functions - Equivalence Classes for $R = \{ (x, y) \ | \ f(x) = f(y) \}$ - Mathe

Let $f: \mathbb{R}^+ \to \mathbb{N}_0$ be given by $f(x) = \lfloor{x}\rfloor$. Define a relation $R \subseteq \mathbb{R}^+ \times \mathbb{R}^+$ by $R = \{ (x, y) \ | \ f(x) = f(y) \}$ ...

Equivalence classes in (R)

We define $A_1 \sim A_2 $ in $M_n(\mathbb{R})$ if there is $G \in Gl_n(\mathbb{R})$ such that $A_1 = G A_2 $. Find a distinguished element in each equivalence class associated to the equivalence re...

multivariable calculus - Is still true that if $\gamma: I \rightarrow \mathbb{R}

I alredy make the proof when $n=3$ and i use the Frenet-Serret formulas, but i´m not sure how to do in $\mathbb{R} ^n$ Class of $C^2$ means twice continuously differentiable, and $\tau$ is the tors...

Show the set of congruence classes mod n is a partition of $\mathbb{Z}$. - Mathe

Let $[x]_{n}=${$ r\in \mathbb{Z}:r-x=nk$ for some $k\in \mathbb{Z}$} be the congruence class of x mod n. In order to show that $\left \{ [i]_{n} \right \}_{i=0}^{n-1}$ forms a partition of the set of

calculus - Is there an injective continuous function from an open set of $\mathb

I saw the following result: " If $A\subseteq \mathbb{R}^{n+p}\to \mathbb{R}^n$ of class $C^1$ in the open set $A$ (with $n,p$ being positive integers), then $f$ is not injective." I wonde...

combinatorics - Choosing $n$ out of $k$ classes limited to $r$ for each class -

As in the title, the problem is to choose $n$ elements from $k$ classes, in a way that from each class there are at most $r$ elements chosen (that is, from $0$ to $r$, inclusive). The specific inst...

C#에서 JSON을 직렬화하는 방법 - .NET

이 문서에서는 System.Text.Json 네임스페이스를 사용하여 JSON(JavaScript Object Notation)으로 직렬화하는 방법을 보여 줍니다. Newtonsoft.Json에서 기존 코드를 이식하는 경우 System.Text.Json으로 마이그레이션 방법을 참조하세요. JSON 출력은 기본적으로 축소 됩니다(공백, 들여쓰기 및 줄 바꿈 문자 제거). 다음은 동기 코드를 사용하여 JSON 파일을 만드는 예제입니다. 앞의 예제에서는 직렬화되...

real analysis - Let $f: A \to \mathbb{R^n}$ be of class $C^r$; $Df(x)$ is non-si

Let $A$ be open in $\mathbb{R^n}$; let $f: A \to \mathbb{R^n}$ be of class $C^r$; assume $Df(x)$ is non-singular for $x\in A$. Show that even if $f$ is not one-to-one on $A$, the set $B=f(A)$ is op...

real analysis - Function from $\mathbb{R n$ to $\mathbb{R n$ with below bounde

Suppose $f :\mathbb{R}^n \rightarrow \mathbb{R}^n $ is of class $C^1$, and $\|f(x)-f(y)\|\geq\|x-y\|$. Prove that $f$ is global invertible, and $f^{-1}$ is also of class $C^1$. We learnt the In...

real analysis - Continuous function $f$ such that $X^n f$ is of the class $C^n$

Let $f: R \rightarrow R$ be continuous on $R$ ( with $f(0)=0$ if necessary) and of class $C^\infty$ on $R \setminus \{0\}$. Assume that for each $n \in N$ the function $g_n(x)=x^n f(x)$, for $x \in...