Consider multiplication group tables modulo $n$ with entries $k_{ij} = (i\cdot j)\ \%\ n$ visualized according to these principles:
Colors are assigned to numbers $0 \leq k \leq n$ from
$\color{black}{\textsf{black}}$ for $k=0$ over
$\color{red}{\textsf{red}}$ for $k=\lfloor n/4\rfloor$ and
$...
This post discusses the integral,
$$I(k)=\int_0^k\pi(x)\pi(k-x)dx$$
where $\pi(x)$ is the prime-counting function. For example,
$$I(13)=\int_0^{13}\pi(x)\pi(13-x)dx = 73$$
Using WolframAlpha, the first 50 values for $k=1,2,3,\dots$ are,
$$I(k) = 0, 0, 0, 0, 1, 4, 8, 14, 22, 32, 45, 58, 73, 90...
$$\int_0^{102}\left(\prod_{k=1}^{100}(x-k)\right)\left(\sum_{k=1}^{100}\frac1{x-k}\right)\,dx$$
I've tried solving this problem but only thing that comes to my mind is the manual integration by multiplication of the expressions which will literally take much longer than the allotted time for com...
Does $$\lim_{x \to - \infty} (\frac{\pi}{2} + \arctan{x} ) \cdot x = - \infty$$?
My logic is that “something“ times "negative infinity" equals negative infinity. Am I right?
In general, not all functors preserve products. But my question is, is it at least true that all functors from Set to Set preserve products?
If not, does anyone know of a counterexample?
One of the most prominent problems of ancient mathematics was the squaring of the circle: to construct the square with the same area as a given circle.
A related problem is linearizing the circle: to find a natural transition between a given line segment of length $L$ and the circle with circumf...
This post discusses the integral,
$$I(k)=\int_0^k\pi(x)\pi(k-x)dx$$
where $\pi(x)$ is the prime-counting function. For example,
$$I(13)=\int_0^{13}\pi(x)\pi(13-x)dx = 73$$
Using WolframAlpha, the first 50 values for $k=1,2,3,\dots$ are,
$$I(k) = 0, 0, 0, 0, 1, 4, 8, 14, 22, 32, 45, 58, 73, 90...
Does $$\lim_{x \to - \infty} (\frac{\pi}{2} + \arctan{x} ) \cdot x = - \infty$$?
My logic is that “something“ times "negative infinity" equals negative infinity. Am I right?
In general, not all functors preserve products. But my question is, is it at least true that all functors from Set to Set preserve products?
If not, does anyone know of a counterexample?
