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quadricorrelator
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Posts posted by quadricorrelator
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Are we going to start discussing vinyl analogue LPs versus digital CDs now?
:D ...ah surface area - the base of life as we know it.... Random noise in analog circuits same same random fluctations in stock market:
It is difficult to escape from the analog world. Even digital circuits are really analog when you get down to the basic devices which make up the circuits.
Digital circuits are made up of transistors, and those are analog devices. The transistors are built up into digital structures, but the designers who do that work are analog designers.
What I find most interesting is that digital circuits are not 100% digital. You think of a digital signal as going between two voltages. But, actually, the voltages are not pure. The voltages have tiny random fluctations on them. This is unavoidable, and the noise is intrinsic to the basic devices. The noise current in a transistor goes as the square root of its operating current. Even a resistor has voltage noise which goes as the square root of its resistance (or can be modeled as current noise inversely proportional to the square root of its resistance).
But, these tiny random fluctuations have a probability distribution, so that big fluctuations are unlikely. However, they do happen occasionally, and they cause errors. A "1" can be mistaken for a "0" etc.
Interestingly enough, the probability characteristics for electrical noise are the same as the random fluctuations in stock prices. They are both modeled as lognormal probability densities.
The electrical fluctuations are so small that you might only get an error 1 out of every 10 billion bits. But, at a 1 Gigabit data rate, this would give an error every 10 seconds! That would be horrible if it were an error in your bank account.
As data rates go up, the problem will get worse and worse. Also, the problem gets worse because the noise fluctations are also proportional to the square root of the bandwidth of the circuit (related to data rate).
The problem is solved partially by allowing errors and then using coding techniques to detect and correct the errors. In the stock market, this kind of redundancy won't work because people don't give back your money, so that you can get another chance to invest.
One last thought: There is one more similarity between stock market statistics and electrical statistics. The volatility of an asset increases as the square root of the amount of time it is held. Luckily, the mean return increases exponentially (compound interest means exponential growth), so if you wait long enough, your mean portfolio value grows faster than its volatility.
We often put resistors in parallel to reduce the noise fluctuations. This is equivalent to diversifying your porfolio in the stock market. The average return stays the same, but the volatility goes down.
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Since we engulf in peacounting: As we talk Alternating Currents, the voltage is at any given time anywhere between zero and 360, depending on which point of the sinus curve you measure. The integral over a full period, however, is 220 to 240 volts.
Shouldn't that be between zero and 340? (240 * 1.414)?
Not sure how serious this discussion is but shouldn't it be 220*1.414? And also what is this integral you are doing all about anyway?
Was wondering that too when I responded earlier that the integral of a sine is zero whether it is a full period or infinite number of periods and nowhere can come up with 220 to 240. If he is referring to zero to 360 degrees it is still zero over that period and the peak is 311 Volts (+ and -) along that axis.
1. Integral over one period of sine: The integral over one period of a sine function is zero. The integral of a curve is the area of the curve above the x axis (horizontal axis) - the area of the curve below the x axis (horizontal axis).
The sine function has the same area above and below the axis so they cancel out.
Integral (sin(kx) dx) from x = a/k to x = a/k + 2*pi/k
= 1/k * Integral (sin(kx) dkx) from x = a/k to x = a/k + 2*pi/k
= 1/k * Integral (sin(u) du) from u = a to u = a + 2*pi (by the change of variables theorem)
= 1/k * (-cos(u)) from u = a to u = a + 2*pi
= 1/k * (-cos(a) - (-cos(a + 2*pi))
= 1/k *( cos(a+2*pi) - cos(a))
= 1/k * (cos (a) *cos(2*pi) -sin(a)*sin(2*pi)) - cos(a))
= 1/k * (cos(a) *1 - sin(a)* 0 - cos(a) )
= 1/k * (cos (a) - cos(a) )
= 0
2. Integration over an infinite number of periods:
The brief answer is that this integral does not converge so it doesn't exist. To see this, you have to go back to the definition of the "improper integral". An improper integral is one whose integration limits go to infinity.
Defintion:
Integral (f(x) dx) from x = 0 to x = infinity
is defined as
limit as u goes to infinity of Integral (f(x)dx) from x = 0 to x = u
But, now we must define the limit of a function as u goes to infinity.
Definition: L = Limit (g(u)) as u goes to infinity if and only if for all E, there exist an H, such that if u>H, then the absolute value of (g(u) - L) is less than E.
Now we can apply these definitions to try to evaluate our integral.
Integral (sin(x) dx) from x = 0 to x = infinity
= limit as u goes to infinity (Integral (sin(x) dx) ) from x = 0 to x = u
= limit as u goes to infinity (-cos(x)) from x = 0 to x = u
= limit as u goes to infinity (cos(u) - cos(0))
= limit as u goes to infinity (cos (u) - 1)
Does such a limit exist? I will assume that such a limit did exist, and the arrive a contradiction. This will show that the original assumption "that the limit does exist" is incorrect.
Suppose the limit was L. Then, for any E, we could find an H such that if u>H, then -E <(cos(u) - 1) - L<E
so that (1+ L)-E < cos(u) < (L+1) + E
This inequality must hold for any value of E, so we can pick an E which is arbitrarily small. Let's pick E = .01, then there exists an H such that u>H implies that
.99 +L < cos(u) < 1.01 + L
so the maximum of cos(u) - minimum of cos (u) can only be .02 for values of u greater than H.
Let us examine some values of u which are greater than H. Let us examine the values of cos(u) for H<u<H + 2*pi. Over this region, cos(u) reaches a maximum of 1, and a minimum of -1. But, we already deduced that the maximum (cos(u)) - minimum (cos(u)) for u>h was .02. This contradiction implies that our original assumption (that the limit exists) is wrong.
3. How to evaluate power delivered to a resistor from a sinusoidal voltage source.
Suppose you have a voltage source across a resistor. Let's say the voltage source is described by Vsin(kt). Let's say the resistor has value R.
From previous messages in this thread, we know that the instaneous power delivered to any device is Power = voltage * current.
The current through the resistor is current = voltage / resistance.
So, the instantaneous power (power at any point in time) = voltage * current
= (voltage * voltage)/resistance
= (V*V*sin(kt)*sin(kt))?R
So the power is changing all the time in this resistor. Sometimes it is 0, but sometimes it is (V*V)/R. We might want to know the average power delivered to the resistor. To do this we must find the area under the power curve for one cycle, and then divide by the cycle time.
The time of one cycle (period) is (2*pi)/k.
Average power= k/(2*pi) * Integral (V*V*sin(kt)*sin(kt))/R dt) from t=0 to t=2*pi/k
= (k/(2*pi))*V*V/R Integral (sin(kt)sin(kt)dt) from t=0 to t=2*pi/k
= (k/(2*pi))*V*V/R*(1/k) Integral (sin(kt)sin(kt)dkt) from t=0 to t=2*pi/k
= (k/(2*pi))*V*V/R*(1/k) Integral (sin(u)sin(u)du) from u=0 to u=2*pi
= (k/(2*pi))*V*V/R*(1/k) Integral ( ((1-cos(2*u))/2)du) from u=0 to u=2*pi
= (k/(2*pi))*V*V/R*(1/k)(1/2) Integral ( ((1-cos(2*u))/2)d2*u) from u=0 to u=2*pi
= (k/(2*pi))*V*V/R*(1/k)(1/2) Integral ( ((1-cos(y))/2)dy) from y=0 to y=pi
(change of variable y=2*u)
= (k/(2*pi))*V*V/R*(1/k)(1/2) ((y + sin(y)/2) from y=0 to y=pi
= (k/(2*pi))*V*V/R*(1/k)(1/2) ((y + sin(y)/2) from y=0 to y=pi
= (k/(2*pi))*V*V/R*(1/k)(1/2) (pi)/2
= V*V/(2*R)
4. Line voltage maximum: The line voltage is not specified in terms of the maximum voltage. When they say 220 volts, they are refering to the RMS voltage. This means that they take the square of the voltage, then they take the mean of that over one cycle, and then they take the square root of that.
So, let's see if we can figure out the peak voltage using this idea.
Suppose our line voltage is Vsin(kt) (k is going to be 2*pi*60)
We want to know what V is. But, the power company tells us only the RMS value.
square root (mean(square(Vsin(kt) = 220.
We need to solve this equation for V.
square root (mean (V*V*sin(kt)*sin(kt)dt))) = 220 from t = 0 to t=2*pi/k
square root (V*V*(k/(2*pi)) Integral (sin(kt)sin(kt)dt))) = 220 from t = 0 to t=2*pi/k
but from the previous calculation we know that this integral is k/4, so
square root (V*V*(k/(2*pi)* (4/k)) = 220
square root V*V/2 = 220
V/1.4 = 220
V = 308
so the voltage swings between -308 volts and +308 volts.
Thank you, I find it interesting, and do care.
I was looking at the previous post really, about the maths genii (?) who tried to beat the market. Of course, like everybody else, they were successful when the market was bouyant. The admission that their maths was incorrect is no surprise, the market movements are so random as to be incoherent.
Maybe their incoherency formulae (better) would work.
Thanks. Since you seem interested, I wanted to mention a link in which economists, and even the Noble Prize winners of the failed company are interviewed about Long Term Capital Management's failure. I found it fascinating. It doesn't go into the mathematical details, but I found it stimulating and dramatic.
http://www.bbc.co.uk/science/horizon/1999/midas_script.shtml
I am studying the mathematics they used right now (the mathematics is useful, but not for beating the stock market). It falls under the category of "Options Pricing Theory". There are many lectures and books about it on the internet. Just do a google search on Black-Scholes.]
Here is a site with many free (and legal) finance books you can download:
http://www.econphd.net/notes.htm
It raises the question in my mind, "If these geniuses can't find a way to win at the stock market, then what change to the rest of us have?" Maybe there is someone out there who has found a system, but why should I think I would be that lucky person?
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Although the level of quadricorrelators maths far out-weighs my own I find it enlightening and the story on the investors comes back to "assume nothing"
Yes I care
quadricorrelator your posts are exercising many a mind - I can imagine some old cogs cranking slowly as the cob-webs clear and their owners reach for a - for those to old to enjoy anyhing but try a Thanks for the welcome. It's good to connect with like minds.
The story of the Long Term Capital Management captured my imagination. I read a riveting book about it called "When Genius Failed" (I bought it at Kinokunya). The core members of the company were shy, socially underdeveloped, and self-confessed nerds, who really didn't even care that much about money (they liked making it, but they didn't care about spending it, and they were mostly interesting in proving their theories were right). They were getting paid more than anyone else in the company too.
It was interesting that the "nerds" were former students of the Nobel Prize winners, and they were in their early thirties at the time. But, in fact, they had far more power and made more money that their professors. The Nobel Prize winners developed the theory, but were muscled out by their students who were more practical. In fact, the conviction of the core traders gave them power to bully executives at brokerage houses like Saloman, Merryl, Bear-Stearns, etc. Eventually they bumped heads with Soros and Buffet (when the company started to fail). Even with all their conviction and supreme financial confidence, they still had trouble finding a date for Saturday night.
According to this book, the hedge fund failed because their predictions were based on historical data, but within three years, the market did something which it had never done before.
Mathematically speaking, they were fitting the probability density curves with a lognormal curve, which is just the "Bell Curve" for IQ etc. This curve drops off very very rapaidly when you go out greater than 5 sigma (5 standard deviations) from the mean. But, the actually market data does fit the curve when you get that far out. But, the theory made use of this part of the Bell Curve. As a result, it failed.
The book points out that Scholes (one of the Nobel Prize winners professors) advisor at University of Chicago, Fama, showed that the market violates the Bell Curve probability density function about every three or four years. According to this book, the guys should have known better.
The same core guys have formed a new hedge fund called JWM Partners (John Merriwether) just down the street from their old company.
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Since we engulf in peacounting: As we talk Alternating Currents, the voltage is at any given time anywhere between zero and 360, depending on which point of the sinus curve you measure. The integral over a full period, however, is 220 to 240 volts.
Shouldn't that be between zero and 340? (240 * 1.414)?
Not sure how serious this discussion is but shouldn't it be 220*1.414? And also what is this integral you are doing all about anyway?
Was wondering that too when I responded earlier that the integral of a sine is zero whether it is a full period or infinite number of periods and nowhere can come up with 220 to 240. If he is referring to zero to 360 degrees it is still zero over that period and the peak is 311 Volts (+ and -) along that axis.
1. Integral over one period of sine: The integral over one period of a sine function is zero. The integral of a curve is the area of the curve above the x axis (horizontal axis) - the area of the curve below the x axis (horizontal axis).
The sine function has the same area above and below the axis so they cancel out.
Integral (sin(kx) dx) from x = a/k to x = a/k + 2*pi/k
= 1/k * Integral (sin(kx) dkx) from x = a/k to x = a/k + 2*pi/k
= 1/k * Integral (sin(u) du) from u = a to u = a + 2*pi (by the change of variables theorem)
= 1/k * (-cos(u)) from u = a to u = a + 2*pi
= 1/k * (-cos(a) - (-cos(a + 2*pi))
= 1/k *( cos(a+2*pi) - cos(a))
= 1/k * (cos (a) *cos(2*pi) -sin(a)*sin(2*pi)) - cos(a))
= 1/k * (cos(a) *1 - sin(a)* 0 - cos(a) )
= 1/k * (cos (a) - cos(a) )
= 0
2. Integration over an infinite number of periods:
The brief answer is that this integral does not converge so it doesn't exist. To see this, you have to go back to the definition of the "improper integral". An improper integral is one whose integration limits go to infinity.
Defintion:
Integral (f(x) dx) from x = 0 to x = infinity
is defined as
limit as u goes to infinity of Integral (f(x)dx) from x = 0 to x = u
But, now we must define the limit of a function as u goes to infinity.
Definition: L = Limit (g(u)) as u goes to infinity if and only if for all E, there exist an H, such that if u>H, then the absolute value of (g(u) - L) is less than E.
Now we can apply these definitions to try to evaluate our integral.
Integral (sin(x) dx) from x = 0 to x = infinity
= limit as u goes to infinity (Integral (sin(x) dx) ) from x = 0 to x = u
= limit as u goes to infinity (-cos(x)) from x = 0 to x = u
= limit as u goes to infinity (cos(u) - cos(0))
= limit as u goes to infinity (cos (u) - 1)
Does such a limit exist? I will assume that such a limit did exist, and the arrive a contradiction. This will show that the original assumption "that the limit does exist" is incorrect.
Suppose the limit was L. Then, for any E, we could find an H such that if u>H, then -E <(cos(u) - 1) - L<E
so that (1+ L)-E < cos(u) < (L+1) + E
This inequality must hold for any value of E, so we can pick an E which is arbitrarily small. Let's pick E = .01, then there exists an H such that u>H implies that
.99 +L < cos(u) < 1.01 + L
so the maximum of cos(u) - minimum of cos (u) can only be .02 for values of u greater than H.
Let us examine some values of u which are greater than H. Let us examine the values of cos(u) for H<u<H + 2*pi. Over this region, cos(u) reaches a maximum of 1, and a minimum of -1. But, we already deduced that the maximum (cos(u)) - minimum (cos(u)) for u>h was .02. This contradiction implies that our original assumption (that the limit exists) is wrong.
3. How to evaluate power delivered to a resistor from a sinusoidal voltage source.
Suppose you have a voltage source across a resistor. Let's say the voltage source is described by Vsin(kt). Let's say the resistor has value R.
From previous messages in this thread, we know that the instaneous power delivered to any device is Power = voltage * current.
The current through the resistor is current = voltage / resistance.
So, the instantaneous power (power at any point in time) = voltage * current
= (voltage * voltage)/resistance
= (V*V*sin(kt)*sin(kt))?R
So the power is changing all the time in this resistor. Sometimes it is 0, but sometimes it is (V*V)/R. We might want to know the average power delivered to the resistor. To do this we must find the area under the power curve for one cycle, and then divide by the cycle time.
The time of one cycle (period) is (2*pi)/k.
Average power= k/(2*pi) * Integral (V*V*sin(kt)*sin(kt))/R dt) from t=0 to t=2*pi/k
= (k/(2*pi))*V*V/R Integral (sin(kt)sin(kt)dt) from t=0 to t=2*pi/k
= (k/(2*pi))*V*V/R*(1/k) Integral (sin(kt)sin(kt)dkt) from t=0 to t=2*pi/k
= (k/(2*pi))*V*V/R*(1/k) Integral (sin(u)sin(u)du) from u=0 to u=2*pi
= (k/(2*pi))*V*V/R*(1/k) Integral ( ((1-cos(2*u))/2)du) from u=0 to u=2*pi
= (k/(2*pi))*V*V/R*(1/k)(1/2) Integral ( ((1-cos(2*u))/2)d2*u) from u=0 to u=2*pi
= (k/(2*pi))*V*V/R*(1/k)(1/2) Integral ( ((1-cos(y))/2)dy) from y=0 to y=pi
(change of variable y=2*u)
= (k/(2*pi))*V*V/R*(1/k)(1/2) ((y + sin(y)/2) from y=0 to y=pi
= (k/(2*pi))*V*V/R*(1/k)(1/2) ((y + sin(y)/2) from y=0 to y=pi
= (k/(2*pi))*V*V/R*(1/k)(1/2) (pi)/2
= V*V/(2*R)
4. Line voltage maximum: The line voltage is not specified in terms of the maximum voltage. When they say 220 volts, they are refering to the RMS voltage. This means that they take the square of the voltage, then they take the mean of that over one cycle, and then they take the square root of that.
So, let's see if we can figure out the peak voltage using this idea.
Suppose our line voltage is Vsin(kt) (k is going to be 2*pi*60)
We want to know what V is. But, the power company tells us only the RMS value.
square root (mean(square(Vsin(kt) = 220.
We need to solve this equation for V.
square root (mean (V*V*sin(kt)*sin(kt)dt))) = 220 from t = 0 to t=2*pi/k
square root (V*V*(k/(2*pi)) Integral (sin(kt)sin(kt)dt))) = 220 from t = 0 to t=2*pi/k
but from the previous calculation we know that this integral is k/4, so
square root (V*V*(k/(2*pi)* (4/k)) = 220
square root V*V/2 = 220
V/1.4 = 220
V = 308
so the voltage swings between -308 volts and +308 volts.
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I am retired. I used to be an analog microchip designer.
Lucky you retired before all that digital stuff took over!!
It seems like analog should be dead, but somehow it is still alive. Many thought analog was on its way out 26 years ago (when I first started). But, the need for analog designers always seems to come up: power supply design for notebook computers is a hot field, cell phone radio frequency amplifiers is a hot field, disk drive design has quite a bit of analog circuitry in it, front end amplifier circuitry for fiber optics requires analog, phase lock loop design for communications, and on and on.
Data received from the world is analog. Such systems always need analog signal processing.
Maxim Corporation, Linear Technology Corporation, and the analog division of National Semiconductor are all analog companies which have been very profitable over a long period of time. But, they have recruiting difficulties.
There is a shortage of analog designers relative to the demand. Universities don't want to teach it because the world needs much greater numbers of digital designers and computer programmers. Students don't want to study it because it is not glamorous (at least not like digital design), it has a lot of mathematics in it, it takes about three or four years of industrial experience before you are independent enough to do a design on your own (partially because it is a mature field it takes time to learn everything already known), and it doesn't fit most people's temperment because it requires a very narrow focus (you spend your career learning more and more about the same basic devices; transistor, capacitor, resistor, and inductor).
But, when I was making my career decision, I didn't even consider digital design. It just wasn't for me. I like the math of analog circuitry. I liked the narrow focus. I liked the idea of going deeper and deeper into the same field.
I think there is quite a bit of job security for analog designers. I don't know how the pay scales compare for equivalent levels of competence.
quadricorrelator Good Post - thank you, I for one found it interesting, proving your never to old to learn something new. One of the most interesting points for me was the "narrow focus" there are many people who try to breakaway from a natural or taught narrow focus not knowing (I include myself) there are fields where it can be advantageous.
mijan24
Thank you for the kind words.
Yes, I tried to fit my personality to the kind of work available. I guess we all must try to find a way we can fit in the work place. Actually, I had great difficulty finding a way to fit in the work place for a long time.
Too Old to Learn: I hope I am never too old to learn. I hope I always enjoy learning. These pleasures will probably out last the enjoyment (or capability to participate in) of physical activities.
Right now I am pursuing my interest in Mathematical Logic, Set Theory, and Manifolds.
I recently became curious in the collapse of the hedge fund "Long Term Capital Management" in 1998. The fund was formed by a bunch of geniuses (two were Nobel Prize winners) who thought they could use purely mathematical techniques (developed by the Nobel Prize winners) to get great returns in the stock market. They were very successful in the first few years, but then their system collapsed very dramatically. They were bailed out by 15 financial institutions, each contributing 300 million dollars in order to prevent a a much more profound problem which might have been very wide spread. You can read about this if you do a google search. It is very dramatic.
I am studying the mathematics behind their methods, even though they failed. It is fascinating to me. I am amazed that advanced mathematics could be applied to finance to make money. It is interesting that the equations are analogous to some of the equations in Quantum Mechanics. They use Brownian motion (a random process describing how a particle suspended in a liquid is bounced around randomly by molecules) to model the random fluctuations in stock prices.
They thought they had found a way to hedge out risk completely use the Black-Scholes formula. Myron Scholes was one of the Nobel Prize winners at the hedge fund.
If I understand correctly, their methods failed because they made false mathematical assumptions. They assumed that future market behavior would be the same as historical market behavior. They tried to fit historical data to a "log normal" curve, but that turns out to be a mistake.
But, I am still learning about it.
Too old to have a romantic relatinnship: Do you have thoughts on this? I suppose it depends on the individual. It is a concern of mine right now.
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I am retired. I used to be an analog microchip designer.
Lucky you retired before all that digital stuff took over!!
It seems like analog should be dead, but somehow it is still alive. Many thought analog was on its way out 26 years ago (when I first started). But, the need for analog designers always seems to come up: power supply design for notebook computers is a hot field, cell phone radio frequency amplifiers is a hot field, disk drive design has quite a bit of analog circuitry in it, front end amplifier circuitry for fiber optics requires analog, phase lock loop design for communications, and on and on.
Data received from the world is analog. Such systems always need analog signal processing.
Maxim Corporation, Linear Technology Corporation, and the analog division of National Semiconductor are all analog companies which have been very profitable over a long period of time. But, they have recruiting difficulties.
There is a shortage of analog designers relative to the demand. Universities don't want to teach it because the world needs much greater numbers of digital designers and computer programmers. Students don't want to study it because it is not glamorous (at least not like digital design), it has a lot of mathematics in it, it takes about three or four years of industrial experience before you are independent enough to do a design on your own (partially because it is a mature field it takes time to learn everything already known), and it doesn't fit most people's temperment because it requires a very narrow focus (you spend your career learning more and more about the same basic devices; transistor, capacitor, resistor, and inductor).
But, when I was making my career decision, I didn't even consider digital design. It just wasn't for me. I like the math of analog circuitry. I liked the narrow focus. I liked the idea of going deeper and deeper into the same field.
I think there is quite a bit of job security for analog designers. I don't know how the pay scales compare for equivalent levels of competence.
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Is the wattage in BKK is it 220 or 110?
don't worry IronWolf, people love to laugh at mistakes!!
Mind you, the wattage question is interesting as the electric companies let you choose what (?) you want, and charge accordingly
so if you want to fit a new aircon or waterheater, you may need to uprate your wattage. That probably means new, heavier wires to compensate
why is the voltage not 240 anyway, anyone??
Since we engulf in peacounting: As we talk Alternating Currents, the voltage is at any given time anywhere between zero and 360, depending on which point of the sinus curve you measure. The integral over a full period, however, is 220 to 240 volts.
Chownah, Quadricorrelator: May I ask what you guys do for a living?
Cheers
raro
I am retired. I used to be an analog microchip designer.
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Simple....if you hook up a battery with some wires to a resistor ( a resistor is something that resists the flow of electrons...duh?) and then you measure the voltage(the battery makes the voltage) and the resistance (the resistor makes the resistance) and the current (how many electrons are going buy in a second is how you measure current....using an ammeter perhaps) and you write these values down....like below
9 volts (a nine volt battery was used)
1 ohm (a one ohm resistor was used)
9 amps (thats a measure of how many electrons were passing...not 9 electrons silly....9 times some really really huge number which is equal to 1 amp)
then you change the battery and you find:
1.5 volts (a single d cell)
1 ohm (the same resistor)
1.5 amps (do you see some coincidence going on here?)
then you change the resistor:
1.5 volts (the same d cell)
0.5 ohm (smaller resistor here)
3.0 amps (does this make sense?)
And to make a long story not much longer....if you did this with alot of different batteries and resistors you would find the if you take the resistance (measured in ohms) and multiply it times the current (measured in amps) that the result would equal the voltage (measured in volts). Or E=IR (don't ask why but E stands for the voltage, I stands for the current and r is the easy one.) Simple, no? This is the explanation so that newbies will understand how Ohm's law is used....the explanation as to why this works is really interesting and truly elegant but to do it justice you really need to get a good physics book and read about it there....REALLY interesting stuff.
So Chow ,if i'm using a 15 AMP skill saw that puts out max power of 2200watts, what would be the average voltage drop if your power cord was about 80' long?
May I ask a question before going further? You say that the saw puts out a max power of 2200 watts. Do you mean that it consumes 2200 watts of electrical power, or that it puts out 2200 watts of mechanical power?
If it is consuming 2200 watts of electrical power (the first assumption above), then you can make calculations to determine the voltage drop in the power cord.
But, if you only know the mechanical output power (the second assumption above), then there is not enough information to answer your question. In this case, you would also have to know the electrical to mechanical conversion efficiency before you could answer the question.
Let me assume that the 2200 watts is the electrical power saw consumption (just for the purpose of going through the calculation).
In this case, you calculate the voltage across the saw is
v = p/i = 2200 watts /15 amps = 147 volts.
But, the voltage at the power outlet is 220 volts. So the drop in the line is 220 volts - 147 volts = 73 volts. This would mean that the power dissipated in the power cord was 73 volts * 15 amps = 1095 watts. The resistance of the cord would be
73 volts/ 15 amps = 4.87 ohms.
These results are unrealistic. I do not believe that an 80 foot power saw cord could have nearly 5 ohms of resistance. I do not believe that the power cord could dissipate 1095 watts. This would be the one of the worst power cord designs in history. The original assumption, that the saw is consuming 2200 watts of electrical power, must be incorrect.
However, if you can find the saw power consumption requirements (maybe it is on the saw somewhere), then we might be able to calculate more realistic results. I imagine that the power saw should come with power consumption requirements, or maybe you can find those requirements on the power saw company website.
I would expect that the drop in the power cord to be a small percentage of the total power used.
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quadricorrelator,
Bedlam is a forum here on TV but you have to have 500 posts before you are given access. This is to weed out the truly obnoxious and unruly posters who usually get eliminated before they reach 500. Mathematics is never discussed.
Back on topic.
At this link:http://www.miskatonic.org/godel.html
I found this which was taken from History of Mathematics, Boyer:
"Gödel showed that within a rigidly logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that are undecidable or undemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can neither be proved or disproved. ..........."
I have always thought that the concept of statement was more general than the concept of formula......formulas being a subset of the set of statements. In most discussion of this type I have seen people use the terms 'propositions' and 'statements' fairly interchangeable but I've never (that I can remember) heard anyone use the term 'formula' at this level of mathematical consideration.
May I mention some sources I use to learn this material?
First I studied, "Introduction to Elementary Logic"
Now I am studying, "First Order Mathematical Logic" by Margaris and "Mathematical Logic" by Kleene.
All three books are Dover reprints so they are relatively cheap for texts. You can buy all three for about 1500 baht.
I found these books at Kinokuniya at the Emporium.
May I also suggest a website which I find very interesting: Metamathematics Proof Explorer Homepage. They start with the basic axioms of set theory and logic, and the derive some of the theorems of mathematics. The proof that 2 + 2 = 4 takes over 20,000 steps. It has over 2000 subtheorems. I think the deepest path back to the axioms is 129 levels.
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quadricorrelator,
Bedlam is a forum here on TV but you have to have 500 posts before you are given access. This is to weed out the truly obnoxious and unruly posters who usually get eliminated before they reach 500. Mathematics is never discussed.
Back on topic.
At this link:http://www.miskatonic.org/godel.html
I found this which was taken from History of Mathematics, Boyer:
"Gödel showed that within a rigidly logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that are undecidable or undemonstrable within the axioms of the system. That is, within the system, there exist certain clear-cut statements that can neither be proved or disproved. ..........."
I have always thought that the concept of statement was more general than the concept of formula......formulas being a subset of the set of statements. In most discussion of this type I have seen people use the terms 'propositions' and 'statements' fairly interchangeable but I've never (that I can remember) heard anyone use the term 'formula' at this level of mathematical consideration.
Thank you for the link. It looks interesting.
statement, sentence, proposition, formula: Statements and propositions and sentences all mean the same thing. Statements/propositions/sentences can be built up from other statements using the logical connectives: implies, and, or, not. The propositional or statement variables can only take on the value of true or false. The statement or proposition can be evaluated using truth tables for the connectives, when each variable is given a true or false assignment.
For example: "p implies q" is a proposition or statement. It's truth or falsity is as follows:
p q p implies q
t t t
t f f
f t t
f t t
But, you could have more complicated statements such as:
(p implies (q implies r)) implies ((p implies q) implies (p implies r))
(this statement is true for any of the eight true/false assignments given to its variables)
You can derive all statements from only one propositional axiom if you pick the right axiom.
The theory of deducing new statements/propositions from the axioms is called "The Propositional Calculus". The theory of determing the truth of statements/propositions/sentences is called "Propositional Logic"
The axioms of propositional calculus are complete. All deducible statements/propositions/sentences are true according to truth tables. And all statements/propositions/sentences which are true according to truth tables are deducible.
Formulas: A formula includes propostitions/statements/sentences, but also includes strings made up of the following additional objects: "for all", "there exists", predicates, functions, non-propositional variables, and constants.
Here is an example of a formula that is not a statement:
(there exists a y) (x < y)
In this case, x and y are non-propositional variables, "<" is a predicate.
You can turn this formula into a statement be preceding it with "for all x" or "there exists x"
You can convert any formula into a statement by adding enough "for all" or "there exists" in front of it. When you do, the statement can be evaluated to be true or false. When you do this, it is called the closure of a formula.
Here is another example (for all x) (x + y = 2)
x and y are variables, 2 is a constant, "+" is a function, "=" is a predicate
If you add the proper formulas to the axioms of the propositional calculus, then you get the full set of axioms for the Predicate Calculus. The Predicate Calculus is complete, but it is much more powerful and general than the Propositional Calculus. It is also complete.
The Predicate Calculus is the basis of Logic. You can create new theories by adding statements (but not formulas) to the axioms of the Predicate Calculus. For example, you can create Set Theory, Number Theory, Group Theory, etc. by adding the proper statements. The theory can be inconsistent or incomplete depending on which statements you use as axioms.
Godel proved one of his incompleteness theorems within Number Theory.
Set Theory contains the axioms for all of mathematics. Number Theory is contained within Set theory (as are all of the mathematical theories).
May I mention a website that I enjoy? It is the Metamathematics Proof Explorer Homepage. I can't find the address right now, but I will send it later.
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It is strange that you are discussing Godel. A few days before your first post I mentioned Godel in the Bedlam forum. I have a question for you. Are you sure that Godel showed that in an axiomatic system there would be statements that could be proven true and false? It was my understanding that what he proved was that in an axiomatic system there would be statements whose truth or falsity could not be determined....and that as a result of this there would be statements that are true but not proveably true as well as statements that are false but not proveably false.
As you mention, Godel did not prove the existence of any statement which could be deduced, and at the same time have its negation also deducible. As you state, one of his results is that there are true statements which are not provable.
I apologize if I gave the impression that such statements exist. If they did exist, then our system of mathematics would be inconsistent (that is the definition of inconsistent).
(You say, "there would be statements whose truth or falsity could not be determined". Are you refering to "decidability" of a logical system? A first order logical system is decidable if there is a decision procedure for determining if a formula is provable. If a system is not decidable then it is not complete.)
I was trying to say that it is impossible to prove the consistency of the basic axioms. This statement is independent of the idea of truth. It is only a statement regarding provability of formulas, so you don't need a definition of truth for this to make sense. The result is: There does not exist any formula (statements are actually less general than formulas, which is why I use "formula" instead of "statement" in the definition of consistency), which can be deduced, and also have its negation deducible from the basic axioms (the axioms of logic are formulas, the axioms of set theory are statements).
May I ask, what is the bedlam forum?
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I'm baffled...ok...no yank-bashing from my side for the next seven days. Will drink Miller Light for chastisement.
So you (resp. Goedel) are suggesting that the whole mathematic is pointless? I somewhat sensed that already back in school...
I think I understand your point. It seems like Godel's Theorem could lead to the conclusion that mathematics is pointless. You might think, "Why should we continue using these assumptions if we don't even know if they are consistent? All hope is lost if we know that we can never prove consistency."
Godel's Theorem was a great psychological blow to some of the leading mathematicians of the time. It was a terrible disappointment for them because they expected that all true statements were provable, and they wanted to prove them. Oddly enough, Godel did not think it should effect the mathematical spirit, or effect progress in mathematics.
An opposing viewpoint might be this: Although we don't know about assumption consistency, mathematics seems to have incredible predictive value in the real world. I don't know of a single case where it has failed. Although, this is still not a proof of consistency because we have not tried all cases (and never can).
Mathematics has great practical use and seems to be perfectly reliable, so that seems to give it some meaning.
A mathematician's point of view might be this: For some reason, most professional research mathematician's continue their work without worrying about these basic assumptions. It doesn't seem to effect their work, and they don't worry about it. Logicians and Set Theoreticians worry about it because it is their job, but not many others can afford to spend time thinking about it, or even learning much about it.
Incorrect Theories can still be useful viewpoint: Incorrect theories can still have value because they might still be approximately correct in many situations. These could be physical theories, personal theories (about relationships, about Thailand, etc.), religious theories, etc.
Example 1 -physical theories: Newton's theories of mechanics are wrong, but we don't notice the problem unless we are moving near the speed of light. It is still our most valuable (and probably our only) tool for understanding mechanical phenomena at low speeds.
Example 2 - personal theories: Theories we make up ourselves have value. We construct theories and make decisions based on those theories. We know the theories might be wrong, but we don't seem to know any other way to make decisions. My belief that "there are no guarantees in life" is an admission that our personal decisions (therefore our theories) might be wrong
Example 3 - religous theories: Some people assume/believe that god is all powerful. This is a useful, but inconsistent assumption. It is a useful assumption because it provides comfort, it may help their faith, and it may help their life. But, it is logically inconsistent.
Limits of Logic: If we use our brains to search for the truth, then we wind up relying on logic. This seems to be a limit of the human brain. The structure of logic may say more about the human brain, then it does about the truth. Logic is a human construct, not necessarily an absolute in the universe.
Logic seems like an absolute because we all agree on it. But, we all agree on it because our brains are all the same. There could be other ways to search for the truth which we can not comprehend or imagine because of our limited brain.
In fact, all logic is limited because it is based on unprovable assumptions (even without Godel this is true). How can an assumption based system claim to be absolute? But, (without non-human intervention of some kind), this is all we ever have.
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amazing...finally we learn something in this forum...just waiting for some lectures on quantum physics and Goedel's incompleteness theorems...Also the concept of "god" is not yet 100% clear to me.
I might even stop Yank-bashing for a while if I a get a sufficient answer...
May I mention a little bit about Godel's Theorems? There is a famous Completeness Theorem, and a famous Incompleteness Theorem. The Incompleteness Theorem shook the foundations of Mathematics when it was presented.
I will try to explain the statement of the Incompleteness Theorem (we can discuss the Completeness Theorem later, but it less dramatic and more difficult to explain in my opinion).
One version of the Incompleteness Theorem states that there are mathematical statements which are true, but we can never prove them using the existing commonly accepted assumptions made within mathematics.
I have to define what is meant by "true" and what is meant by "provable" for this to make sense. I have to explain how a statement can be true without being provable.
Let me give an example of a statement which may or may not be true: "The sum of the first odd numbers is a perfect square." We can test a few examples of this statement.
1 + 3 = 4 = 2 *2 which is 2 squared - a perfect square
1 + 3 + 5 = 9 = 3 *3 which is 3 squared - a perfect square
1 + 3 +5 + 7 = 16 = 4 * 4 which is 4 squared - a perfect square
It seems like this statement might be true because it is true for the first three case, but we haven't tested every case. We can't test every case because the list of cases to test is infinite. If we thought we had tested every case, there would always be more left to test.
But, the statement could still be true. We just don't know if it is true using this method of testing.
We might be able to use the basic assumptions in mathematics and construct a proof of this statement from the assumptions. In fact, it is possible to do construct such a proof for this statement (one method is to use the principle of induction).
But, maybe there are some statements which seem true, but we can't test them, and we can't find a proof for such a statement. Godel showed that there are statements which are true, but we can never prove them.
There are two concepts here. One concept is whether or not the statement is true. The other concept is whether or not the statement can be proved from basic mathematical assumptions. It is possible to have one (truth of a statement) without the other (being able to prove it).
What's worse is this: If, somehow, we discovered one of these statements, and then added this statement to our list of basic mathematical assumptions, it still would not solve the problem. There would still unprovable, but true statements within mathematics. In fact, you can not find a finite list of assumptions from which all true statements can be proven (Mathematics can not be finitely axiomatized).
Let me back up a little bit. Mathematics starts with certain assumptions. In fact, the assumptions can be broken into the assumptions of Logic, and the assumptions of Set Theory. All of mathematics (algebra, calculus, topology, differential equations, geometry, etc.) has been developed from these assumptions. But, it is a finite number of assumptions. It is impossible to prove all true statements from any finite list of axioms according to Godel's Theorem.
Unfortunately, the bad news does not end here. It gets worse. No one knows if the basics axioms of mathematics (the logical axioms added to the set theory axioms) are even consistent. By inconsistency I mean the following: It is possible that we can derive a formula and and its opposite using the basic assumptions of mathematics (the ones accepted by mathematicians). It is possible we can prove that a formula is true, but also find another proof that the formula is false using the same universally accepted assumptions.
We just don't know if our assumptions are consistent. And, it even gets worse. Godel showed that we can never show consistency of these assumptions. So we don't know if the assumptions are consistent, and we can't ever know.
All statements within the framework of commonly accepted axioms of logic and set theory, may be based on a inconsistent assumptions.
The consequence of inconsistent assumptions is complete disaster. Within inconsistent systems, you can prove that every formula is true. The system is meaningless (incidentally, this notion of inconsistency resolves some of the paradoxes such as God being able to "create a rock so heavy that even he can't lift it". The assumption of total omnipotence is inconsistent. The assumption of total omniscience eliminates free will).
The completeness theorem is less dramatic and is actually quite good news.
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I will try to explain the intuitive idea behind Ohm's law. I will use an analogy first.
Suppose you had a pipe with water running through it. A greater volume of water will flow through the pipe per second if there is a greater pressure difference across the ends of the pipe.
You can think of the pressure difference across the pipe as the voltage being applied across your appliance. You can think of the water flow rate as the current. In an electrical circuit, it is electrons that flow (not water).
So, if you double the water pressure difference, then the water flow rate doubles. That makes intuitive sense. And, if the voltage across a circuit doubles, then the current (electron flow rate) also doubles.
The purpose of this explanation was to give an intuitive idea of the meaning of voltage and current, by using an analogy.
If you understand the idea of voltage and current, then you can go a step further. Let's do this by asking the question, "Given a pressure difference across a pipe, what is the water flow rate?" The answer depends on the diameter of the pipe, and the kind of obstructions inside the pipe. You need a greater pressure difference to achieve the same water flow rate for a given obstruction level. The ratio of pressure difference to water flow rate is a measure of the obstruction level. A lower ratio means that you need less water pressure difference to get the same flow rate. This means there is less obstruction.
You can think of electrical obstruction as analogous to the physical obstruction in a water pipe. The ratio of voltage across the appliance, to the current flowing through the appliance is the electrical obstruction level. Normally, the electrical obstruction level is called resistance.
You can state this in symbolic terms by saying that the V/I = R
V = The voltage difference across the appliance
I = The charge flow rate or current through the appliance
R = The resistance, or level of electrical obstruction
This formula is Ohm's Law.
I want to mention one qualification about this formula. This formula is only true if the resistance, (electrical level of obstruction) is both linear and memoryless. Some common electrical components, such as a capacitor or inductor, are not memoryless. Others, such as a diode are not linear (or memoryless). If the device is linear and memoryless, then it is called a resistor. Many household appliances (such as a light bulb, or an iron) behave as resistors, so you can safely apply Ohm's law.
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A man's true nature is not to be monogamous. It is a man's nature to spread his seed into anything he can. That's just the way it is and our bodies will never change. Over time, people have placed these rules of monogamy on ourselves to show loyalty to our partners. It never started out this way but it has become this way.
I do believe a person should remain loyal to one he/she loves or cares about. But at the same time, it is also possible for one to love or care deeply about more than one person at a time too. Lots of grey area in this matter IMO.
Have I ever cheated? No. Have I come close before? Yes! I don't feel bad about it because the person was someone I care about and have real feelings for.
May I offer my opinion on this?
I share part of your view. I think that part of man's nature is to spread his seed. But, there are many other parts of his nature which conflict and compete with the "seed spreading" part of his nature.
For example, part of man's nature is to remain loyal to those he loves, and to avoid hurting those we love, and to reduce the chance of destroying the family unit, and to reduce the chance of spreading disease, etc. etc. It is impossible to completely satisfy all parts of our nature simultaneously.
In other words, I believe that believe that man has many parts of his nature which conflict. We can never fully be at peace because of this.
The proper balance for an individual depends on his goals and his situation at the time.
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Power is defined at the the derivative of Energy with respect to time (dE/dt). You can calculate the power by first calculating the energy in terms of voltage and current. So, must first define voltage and current.
Voltage between a point in space, A, to a point in space, C, is the path integral over the electric field between those two points. If the field is conservative (the curl of the E-field is 0), then the integral is path independent.
The electric field at a point anywhere in space is defined as the force per unit charge on an infintesimal test charge at that point in space.
Current is the rate of change, q, passing through a point with respect to time = dq/dt.
So, E = (Path integral of the Force per unit charge from A to C ) * q
P = dE/dt = d((Path integral of the force per unit charge from A to C ) * q)/dt
= (Path integral of the force per unit charge from A to C ) * dq/dt
= (Path integral of the electric field from A to C ) * dq/dt
= V * I (voltage times current)
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"No worries quadri I was actually having a snipe at Old Croc more than you..
I have always liked Mathematicians, big up respect yo."
Thanks. It's good to hear a few kind words once in a while.
(actually, I am not a mathematician, but just enjoy studying it)
-q
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I am guilty of showing off at times, and insecurity (much of the time), and many other social crimes which seem to annoy people (but not cause harm). I suppose those qualify as quirks too.
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"Quadricollerator, please don't take this as an insult, but i think your chances of long term happiness will be significantly improved by seeking proffesional help."
Thanks. No insult taken. I've tried seeking professional help many times (for extended time periods in some cases), but could not get relief.
My thinking is this: People with emotional/mental disorders can be helped by the field of psychiatry to various degrees. Some people are lucky enough to be happy and emotionally stable (due to heredity and environment) and don't need help. Others are unhappy, but can improve with professional help and/or psychopharmacology. But, some can not be helped,or may even be hurt by professional intervention. I believe I fit into that category. The most unfortunate group are so unhappy that only suicide can provide relief from unbearable and excrutiating emotional pain.
So far, the best I can do is to pick the the least destructive object of obsession. In my case it is seems to be immersion in foundational thinking, and to pick an environment which suites me (Thailand).
-q
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It's not made up (I am not sure what "Pi movie" means").
Two months ago I was deeply disturbed and upset by the fact that the definition of an interpretation of first order logical systems relied on the definition of a set, but sets can not be defined without using set theory axioms, and those consitute a first order logical system. This seemed circular to me and drove me crazy. It took two months to resolve this conflict.
I believe that this is resolved by realizing that Interpretation Theory is developed within the framework of Set Theory. As such, it is an axiomatic theory itself, and does not require the existence of an external object called a set. It isn't explained this way in the sources I had at the time, so I had to go further (at least I think it is the correct answer).
I can't let go of this subject. Some collections of objects are too big to be sets. That is, inconsistencies arise if you view sets as collections of objects. For example, the set of all sets which don't contain themselves is such a set which causes an inconsistency. If this set doesn't contain itself, then it must be a member of itself by definition, and if it does contain itself then it can't contain itself. The resolution was the development of axiomatic set theory. But it is still not enough for me because there still exist collections of sets called classes which are larger than any set, and those objects are not expressed within Zermelo-Frankel-Cantor Set theory.
I think I can't be happy until I understand the von Neumann-Bernay-Godel axioms, which contain a theory of these larger objects (without any known inconsistencies).
It is an inward psycho-epistimilogical spiral which continually intensifies. Originally I was concerned with tensor analysis on differential manifolds, but that degenerated into an orgy of foundational worry and obsession leading to ZFC and NBG theory.
Maybe it is beyond quirkiness or eccentricity.
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Thanks for the website.
Actually, it's too late. My brain already exploded three times. But, for the past few years, its been O.K.
I have failed to subdue OCD. Instead I try to harness it and focus on something useful.
I have to do all my math proofs by going back to the Zermelo-Frankel Set Theory Axioms and use only formal deductions of the Predicate Calculus.I obsess about the consistency of the Zermelo-Frankel Axioms as this has never been resolved (and it has shown that it can not be resolved). As such, no model has been found for set theory, and yet we make statements about sets all the time. A deduction of a formula and its negation from these axioms would be destabilize me completely.
I constantly worry that countable models exist for the Real Numbers due to the Lowenheim-Skolem Theorem, but the Real Numbers consitute an uncountable set.
Have a read before your brain explodes:
http://www.obsessivecompulsivedisordernews...e-disorder.html
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I have to do all my math proofs by going back to the Zermelo-Frankel Set Theory Axioms and use only formal deductions of the Predicate Calculus.
I obsess about the consistency of the Zermelo-Frankel Axioms as this has never been resolved (and it has shown that it can not be resolved). As such, no model has been found for set theory, and yet we make statements about sets all the time. A deduction of a formula and its negation from these axioms would be destabilize me completely.
I constantly worry that countable models exist for the Real Numbers due to the Lowenheim-Skolem Theorem, but the Real Numbers consitute an uncountable set.
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It is difficult to give advice with any level of confidence because I don't know all the details. But, I will say this: I very very rarely ever lend money or offer charity. My gifts, donations, etc. go to those who I care deeply about. Giving charity to strangers is taking food out of the mouths of the people you do care about (even if those you care about are your offspring or wife whom you have yet to meet).
My reasoning is as follows:
1. Ninety percent of the time you will never see the borrowed money again. How did they get themselves in a position where they can't raise 20,000 baht?
2. I don't want to get involved with collateral or anything like that, it normally turns into a big mess. Are you really going to have the guts to keep the collateral when they don't pay up? And if you do keep the collateral, the relationship is most likely over anyway.
3. If you do lend money, it sets a precident for them to ask again in the future. Only next time they will ask for more.
4. If you lend money once, and you refuse a second time, then you will be demonized and labeled a cheap charlie, etc. It's best never to get started.
Based on my experience here in Thailand, I will say that it is almost never a good idea to lend money or give charity. It raises expectations and you will be asked again. If you don't give again, then you become the target of abuse. If you never gave in the first place, then no one bothers you too much. In summary, "The road to ###### is paved with good intentions".
My daughter's teacher has asked me to lend her some money, through my wife, of course.I can't see her running away with the amount of 20,000 baht.
She said she would pay 3% a month interest, which is well below the "normal" rate. I know that some of the local money lenders charge 20% per month.
I am going to give her the cash but ask for something in writing.
I think that she may well be lending it to someone else for a higher rate of interest.
What would you do?
I don't want to piss off the school, making it hard for my daughter.
My daughter's teacher has asked me to lend her some money, through my wife, of course.I can't see her running away with the amount of 20,000 baht.
She said she would pay 3% a month interest, which is well below the "normal" rate. I know that some of the local money lenders charge 20% per month.
I am going to give her the cash but ask for something in writing.
I think that she may well be lending it to someone else for a higher rate of interest.
What would you do?
I don't want to piss off the school, making it hard for my daughter.
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A nice name would be
Yitmar Boobitzsky
:D 
Whats The Wattage In Bkk?
in Bangkok
Posted
I was concerned with "thermal noise" in resistors. Thermal noise depends on the value of the resistor (I don't know about "contact" noise). I do not know the about the physics of noise (yet). But, I do know its characteristics.
You can model resistor noise as a voltage source (a battery) in series with the resistor. The voltage of this battery is a random process whose frequency spectrum is constant at all frequencies. But all circuits are have limited bandwidth, so the amount of noise is limited as by this as well. The stray capacitance across the resistor will limit the bandwidth of any signal flowing through it. The wire connected to the resistor will have inductance and that can play a role as well (it can even resonate with the stray capacitance).
The probability density function for the random resistor voltage at any point in time is a Bell curve (normal curve). Its standard deviation is proportional to the square root of the resistance, the temperature, and the bandwidth of the circuit.
You can equivalently model the noise source as a current source in parallel with the resistor. In this case the current goes inversely with the square root of the resistance.
The effect of noise on overall circuit performance depends on how you are using the resistor.
Every resistor and every transistor in the circuit degrades circuit performance to some extent. You have to add up the contributions from each noise source to find the overall effect.
In the design of highly sensistive, high speed photodiode amplifiers for fiber optic communications, I found that I wanted to use large resistors. The voltage noise from the resistor goes as the square root of the resistance, but the voltage signal was directly proportional to the resistance since this voltage was generated by a current going through the resistor (ohm's law determined the voltage).
The noise increased as the square root of the resistance, but the signal was amplified in proportion to the resistance, so it was best to use the biggest resistor possible.
In this case, I the amplifier was analog, although I was amplifying a digital signal. Bit errors were made according to the amount of noise in the circuit.