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Thanks for the replies! analogdesign, can you describe what your job entails? What is involved in the projects that you and your coworkers work on? For instance, is it more common to design a specific circuit board, or to engineer how all the parts fit together to make the instrument function...

I've been considering a change in career direction recently, and have been thinking about a career in developing scientific instrumentation. I know that "scientific instrumentation" is vague and encompasses a lot, hence the reason I am posting -- to try sharpen my understanding of exactly what...

In 2 dimensions, is the geodesic deviation equation governed by a single scalar, independent of the direction of the geodesics? That is, if ξ is the separation of two nearby geodesics, do we have d^2 \xi/ds^2 + R\xi = 0 where R is a scalar that is completely independent of the direction of the...

Interpreting "diverging thermal fluctuations" as equivalent to "infinite correlation length" doesn't make sense with another article I am reading which says: "What drives the correlation length to infinity are thermal fluctuations, which become very large close to criticality." If large "large...

Ok, that makes sense. It brings up another question for me though. Why do people call a system "scale invariant" when the correlation length diverges? The correlations still drop off (with distance) via a power law, right? So if I zoom out they will change, and so don't seem "invariant".

According to wikipedia: "As for a classical second order transition, a quantum second order transition has a quantum critical point (QCP) where the quantum fluctuations driving the transition diverge and become scale invariant in space and time." I am confused about what this means. Why do the...

Suppose we apply a uniform field to an infinite conducting slab (i.e. like an infinite parallel plate capacitor, but the interior is included as part of the conductor). What is the resulting field? The simple answer is that a surface charge develops on the boundary planes of the conductor so as...

To preface my question, I know it is related to the Gibbs paradox, but I've read the wikipedia page on it and am still confused about how to resolve the question in the particular form I state below. Suppose a completely isolated ideal gas consisting of identical particles is confined to a...

I don't get the point of equations of state since they seem to me to just indicate that we defined too many state variables. Why not just trim down our set of state variables and do away with the equations of state (i.e. for an ideal gas, just notice that P and V are sufficient to describe the...

So while there is some notion of distance in classical spacetime, you cannot speak of a distance between two arbitrary spacetime points, right? You can only speak of the distance between points on the same time slice (this would be the spatial metric) or points at the same position slice (this...

So then is it not possible to think of classical space and time as a single spacetime since we cannot assign a definite meaning to the distance between spacetime points (independent of reference frames)?

In special relativity we have the invariant spacetime interval ds2 = dx2 - c2dt2. If we think about classical (non-relativistic) space and time as one spacetime in which the transformation between reference frames is given by the Galilean transformation, is there a corresponding spacetime...

I am reading 'Classical Dynamics: A Contemporary Approach' by J. Jose, and I am confused about a step in the author's development of potential energy for a system of many particles. He begins by writing down a term equivalent to the total change in kinetic energy of the system: \sum_i...

Both posts were helpful. Thanks.

I am trying to follow the following reasoning: Given a known matrix A, we want to find w that maximizes the quantity w'Aw (where w' denotes the transpose of w) subject to the constraint w'w = 1. To do so, use a lagrange multiplier, L: w'Aw + L(w'w - 1) and differentiate to...

A color image is essentially 3 grayscale images, right? Can you just compress each of the three matrices individually using the methods available for grayscale images?

Given a matrix A (for simplicity assume 2x2) we can use the Cayley-Hamilton theorem to write: A^k = a_0 + a_1 A for k>= 2. So suppose we have a given k and want to find the coefficients a_0,a_1. We can use the fact that the same equation is satisfied by the eigenvalues. That is, for any...

Do you mean that, for a linear system, if the input and state variables are zero then the output must also be zero? My textbook defines a linear system as a system which satisfies the following (x denotes state variable, u input, y output): Given \begin{array}{l l} x_1(t_0) \\ u_1(t)...

I am reading "Linear System Theory and Design" by Chen and he says (in what follows g(t,tau) is the impulse response function): If a system is causal, the output will not appear before an input is applied. Thus we have Causal <==> g(t,tau) = 0 for t < tau. However, this seems incorrect...

I just finished up applying to physics PhD programs for fall 2013. I was trying to think whether there was anything else I could do to increase my chances of admission. Would visiting the schools prior to acceptance increase my chances of getting in, or would it be perceived as annoying? I...

I will look into this. You might have a point. Also, I was looking around a bit and found that the university of pittsburgh has a program I might be interested in. So I figured I would post it here for future reference for others who find themselves in a similar position to myself...

That is probably what I would say to cytochrome about pursuing it on the side. I can pursue it on the side, but I will be scratching the bottom of the barrel for time. It is also somewhat depressing when none of your colleagues are very interested in the same questions as you. This is a good...

I am currently applying to graduate programs in physics. I have a strong application: solid grades, high physics gre, research experience, and good recommendations, so I think I have some pretty good chances of getting in. But when looking at grad programs, I am continually finding myself...

I think you may be confused about what the stabilizer is. Suppose you have a group G a set S (this set is not necessarily a group, it is just a bunch of elements) There need be no connection between the elements of S and the elements of G. That is, S is just a collection of elements that...

That makes sense. But I will need to think about this for a bit to see that it is equivalent to the definition in terms of cosets.

I followed you up to here: But then I lost you here: I don't understand what you mean by substituting an equivalence class (i.e. [x]) for x in p(x). I am thinking about the x's purely as formal variables. In my thinking, we could remove them and just use n-tuples with a special definition...

I just had a discussion with someone who said he thought about quotient rings of polynomials as simply adjoining an element that is a root of the polynomial defining the ideal. For example, consider a field, F, and a polynomial, x-a, in F[x]. If we let (x-a) denote the ideal generated by x-a...

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Homework Statement Show that the ideal (3, x^3 - x^2 + 2x -1) \text{ in } \mathbb{Z}[x] is not principal. (The parentheses mean 'the ideal generated by the elements enclosed in parentheses') 2. The attempt at a solution I came up with a solution (see attachment), it is just rather...

Thanks for the quick response!

I was just looking at wikipedia's article on ring homomorphisms (http://en.wikipedia.org/wiki/Ring_homomorphism) and I am a little confused. If you look at the definition they give for ring homomorphism, they require only that addition and multiplication is preserved over the homomorphism...

Homework Statement Let D be a division ring, C its center and let S be a division subring of D which is stabilized by every map x -> dxd-1, d≠0 in D. Show that either S = D or S is a subset of C. 2. The attempt at a solution I haven't actually started working on it yet because I am not...

I wonder whether it is possible to read the signals from these nerves (by placing a electrode or something on your arm) and interpret them so as to be able to track your hand.

What are the nerve's being stimulated by?/What information are they transmitting? Are they transmitting information about how tense my various muscles are and then from that information my brain is calculating the orientation and extension of my arm and hence the location of my hand?

If I close my eyes (so as to remove my sense of sight) and spread my fingers so they do not touch (so as to remove sense of feeling) and then move my hand around, I still have a sense of where in space it is located. How is my brain figuring out the location of my hand under these conditions?

So its not really that it needs more fuel, it actually needs the same exact amount of fuel (in vapor form), it is just that it takes extra liquid fuel to produce an equivalent amount of evaporated fuel on a cold day?

Why does the choke help an engine start when it is cold? I know that the choke restricts the air flow and thus makes the mixture of air/gas entering the engine more concentrated with fuel, but why would that make it easier for the engine to start on a cold day? Shouldn't the engine be easiest to...

I think this is a good question. If you will allow me to rephrase what you are saying: (1) We claim that the numbers (both rational and irrational) correspond to points on the number line. (2) The way to find out what point a given number corresponds to is to by looking at its decimal...

Draw two points on the number line as follows: Draw a point 1 unit along the number line. Label it point A. To draw the second point, construct a square which has the line segment extending from the origin to A as one of its sides. Draw the diagonal of this square, and then rotate the diagonal...

To get an intuitive feel for irrational powers, I like to think about it as follows: I think it is helpful to first consider the question 'why is anything raised to the 0th power 1?' The answer is that, if you look at f(x) = y^x defined for all non-zero x, you will note that f(x) becomes...

By 'true' I definitely do not mean descriptive of physical reality. I am also not sure I want to restrict my definition to logical consistency. The best way I can get at what I mean by 'true' is by explaining what I mean by 'not true.' Mathematics that is 'not true' would be ontologically...

Though we chose our axioms so as to attempt to capture the behavior observed in the physical world, I do not think that the physical world contains the mathematics we come up with. Take euclidean geometry for instance. I do not think there are any such things as lines, circles, or spheres in the...

Can you explicitly lay out how you translated the statement used in the liar paradox ("this statement is false") into "P if and only if not P"? I am a little confused because you mentioned before that the liar paradox is not a statement. However, "P if and only if not P" is a statement, albeit a...

I guess I agree with you. It would be ridiculous for mathematicians to wait around for solid foundations...you may never get anywhere. I guess I am just unsettled by the recognition that we do not ultimately know whether the mathematics we are doing is true; on that question, we basically make...

SteveL27, thanks for all of those links! They were quite interesting, if not disturbing. I think this is a bit of an oversimplification. The vast majority of mathematics today is much more abstract and much more distant from the intuitively obvious idea that 1 + 1 = 2. Maybe you can keep 1 + 1...

I am not familiar with formal logic/ZFC, but your response is very good and so I would like to interact with it as best as I can at the moment (I will be taking formal mathematical logic next spring). I would guess that there is not really a proof that there is not a statement satisfying the...

It seems to me (though I would be *extremely* glad to be proven wrong here) that in mathematics we often blindly assume that the theorems we attempt to prove/disprove are either true or false. Such an assumption is implicit in every proof by contradiction. We eliminate the possibility of the...

I noticed you included Penn State on your list. I recently applied to Penn State and, if I am remembering correctly, they ask you to supply your GPA calculated on the basis of your junior and senior year. You could double check with them about whether that is correct.

bublik13, I know you are waiting for a response from mathwonk, but in the meantime I want to chime in here. When I was an undergrad, I recall feeling the same thing. I was a physics major, and was constantly considering switching to math. For some reason, though, I had this idea that you had...

Looking over UGA's program has made me doubt whether I want to go to lehigh. It looks like they have a larger range of specialties to choose from. They have people working in algebra, geometry, topology, mathematical physics, and analysis. I guess that is one of the benefits of a large...