Have a look at the open access textbook Quantum Computing for the Quantum Curious by Hughes et al.
Description from the Springer website:
"This book is open access, which means that you have free and unlimited access.
Demystifies quantum computing, using only high school physics.
Bridges the...
Use the discount code CYBERWEEK19 to purchase English IT eBooks from Springer online for only 7 Euro. Go to the checkout and enter the code. Afterwards, you can proceed browsing the book list and it automatically shows if the code was applied.
Note: At the checkout read again if the discount...
Springer math books are 40% off till Nov 30. Coupon code is MATH19PE.
I bought Linear Algebra by Axler , the analysis books by Pugh and Abbott.
Any further recommendations for good books?
You are asking about the difference between "Proof by contraposition" and "Proof by contradiction", and here is an example.
To prove p \rightarrow q:
- In proof by contraposition you start by assuming that \neg q is true and derive the statement \neg p. Here, the path is clear, i.e. you start...
Go has been considered one of the games in which the human is superior to any AI. However in a first match Google's AI has beaten world Go champion Lee Se-dol. Earlier this year the AI beat the European Go champion.
Try to get an intuitive feeling for what the Kronecker-Delta \delta_{ij} does.
Consider this sum:
\sum_{j=1}^{n} c_j = c_1 + c_2 + \dots + c_n
Now observe what happens if we multiply with the Kronecker-Delta:
\sum_{j=1}^{n} \delta_{ij} c_j = \delta_{i1} c_1 + \delta_{i2}c_2 + \dots +...
For a sequence (a_n)_{n \in \mathbb{N}} you can form the series \sum_{n=0}^\infty a_n = a_0 + a_1 + a_2 + \dots.
If you take the sequence a_n = 1 and plug it into the expression above,
then you get the series \sum_{n=0}^\infty a_n = \sum_{n=0}^\infty 1 = 1 + 1 + 1 \dots .
Or consider the...
This nature article mentions that other interferometers such as Geo600 in Germany and Virgo in Italy were not operating at the time . Is it known whether Geo600 would have detected the gravitational waves?
What happens with gravitational waves? Do they exist forever or can they be absorbed or...
Here is a proof mentioned on math.stackexchange using the "sign-preserving property" of continuous functions:
http://math.stackexchange.com/a/543800
And two more links for the proof of the sign-preserving property:
https://math.la.asu.edu/~dajones/class/371/ch4.pdf...
One way to understand this is to interpret it as a two-player game, let's say between you and me. The game works as follows:
I give you a number \epsilon > 0, and your task is to find a number \delta > 0 such that a certain condition is fulfilled.
You can read about this two-player game...
Here is a definition of what the limit means when it involves infinity:
\lim_{x \to +\infty}f(x) = +\infty
if for every number M>0 there is a corresponding number N such that
f(x)>M whenever x>N.
Intuitively this means, if I give you a positive number M, then you can find a number N such that...
Here are some nice videos that explain the time and frequency domain:
Intro to Fourier Transform, see also the website http://www.thefouriertransform.com/#introduction
Fourier Transform Introduction
Fourier transform explained with an oscilloscope
Approved textbooks by The American Institute of Mathematics (the books are open-access):
1) A Gentle Introduction to the Art of Mathematics by Joe Fields
2) Mathematical Reasoning: Writing and Proof by Ted Sundstrom
Do you mean by surface the purple curve? If yes, you could rotate this curve around the "x-axis" as shown here. You would have to adjust the jpeg to make the balloon symmetrical around the x-axis.
The idea is to http://calculus-geometry.hubpages.com/hub/Disk-Method-Volume-of-Solid-of-Revolution...
The purple vector is the projection of \vec{u} onto \vec{v}.
Now, you have to think of how the purple vector is defined. It has (i) a length and (ii) a direction.
(i) The length of the purple vector is | \vec{u}| \text{cos}(\theta).
(ii) The direction of the purple vector is the same as that of...
In case you are asking for the derivation of the formula I've drawn a sketch (see attachment). You can see that it is indeed just expressing the cosine in terms of the dot product.
A differential equation is an equation that contains a function f(x) and one or more derivatives of f(x).
Example 1:
f(x) = -f ''(x)
This is a differential equation since it contains f(x) and the second derivative f ''(x).
The goal is to find a function f(x) that fulfills the differential...
You can use the dot product. For example, if you have a vector v and want to find vector c that is orthogonal to v, then use the dot product <v,c> and set it equal to 0.
Example:
v = (4,2,3)
c = (x,y,z) = ?
(i) Set the dot product to zero:
<v,c> = 4x + 2y + 3z = 0
(ii) Choose some...
You can also find n by "brute force", i.e. check if f(1), f(2), f(3), ... is divisible by 17. Use http://www.wolframalpha.com/ or write a small program in your programming language of choice.
I observed triboluminescence by using adhesive tape. However, I found that not every tape showed the effect. Even tapes of the same brand showed different behavior (they only differed in size). I also recommend staying in a very dark room for a couple of minutes such that your eyes adapt to the...
Problem solving is about asking the right questions. Polya, a famous mathematician, described these questions in his book "How to solve it". A summary can be found here:
http://www.math.grin.edu/~rebelsky/ProblemSolving/Essays/polya.html
http://www.math.utah.edu/~pa/math/polya.html
Here are some articles on vectors using C++ in game programming:
Math for game programmers 02 – Vectors 101
C++ Vector Class
Youtube video:
Advanced C++ Tutorials 1 : Vectors & Physics : Convert Polar to Cartesian
Advanced C++ Tutorials 2 : Vectors & Physics : Cross Product
0.999... = 1 is not a definition. It is a result as mentioned by haruspex.
0.999... is defined as \sum_{k=1}^{\infty}9/10^k, i.e. it is the limit of the sequence s_n = \sum_{k=1}^{n}9/10^k.
s1 = 0.9
s2 = 0.99
s3 = 0.999
...
One can show that this sequence converges to 1, i.e. if I...
I've been playing the piano for about 3 years. I love it too! I find it super relaxing. The best thing about it is since I've started playing the piano I learned so much more music. I learned about classical music and jazz.
Here is for example an inspiring talk about classical music...
Meaning of "in partial fulfillment"
In a thesis one often finds this sentence:
"A thesis submitted in partial fulfillment of the requirements for the degree of... ", e.g. here:
http://dmg.caup.washington.edu/pdfs/Thesis.HunterRuthrauff.2012.pdf
What does "in partial fulfillment" mean, in...
The basic idea behind a Fourier transformation is to represent a function f(x) as a sum of other functions b1(x), b2(x), ..., bn(x):
f(x) = c1*b1(x) + c2*b2(x) +...+ cn*bn(x)
The c's are the coefficients.
Let's take a very easy example:
Suppose you are given the functions
b1(x) =...
I recommend looking at the factorial function. It can both be implemented iteratively and recursively.
As already mentioned iteration involves using for loops. The recursive approach uses a function that calls itself.
This sounds a little strange but let's have a look at how 3! is calculated...
I think the confusion stems from two things:
1. Solving for x in the equation x^2 = a.
2. Using the square root function.
If you are given the equation x^2 = a, then there are two solutions, namely a positive and a negative: x = \sqrt{a} and x = -\sqrt{a}.
However, if you are using the...
The moment when he was spinning was very scary. Here is an interview of him:
http://kurier.at/video/nachrichten/1900974173001-felix-baumgartner-ich-hatte-traenen-in-den-augen.php
In German
Baumgartner: "It really started to get violent. For a few seconds I thought I would lose my...