User Nash - Mathematics Stack Exchange

7 votes

1 answer

220 views

Let $X=\mathbb{D}^2/\sim$, where $(\cos(\theta),\sin(\theta))\sim(\cos(\theta+\frac{2\pi}{3}),\sin(\theta+\frac{2\pi}{3}))$, $\theta\in \mathbb{R}$

asked Jun 24, 2019 at 3:28

5 votes

1 answer

238 views

The succession of the coefficient theorem is split but not naturally

asked Aug 20, 2019 at 20:52

4 votes

2 answers

760 views

Show the quotient space of a finite collection of disjoint 2 simplices obtained by identifying pairs of edges is always a surface, locally homeomorp

asked Jul 13, 2019 at 22:10

4 votes

0 answers

66 views

$f_n:C_n\to D_n$ is injective for all $n\in\mathbb{Z}$ but $f_:H_n(C_)\to H_n(D_*)$ is not injective for some $n\in \mathbb{Z}$.

asked May 16, 2019 at 1:36

4 votes

2 answers

817 views

Show that the ring of polynomials with coefficients in a field, and in infinitely many variables, is not Noetherian

asked Aug 21, 2018 at 2:58

4 votes

2 answers

482 views

$\mathbb{C}$ is isomorphic to $\bar{\mathbb{Q}}$?

asked Sep 21, 2017 at 17:42

4 votes

1 answer

465 views

Show that if $\{a_1, ... ,a_n\}$ is a $p$-basis for $K/F$, then $[K: F] = p^n$.

asked Nov 10, 2017 at 3:21

4 votes

1 answer

318 views

Suppose that $q: X\to Z$ and that $p: X\to Y$ are covering spaces. Suppose there is a continuous function $r: Y\to Z$ such that $r\circ p=q$.

asked Apr 8, 2018 at 16:52

4 votes

2 answers

1k views

Find the universal covering of $\mathbb{R}P^2\vee\mathbb{R}P^2$

asked Apr 30, 2018 at 18:24

4 votes

3 answers

3k views

Using taylor series expansion to approximate the derivative of a function

asked May 21, 2018 at 20:48

3 votes

3 answers

445 views

Show that $f_n$ is continuous at $0$

asked Apr 8, 2018 at 23:13

3 votes

1 answer

156 views

Let $X$ be the space obtained from $\mathbb{R}^3$ by removing the axes $x,y$ and $z$. Calculate the fundamental group of $X$.

asked Mar 18, 2018 at 19:09

3 votes

2 answers

607 views

Let $f: \mathbb{R}^3\to \mathbb{R}^3$ be given by $f(\rho, \phi, \theta) = (\rho\cos\theta \sin \phi, \rho \sin \theta \sin \phi, \rho \cos \phi).$

asked Jan 3, 2018 at 2:54

3 votes

1 answer

176 views

Find all the whole solutions of the equation: $15x+12y+30z=24$

asked Mar 13, 2018 at 0:41

3 votes

1 answer

329 views

If $X$ is Hausdorff and $\sim$ is an equivalence relation in $X$, then $X/\sim$ endowed with the quotient topology is also Hausdorff

asked Oct 20, 2017 at 20:16

3 votes

1 answer

357 views

If $X$ is arc-connected and $f:X\rightarrow Y$ is a continuous function, then $f(X)$ is also arc-connected

asked Oct 21, 2017 at 18:44

3 votes

1 answer

8k views

Find all ideals in $\mathbb{Z}$ and also in $\mathbb{Z}_{10}$

asked Nov 11, 2016 at 2:38

3 votes

1 answer

1k views

Prove that $C_$ is acyclic if and only if the trivial homomorphism $f_: C_\to D_$ is a homotopic equivalence.

asked Jun 4, 2018 at 17:45

2 votes

1 answer

228 views

Homological groups of the $T^2 = S^1 \times S^1$ and quotient out the circle $S^1 \times \lbrace x \rbrace$ for some point $x \in S^1$

asked Jul 23, 2019 at 21:19

2 votes

1 answer

403 views

If $f,g:(X,A)\to (Y,B)$ are homotopic functions under a homotopy $H:X\times [0,1]\to Y$ then $f_=g_:H_n(X,A)\to H_n(Y,B)$. [duplicate]

asked Jul 2, 2019 at 23:07

2 votes

0 answers

96 views

Let $T$ be a linear operator in a vector space $V$ such as $T$ admits an ajoint. Prove that if $T^*T=0$ then $T=0$.

asked Nov 2, 2022 at 22:18

2 votes

1 answer

56 views

Finding a 3 dimensional manifold that integrates involutive distribution given by $X=∂/∂x + z∂/∂y$, $Y =∂/∂z + x∂/∂w$ and $Z =∂/∂w - ∂/∂y$

asked Oct 4, 2023 at 13:17

2 votes

1 answer

762 views

Let $R$ be a ring with unity such that for each $a ∈ R$ there exists $x ∈ R$ such that $a^2x =a$.

asked Nov 7, 2016 at 17:46

2 votes

2 answers

959 views

Let $R$ be a ring. Define a circle composition $\circ$ in $R$ by $a \circ b =a+b-ab$, $a, b \in R$.

asked Nov 7, 2016 at 22:31

2 votes

1 answer

779 views

If $f$ is a entire function such that $f(z+n+im)=f(z)$ for all $z\in \mathbb{C}$ and for all $n,m \in \mathbb{Z}$, then $f$ is constant.

asked Nov 18, 2017 at 1:03

2 votes

1 answer

373 views

Let $X=[0,1]^{[0,1]}=\prod_{\alpha\in [0,1]}[0,1]$ with the product topology, show that $X$ is not sequentially compact.

asked Nov 26, 2017 at 22:49

2 votes

0 answers

33 views

The use of $0$ for the scalar $0$ as well as for the zero vector should cause no confusion, in general.

asked Dec 14, 2017 at 18:03

2 votes

2 answers

324 views

Show that $X$, with the usual addition and the usual multiplication by real numbers, is a real vector space of dimension $n + 1$. Find a basis

asked Dec 30, 2017 at 4:15

2 votes

2 answers

189 views

For any $n\times n$ matrix $A$, there corresponds a vector $x\neq 0$ such that $\|Ax\|=\|A\|\|x\|$

asked Apr 16, 2018 at 23:23

2 votes

3 answers

1k views

If $f:\mathbb{R}^2\to\mathbb{R}^1$ is of class $C^1$, show that $f$ is not one-to-one.

asked Apr 23, 2018 at 0:31

Stack Exchange Network

121 Questions

Let $X=\mathbb{D}^2/\sim$, where $(\cos(\theta),\sin(\theta))\sim(\cos(\theta+\frac{2\pi}{3}),\sin(\theta+\frac{2\pi}{3}))$, $\theta\in \mathbb{R}$

The succession of the coefficient theorem is split but not naturally

Show the quotient space of a finite collection of disjoint 2 simplices obtained by identifying pairs of edges is always a surface, locally homeomorp

$f_n:C_n\to D_n$ is injective for all $n\in\mathbb{Z}$ but $f_:H_n(C_)\to H_n(D_*)$ is not injective for some $n\in \mathbb{Z}$.

Show that the ring of polynomials with coefficients in a field, and in infinitely many variables, is not Noetherian

$\mathbb{C}$ is isomorphic to $\bar{\mathbb{Q}}$?

Show that if $\{a_1, ... ,a_n\}$ is a $p$-basis for $K/F$, then $[K: F] = p^n$.

Suppose that $q: X\to Z$ and that $p: X\to Y$ are covering spaces. Suppose there is a continuous function $r: Y\to Z$ such that $r\circ p=q$.

Find the universal covering of $\mathbb{R}P^2\vee\mathbb{R}P^2$

Using taylor series expansion to approximate the derivative of a function

Show that $f_n$ is continuous at $0$

Let $X$ be the space obtained from $\mathbb{R}^3$ by removing the axes $x,y$ and $z$. Calculate the fundamental group of $X$.

Let $f: \mathbb{R}^3\to \mathbb{R}^3$ be given by $f(\rho, \phi, \theta) = (\rho\cos\theta \sin \phi, \rho \sin \theta \sin \phi, \rho \cos \phi).$

Find all the whole solutions of the equation: $15x+12y+30z=24$

If $X$ is Hausdorff and $\sim$ is an equivalence relation in $X$, then $X/\sim$ endowed with the quotient topology is also Hausdorff

If $X$ is arc-connected and $f:X\rightarrow Y$ is a continuous function, then $f(X)$ is also arc-connected

Find all ideals in $\mathbb{Z}$ and also in $\mathbb{Z}_{10}$

Prove that $C_$ is acyclic if and only if the trivial homomorphism $f_: C_\to D_$ is a homotopic equivalence.

Homological groups of the $T^2 = S^1 \times S^1$ and quotient out the circle $S^1 \times \lbrace x \rbrace$ for some point $x \in S^1$

If $f,g:(X,A)\to (Y,B)$ are homotopic functions under a homotopy $H:X\times [0,1]\to Y$ then $f_=g_:H_n(X,A)\to H_n(Y,B)$. [duplicate]

Let $T$ be a linear operator in a vector space $V$ such as $T$ admits an ajoint. Prove that if $T^*T=0$ then $T=0$.

Finding a 3 dimensional manifold that integrates involutive distribution given by $X=∂/∂x + z∂/∂y$, $Y =∂/∂z + x∂/∂w$ and $Z =∂/∂w - ∂/∂y$

Let $R$ be a ring with unity such that for each $a ∈ R$ there exists $x ∈ R$ such that $a^2x =a$.

Let $R$ be a ring. Define a circle composition $\circ$ in $R$ by $a \circ b =a+b-ab$, $a, b \in R$.

If $f$ is a entire function such that $f(z+n+im)=f(z)$ for all $z\in \mathbb{C}$ and for all $n,m \in \mathbb{Z}$, then $f$ is constant.

Let $X=[0,1]^{[0,1]}=\prod_{\alpha\in [0,1]}[0,1]$ with the product topology, show that $X$ is not sequentially compact.

The use of $0$ for the scalar $0$ as well as for the zero vector should cause no confusion, in general.

Show that $X$, with the usual addition and the usual multiplication by real numbers, is a real vector space of dimension $n + 1$. Find a basis

For any $n\times n$ matrix $A$, there corresponds a vector $x\neq 0$ such that $\|Ax\|=\|A\|\|x\|$

If $f:\mathbb{R}^2\to\mathbb{R}^1$ is of class $C^1$, show that $f$ is not one-to-one.