국제단위계 - 위키백과 한국어

(셀시우스 , degree Celsius) ℃ K - 273.15 광선속 루멘 (lumen) lm cd·sr... 10 n 접두어 기호 배수 십진수 10 30 퀘타 (quetta) Q 백양 1 000 000 000 000 000 000 000 000 000 000 10 27 론나 (ronna) R 천자 1 000 000 000 000...

Degree Theory (1) - Definition of Degree :: jjycjn's Math Storehouse

$\mathbb{R}^n$의 bounded open set $ \Omega$를 생각하자. 그리고 $f:\bar{\Omega} \to \mathbb{R}^n$이 다음을 만족하는 mapping이라 하자.(1) $f \in C^1(\Omega) \cap C^0(\bar{\Omega})$.(2) $y \notin f(\partial \Omega)$를 만족하는...

algebra precalculus - Do almost all polynomials of degree $n$ in $\mathbb R[x]$

It follows from the fundamental theorem of algebra that, if $P(x)\in\mathbb R[x]$ is a nonzero polynomial of degree $n$, then $P(x)$ has at most $n$ distinct roots in $\mathbb C$ (to be precise, it...

abstract algebra - Total number of roots of a polynomial of degree $n$ over a ri

Let $R$ be a commutative ring with unity and $f$ be a polynomial of degree $n$ over $R$. Under what conditions on $R$, does $f$ has at most $n$ roots ? I am asking because in $\mathbb{Z}/12\mathbb...

Bound by polynomials of degree r + 1

I would like to bound $\sum_{k = 1}^n k^r$ by polynomials in $n$ of degree $r + 1$. I assume that $r\in\mathbb{R}$. I found the upper bound: $\sum_{k = 1}^n k^r < n^{r+1}$, but need the lower bo...

number theory - Given $p(x)$ is a polynomial of degree $n$, and $r$ to be its ro

Let $p(x)=a_0x^n+a_1x^{n-1}+\cdots+a_n,a_0 \ne 0$ to be univariate (1 variable) polynomial of degree $n$. Let $r$ be its root, i.e. $p(r)=0$. How can I prove that: $$|r| \le \max\left(1, \sum_{i=...

real analysis - Show that $\lim_{x\to \alpha_{i}^{\pm}}\frac{p_n(x)}{p_{n+1 x

Let $p_n\in \mathbb{R}[x]$ of degree $n$ with positive leading coefficient, having $n$ simple real roots. Let $\alpha_1<\dots <\alpha_{n+1}$ denote the real roots of $p_{n+1}$. If $p_n/p_{n+1...

derivatives - A proof that $\displaystyle\sum_{i=1}^{n}P''(r_i)/P'(r_i)=0$ for a

Let $P(x)$ be a degree $n$ polynomial with distinct roots $r_1, r_2, \cdots r_n$. Prove that $$\displaystyle\sum_{i=1}^{n}\frac{P''(r_i)}{P'(r_i)}=0$$ My proof: We can equivalently rewrite the

real analysis - Sequence of polynomials $\{P_n\}$ of bounded degree such that $\

I was wondering that: Suppose that $\{P_n\}$ is a sequence of polynomials with degree $\le p$ such that for every test function $\varphi\in C^\infty_c(\mathbb{R}^d)$, the sequence of integrals $\le...

linear algebra - Prove that $\{1, (x-2), (x-2 2, ... , (x-2 n\}$ is a basis of

Let $V$ be the real vector space of all polynomial functions from $\mathbb{R}$ to $\mathbb{R}$ at most $n$ degree. That is, the space of all functions with form $f(x)=c_0+c_1x+...+c_nx^n$ with $c_i...