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1 Kommentar 24.4.07 13:29, kommentieren




1)f(xy)=f(x)+f(y) => f(x)=f'(1)ln abs(x) (wenn f(x) differenzierbar!)

  Wenn f(x) fuer x=0 definiert ist, dann ist f(x)=0

2)f(x+y)=f(x)+f(y) => f(x)=cx  fuer alle x \el\ \IQ

  Wenn kontinuirlich, monoton steigend oder begrenzt fuer alle x

3)f(x+y)=f(x)f(y) => f(x)=exp(cx) (\forall\  a, f(a)!=0)

  Wenn f(a)=0 fuer ein a gilt, dann ist f(x)=0

4)f(xy)=f(x)f(y) => f(x)=x^c, f(x)=0

5)f((x+y)/2)=(f(x)+f(y))/2 => f(x)=cx+a


Allgemeines Vorgehen:

Setze x=+-y, +-1, 0, +-f(y), +-f(0), +-a oder +-f(a), wenn f(a)=0

Untersuche ob f(-x)=f(x) oder -f(x)

Untersuche ob f(tx)=t^n f(x)




Sachen wie z.B. f(2x)=2f(x) oder f(2x)=4f(x).

Versuche per Induktion zu zeigen, dass f(nx)=nf(x) bzw. =n^2f(x)


Sachen wie z.B. f(nx)=nf(x) oder f(nx)=n^2f(x) fuer x \el\ \IZ

Setze x=m/n <=> nx=m*1 => f(nx)=f(m*1) <=> nf(x)=mf(1) => f(x)=(m/n)f(1)=xf(1)


Sachen wie z.B. f^2(x)=f(x+y)f(x-y). Multiplikation von Funktionen

Logarithmus anwenden. g(x)=ln(f(x)) => 2g(x)=g(x+y)+g(x-y)


Sachen wie z.B. f(xy)=f(x)+f(y). Multiplikation in Funktionswerten.

Setze x=exp(u). =>f(exp(u+v))=f(exp(u))+f(exp(v))

g(u)=f(exp(u)) => g(u+v)=g(u)+g(v)




Der ggT(a,b) kann dargestellt werden als ggt(a,b)=ax+by mit x,y \el\ \IZ

Spezialfall: a,b koprim => ax+by=1 hatte ganzzahlige Loesungen



n!+2, n!+3,...,n!+n sind (n-1) aufeinanderfolgende, zusammengesetzte, ganze Zahlen

Alle Primzahlen > 3 haben die Form 6n+-1

Alle paarweise primen Tripel ganzer Zahlen mit x^2+y^2=z^2 sind gegeben durch:

x=abs(u^2-v^2), y=2uv, z=u^2+v^2  ggT(u,v)=1, u!=v mod 2

a==b mod m <=> m|a-b <=> a-b=qm <=> a=b+qm

a==b mod m \and\ c==d mod m => a+-c==b+-d mod m \and\ ac==bd mod m

=> a^k==b^k mod m \and\ f(a)==f(b) mod m, f(x)=(a_n)x^n+(a_(n-1))x^(n-1)+...(a_1)x+a_0


Teilbarkeitsregel: ggT(c,m)=1 \and\ ca==cb mod m => a==b mod m

Fermat's kleiner Satz:

a \el\ \IZ, p \el\ \IP => a^p==a mod p

Wenn ggT(a,p)=1 => a^(p-1)==1 mod p



\phi2(m)=Anzahl der Zahlen von {1,2,...,m}, welche prim zu m sind

ggT(a,m)=1 => a^\phi2(m)==1 mod m


Gausklammern bzw. floor:

n \el\ \IZ, x \el\ \IR, n<=x<n+1 => floor(x)=


m,n \el\ \IZ => floor((x+m)/n)=floor((floor(x)+m)/n)

Spezialfall: floor(floor(x)/n)=floor(x/n)

floor(x+1/2)= "runden"


Die Primzahl p teilt n! mit multiplizitaet x



Nuetzliche Faktorisierungen:




Identitaet von Sophie Germain:




Unendlicher Abstieg

Bsp: a^2=ab+b^2

Leicht laesst sich zeigen, dass a und b gerade sein muessen.

Also: a=2a_1, b=2b_1 =>

1 Kommentar 20.4.07 12:20, kommentieren


18.4.07 14:42, kommentieren


Is The Nuclear Option A Viable Option?

Nuclear power supplies a sixth of the world’s electricity.
Along with hydropower (which supplies slightly more than a sixth), it is the major source of “carbon-free” energy today.
The fossil-fuel alternatives have their drawbacks. Natural gas is attractive in a carbon-constrained world because it has lower carbon content relative to other fossil fuels and because advanced power plants have low capital costs. But the cost of the electricity
produced is very sensitive to natural gas prices, which have become much higher and more volatile in recent years. In contrast, coal prices are relatively low and stable, but coal is the most carbon-intensive source of electricity. The capture and sequestration of carbon dioxide, which will add significantly to the cost, must be demonstrated and introduced on a large scale if coal-powered electricity is to expand significantly without emitting unacceptable quantities of carbon into the atmosphere. These concerns raise doubts about new investments in gas- or coal-powered plants. In an open fuel cycle, also known as a once-through cycle, the uranium is “burned” once in a reactor, and spent fuel is stored in geologic repositories. The spent fuel includes plutonium that could be chemically extracted and turned into fuel for use in another nuclear plant. Doing that results in a closed fuel cycle, which some people advocate.
A longer-term option could involve recycling all the transuranics (plutonium is one example of a transuranic element), perhaps in a so-called fast reactor. In this approach, nearly all the very long lived components of the waste are eliminated, thereby transforming the nuclear waste debate. Substantial research and development is needed, however, to work through daunting technical and economic challenges to making this scheme work.

Pros -
1. Little Pollution As demand for electricity soars, the pollution produced from fossil fuel-burning plants is heading towards dangerous levels. Burning coal produces carbon dioxide, which depletes the protection of the ozone. The soft coal, which many power plants burn, contains sulphur. When the gaseous by-products are absorbed in clouds, precipitation becomes sulphuric acid. Coal also contains radioactive material. A coal-fired power plant emits more radiation into the air than a nuclear power plant.
The world's reserves of fossil fuels are running out. The sulphurous coal which many plants use is more polluting than the coal that was previously used. Most of the anthracite, which plants also burn, has been used up. As more soft coal is used, the amount of pollution will increase. According to estimates, fossil fuels will be burned up within fifty years. There are large reserves of uranium. This is still a more lengthy solution to the current burning of coal, gas, and oil.
2. Reliability Nuclear power plants need little fuel, so they are less vulnerable to shortages because of strikes or natural disasters. International relations will have little effect on the supply of fuel to the reactors because uranium is evenly deposited around the globe. One disadvantage of uranium mining is that it leaves the residues from chemical processing of the ore, which leads to radon exposure to the public. These effects do not outweigh the benefits by the fact that mining uranium out of the ground reduces future radon exposures. Coal burning leaves ashes that will increase future radon exposures. The estimates of radon show that it is safer to use nuclear fuel than burn coal. Mining of the fuel required to operate a nuclear plant for one year will avert a few hundred deaths, while the ashes from a coal-burning plant will cause 30 deaths.
3. Safety Safety is both a pro and con, depending on which way you see it. The results of a compromised reactor core can be disastrous, but the precautions that prevent this from happening prevent it well. Nuclear power is one the safest methods of producing energy. There are a number of safety mechanisms that make the chances of reactor accidents very low. A series of barriers separates the radiation and heat of the reactor core from the outside. The reactor core is contained within a 9-inch thick steel pressure vessel. The pressure vessel is surrounded by a thick concrete wall. This is inside a sealed steel containment structure, which itself is inside a steel-reinforced concrete dome four feet thick. The dome is designed to withstand extremes such as earthquakes or a direct hit by a crashing airliner. There are also a large number of sensors that pick up increases in radiation or humidity. An increase in radiation or humidity could mean there is a leak. There are systems that control and stop the chain reaction if necessary. An Emergency Core Cooling System ensures that in the event of an accident there is enough cooling water to cool the reactor.
Also, a nuclear reactor emits virtually no carbon dioxide (CO2), the main greenhouse gas released from human activities (though of course building the power station produces a lot of CO2). They say nuclear power is safe, and that the 1957 Windscale fire in the UK, Three Mile Island in the US in 1979, and even Chernobyl have killed massively fewer people than the oil and coal industries. Beyond that, they say modern reactors are inherently far safer than those built 20 or 30 years ago, reducing a small risk still further. Supporters say uranium prices have remained steady for decades, meaning nuclear energy is far more secure than fossil fuels can ever be.
Cons -
1. Meltdowns If there is a loss of coolant water in a fission reactor, the rods would overheat. The rods that contain the uranium fuel pellets would dissolve, leaving the fuel exposed. The temperature would increase with the lack of a cooling source. When the fuel rods heat to 2800°C, the fuel would melt, and a white-hot molten mass would melt its way through the containment vessels to the ground below it. This is a worst case scenario, as there are many precautions taken to avoid this. Emergency water reservoirs are designed to immediately flood the core in the case of sudden loss of coolant. There are normally multiple sources of water to draw from, as the low pressure injection pumps, containment spray system, and refuelling pumps are all potentially available, and all draw water from different sources.
2. Radiation Radiation doses of about 200 rems cause radiation sickness, but only if this large amount of radiation is received all at once. The average person receives about 200 millirems a year from everyday objects and outer space. This is referred to as background radiation. If all our power came from nuclear plants we would receive an extra 2/10 of a millirem a year. The three major effects of radiation (cancer, radiation sickness and genetic mutation) are nearly untraceable at levels below about 50 rems. In a study of 100,000 survivors of the atomic bombs dropped on Hiroshima and Nagasaki, there have been 400 more cancer deaths than normal, and there is not an above average rate of genetic disease in their children.
3. Waste Disposal The by-products of the fissioning of uranium-235 remains radioactive for thousands of years, requiring safe disposal away from society until they lose their significant radiation values. Many underground sites have been constructed, only to be filled within months. Storage facilities are not sufficient to store the world’s nuclear waste, which limits the amount of nuclear fuel that can be used per year. Transportation of the waste is risky, as many unknown variables may affect the containment vessels. If one of these vessels were compromised, the results may be deadly.
There is also an inevitable link between civil and military atoms, they retort. If we say we need them to stave off climate change, then so can countries like Iran and North Korea - and there is no impermeable barrier between electricity and bombs. They say nuclear energy is economic only under a very restricted analysis - by the time you have factored in the costs of construction, insurance, waste disposal and decommissioning, you need huge subsidies. And, opponents ask, what happens to the waste? The only answer we have come up with so far entails storing the most radioactive waste under guard for millennia, until it has decayed to safe levels. Certainly nuclear power would provide energy to a centralised supply system. But it would do nothing directly to reduce CO2 from transport, unless it made the advent of the hydrogen economy likelier. Also, given the long planning and construction lead times, it would be a good decade or so before we saw any new power stations, even if we decided to go ahead today.
On April 25th -26th, 1986 the World's worst nuclear power accident occurred at Chernobyl in the former USSR (now Ukraine). The Chernobyl nuclear power plant located 80 miles north of Kiev had 4 reactors and whilst testing reactor number 4 numerous safety procedures were disregarded. At 1:23am the chain reaction in the reactor became out of control creating explosions and a fireball which blew off the reactor's heavy steel and concrete lid. The Chernobyl accident killed more than 30 people immediately, and as a result of the high radiation levels in the surrounding 20-mile radius, 135,00 people had to be evacuated.
One of the causes was a design fault in the reactor - The reactor type used at Chernobyl suffers from instability at low power and thus may experience a rapid , uncontrollable power increase. Although other reactor types have this problem they incorporate design features to stop instability from occurring. The cause of this instability is:
§ Water is a better coolant than steam
§ The water acts as a moderator and neutron absorber (slowing down the reaction) whilst steam does not.
Another was that there was a violation of procedures - While running a test of the reactor numerous safety procedure were violated by the station technicians.
§ Only 6 - 8 control rods were used during the test despite there been a standard operating order stating that a minimum of 30 rods were required to retain control.
§ The reactor's emergency cooling system was disabled.

‘Technical Stuff’!!

1 Kommentar 21.2.07 08:56, kommentieren


The points with a concentration of 0%, 1%, 3% and 4% can be connected by nearly a straigt line. But 2% is above this line. Between 0% and 1% the gradient is 6.8. Between 1% and 2% the gradient is 10. Between 2% and 3% the gradient is 4.8. Between 3% and 4% the gradient is 6.9. Between 4% and 5% the gradient is 11.4.
For molecules to react they have to be very close together. That’s normally just the case when they collide. And then there still must be certain circumstances for them to react. One of these circumstances is that the molecules collide with the right angle. Another one is that they collide with enough energy so that they can break some of their bonds to form new bonds. This minimum amount of energy is called activation energy. If the molecules collide with less energy they will just bounce back without any reaction. So the activation energy is needed to break bonds in the molecules.
The molecules collide randomly. That means that the chance that they collide with enough energy and with the correct angle is low. Some reactions are slow under “normal” circumstances or don’t even occur. To increase the rate of reaction one can increase the number of collisions and the energy the molecules collide. Or one can bring the molecules in position so that they are close together and that it is more likely that they collide with the correct angle. This is done by catalysts. Very good for that are biological catalysts, which are known as enzymes.
Enzymes consist of a unique sequence of amino acids. This sequence is known as primary structure. The enzymes don’t lie flat but they fold themselves in a three-dimensional shape which is hold together by bonds formed by the side groups of enzymes. Obviously the unique primary structure gives the enzyme a unique tertiary structure.
Enzymes have a part called active site. A small number of molecules with a certain shape called substrates can bind to this part. Reactions between these molecules are catalysed by this enzyme. When substrates bind to the active site they form together with the enzyme an Enzyme-Substrate-Complex (ESC). In the ESC the enzyme wraps itself around the subtrates, so the enzyme changes its shape.
This causes the substrates to come into a position where it is likely that they collide with the correct angle and with enough energy. So it is much more likely that the substrates react. When the substrates reacted they change their shape and so they stop bonding to the active site. So they leave the enzyme after the reaction. The enzyme is flexible and changes its shape back to its original shape. So it can again form an ESC and catalyse reactions.
The more enzymes there are the more ESC can be formed and so more products can be formed in a certain time. That means that the rate of reaction increases.
Doubling the number of enzymes doubles the number of ESC and so also doubles the number of products. And finally the rate of reaction doubles. That means that the rate of reaction is proportional to the number of enzymes and/ or to the concentration of enzyme.
I expected that the rate of reaction is proportional to the enzyme concentration. In the experiment we got for 3% twice a faster rate of reaction than for 4%. So we repeated 3% twice and 4% once. The result we got then were better. The graph shows nearly a straight line but 2% and 5% seem to be too high. That is caused by errors.
The errors come from 3 sources: The procedure, the equipment and human error.
Errors in the procedure were:
-         We found our endpoint of the reaction by comparing the colour of the mixture to a colour table. But comparing colours by eye isn’t very accurate. So we stopped the time at different stages of the reaction.
-         We found our endpoint of the reaction with the help of pH. During the reaction the pH was changing. The problem is that the pH affects the rate of reaction.
An enzyme has an optimum pH (for urease this optimum pH is 7.4). That means with this pH it works better than with every other pH. That is because the pH affects the shape of an enzyme and the shape is crucial for the work of an enzyme. The lower the concentration of the enzyme is the bigger is the effect of the pH on the rate of reaction.
That means that the pH affected the reaction in the mixture with a urease concentration of 1% more than the reaction in the mixture with a urease concentration of 5%.
For me this seems to be the biggest error in the experiment.
-         The water temperature wasn’t constant during the experiment.
The temperature affects the rate of reaction. So it has to be kept constant.
But the temperature didn’t vary a lot and so it just slightly affected the results.
-      We did the experiment just for 5 concentrations and repeated each concentration just 3 times. By using more concentrations and repeating them more often we would get better and more accurate results.
Errors from the equipment were:
-         The urease didn’t dissolve completely in the water. At the bottom of the beaker was a lot urease. So the concentration varied from the concentrations we wanted to have. That could explain that twice the reaction with an enzyme concentration of 3% was faster than the reaction with an enzyme concentration of 4%

1 Kommentar 7.12.06 16:09, kommentieren

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