Let $a,b,c$ be given nonnegative integers with $gcd(a,b,c)=1$. Consider a given positive integer $n$ and positive integers $i,j$.
Let $f_n(a,b,c)$ be the number of distinct solutions to $1<ai + bj + cij<n$.
As an example $f_n(1,1,2) = n-(\pi(2n+1)-1)$ where $\pi$ is the prime counting function....
@PeterTamaroff Do you mind if I just give the general idea for your question about $\int_0^\infty t^n e^{-e^t}\,dt$? I think the details should be similar to the usual ones after the correct scaling is chosen.
@anon well but writing $f\cdot x$ for $f(x)$ seems strange for me too (I know the riesz markov representation theorem it is a pure notational question)
@DominicMichaelis That's probably best anyway. It's not like there is some associative law for expressions like $(x^\ast \cdot x )\cdot A$ for most $A$.
I have great trouble in the exercises of linear groups.
@DominicMichaelis Though easy, it's not necessary to show that $A$ is diagonalizable. It's not hard to show that, if $\lambda_1,\dotsc,\lambda_n$ are all eigenvalues of $A$ with replicates, then $f(\lambda_1),\dotsc,f(\lambda_n)$ are all eigenvalues of $f(A)$ with replicates if $f$ is a polynomial.
actually i am given f(x) is a minimal polynomial. And I am given a 4*4 matrix. I am to find the rank of matrix f(A). I don't know how to proceed. why f(A) is called as matrix. wouldn't it be a polynomial. I know I could be sounding stupid. But I would appreciate if you could point any reading material to me. Thanks.
i am looking at the previous year question of an entrance test (graduation-based)
i want to get admission in a college to study further math, but entrance test is tough, coaching classes are expensive. i am trying to self-study. i looked at the question and read about the relevant definition on net. how m unable to solve the question
@TobiasKildetoft I think i am in really deep water everytime I try to look for results like that, I see documents on "algebraic spaces" and "stacks", etc
and then I know i'm in deep water
@TobiasKildetoft but today man
We used like Yoneda 5 times
@TobiasKildetoft I think I am beginning to get a grip on yoneda and it's power
@TobiasKildetoft thank god he said no need to write up anything this week as I have a complex analysis assignment due on tuesday and a functional analysis assignment on friday :D
Use the adjoint representation of $SL_2(\mathbb C)$ to define an iso $SL_2(\mathbb C)/\{\pm I\}\approx SO_3(\mathbb C)$
And some with a bit topology:
$O_{2,1}$ has four connected components
Double coset $TPT$ is homeomorphic to a torus, where $T$ is the subgroup of $SU_2$ of diagonal matrices, and $P\in SU_2$, and none of the entries of $P$ is zero.
Come back to the problems. Adjoint representation furnishes a homomorphism $SL_2(\mathbb C)\to O_3(\mathbb C)$. However, I cannot see any clear idea that its image is $SO_3$.
@robjohn did you do it mathematically or computationally? A friend of mine, a good mathematician, only did it computationally after 1 hour from the moment I gave the problem.
@robjohn I noticed you added an application of Laplace to the end of your answer on Peter's question so I refined my bound at the end of mine in response ;)
Some time ago, stumbled out of an integral:
$$\int_{0}^{\infty}\frac{1}x{}\left (\frac{\sinh ax}{\sinh x}-ae^{-2x}\right )dx=\ln\frac{\pi\cos\frac{a\pi}{2}}{\Gamma^2(\frac{a+1}{2})};\left | a \right |<1$$
I have no idea where to start?
Could someone please give me a hint (or a link) on how to prove that: $$\sum_{k=1}^{n}\cos(kx)=\frac{\sin\left ( n+\frac{1}{2} \right )x}{\sin\left ( \frac{x}{2} \right )}-\frac{1}{2}$$
@robjohn Well, I think a bit differently. When I understand the profound beauty of a question then I definitely say I like it, or even more, I love it.
$\sigma_0(n)(=\tau(n))=O(\sqrt{n})$ as $n \rightarrow \infty$. Why? I guess this also means (via $\varphi(n)+\tau(n)=n$) that $\varphi(n)=O(n-\sqrt{n})=O(n)$ as $n \rightarrow \infty$ (probably abusing O notation there).
Hilbert's 10th Problem is a fun read. Ostensibly about how diophantine equations in general are unsolvable, but the key result still has me stumped: There exists a polynomial such that, for all integers $x,y,z$, $p(x,y,z,w_1,\dots,w_n)=0$ has a solution if and only if $z=x^y$. This still strikes me as strange - we can express integer exponentiation in terms of simple polynomials.
I've read and understood the proof - it's "elementary." But I still find it unbelievable.
Once you have exponentiation, you easily get all the other types of numbers and functions. You can get factorials, for example, and from there find a polynomial $q$ such that $q(x,w_1,...,w_n)$ as a solution if and only if $x$ is prime.
Let $a,b,c$ be given nonnegative integers with $gcd(a,b,c)=1$. Consider a given positive integer $n$ and positive integers $i,j$.
Let $f_n(a,b,c)$ be the number of distinct solutions to $1<ai + bj + cij<n$.
As an example $f_n(1,1,2) = n-(\pi(2n+1)-1)$ where $\pi$ is the prime counting function....
Some time ago, stumbled out of an integral:
$$\int_{0}^{\infty}\frac{1}x{}\left (\frac{\sinh ax}{\sinh x}-ae^{-2x}\right )dx=\ln\frac{\pi\cos\frac{a\pi}{2}}{\Gamma^2(\frac{a+1}{2})};\left | a \right |<1$$
I have no idea where to start?
