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  1. B

    Will there ever be a new CPU manufacturer?

    Currently in desktop CPU manufacturing, there are only 2 companies that hold practical all of market. I have read it is because Intel and AMD own licenses to x86 and x64 architecture ( Lets assume VIA is non-existent). Now if a new company wants to make CPUs they need to make some new CPU...
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    News RIP Chester Bennington of Linkin Park

    I am very shocked to hear that lead singer of Linkin Park had commited suicide on Thursday. I used to listen to them alot back when they released their third album, Minutes to Midnight, in 2006-07. They are one of my most favorite bands and Chester in my opinion was a very versatile singer. I...
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    Potential of a cylinder.

    1. Homework Statement 2. Homework Equations 3. The Attempt at a Solution The position of the point (where V is to calculated) on the z-axis would be ##u = z_0 + l/2##. So in cylindrical coords, $$V(u) = \int_V {k \rho \over (s^2 + (u -z)^2)^{1/2}} dV = k \rho \int_0^L \int_0^{2\pi}...
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    B Collision between moving walls

    https://www.physicsforums.com/attachments/upload_2017-6-17_4-22-26-png.205585/?temp_hash=f9f8d75085046fd530ad1071794d65c1 I have problem with the solution given of the (b) part of the question. The given solution is : I did not understand why the value of ##\Delta T = 2x/v## even when...
  5. B

    Rising ring

    1. Homework Statement 2. Homework Equations 3. The Attempt at a Solution For ##0 \ge \theta \ge \pi/2## Forces on the ring, ##Mg + 2N\cos \theta = F\qquad 1## Forces on the beads ##mv^2/R = - N + mg \cos \theta## By conservation of energy when the bead has fallen through some...
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    Damped Oscillators

    1. Homework Statement 2. Homework Equations 3. The Attempt at a Solution After the release the block will move towards right and friction will be towards the left. ##M\ddot x = f - kx## Solving for ##x##, ##x = A\cos (\omega t) + B\sin(\omega t) + f/k## Initial conditions are ##x(0)...
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    B About vector spaces

    I am confused why is space over field ##R## not over field ##C## ? The entries in each vector is an element of ##\Bbb C## not ##\Bbb R##.
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    B Why does every subfield of Complex number have a copy of Q?

    Why does every subfield of Complex number have a copy of rational numbers ? Here's my proof, Let ##F## is a subield of ##\Bbb C##. I can assume that ##0, 1 \in F##. Lets say a number ##p \in F##, then ##1/p \ p \ne 0## and ##-p## must be in ##F##. Now since ##F## is subfield of ##\Bbb C##...
  9. B

    B Why the hate on determinants?

    Why do most books on linear algebra have something like "Determinants are useless now".I have seen this in Strang, Friedberg and Axler's book. Are determinants of no use in Maths ? which tool has taken its place in algebra ? And why this happened ?
  10. B

    Number of matrices

    1. Homework Statement Let ##A = \begin{bmatrix} a&b\\c&d \end{bmatrix}## such that ##a+b+c+d = 0##. Suppose A is a row reduced. Prove that there are exactly three such matrices. 2. Homework Equations 3. The Attempt at a Solution 1) ##\begin{bmatrix} 0&0\\0&0 \end{bmatrix}## 2)...
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    B Associativity of Matrix multiplication

    ##\begin{align}[A(BC)]_{ij} &= \sum_r A_{ir}(BC)_{rj} \\ &= \sum_r A_{ir} \sum_s B_{rs}C_{sj}\\ &= \sum_r\sum_s A_{ir}B_{rs}C_{sj}\\ &= \sum_{s} (\sum_{r} A_{ir} B_{rs}) C_{sj} \\ &= [(AB) C]_{ij}\end{align}## How did it went from ##2## to ##3##. In general is there a proof that sums can be...
  12. B

    B ##AB = I \implies BA = I##, for square matricies ##A,B##

    Let ##(AB)_j## be the jth column of ##AB##, then ##\displaystyle (AB)_j = \sum^n_{r= 1} B_{rj} \alpha_r## where ##\alpha_r## is the rth column of ##A##. Also ##(BA)_j = B \alpha_j \implies A(BA)_j = \alpha_j## susbtituting this in the sum ##\displaystyle (AB)_j = \sum^n_{r = 1} B_{rj}A(BA)_r##...
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    B Invertibility of a Matrix

    If ##A## is ##m \times n## matrix, ##B## is an ##n \times m## matrix and ##n < m##. Then show that ##AB## is not invertible. Let ##R## be the reduced echelon form of ##AB## and let ##AB## be invertible. ##I = P(AB)## where ##P## is some invertible matrix. Since ##n < m## and ##B## is ##n...
  14. B

    Prove that a matrix can be reduced to RRE and CRE

    1. Homework Statement Let ##A## be an ##m \times n## matrix. Show that by means of a finite number of elementary row/column operations ##A## can be reduced to both "row reduced echelon" and "column reduced echelon" matrix ##R##. i.e ##R_{ij} = 0## if ##i \ne j##, ##R_{ii} = 1 ##, ##1 \le i \le...
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    Inverse of a matrix

    1. Homework Statement Find the inverse of ##A = \begin{bmatrix} 1 & \dfrac12 & & \cdots && \dfrac1n \\\dfrac12 & \dfrac13 && \cdots && \dfrac1{n+1} \\ \vdots & \vdots && && \vdots \\ \dfrac1n & \dfrac1{n+1} && \cdots && \dfrac1{2n-1}\end{bmatrix}## 2. Homework Equations 3. The Attempt at a...
  16. B

    B Help understanding a proof

    For a ##n\times n## matrix A, the following are equivalent. 1) A is invertible 2) The homogenous system ##A\bf X = 0## has only the trivial solution ##\mathbf X = 0## 3) The system of equations ##A\bf X = \bf Y## has a solution for each ##n\times 1 ## matrix ##\bf Y##. I have problem in third...
  17. B

    B What do "linear" and "abstract" stand for?

    What does "linear" in linear algebra and "abstract" in abstract algebra stands for ? Since I am learning linear algebra, I can guess why linear algebra is called so. In linear algebra, the introductory stuff is all related to solving systems of linear equations of form ##A\bf{X} = \bf{Y}##...
  18. B

    B Proof of elementary row matrix operation.

    Prove that interchange of two rows of a matrix can be accomplished by a finite sequence of elemenatary row operations of the other two types. My proof :- If ##A_k## is to be interchanged by ##A_l## then, ##\displaystyle \begin{align} A_k &\to A_l + A_k \\ A_l &\to - A_l \\ A_l &\to A_k + A_l...
  19. B

    Linear ordinary differential equation.

    1. Homework Statement ##\dfrac{dy}{dx} + y = f(x)## ##f(x) = \begin{cases} 2 \qquad x \in [0, 1) \\ 0 \qquad x \ge 1 \end{cases}## ##y(0) = 0## 2. Homework Equations 3. The Attempt at a Solution Integrating factor is ##e^x## ##e^x\dfrac{dy}{dx} + e^x y = e^x f(x)## ##\displaystyle...
  20. B

    A simple DE.

    1. Homework Statement ##(2x + 3y + 1)dx + (4x + 6y + 1) dy = 0## ##y(-2) = 2## 2. Homework Equations 3. The Attempt at a Solution Let ##z = 2x + 3y## then ##z^\prime = 2 + 3y^prime## ##\displaystyle \dfrac{(z + 1)}{2z + 1} + \dfrac13\left({dz \over dx} - 2\right) = 0## ##\dfrac{dz}{dx}...
  21. B

    Equation of states for a gas that forms dimers

    1. Homework Statement Show that to a first approximation the equation of state of a gas that dimerizes to a small extent is given by, ##\dfrac{PV}{RT} = 1 - \dfrac{K_c}{V}## Where ##K_c## is equilibrium constant for ##A + A \iff A_2## 2. Homework Equations 3. The Attempt at a Solution...
  22. B

    What is initial slopes of the plot Z versus P ?

    1. Homework Statement Use the van der waals constant for ##H_2## and ##O_2## to calculate the initial slopes of the plots of compressibility factor Z versus P. 2. Homework Equations 3. The Attempt at a Solution Using virial expansion for van der waal gas in terms of ##P## I get ##Z = 1 +...
  23. B

    Inverted garbage can

    1. Homework Statement An inverted garbage can of weight ##W## is suspended in air by water from a geyser. The water shoots up the ground with speed ##v_0##, at a constant rate ##dm/dt##. The problem is to find the maximum height at which garbage can rides. 2. Homework Equations 3. The...
  24. B

    Terminal velocity of a raindrop

    1. Homework Statement A raindrop of initial Mass ##M_0## starts to fall from rest under the influence of gravity. Assume that the drop gains mass from the cloud at a rate proportional to the product of its instantaneous mass and its instantaneous velocity ##\dfrac{dM}{dt} = kMV##, where ##k##...
  25. B

    I Field between Parallel Plates in a Capacitor

    I guess it is a trivial fact that field must be ##(\phi_1 - \phi_2)/s## but I don't get how ? is there a derivation for it ?
  26. B

    Calculus Translation of a German book about ODEs

    I need a translation of "Differentialgleichungen : Losungsmethoden und Losugen", I guess it is written in German. This book was referenced in Shepley L. Ross' book on ODE. If the English translation is not unavailable, I am fine with a book that contains a "list" of special differential...
  27. B

    Pulley system on a big block

    1. Homework Statement Find the accelaration of ##M_1## in the given system if ##F = 0##. 2. Homework Equations 3. The Attempt at a Solution ##x_3 -x_1 = k \iff \ddot x_3 = \ddot x_1## and ##h - y_3 + x_3 - x_2 = l \iff \ddot y_3 + \ddot x_2 = \ddot x_3 \qquad (*)## h is the...
  28. B

    Tension in a rope hanging between two trees

    1. Homework Statement A uniform rope of weight ##W## hangs between two trees. The ends of the rope are same height, and they each make an angle ##\theta## with the trees. Find : a): The tension at the either end of the rope. b): The tension in the middle of the rope. 2. Homework...
  29. B

    Help in understanding a question about a pebble and a wheel and friction...

    1. Homework Statement A wheel of radius R rolls along the ground with velocity V. A pebble is carefully released on top of the wheel so that it is instaneously at the rest on the wheel. Show that in the case ##V < \sqrt{Rg}## and the coefficient of friction is ##\mu = 1## the pebble starts to...
  30. B

    Spinning block with decreasing radius

    1. Homework Statement A mass ##m## whirls around on a string which passes through a ring. Given that gravity is not present and initially the mass is at a distance ##r_0## from the center and is revolving at angular velocity ##\omega_0##. The string is pulled with constant velocity ##V##...
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