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A very handy tool that allows you to create key mappings.KEYMACRO supports the following mappings:
1) SCROLL UP: PageUp
2) SCROLL DOWN: PageDown
3) HOME: Home
4) END: End
5) CTRL + HOME: Home
6) CTRL + END: End
7) CTRL + PAGE UP: Next Page
8) CTRL + PAGE DOWN: Previous Page
9) SHIFT + CTRL + HOME: Home
10) SHIFT + CTRL + END: End
11) SHIFT + CTRL + PAGE UP: Next Page
12) SHIFT + CTRL + PAGE DOWN: Previous Page
13) SHIFT + PAGE UP: Next Page
14) SHIFT + PAGE DOWN: Previous Page
15) CTRL + SHIFT + PAGE UP: Next Page
16) CTRL + SHIFT + PAGE DOWN: Previous Page
17) CTRL + PAGE UP: Previous Page
18) CTRL + PAGE DOWN: Next Page
19) SHIFT + CTRL + PAGE UP: Previous Page
20) SHIFT + CTRL + PAGE DOWN: Next Page
21) CTRL + SHIFT + PAGE UP: Next Page
22) CTRL + SHIFT + PAGE DOWN: Previous Page
23) HOME: Home
24) END: End
25) CTRL + HOME: Home
26) CTRL + END: End
27) CTRL + PAGE UP: Next Page
28) CTRL + PAGE DOWN: Previous Page
29) CTRL + SHIFT + HOME: Home
30) CTRL + SHIFT + END: End
31) CTRL + PAGE UP: Previous Page
32) CTRL + PAGE DOWN: Next Page
33) CTRL + SHIFT + PAGE UP: Previous Page
34) CTRL + SHIFT + PAGE DOWN: Next Page
35) CTRL + HOME: Home
36) CTRL + END: End
37) CTRL + PAGE UP: Previous Page
38) CTRL + PAGE DOWN: Next Page
39) CTRL + SHIFT + PAGE UP: Previous Page
40) CTRL + SHIFT + PAGE DOWN: Next Page
Use this application to record all actions made in the program.Add the new entry by clicking on “Add action”. Then select the action to record in the “Action” menu.
Important: use the “Ignore case” option before choosing the option.
Configure as many action as you want for example you can create the following command: CTRL + SHIFT + 384a16bd22

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– A powerful and versatile application for creating and modifying 3D models.
– Transform, reshape, morph and paint 3D objects.
– Create textures from personal images and subdivide objects.
– Add many additional effects to your 3D objects.
– Allows creation of new 3D objects.
– Intuitive user interface and support documentation.
– Powerful and versatile 3D applications suite.
– 3D modeling software for professional and novice 3D artists.
– Provides additional features such as collision detection, assembly, rendering and shaders.
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How to find the solution to a recurrence relation?

I want to find the solution to the following recurrence relation:
$T(n) = 5 \cdot T(n/2) + O(1)$ where $n \geq 2$
My first idea was to find an upper bound for the running time:
$\sum_{i=1}^\infty \frac{i}{n^i} = \frac{n}{n-1} \leq \frac{2}{n}$
By using the master theorem I can derive $n \leq 2$ which is not useful at all. How would you solve this?

A:

Hint: $n$ is of the order $\log n$ and we have
$$
\frac{T(n)}{n}=5\frac{T(n/2)}{n/2}+\frac{O(1)}{n}
$$
The left hand side is of the order $\log n$, the right hand side is of the order $1$, so you want to show that
$$
T(n/2)=\frac{O(\log n)}{n}
$$
Suppose we have a look at
$$
T(n)=5T(n/2)+O(1)
$$
What happens when you divide both sides by $5n$ and take the limit?

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