woelen

Computing a logarithm of a general number without special properties is quite a tedious job when it needs to be done by hand. If you want to have it done by means of a piece of software, then the task is not that difficult. The way, this is done in numerical software (and also in hardware like math-coprocessors) is to take an interval [1, 2>. A very good approximating polynomial function in x is determined for the log value in this interval. These polynomials are published in tables, but one can also dermine themselves by means of interpolating software. Once you have such a polynomial, it is easy to compute log(x) in that interval. If you want to obtain the log of any number z, then that number must be written as z0*2ⁿ, with z0 in the interval [1, 2>. E.g. for 7, we write 1.75*2². Now for general z, we have log(z) = log(z0) + n*log(2). In binary computers, this type of algorithms can be very fast. Determining n is easy, due to the way numbers are represented in the computer hardware. The value of log(2) can be stored as a constant to the desired precision. With this algorithm, computation of log() functions is reduced to some multiplications and additions and a little (possibly) machine-dependent bitstuffing to determine n. There is one flaw in this method, and that is that the relative precision of logarithms near for x near 1 is quite bad. The reason of this is that log(1+k) = k + O(k²). A number near 1, with k close to zero itself has full precision, but for the number k, there only remains a low precision left. For this reason, a more complete set of functions is offered by many math libraries, which also include a function log1(k), which is log(1+k), but now one can supply a value k at full precision and also gets an answer at full precision. For the same reason, many libraries also supply a function exp1(x), which is exp(x) - 1. For x close to 0, otherwise one would loose a lot of bits of precision, due to the term 1, which swamps all bits of precision for small x.

gib65, if you rally go for long windedness, then lookup the solutions for third and fourth degree equations. Just as their is a formula for second degree equations a*x² + b*x + c = 0, there also are formula's for third degree equations a*x³ + b*x² + c*x + d = 0 and fourth degree equations. Especially the latter one is EXTREMELY longwinded, but it is a closed form expression in the coefficients a,b,c,d,e for a fourth degree equation. http://en.wikipedia.org/wiki/Cubic_equation http://en.wikipedia.org/wiki/Quartic_equation The formula for the solution of the quartic equation is so longwinded, that no textbook or website simply gives the formula, but they only give an outline (recipe) for how to solve the equation by means of a few steps.

This equation A = NOT A remains a simple equation. Whether this has any solution or not depends on the set in which A may be. If A only may be in the set {0, 1}, then this equation has no solution. You could of course extend the set in which A may exist. Simply by adding a special element with the property Λ = NOT Λ, you can make this equation solvable. Now the set of possible values is extended and it contains at least {0, 1, Λ} and you can solve all kinds of logical equations, with NOT's in it. It however, requires you to introduce a new symbol Λ. You could also make rules for what is 1 AND Λ, 1 OR Λ, 0 AND Λ, 1 AND Λ, Λ AND Λ, etc. In this way, a new "boolean" algebra is created and one can solve all older solvable equations, plus a whole bunch of new equations. Just play around with it and you can grasp the idea. Now a more interesting, but still very basic, example: Suppose we have a set {0, 1, 2, 3, 4}, with operators ┼, ●. The ┼ is addition, but the result always brought back to {0, 1, 2, 3, 4} by going modulo 5. E.g. 1 ┼ 4 = 0, 4 ┼ 3 = 2. The ● is multiplication, but also modulo 5, e.g. 3●4 = 2, 2●2=4, 2●3=1. Now suppose we want to solve the equation x●x = 4. This has two solutions, being 2 and 3 (3●3 is 9 modulo 5, which is 4). Now suppose we have the equation x●x = 2 and x must be in the set {0, 1, 2, 3, 4}. This equation has no solution. Again, we can introduce a new symbol , with the abstract property Λ●Λ = 2. Now we extend the solution space to the set {0,1,2,3,4,Λ}. Now we created a new number system and with the operators ┼, ● we can have expressions like 2┼Λ, Λ●4, etc. By introducing this new symbol and keeping it as a purely abstract entity, with the property Λ●Λ=2, we add a lot of new arithmetic to our simple system and we allow the solution of all equations of the form x●x=b, with b any number from {0, 1, 2, 3, 4}. E.g. 2●Λ is a solution for x●x= 3. (2●Λ●2●Λ = 4●Λ●Λ = 4●2 = 3), the other solution for this is 3●Λ. The original solution space has 5 elements, the extended solution space has 25 elements. YT's example of extended boolean algebra has an extended solution space with 4 elements, 0, 1, Λ, 1+Λ, with the appropriate definition of "+". So, what does this short digression say. YT's example is not anything about complexity, it is just a matter of selecting the solution space in which the equation can be solved. The technique of extending a solution space is common in mathematics. Many equations, which cannot be solved in a certain solution space can be solved in an extension of that solution space. There are quite some problems in mathematics, physics and cryptology that can be solved in this way, which otherwise could not be solved. There is a whole branch of mathematics, devoted to this subject, which I just touched upon. I have tried to explain this in layman's terms and it is a fascinating branch of mathematics. It is the theory of algebras (not to be confused with what high school pupils call 'algebra'). Again, here you see a nice example of simple equations, but with a more "complex" language, allowing us to express things, which cannot be expressed in a simpler language.

This type of circuits, as described by YT is called "time dalay oscillator". Any physical gate has a certain time delay between input and output, hence the terminology. This kind of oscillators is related to the class of oscillators, called "phase shift oscillators". The latter are common in analogue electronics, where an n-th order linear circuit is fed back. The dynamics of such feedback circuits, btw. also can be described by complicated formulae with lots of terms (but I would not call that complex, I would only call such formulae tedious and error prone ).

And it is cheap, non-toxic, non-corrosive and non-flammable.

Indeed, there are quite some elements, which are very similar and as a whole are not very interesting. To me, the alkali metals are not interesting. The free metals are to some extent, but the ions are not. Really boring stuff. As energetic as a dead dog and as colorful as a grey sky. After Cr,V, the set of non-metal elements in the right upper corner are my favorite group (except F and the inert gases). It really is remarkable what can be done with these elements and combinations of them. I've made the weirdest compounds in this area, compounds like ONBr, ON-SCN, S2Br2, PBr5, SeBr4. A lot of things are still to be discovered there also. Also, quite energetic and colorful compounds can be made this way, often with colored vapors or gases (I also really think the idea of colored gases is quite special).

What do you mean with complex? A complex formula can be complex in the sense of a lot of symbols. It can also be complex in the sense of the underlying concepts. These are equations from electromagnetism, telling something about electromagnetic waves propagate through space and how the two phenomena are related to each other. This may seem very complex to you, but why are they complex? Probably because of the unfamiliarity with the math involved. http://en.wikipedia.org/wiki/Maxwell's_equations The same equations can be summarized in a simpler formula (thanks to Atheist ): [math]\square^2A_a = -\mu_0J_a[/math] This equation seems much simpler, but it describes the same equations as those in the Wiki-page, but using a more advanced mathematical language, which allows one to express things with fewer symbols. It is not important for you to know the meaning of these formulas, but these sets of formulas nicely demonstrate that the concept of complexity is not easily defined. The more advanced the language, in which things can be expressed, the more compact things can be expressed. Things which are overly complex, when described in one language, can look much simpler (and easier to memorize) in a more advanced language. Of course, this is at a cost. One needs to learn (and understand) the more advanced language. Complexity is shifted from a practical level to a meta-level.

Chromium is one of my favorite elements. It is so colorful and its chemistry is increadibly rich. If I would call chromium the king of elements, then its neighbour, Vanadium, would be the queen. This element is as colorful as chromium, but it is easier to understand. Or would I have to exchange things, being chromium the queen and vanadium the king? Women are more complex, isn't it ?

One wet oxidizer which does dissolve sulphur is a solution of the salt O2PtF6 in anhydrous HF. This salt has the cation O2(+) and the anion PtF6(-). O2(+) is a sufficiently strong oxidizing cation, which does oxidize sulphur. It also oxidizes water, hence the need of anhydrous HF. Just pick up a few liters of this at the local hardware store and try it with your flowers of sulphur

No, this does not work. Solid sulphur is quite inert and does not react with H2O2, not even with the strong stuff. Only the strongest "wet" oxidizers are capable of oxidizing sulphur at room temperature. Even concentrated HNO3 cannot do the job at room temperature. When it is heated, then the reaction proceeds slowly.

I think the entire Western world will have to step back quite a lot in material welfare. Till very recent times, the growth of Western economies and material welfare was at the cost of the rest of the world. We took resources from all over the world and added value in our own countries. The people in the other countries, however, did not gain anything from this. The big winners were the companies taking away the resources and of course also all other people, adding value to these raw resources. Now we see a development that people in other parts of the world also can do the adding of value. There is less work for people in the USA, West-Europe and Canada. In the Netherlands, this effect can be seen already. Also highly educated people loose their jobs and find no similar job. Many people work at a lower level than they used to do, and also earn less money. Also, people from Eastern Europe (Poland, Czech republic, and since a short time also people from Romania) are coming here to do all kinds of jobs and not only the lowest level jobs. They are not paid as well as their Dutch collegues, but do the same work at the same quality. Dutch workers loose jobs due to this, because they are more expensive. Nowadays, still Western companies do the adding of value, but it is done more and more in other parts of the world. We all see the outsourcing effect in IT-business, but also in chemistry in the manufacture of refined specialty products. The next step will be that 3th-world companies will take over this business and that countries do not sell their resources anymore to Western companies for a bargain, but they can use their resources on their own in their own adding of value process. They then will export the refined/added value products. So, I think that the average welfare will not change over the world, but that it will be divided more evenly. I personally think it is not a bad development. Of course it hurts if you cannot afford a big MPV anymore and have to take a smaller car, or have to go out less frequently, but in the long run we will all benefit. We still can afford a car and still can go out, but we have to work a little harder for it and we have to step back a bit.

Cheating in the sense of having a n-some party in a drunken mental condition only can lead to one conclusion: dump her. Try to forgive, which will be hard enough, and try to live on, which also may be quite hard after such a bad experience, but don't continue with that girl. I think ParanoiA is right in this situation. Such a girl is not worth a durable relation and I would definitely not want to be married to a wife, who is acting like that. The fact alone that she is going to a drunken party without her boy friend (to my opinion even going to a drunken party together is a bad thing) is a sufficient strong indication that it is time to quit. Ecoli, with many things people can have more chances, but with this, it is: "the first failure will be the final failure".

Ryan, in fact your idea is not that strange at all. It can be expressed mathematically also. The rate at which time elapses and the speed at which you move are related as follows: c² = v² + c²*(dt/dt0)² Here dt0 is the rate of flow of time for an object, which is at rest, relative to you, dt is the rate of flow of time for the object you observe, which has a velocity, equal to v. So, you see, that if v is increasing, then the rate of flow of time t must decrease. For v = 0, dt is equal to dt0 and time flows at maximum rate (the rate, we observe intuitively, and which intuitively is absolute). For v = c, dt is equal to 0. This formula also shows that the maximum velocity, which can be achieved is c, otherwise we would need to introduce complex arithmatic, which has no physical meaning here.

The key to understanding this is the concept of "Lorentz transformation". It is a transformation between two coordinate frames, which preserves the speed of light. In Newtonian mechanics, we have something called Galilei transform. It relates coordinate frames of different observers, who are moving relative to each other. This transformation leads to addition of velocities as simple linear additions. E.g. you are moving with a velocity v1, and you fire an arrow, which moves with speed v2, relative to you. A fixed observer (whatever "fixed" may be), observes an arrow, moving with speed v1+v2. In reality, however, velocities may not simply be added (or subtracted if they are in opposite direction). A more accurate description is that velocities are "added", using Einstein addition, instead of normal addition. So, instead of writing v1+v2, we must write v1┼v2, with ┼ being the Einstein addition. The Einstein addition is as follows for velocities in the same direction: v1 ┼ v2 = (v1 + v2)/(1 + v1*v2/c²), where c is the speed of light. This has the following properties: v1 ┼ c = c c ┼ v2 = c v1 ┼ v2 < v1 + v2 For v1, v2 much smaller than c, v1 ┼ v2 is VERY close to v1 + v2. That is why in daily life we never notice the properties of the Einstein addition of velocities. We are so much trained by our daily life experiences that velocities simply add up, that any deviation of that is totally alien to us. Even for the fastest rockets we have, v1 ┼ v2 still is very very close to v1 + v2. The following link may be helpful for you: http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/veltran.html

We can't know for sure. Funny to see all those people struggle with this stuff all over again. Now we have 1) (Young earth) creationists 2) Evolutionists And I came accross yet another theory, a devolutionist. No young earth. It also supports change of species, just like in evolution theory, but over the aeons species are degenerating (going to a less and less advanced state) instead of becoming more and more advanced. What it is worth, I don't know, but it is worth mentioning here, just as a curiousity. It is just a demonstration that there are so many alternative theories and we cannot know for sure which is true. http://www.evolutionisdegeneration.com/index.asp?PaginaID=1102 Don't understand me wrong. I'm not saying that this theory is the answer, but my point is that there are so many theories about our origins, and that Dawkins cannot state that all other points of view besides his point of view are ignorant, stupid, unscientific or whatever nice qualifications he has. It tells more about him, than about the people who question eveolution theory.

Dawkins is not worth discussing at all. Fundamentally, his quackery is not different from the quackery of some reli-fundies from the deep south of the USA. The circle nicely closes at the extreme ends of the rope. Reli-fundie and anti-reli-fundie meet each other at the extreme end. Such a shortcircuit can lead to a nice fight with lots of fireworks . Crackpots make a lot of crackling noise.

Yes, I realized that, while giving the answer. I already told in my previous post: But I also explained that for large N, these have the same limit. I could have expressed it a little bit more careful. They are not equivalent, as I may have suggested in my previous post, but for large N, their limit is the same, and hence I considered the question answered. But it is good that you pointed out this non-equivalency explicitly. In other cases, such non-equivalencies may indeed result in different answers.

What reductor do you think will be used? Do you really think that the iron is a catalyst only? If you want to reduce -NO2 to -NH2, then you need something to reduce with.

What do you want to tell on the website? Should it explain theory? Should it contain research results from you or your faculty? What audience do you have in mind? Before starting with a website, first think about all these things. Building a good website takes a LOT of time and effort. I know this, because I made a website myself on science (and a lot of chemistry) and it has taken me 20 months already before I had what I have now. Have a look at it: http://www.oelen.net/science

In order to solve this problem the assumption is made that the thickness of the wood can be neglected, so we simply work with width w and not with w plus a few mm for the wood's thickness. First compute the height as a function of w. The area of the base is 3w². The total volume must be 1.5 m³. So, the height is 1.5/(3w²) = 0.5/w². Now, what is the area of the sides? There are four sides. Each side has a height of 0.5/w² and two of them have a length w, and two of them have a length 3w. So, the total area of wood for the sides is 2*3w *(0.5/w²) + 2*w*(0.5/w²), which is 3/w + 1/w = 4/w. So, the cost of the box is 5*(4/w) + 8*3w². = 20/w + 24w². E.g. for a box with a width of 2 meters, the cost would be 20/1 + 24*1*1 = 20+24 = 44 dollars. This is the answer to question (a). Now try to obtain the answers for (b) and ©. You have seen the reasoning and now the others should be possible for you.

Martin you have made your point now with the Dawkins stuff! We know your position. We know it. Really, we know it. No need to repeat all over again.

The reaction between citric acid and an oxidizer is very complicated. Acetone is one of the reaction products, but there are many more. Also CO2 is formed. The reaction requires acid as well. Writing a chemical equation of this reaction does not make real sense. Sometimes it hardly makes sense to write such equations, because the reactions are not clean and simple, but many reactions occur at the same time in unknown ratios.

Shmoe, again thanks for your explanation. The error analysis still was quite cumbersome for me and involved a lot of work. Your explanation is compact and can be understood well. This does not seem a difficult question to me, but if I am missing something, then please let me know. Take two functions A(N) and B(N). A selects all numbers, less than or equal to N and counts the selected numbers. B selects a fraction of all numbers less than or equal to N for all N. The function A(N) is equal to N. The function B(N) is equal to b*N, with 0 < b < 1. Now, if we take A(1) + A(2) + .... A(N), we come to ½N² + O(N). If we take B(1) + B(2) + .... B(N), we come to b*½N² + O(N). The probability that a number is selected, less than N equals 1 for function A, for function B it equals b*½N² / ½N², which equals b. Now take B(N) equal to phi(N). The sum now equals (3/pi²)*N² . The probability is equal to (3/pi²)*N² / ½N² = 6/pi². I did not check my answer numerically, by taking a lot of numbers, but I have a strong evidence that this is the correct answer. The answer somehow surprises me, however. Intuitionally I would expect this chance to be much lower. So, more than 60% of all pairs of randomly selected very large numbers are relatively prime to each other! Actually, I have answered a slightly different question. You asked for picking two random numbers, less than n. I answered to the problem of given a number N, what is the chance that the a random number less than N is relative prime to N. These questions are equivalent to my opinion. Suppose I pick two numbers N1, N2, both less than N in a random way. These numbers still are of order N. Now take the largest one of these two and determine the chance that the other one is relative prime to it. This works, because N1, N2 still are of order N and we still may assume the limit situation for N-->infinity.

How is this going to put people at risk? As I understand it, it is not breaking in at the lights, making your one green and the others red. It is just making yourself detected by the sensor system of the lights. But I agree with Sayonara that contacting the relevant authorities may be a more useful solution, which also helps other people. On the other hand, such authorities tend to be sooooooo slooooooow.....

I do not know how these traffic lights work. If they simply cycle through all lights around the intersection, then such a thing is of no use at all. Sometimes, the cycles can be very complicated (and annoying). E.g. ABABCABABC...., where C is some smaller parallel street, or for traffic to the left. But still, the cycle is fixed and does not depend on actual traffic. The more intelligent traffic lights do have sensors, but I do not know how these work. I can imagine that they miss bikes and other smaller participants of traffic. I would say, buy a few of these magnets and attach them to the frame (outside). The nickel-plated ones are corrosion resistant and keep up very well in all kinds of weather. Where I live, this issue is solved in another way. Cars and other large traffic are detected by sensors, but people on bikes and pedestrians have to push a button, telling that they are there.