In mathematics, a bump function is a function f : Rn → R on a Euclidean space Rn which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. These are also called test functions [φ]. The space of all bump functions on Rn is denoted
C
0
∞
(
R
n
)
{\displaystyle C_{0}^{\infty }(\mathbf {R} ^{n})}
or
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The question is :
Does there exist a non-negative continuous function $f : [0,1] \longrightarrow \mathbb R$ such that $\int_{0}^{1} f^{n} (x)\ dx \rightarrow 2$ as $n \rightarrow \infty$?
Is it true or not?If it isn't then how can I find example?If it is true then please give me a hint.Then I w...
The question of usefulness of mathematics in everyday life is a cliche, and I am not asking that.
What are some objects$^*$/algorithms/other curious stuff/tricks, which has surprisingly deep mathematical principle governing them ?
($^* \rightarrow$ objects means concrete touchable stuff th...
Given two matrices A and B
$$
A =
\begin{pmatrix} 1 & 0 & 0 \\ 1 & 3 & 0 \\ 1 & 2 & 1 \end{pmatrix}, \quad
B =
\begin{pmatrix} 3 & 0 & 4 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}
$$
I have checked basic properties such as trace, charatersitic polynomial and are same .
I also realise that i ne...
Many of you may recall "An obvious pattern to $i↑↑n$ that is eluding us all?", an old question of mine, and just recently, I saw this new question that poses a simple extension to tetration at non-integer values:
$$a↑↑b=\begin{cases}a^b,&b\in[0,1]\\a^{a↑↑(b-1)},&b\in(1,+\infty)\\\log_a(a↑↑(b+1)),&...
I'm currently studying an introduction to Riemannian geometry i.e. connections, curvature and isometric immersions (the Gauss, Codazi and Ricci equations).
I find the introduction to Riemannian geometry interesting, but whenever I look at some theorems beyond the introductory topics they seem qu...
Suppose we have two unit vectors $j_1$ and $j_2$. Let's also define a two-dimensional plane orthogonal to each of them: let's call the first $P_1$ and the orthogonal plane to $j_2$ $P_2$. Now, suppose I have two other vectors $v_1$ and $v_2$. I want to show that the length of the projections of $...
