timo

As for a "proof" (more an explanation in fact): Both sides of the equation are the same (that is the definition of an equation). So you add the same value to both sides of equation (1), merely represented in different forms (namely the left- and the right-hand side of equation 2). So on the left-hand side of equation (1) you have a value and add some other value, and on the right-hand side you add the same two values. After addition, both sides of course are equal again. Note that the same reasoning applies for multiplying an equation with a constant: Both sides are equal before multiplication, so they are after both have been multiplied with the same constant. (*) EDIT: Above is really just the very elaborated form of what DrRocket said. It is not coincidence that by adding the equations you successively eliminate variables. You explicitly chose your factors such that that happens. It is also not a complete coincidence that this method works: You are not doing this on arbitrary equations but (usually) only on linear equations of the type ax+by+... = d (i.e. no equations like tan(x) + 5y² = d). (*) Careful: Two sides being equal after a multiplication does not always imply that they were equal before the multiplication, since [math] 1\neq 2 [/math] but [math] 0 \cdot 1 = 0 \cdot 2 [/math].

You should try to learn graduate level spelling. (scnr)

Well, two comments here: 1) I didn't say "a proper definition of the terms used is an integral part of mathematical expressions" just for fun. Since I am familiar with standard QM lectures I have a good idea what [math]| q_k \rangle[/math] and [math]q_k[/math] are supposed to be, namely an orthonormal basis of eigenstates and their eigenvalues, respectively. But I think saying this should have been part of your post, not my reply. 2) [math] \langle i|Q| j \rangle = \langle i| \left( \sum_k q_k | q_k \rangle \langle q_k | \right) | j \rangle = \sum_k q_k \langle i | q_k \rangle \langle q_k | j \rangle[/math]. Or in other words: No your expression is not correct, you are missing a summation. Similarly, it is not correct to say that [math] \langle i | q_k \rangle [/math] is either one or zero, and especially "k=i" makes little to no sense (it would make sense if the expression had been [math]\langle q_i | q_k \rangle[/math], but it apparently isn't).

Hi Patrick, welcome to sfn. There is a code for latex which is [ math] latex here [ /math] (as you may guess: without the space in the tags). Maybe you can use it to rewrite your question in the form of proper mathematical expressions, because this way it is hard to follow what you are trying to say/ask. For this, note that a proper definition of the terms used is an integral part of mathematical expressions.

Obviously, in this form the number of opening and closing parentheses in your relation between the letters E, M, C, and S does not match, which makes it very dubious. But the more fundamental issue here is: what is your point?

I don't really plan to justify my children's existence to them. Or are you asking why I would want to have children if they are going to die, anyways? Speaking for me, and I think also for many other atheists, having children is one way of making a footprint in the world, which can perhaps be considered the atheistic equivalent of an afterlife. By the way: why would someone believing in an afterlife desire children and go through all the hassle to raise them rather than just drowning them in a lake directly after birth? I think the desire to reproduce (physically, ideologically, ethically) is rather decoupled from the belief in an afterlife.

As are nematic, isotropic, lamellar, liquid ordered, liquid disordered, jammed, cat, hot, and waste.

Without additional restrictions that is definitely wrong. Take the solutions of "x-i=0" and "(x-3)*(3x-i)=0" for example. It is pretty trivial to check if your indeed found the roots by just plugging your candidate solution into the original polynomial. I don't understand what you are saying there. Five minutes by hand, twenty seconds with Wolfram Alpha. You probably calculated something like (4x^3 + 23x^2 +34x -10) * ( x +3 - i) rather than (4x^3 + 23x^2 +34x -10) / ( x +3 - i). But it seems to me that you are not understanding the reason behind doing this in the first place: Any 3rd degree polynomial can be written in the form p(x)=(x-a)(x-b)(x-c). The idea is to find this form, since then solving p(x)=0 is trivial.

Well, if your calculations are correct then that should of course be the correct solution. I somehow wonder why you think -3-i was a zero? Is that generally true for polynomials with real-valued coefficients that roots must be in some way be related via complex conjugation? What I meant is that you should have found that (4x^3 + 23x^2 +34x -10) / ( x +3 - i) = 4x^2 + (11+4i)x - (3+i), meaning that you are looking for the solutions of (4x^3 + 23x^2 +34x -10) = [4x^2 + (11+4i)x - (3+i)] * [ x +3 - i ] = 0. A product is zero if one of its factors is zero. The two factors are a first and a 2nd order polynomial, respectively. For both cases you should know how to find the zeros.

Imagine you had the polynomial written in the form (x-a)(x-b)(x-c) and were supposed to find the x where it assumes the value zero. That would be rather simple. In this case you know that your polynomial can be written in the form (x+3-i)(ax^2 + b^x + c). Do so by polynomial division and the solve your problem. May I ask: Where have you encountered this problem?

Pah! We all know that these votes are all from other moderators - no sane person could want any of these original thinkers, intellectually and socially extremely skilled savants to be banned.

What you seem to be misunderstanding is that your personal feelings about what the terms used in physics could mean do not necessarily equate with what the terms actually mean in physics (note that this section of this "science forum" is called "physics"). "Everything is just energy" is a prominent layman expression, but technically it is plain dead wrong as actually said in previous posts of this thread (if "energy" and e.g. "electron" are supposed to have the meaning they have in physics). I am not objecting to what you think about photons (I do not particularly care about that), I am objecting to your style of writing that sounds as if it was an informed opinion while from what you write it is obvious that you have never attended a quantum field theory or particle physics lecture. Just write "I think" rather than "surely". No offense meant - I just believe that such subtleties are important for other readers of this thread that cannot tell someone's personal view from mainstream physics' view. In the context in which one speaks about photons a particle can be considered a minimum excitation of the field with respect to the vacuum. Not sure if that can be considered a "definition", though. But it's really not been my point that I think that your ideas about particles are wrong (see above).

You seem to start quite a few statements with "surely ..." that are contradicting mainstream physics.

Nothing in particular. I merely remember that in particle physics Neutralino WIMPs were being sold as the most viable candidate for dark matter a few years ago. But this is of course a very biased view on the issue as particle physicists tend to love supersymmetry. I have not had any contact with the topic for several years now, so I am not up to date - hence my question.

Why so?

I really like econophysics, it is the fun part of statistical physics (the modeling and playing with complex systems) without the boring part (the physics). They sometimes even predict experimentally testable things. But I feel like pointing out that this "study" is a computer simulation based on rather ad-hoc assumptions. Sidenote: The "Universita di Catania" translates to "University of Catania", not to "Cornell University" .

a) The proper term is "center of mass", not "center of gravity". b) When a car drives through a curve then there are two relevant forces acting on it, that can be considered as acting on the center of mass: gravity and the centrifugal force (the latter is not a gravitational force which is why the term "center of gravity" that you guys all seem to like so much is inappropriate - as I already mentioned in my first post). c) The gravitational force points down, the centrifugal force horizontally towards the left or the right of the car (depending on the curve direction). d) The point around which the car could tip is the contact point of the appropriate wheel with the road. e) Gravitational force tries to tip the car in the one direction, centrifugal force into the other. Stronger torque wins. I think that should be enough information for you to draw the diagram yourself.

In addition to what IA said, there is another effect that makes the lower center of mass more stable with regards to tipping: For the same given centrifugal force (assuming tipping while driving through a curve) the torque is lower.

Well, I guess "precision timing" would be a nice example, not strictly for "major contribution to our understanding" (whatever that may be exactly) but at least for the related "major contribution to our civilization".

I wonder why you and Klaynos are so dismissive of the question. I'd rather like to see concrete examples, too.

There has been a variety of conventional objects being proposed as being dark matter, including lonely planets, black holes, neutrinos, neutron stars, ... . Except for the neutrinos they go under the name of Massive Astrophysical Compact Halo Objects (MACHOs). They are mentioned on slide 9 of the talk I linked. I think they appear too seldom to account for dark matter, but you can just have a look at the paper that is mentioned if you are really interested.

The point of dark matter is that several phenomena that do not match how we expect nature to behave could be explained by assuming some (invisible) additional matter content. The point is not that an elementary particle with no electric or color charge is something so great that we want to give it an extra name. Dark matter must not have an electric charge, but that doesn't make anything with that property dark matter. The addition of neutrinos alone would not suffice to explain the conflicting phenomena. Or in other words: based on having the characteristics attributed to dark matter you can call neutrinos being dark matter without being completely incorrect - but it seems like a pointless statement to me.

Assuming a sufficiently long separation time such that the effects caused by separating the clocks and bringing them back together for comparison can be ignored the theory of relativity predicts a difference of zero tics per 24 hours in your scenario. But I wonder: Why do you even care?

The physicist in me wants to tell you about torque. The pedant wants to rip your head off for statements like "has a center of gravity of two feet" (which presumably means "has a center of mass two feet above ground") and assuming that "stability of a car" was a well-defined term (while in reality the location of a car's center of mass is rather irrelevant when I drop a large rock on it).

Everything related to a Big Bang scenario would come to my mind, e.g. understanding the cosmic microwave background.