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...
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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...
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...
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
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 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...