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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...

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...

Homework Statement Homework Equations 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} \int_0^R {k...

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...

Homework Statement Homework Equations 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 angle...

Homework Statement Homework Equations 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) = x_0, \dot...

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##.

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##...

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 ?

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. Homework Equations The Attempt at a Solution 1) ##\begin{bmatrix} 0&0\\0&0 \end{bmatrix}## 2) ##\begin{bmatrix}...

##\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...

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##...

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...

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...

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}## Homework Equations The Attempt at a Solution...

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...

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}##...

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...

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## Homework Equations The Attempt at a Solution Integrating factor is ##e^x## ##e^x\dfrac{dy}{dx} + e^x y = e^x f(x)## ##\displaystyle ye^x = \int...

Homework Statement ##(2x + 3y + 1)dx + (4x + 6y + 1) dy = 0## ##y(-2) = 2## Homework Equations 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} =...

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## Homework Equations The Attempt at a Solution Using...

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. Homework Equations The Attempt at a Solution Using virial expansion for van der waal gas in terms of ##P## I get ##Z = 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 The Attempt at...

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## is...

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 ?

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...

Homework Statement Find the accelaration of ##M_1## in the given system if ##F = 0##. Homework Equations The Attempt at a Solution [/B] ##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 height of...

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. Homework Equations...

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...

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## starting...