Ok, I found another approach.
As we can see, the blue curve (3*cos(theta)) exists in the first and fourth quadrants. But the red curve (1+cos(theta)) exists in all four quadrants. So we can divide the region in question into four segments:
If I calculate it like this, then
And the total area:
Well... It certainly looks plausible. But am I correct or not?
(Note that the upper limits of integration are different.) Now calculate the half-area outside the first curve and inside the second curve:
Why??? And I do not understand where did pi/2 come from?
The half-area inside the first curve and outside the second is then given by
Sorry, but I do not think so....
But if we take that approach, then we can try:
So which answer is a correct one???
Find the area of the region insideand outside
I understand that this should be calculated by a double integral ofand . And I see that the area I am looking for can be calculated by halves (one half above x-axes and one below).
We have a linear transformation from a matrix space, to a real space defined as a product of all elements in the matrix.
How can we define a matrix representing such transformation with respect to the standard basis?
The 'standard basis' here is a set of matrices:
So I can write something like that:
Here is what I understand:
Since the matrix representing T is in "the most appropriate" basis and it is diagonal, it have to be
So how do I find b2 an b3?
Homomorphism? I guess not. At least what I read in my lecture notes does not give me any clues on how to solve the problem.
Ok. Taking a step back. According to my notes:
- Let T:V->W be a linear transformation.
- If T is invertible, we call it an isomorphism.
So to prove thatis isomorphic, I need to show that it is invertible. Yes?
Well, maybe the trick is in the properties of invertible matrices?
How to start the proof?
I am thinking about something like this: Let, then by definition of matrix invertibility . So .
But there is an issue of order, it does matter in the matrix product and I do violate it in this proof. So it must be wrong. But I do not see how to fix it.
We have two spheres with centers (x_1, y_1, z_1) and (x_2, y_2, z_2). And radii r_1 and r_2.
What is the volume of solid intersection of these two spheres?
I am not sure how to approach this question.
I know that a point will belong to the solid iff:
It should be something like a sum of all points which satisfy the inequality:
After a second look at all these... I have a feeling that on average, we would have exactly half of the bits inverted. So we can say that if we increase number of bits, on average, the randomized inversion will require half the power than full inversion.
Is this correct or am I oversimplifying it?
We have two three digit binary numbers. In one number all digits do invert every single step (does not matter what the initial value was, lets say we have a 000-111).
In another number, each digit inverts on random.
Each inversion requires some work to be done. So for the first number, in 10 steps we require 3*10=30 energy.
If none of the digits in the second number was inverted (this can happen), then in the same 10 steps we require 0*10=0 energy. If all of the digits in the second number was inverted (this also can happen), then we will require 3*10=30 energy.
But since the second number is under randomizer we will usually have less energy spent on it in comparison with the first number.
The question is: what is the average reduction of energy which would be required by the second number in comparison with the first one?
My current train of thought is:
There are four possible cases:
0 inverted - 2^3 combinations
1 inverted - (2^3)^3 combinations
2 inverted - (2^3)^3 combinations
3 inverted - 2^3 combinations
At each step one of those four cases can happen.
So the probability to spend get any single from combination A to combination B is
So on average we supposed to get
Is this correct?
How to find n in the equation:
By trial and error I found that n=164. But is it possible to express n for a direct calculation? How did people solve such problems before the age of computers?
The problem is:
For which positive integers k is the following series convergent?
For series to be convergent the next inequality should be true (by the Ratio Test):
And now I simplify:
But since k is a constant this limit will never be less than 1. Therefore the series divergent for all possible k.
Did I make a mistake somewhere? Textbook is looking for a convergent series...