Dapthar

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From a mathematical standpoint' date=' I think that Minkowski space shares enough properties of Euclidean 4-space for the Banach-Tarski 'paradox' ([i']It's not a paradox, it's just goes against one's intuition.[/i]) to still hold. I believe that you can still create sets that are not Lebesgue measurable in Minkowski space as well, and aside from the Axiom of Choice, this is all one needs to make Banach-Tarski 'work'. Physically, I don't believe that non- Lebesgue measurable subset of a physical object is possible, so in 'real life Minkowski space', Banach-Tarski wouldn't work. The Banach-Tarski 'Paradox'. A very nice introduction (i.e.' date=' not too mathematically detailed[/i']) to Banach-Tarski is located here: http://www.kuro5hin.org/story/2003/5/23/134430/275

The Matrix Revolutions

But it's a school in Canada. That's like being the sharpest butter knife in the drawer.

Of course this is rather late, but if you can cook, I suggest you have a Caesar salad. Recipe link: http://www.reluctantgourmet.com/caesar.htm If you're a vegan, you can leave out the egg. I frankly don't think it adds much to the salad myself. Both of them are fast food restaurants in the US. Links: http://www.pandaexpress.com/default.asp http://www.jackinthebox.com/

I don't get why you start looking for specific solutions past this point, since there are an infinite number of them. After you get that the following system must be true: [math]1 - ab + bc - b = 0[/math] [math]c^2 + a - ac - bc = 0[/math] I would just do the following: [math]1 - ab + bc - b = 0[/math] [math]\iff 1 + b(-a + c - 1) = 0[/math] [math]\iff -1 = b(-a + c - 1)[/math] [math]\iff \frac{-1}{-a + c - 1} = \frac{1}{a - c + 1}=b[/math] Also, [math]c^2 + a - ac - bc = 0[/math] [math]\iff c^2 + a - ac = bc[/math] [math]\iff c + \frac{a}{c} - a = b[/math] Thus, since [math]b = b[/math]a solution exists if and only if [math]c + \frac{a}{c} - a = \frac{1}{a - c + 1}[/math]. Of course, if the denominator in that last equality, or the right hand side, or [math]c[/math] is zero, you have to go back and work out a special case.

I took it about 6 years ago myself, but I don't think much has changed since then, aside from the addition of slope fields to the BC exam. By the way, are you taking the AB or the BC exam? Remember the "correction for guessing" penalty on the multiple choice section, and try to write something down for every free response question, because you'll surely get a zero for a free response question if you leave it blank. Also, as far as I know, there is no additional penalty for incorrect answers to free response questions. Finally, bring a backup set of batteries for your calculator.

Yup. Yup. This is redundant, since if the vector in question is a linear combination of the two basis vectors, this relation is automatically satisfied.

I highly doubt this is true. If it was, how do you explain the majority of Americans doing nothing to halt despicable causes such as slavery from the 1600s to the early 1800s, the forced appropriation of Native American land in the 1800s? I propose a much more cynical premise: As long as whatever changes the government makes doesn't cause a significant drop in the average citizen's quality of life, then they will do nothing to oppose it.

One such movie that I highly recommend is Stand and Deliver. It is a film that centers around a dedicated teacher who is determined to teach Calculus to a group of inner-city youth who have 'slipped through the cracks' of the public school system. I realize it sounds like a rather 'corny' movie, however, Edward Olmos (the teacher) gives a solid performance, and the students are portrayed by decent actors as well. Also, the movie is based upon a true story, and the status of the real-life counterparts of the students portrayed in the film, as well as other students who took this man's class, are mentioned at the end of the movie.

Or end up going insane. Same difference.

Yes. For example, let's consider Electrical Engineering. There are EE jobs available everywhere, so if you don't like a company, you can always pack up and leave. The average starting pay (in the US, anyways) is pretty decent. A person with a B.S. in EE in 2003 was getting $49,794 a year to start. (Source: http://stats.bls.gov/oco/ocos031.htm) I would just like to take a moment to mention that the above source is an excellent resource for salary data for almost every profession imaginable. (And no, the Man did not pay me to say that.) The alphabetical index of jobs located here: http://stats.bls.gov/search/ooh.asp?ct=OOH For the most part, the hierarchy of pay for people with a Bachelor's degree in a tends to go as follows (numbers listed are average first year salaries): (Please see inset graphic below) (Source: http://money.cnn.com/2005/04/15/pf/college/starting_salaries/ ) Both are grossly exaggerated by the media. Most engineering professions are rather stable career choices, so yes, I think they are good jobs. You have to be comfortable with Math before Calculus. Most engineering professions use Calculus I, II, and Multivariable Calculus, along with Linear Algebra and Differential Equations. In addition, each field of engineering has a penchant for some subfield of Mathematics. For EE it is Complex Analysis and Fourier Analysis. For Aerospace Engineering, it's PDEs. However, everything beyond and including Calculus is taught during the course of a B.S. Aside from the Math prerequisites, you just have to have a genuine interest in the material, and a desire to learn. The 'tinkering' aspect is not universal among engineering disciplines. For example, most fields in EE have very little 'hands-on' interaction. The same goes for Aerospace Engineering, Chemical Engineering, and Computer Engineering. You don't have to be. You learn to deal with the necessary details during the course of your study. You don't have to be. Most of the difficulties in engineering lie in understanding the theory, and knowing how to appropriately apply it. I think you should give it a try, take an introductory course or two in Electrical Engineering, Mechanical Engineering, Aerospace Engineering, or Computer Engineering, and see if they interest you. Alternatively, you could just do some research online into each of these fields, and see what sparks your interest.

It might help if you told us the make and model of the mouse, as well as if it connects to your computer via a USB or PS/2 port. Without this information it will be very difficult for anyone here to help you.

I've experienced similar difficulties during my high school years, except with English teachers. I suggest the following course of action: First, ask your friend if you can borrow their journal, so you can make a few copies to use as evidence. Second, make sure your parent/guardian is informed of the details of your difficulties, show them the photocopies, and let them review your case. If you can convince them that your claims are valid, then ask your parents/guardians to set up an appointment with your teacher. In my experience, high school teachers won't give the time of day to students, but concerned parents/guardians make them stand up and notice. Bring the photocopies with you. Finally, this may be hard to do, by try to take an objective look at your situation, and make sure that it is worth fighting for. If this is an isolated incident, it is best not to pursue it any further. However, if this incident is indicative of a pattern of behavior, then by all means, go for it. Also, remember that as RedAlert and Bettina mentioned, getting angry tends to not further your cause. I know that sometimes the things people do, high school teachers in particular, can make you angry, and you feel a great deal better by just telling them off. However, while it makes you feel better in the short run, worsens your long term prospects of a fair grade. You have to remain calm and collected, for that is the best way to advance your agenda. Here's a technique that I used which helped me is that every time a teacher said or did something that agitated me: I didn't say anything for a second, and thought about how they would simply repeat that behavior if they knew it 'pushed my buttons'. Denying them that satisfaction allows you to claim the high ground, and never have your claims dismissed because a teacher clams you're being 'irrational'. Remember, logic and cooler heads tend to prevail.

Then you would have to get a DNA sample from every child at birth. Even if people were willing to agree with such a gross invasion of privacy, this system still wouldn't catalog the DNA of children who were not born in a hospital.

One can get around this by injecting themselves with synthetic testosterone. (I first learned this from an episode of Law & Order.) (Source: http://www.ncjrs.org/pdffiles/163390.pdf) The above PDF is about 220 kB, and is a study by the US Department of Justice on child molestation. The section I am referring to is quoted below:

No problem.

The majority of companies that I have investigated that are working on holographic technology are trying to make the holograms larger' date=' cheaper, faster. Here's some more reference links on the subject, as well as some current products: [list'][*]A research paper on the subject, with links to videos (The PDF at this link is approximately 812 kB. The videos linked in the article are about the same size.) [*]A prototype product from Texas Instruments [*]Recent developments in holographic technology

I'm still reading the website at the moment' date=' so I'll hold off on commenting about it until I finish. Nope, far from it. I'm currently a student who's almost finished with their Baccalaureate studies (in EE and Math), and I have no plans to pursue studies in medicine. After rereading your post, I see that I misread the portion you are referring to. My mistake. I agree, but one should not try to determine potential allergens on their own, as potentially fatal reactions may result. I still content that a child should have an allergen test performed by a doctor before they begin any sort of treatment regiment, holistic or otherwise. I agree. I have investigated and used some rudimentary holistic methods myself. I may have already mentioned that via diet and exercise, I no longer experience asthma attacks. In addition, a few years ago, even though my weight was normal, my cholesterol was dangerously high, (280+), and I was told that I would need to begin a drug treatment regimen to avoid potential damage to my blood vessels. However, I resolved to not resign myself to a daily prescription regimen, and 6 months later, via alterations in my diet, and a modified exercise regimen, I had lowered my cholesterol to acceptable levels (< 200). The problem is that doctors usually inform their patients of the dangers of prescription drugs, while vitamins, for the most part, do not have the potential side effects listed on the bottle. I agree, but as I mentioned above, only one who takes the time to research vitamin supplements know about the side effects. In addition, most people intrinsically assume that vitamins are safe, and that 'more is better', so they very well could take 2-3 multivitamins per day. Since multivitamins usually have 100% of the recommended daily amount of a fair amount of vitamins, there is a real potential for overdose in such a situation While this incident is unfortunate, it was most likely due to the doctor not properly investigating your risk for side effects. As such, it is an example of the system not working properly. I contend that, in large part, the 'traditional' system of medicine tends to benefit the patient more often than not. On a related note, if you haven't seen the movie John Q, I highly recommend it. It is a rather extreme example of the medical system gone awry, and a fairly good movie as well. I will try to find this article, but I'll reserve my comments until after I read it. Common side effects of Cystic Fibrosis drugs As the data at the link indicates, most of the common side effects are rather mild, compared to the potential symptoms of the disease itself. Actually, I'm interested in specifically what type research that you suggest they pursue, since, as you mention, most of the necessary ingredients for a holistic regimen are readily available.

Yes, rudimentary holographic projectors have been around since the early 1990's, however, it takes a great deal of time to 'render' the hologram. Thus, unless better 'rendering' techniques are devised, real-time holographic video conferencing will not be feasible until computers become much faster. (Source: http://xenia.media.mit.edu/~lucente/holo/holovideo.html)

If all the people are in the same room, they are most likely are all reacting to the same stuffy conditions in the room. As for yawns being 'transmitted' over the phone, my guess is that its probably just psychological. (Source: http://www.straightdope.com/classics/a2_309.html)

If you change to '[math]u[/math]limits', you don't need to substitute the functions of [math]x[/math] back in at the end, just evaluate the integral as one normally would.

(I think that there might be a small problem with your notation, since if the sets in question are open, there is, in general, no maximum or minimum distance between them. You would have to use [math]\sup[/math] and [math]\inf[/math] instead of [math]\min[/math] and [math]\max[/math] respectively.) As with most things in Topology, it really helps to have an intuitive notion of what a definition 'actually means'. In this case, here's what I have as my 'internal definition' of [math]h(A,B) = r[/math] "Every point in [math]A[/math] is at most [math]r[/math] units away from a point in [math]B[/math]." Thus, with this in mind, we can formulate an idea for the proof. (You should be able to handle the details on your own.) So, what does [math]h(A\cup B, C\cup D) = r_1[/math] mean? It means that: Every point in [math]A[/math] and [math]B[/math] is at most [math]r_1[/math] units away from some point in [math]C[/math] or [math]D[/math]. With this in mind, we can easily see that (i.e. you have to manipulate the definitions to prove this): [math]h(A\cup B, C\cup D) \leq \max\{h(A, C\cup D),h(B,C\cup D)\}[/math]. However, it is also clear that (again, fill in the details): [math]\max\{h(A, C\cup D),h(B,C\cup D)\} \leq[/math][math] \max\{h(A, C), h(B,D)\} [/math] (This part is simply a basic application of the definition of Hausdorff distance.) Technically, [math] h(A\cup B, C\cup D) \leq \max\{h(A,D), h(B,C)\} [/math] as well, since the problem is 'symmetric with respect to the sets'.

I'm working on it, and I'll post a reply in a few minutes.

Matt and Nicoco are right. Your calculations implicitly assume that the standard basis was the basis the problem was dealing with, so your matrix is the standard matrix. This assumption is most likely correct, since the problem doesn't seem to mention that an alternate basis should be used.