most of the components will use the timestep that you define in the SimulationControl object. The .eso and .sql file contain the output variables that you requested, and its size is a function of both the frequency and the number of variables requested.

If you first requested the variables to be yearly (RunPeriod) you would get a single reported value. Changing them to hourly would give you instead 8760 values. However, this should not be a problem if you are requesting a single variable, such as the Facility Total Building Electric Demand Power. However, if you requested X variables, now you have X times 8760 results.

Check how many variables you are requesting, and delete any of them that you do not need for the time being. It will reduce your file size.

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Thanks a lot. It really helps! Then I guess it means I have no chance to get the hourly result if I chose a different timestep.

( 2019-07-18 10:46:41 -0500 )edit
1

If you run EnergyPlus with a sub-hourly timestep, you can still request outputs with an hourly frequency.

( 2019-07-18 11:22:53 -0500 )edit


text:
For a while now I’ve been thinking that, yes, ggplot2 is awesome and offers a lot of geoms and stats, but it would be great if it could be extended with new user-generated geoms and stats. Then I learnt that ggplot2 actually has a pretty great extension system so I could create my own geoms I needed for my work or just for fun. But still, creating a geom from scratch is an involved process that doesn’t lend itself to simple transformations.

Finally, I thought of a possible solution: create a generic stat –a tabula rasa, if you will– that can work on the data with any function. Natively ggplot2 offers stat_summary(), but it’s only meant to be used with, well, summary statistics. What I wanted was something completely generic and this is my first try.

Below is the code for stat_rasa() (better name pending). It works just like any other stat except that it works with any function that takes a data.frame and returns a transformed data.frame that can be interpreted by the chosen geom.

# ggproto object
StatRasa <- ggplot2::ggproto("StatRasa", ggplot2::Stat,
compute_group = function(data, scales, fun, fun.args) {
# Change default arguments of the function to the
# values in fun.args
args <- formals(fun)
for (i in seq_along(fun.args)) {
if (names(fun.args[i]) %in% names(fun.args)) {
args[[names(fun.args[i])]] <- fun.args[[i]]
}
}
formals(fun) <- args

# Apply function to data
fun(data)
})

# stat function used in ggplot
stat_rasa <- function(mapping = NULL, data = NULL,
geom = "point",
position = "identity",
fun = NULL,
...,
show.legend = NA,
inherit.aes = TRUE) {
# Check arguments
if (!is.function(fun)) stop("fun must be a function")

# Pass dotted arguments to a list
fun.args <- match.call(expand.dots = FALSE)$... ggplot2::layer( data = data, mapping = mapping, stat = StatRasa, geom = geom, position = position, show.legend = show.legend, inherit.aes = inherit.aes, check.aes = FALSE, check.param = FALSE, params = list( fun = fun, fun.args = fun.args, na.rm = FALSE, ... ) ) }  For example, let’s say we want to quickly glance at detrended data. We then create a very simple function Detrend <- function(data, method = "lm", span = 0.2) { if (method == "lm") { data$y <- resid(lm(y ~ x, data = data))
} else {
data$y <- resid(loess(y ~ x, span = span, data = data)) } as.data.frame(data) }  and pass it to stat_rasa() library(ggplot2) set.seed(42) x <- seq(-1, 3, length.out = 30) y <- x^2 + rnorm(30)*0.5 df <- data.frame(x = x, y = y) ggplot(df, aes(x, y)) + geom_line() + stat_rasa(geom = "line", fun = Detrend, method = "smooth", color = "steelblue")  We can get better legibility and less typing by creating a wrapper function with a more descriptive name. stat_detrend <- function(...) { stat_rasa(fun = Detrend, ...) } ggplot(df, aes(x, y)) + geom_line() + stat_detrend(method = "lm", color = "blue", geom = "line")  Another case could be calculating contours from an irregular grid. Since ggplot2::stat_contour() uses grDevices::contourLines(), it needs values defined in a regular grid, but there’s a package called contoureR that can compute contours from irregularly spaced observations. With stat_rasa() we can integrate it with ggplot2 effortlessly by creating a small function and using geom = "path". IrregularContour <- function(data, breaks = scales::fullseq, binwidth = NULL, bins = 10) { if (is.function(breaks)) { # If no parameters set, use pretty bins to calculate binwidth if (is.null(binwidth)) { binwidth <- diff(range(data$z)) / bins
}

breaks <- breaks(range(data$z), binwidth) } cl <- contoureR::getContourLines(x = data$x, y = data$y, z = data$z,
levels = breaks)

if (length(cl) == 0) {
warning("Not possible to generate contour data", call. = FALSE)
return(data.frame())
}
cl <- cl[, 3:7]
colnames(cl) <- c("piece", "group", "x", "y", "level")
return(cl)
}

stat_contour_irregular <- function(...) {
stat_rasa(fun = IrregularContour, geom = "path", ...)
}

set.seed(42)
df <- data.frame(x = rnorm(500),
y = rnorm(500))
df\$z <- with(df, -x*y*exp(-x^2 - y^2))

ggplot(df, aes(x, y)) +
geom_point(aes(color = z)) +
stat_contour_irregular(aes(z = z, color = ..level..), bins = 15) +
scale_color_viridis_c()

And voilà.

There’s always things to improve. For example, the possibility of using a custom function to compute parameters that depend on the data, but I believe that as it stands covers 80% of simple applications. I should also use a better name, but naming things is hard work.


text:
# zbMATH — the first resource for mathematics

$$n$$-valued groups: theory and applications. (English) Zbl 1129.20045
The author gives a survey of his most important results in algebraic and topological theory of $$n$$-valued groups. The basic definitions are richly exemplified by $$n$$-valued groups having applications in various fields of research, e.g., a class of $$n$$-valued groups as deformations of a finite group. Algebraic representations of $$n$$-valued groups and their representations on graphs are also given. Using the method of generalized shift operators developed in his previous papers, some formal two-valued groups over the field of complex numbers are defined. A new definition of integrability of multivalued dynamics with discrete time taking action of $$n$$-valued groups for the base is also given. An $$n$$-valued group defined on a compact Hausdorff space $$X$$ such that the ring of functions $$C[X]$$ is endowed with an $$n$$-Hopf structure is described.

##### MSC:
 20N15 $$n$$-ary systems $$(n\ge 3)$$ 17A42 Other $$n$$-ary compositions $$(n \ge 3)$$ 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16S34 Group rings 05E30 Association schemes, strongly regular graphs


text:
# Problem involving calculus

1. Sep 20, 2010

### the_morbidus

1. The problem statement, all variables and given/known data
A new cottage is built across the river and 300 m downstream from the nearest telephone relay station. The river is 120m wide. In order to wire the cottage for phone service, wire will be laid across the river under water, and along the edge of the river above ground. The cost to lay wire under water is $15 per m and the cost to lay wire above ground is$10 per m. How much wire should be laid under water to minimize cost?

2. Relevant equations
Pythagoras theorem a^2 +b^2 = c^2

3. The attempt at a solution

ok so i think the wire going across the river will create a triangle

|\........| ^
|..\......| |
|....\....| x
|......\..| |
|........\| V
|.........|
|.........| 300-x
|_120 _| won't allow me to make a proper diagram but hope you guys get the picture so try to imagine it without all the dots on it.

so the equation for the hypotenuse for the lenght of wire under water will be sqrt(x^2 +120^2) = sqrt(x^2 +14400)

the equation for the lenght of wire above ground is 300-x

the function for the price for the total wire usage is P= sqrt(x^2 +14400)/15 + (300-x)/10
P=1/15 (x^2+14400)^1/2 + 1/10 (300-x)
now for the derivative of the function

P'=1/15 *1/2 (x^2 +14000)^-1/2 *2x +1/10(-1)
P'=x/(15(x^2 +14400)^1/2) - 1/10
now to find the value for x when the function equals 0

x/(15(x^2 +14400)^1/2) - 1/10=0
x/15(x^2 +14400)^1/2 = 1/10
10x=15(x^2+14400)^1/2
(10x)^2=(15(x^2+14400)^1/2)^2
100x^2=225(x^2+14400)
100x^2=225x^2 +3240000
-125x^2=3240000
x^2=3240000/-125
x^2=-25920
x=sqrt(-25920) see the problem is the negative answer so i know i have made a mistake on my approach, because i was thinking once i have obtained the value of x i could add it so the equation sqrt(x^2 +14400) and i would obtain the minimum amount of wire laid under water to minimize the cost.

Can anyone help me?? thanks!!!

Last edited: Sep 20, 2010
2. Sep 20, 2010

### Dick

If the length underwater is L meters, then the cost of laying the cable underwater is 15*L, not L/15. You've set it up so the cost of laying a meter of cable underwater is \$(1/15). That makes it cheaper to lay the cable underwater than to string in on land, so the cheapest way is to make it ALL underwater. That's why you aren't getting a critical point in a realistic place.

3. Sep 20, 2010