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    How do I solve this stochastic differential equation?

    Homework Statement I am trying to solve this \begin{align} d X_t = - b^2 X_t (1 - X_t)^2 dt + b \sqrt{1 - X_t^2} dW_t \end{align} where $b$ is a constant. Note that I have the answer here and can provide it if necessary. But I want to know how one would come up with it. Homework Equations...
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    Need some help with Mathematical Finance

    Can anybody provide some resources (books, websites etc. - preferably websites or course notes) for a (somewhat rigorous) introduction to Mathematical Finance? Basically, I want to have the background to understand stuff like this
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    Questions about Laplace's equation and Green's functions

    Homework Statement Questions 3(iii) and 3(iv) Homework Equations The Attempt at a Solution 3(iii) So here we "guess" that the solution is f = f(r) (f for phi). Then we just have...
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    Questions about complex analysis (Cauchy's integral formula and residue theorem)

    Here are the links: I've been thinking about...
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    Questions about complex analysis (Cauchy's integral formula and residue theorem) [Broken] Solutions are here. [Broken] My first question is about 3(ii), the proof of Cauchy's integral formula for the first derivative. The proof here uses the deformation...
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    Good book for vector/multivariable calculus

    What's the best book for multivariable calculus? I'm a second year undergraduate student in Mathematics. Here is the content: Functions from Rn to Rm: differentiation, contractions, Newton’s method, inverse function theorem, implicit function theorem, higher derivatives...
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    Questions about conic sections

    Thank you. For 3, how about degenerate conics? How can I find the focii and directrices from the standard equation?
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    Questions about conic sections

    Given the equation of a conic section, how can I: 1) find its focii 2) find the equations of its directrices 3) find out what type of conic it is, without using either the arduous matrix method or the equally arduous rotation method To be honest, I don't really like conic sections...
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    Functions questions

    Thank you. I will try those things. Drawing a triangle for the last one seems like a good idea. I thought of that but are we allowed to use H(-x)? The question says that it should be in terms of a(x), b(x) and H(x). Doing that, won't you get tan t = x/2 hence t = arctan (x/2)? I managed to...
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    Functions questions

    Homework Statement I'm having trouble with questions 3, 4, 8, 9 & 10. Homework Equations The Attempt at a Solution For 3 I got (b-dx)/(cx-a) as the inverse and \frac{(x(bc-ad))}{(acx + bc + cdx + d^2)} as the composed function. The bc -...
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    Questions about electromagnetic induction

    Why is there an iron core in a transformer? Is it necessary? In a transformer, how does the changing magnetic field on the primary induce a current on the secondary? Can we think of the changing magnetic field causing a changing force on the secondary and causing it to move back and forth...
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    Question about stationary waves

    Thank you. So what does "constant phase difference" mean?
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    Question about stationary waves

    I don't get it. Don't the waves need a constant phase relationship for a stationary wave to be formed? So they are in phase and this is the constant relationship, so why will they ever be out of phase? Aren't all the points in phase? Although I guess it makes sense because they are both...
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    Fleming's left hand rule

    I think I know what Fleming's left hand rule but how does Fleming's right hand rule come about and why is it opposite for generators? Is Fleming's right hand rule just the left hand rule for electron flow (instead of current) or would it be more proper to use the left hand rule in that case but...
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    Question about stationary waves

    Also, theoretically, do the two waves have to be exactly in phase to produce a stationary wave? And are anti-nodes in phase and nodes out of phase or are they both in-phase (otherwise we wouldn't get a stationary wave) Finally, why is the distance between two adjacent nodes or anti-nodes...
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    Question about stationary waves

    So a stationary wave is produced when both waves are in phase right? Anti-nodes are where both waves are at the peak and nodes are where both waves have 0 amplitude. Is that correct?
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    Question about stationary waves

    I have a question about stationary waves. Anti-nodes are where waves are in phase and nodes are where the waves are out of phase, right? But don't the waves have to be in phase for a stationary wave to be produced (so there wouldn't be any nodes)? Or do they only have to be coherent?
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    Question about perfectly competitive labor market

    Explain why, in a perfectly competitive labor market, the total net advantages of all occupations would be equalized. (10) In a perfectly competitive labor market, all workers have the same wage. So if they have the same wage how can net advantages be equalized? Jobs which have bad working...
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    Question about analogies

    Is the answer I = \frac{Q}{RC}. I neglected the negative like they did in the question. Why do they get rid of the negative when we talk about activity? Is it because we need it when using differential equations because it is decaying but we leave it when not using the differential equation...
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    Question about analogies

    Is the A in A = \lambda N equal to \frac{dN}{dt}?
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    Question about analogies

    Homework Statement The discharge of a capacitor through a resistor is analogous to radioactive decay. Write down the equation for capacitor discharge which is analogous to A = \lambda N. Explain the analogy between A and the subject of your equation. Homework Equations The...
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    Circular motion equations

    I've now learned a bit about polar co-ordinates. How does it work with them?
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    Question about charge in a circuit

    I'm unsure about the equation Q = It (charge = current * time). Say we have a perfect circuit with minimal resistance. The current will stay constant and never run out. Work doesn't need to be done because no resistance needs to be overcome. But then, as time goes on, the charge increases...
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    Circular motion equations

    But if we do it with vectors for non-constant angular velocity isn't it kind of the same as the way I did in the first post (take x and y separately and combine them with Pythagorean theorem)?
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    Circular motion equations

    I haven't learned about complex numbers and polar co-ordinates yet. I find it interesting that complex numbers (which seem abstract and not that relevant in the real world) are good at describing circular motion. I thought they were only used in really advanced level physics. For uniform...
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    Circular motion equations

    Is this derivation correct? I managed to derive v = r \omega and (I think) a = \omega^2 r. I did a^2 = \ddot{x}^2 + \ddot{y}^2 to get eventually a^2 = r^2(\dot{\theta}^4 + \ddot{\theta}^2). Then I said that, for uniform circular motion, the angular velocity \dot{\theta} is a constant...
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    Energy stored on a capacitor [Broken] Note: the energy used by the cell to charge the capacitor, W = QV, but the energy stored on the capacitor = 1/2 QV. So half the energy is lost in the circuit as heat energy as the...
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    Energy stored on a capacitor

    Why is it E = \frac{1}{2} C V^2? I mean why is it half? It seems to arise from integration but can't we say that Q = CV and then multiply both sides by V to give QV = CV^2? The left hand side is equal to work done because voltage is defined by work done/charge.
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    Need help on circle/triangle question

    Thanks. I managed to do that part. Are the co-ordinates of P P (1 + \csc \theta + \cot \theta, 0)? I can't do part (ii). For the explanation I'm unsure. All I know is that angle PRO (in the B area) is \frac{\pi}{2} - \theta. I got A(\frac{\pi}{4}) = 1 + \sqrt2 -\frac{3\pi}{8}. Is that correct?
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    Need help on circle/triangle question [Broken] Question 4 on page 12. Could someone give me some hints on how to do this? I have no idea how. All I know is that we could draw a line from the center of the circle to Q, which I think would be perpendicular to that tangent. Also...