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Mathematics : Recent Questions and Answers (Page 3)

Find answers to your questions about Mathematics or help others by answering their Mathematics questions.

Decomposition of a linear operator to a partially orthogonal operator and a semi-definite self-adjoint operator

$DeclareMathOperator{A}{mathscr{A}}$$DeclareMathOperator{B}{mathscr{B}}$$DeclareMathOperator{C}{mathscr{C}}$$DeclareMathOperator{kernel}{mathrm{Ker}}$$DeclareMathOperator{diag}{mathrm{diag}}$$DeclareMathOperator{span}{mathrm{span}}$$DeclareMathOperator{real}{mathbb{R}^2}$$DeclareMathOperator{rank}{text{rank}}$ The question is:Let $A$ be a linear operator on the $n$-dimensional Euclidean...

Asked on 01/07/2022 by Zhanxiong

1 answer

Fourier expansions of Eisenstein series as a Poincare series for the Fuchsian group

In Miyake's book, Modular Forms, Ch 2.6, thm 2.6.9, there is a statement which relate to Fourier expansion of the Eisenstein series. Let $Gamma$ be a Fuchsian group, ...

Asked on 01/07/2022 by LWW

1 answer

Is eigenvalue multiplied by constant also an eigenvalue?

Let $A$ be an $n × n$ matrix. If $lambda$ is an eigenvalue of $A$ and $c$ is a nonzero scalar, then $clambda$ is...

Asked on 01/07/2022 by Ruby Cho

1 answer

Can someone explain the proof of the following linear differential equation

I was working through a robotics book, and came across the following equation, solution, and proof where $A$ is a $n times n$ constant matrix, and $x_0...

Asked on 01/07/2022 by Lucas G

3 answer

Matroid induced by a matrix where a circuit's nullspace is spanned by a non-negative vector

Let $A = [a_1, dots, a_n] in mathbb{R}^{m times n}$, $[n] = {1, dots, n}$, and $mathcal{I} subset mathcal{P}([n])$ be the set of all $I in...

Asked on 01/07/2022 by kaba

1 answer

Area between parabola and a line that don't intersect? 0 or infinity

Came across a problem on social media,Find the area of the region bounded by a parabola, $y = x^2 + 6$ andline a line $y = 2x...

Asked on 01/05/2022

1 answer

Positive integer solutions to $frac{1}{a} + frac{1}{b} = frac{c}{d}$

I was looking at the equation $$frac{1}{a}+frac{1}{b} = frac{c}{d},,$$ where $c$ and $d$ are positive integers such that $gcd(c,d) = 1$.I was trying to find positive...

Asked on 01/05/2022

3 answer

Are all complex functions onto?

I am not sure whether this question even makes sense. But I was just wondering whether all inverse operations of functions defined in complex numbers will stay inside complex numbers....

Asked on 01/05/2022 by Truth-seek

4 answer

Proving Euler’s Totient Theorem

I’m trying to self-learn some number theory. Currently, I’m trying to prove Euler’s Totient Theorem (that is, $a^{varphi(n)} equiv 1 pmod{n}$ for $a,ninmathbb{Z}$, where $varphi$ denotes Euler’s...

Asked on 01/05/2022

3 answer

Why is identity map on a separable Hilbert space not compact? False proof.

Why is identity map on a separable Hilbert space not compact? False proof. Let $e_n$ be the orthonormal basis. Then the projection map onto $H$ is defined by...

Asked on 01/05/2022

1 answer

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