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    Conditional probability of several events? (sports-related)

    I'm casually working on determining the probability of a team in a given sport (let's say football) reaching at least a certain level in a season. There are two main parts to this: How many games they won in the season, and how far they got in the playoffs. I'd like to assign one final number...
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    Trying to understand the oscillation of electrons in the magnetic fiel

    Working on understanding the physics of how an electron oscillates along the Earth's magnetic field. I understand that an electron will spiral around the magnetic field line, that's easy to tell from the Lorentz force. What I don't understand is what causes the oscillation. My best guess is...
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    A continuous function having an inverse <=> conditions on a derivative?

    Sorry for the poorly-worded title. I help tutor kids with pre-calculus, and they're working inverse functions now. They use the "horizontal line test" to see if a function will have an inverse or not by seeing visually if it's one-to-one. I was thinking about what that might imply. If a...
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    Covariance between x and f(x)?

    Homework Statement As part of an assignment, I have to determine propagated error of some function: f(x,t) I did it first with x & t being completely uncorrelated, but now I'm given x as a function of t, x(t), and have to do the same. Homework Equations I know the linear approximation for...
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    How to evaluate this integral to get pi^2/6:

    \int_0^\infty \frac{u}{e^u - 1} I know that this integral is \frac{\pi^2}{6}, just from having seen it before, but I'm not really sure if I can evaluate it directly to show that. I know that: \zeta(x) = \frac{1}{\Gamma(x)} \int_0^\infty \frac{u^{x-1}}{e^u -1} du Does the value...
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    Proving that 'volume' and 'surface' of hypersphere go to 0 as n -> infinity?

    Homework Statement I'm supposed to find the equations of a hypersphere in n-dimensions (meaning the set of points within the radius R), as well as of its surface (the set of points at exactly radius R). I've already found the equations, and now need to show that both go to zero as n goes to...
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    Perturbation with equations of motion for air resistance

    Homework Statement "A ball is tossed upwards with speed V_0. Air resistance is -mkv^2 and there's gravity too. Find the the time it takes the ball to reach the maximum height. Do not solve the equation of motion exactly. Use the perturbation method on the equation of motion. Solve the equation...
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    Estimate mass of neutrino given distance & KE

    Homework Statement A supernova 1.54\times10^21 m away sends out neutrinos, and a detector on Earth detects two, ten seconds apart. The first one (A) that comes has a kinetic energy of 30 MeV, the second (B) has a kinetic energy of 10 MeV. Using this, I'm supposed to come up with an upper...
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    Finding total energy as a function of the Fermi Energy

    Homework Statement "The numerator of this fraction: \overline{E}=\frac{\int \! E N(E)D(E)dE}{\int \! N(E)D(E)dE} (N(E) is the number of particles in an energy state, D(E) is the density of states) is the total (as opposed to the average particle) energy, which we'll call U_{total} here...
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    For what Z values does an atom begin to differ from a nonrelativistic model?

    Homework Statement "Relativistic effects are rather small in the hydrogen atom, but not so in higher-Z atoms. Estimate at what value of Z relativistic effects might alter energies by about a percent and whether it applies equally to all orbiting electrons or to some more than others. For this...
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    Finding all of the right cosets of H in G

    Homework Statement "Write out all the right cosets of H in G where G = (a) is a cyclic group of order 10 and H = (a^2) is the subgroup of G generated by a^2." Homework Equations - If G = (a), then G = {a^i | i=0,-1,1,-2,2...}. - A right coset is the set Hb = {hb | h is in H} - Order of...
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    Find the limit using Maclaurin series:

    Homework Statement \lim_{x\to0}[\frac{1}{x^2} - \frac{\cos(x)}{\sin(x)^2}] I'm supposed to use Maclaurin series to evaluate this limit. The instructions suggest, as a hint: "First combine the fractions. Then find the first term of the denominator series and the first term of the numerator...
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    Which momenta can never be measured?

    Homework Statement "A particle is described by: \psi(x) = \left\{ \begin{array}{lr} C & : \left|x\right| \leq +\frac{1}{2w}\\ 0 & : \left|x\right| > \frac{1}{2w} \end{array} \right. What momenta can never be measured?" Homework Equations \Delta p...
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    Range of wavelengths from a laser pulse?

    Homework Statement "A 1 fs pulse of laser light would be 0.3 um long. What is the range of wavelengths in a 0.3 um long pulse of (approximately) 600nm laser light?" Homework Equations (delta omega)(delta t) >= 1/2 c = (lambda)(frequency) The Attempt at a Solution I replaced (delta...
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    Construct a sequence whose set of limit points is exactly the set of integers?

    Homework Statement "Construct a sequence whose set of limit points is exactly the set of integers?" The Attempt at a Solution I need a sequence that will have an infinite number of terms that arrive at each of the integers, right? And since the sequence is indexed by the natural numbers...
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    Showing that there's a Cauchy sequence where Xn<X for all n?

    Homework Statement "If x is a real number, show that there exists a Cauchy sequence of rationals Xl, X2,... representing X such that X n < x for all n." Homework Equations - All Cauchy sequences are convergent - All Cauchy sequences are bounded. The Attempt at a Solution These proofs...
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