User Bach

13 votes

3 answers

4k views

What is the intuition behind the law of quadratic reciprocity?

asked Feb 6, 2020 at 0:30

10 votes

2 answers

312 views

Prove $ \sum_{i=0}^{n}(-1)^{i}q^{(i+1)i/2}{n\choose i}_{q}{n+m-i\choose n}_{q}=1 $

asked Apr 6, 2023 at 8:41

8 votes

3 answers

935 views

Suppose $ \alpha, \beta>0 $. Compute: $ \int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx $

asked Oct 22, 2018 at 13:30

7 votes

1 answer

421 views

Differentiability of $ G(x)=\int_{\mathbb R} e^{tx}f(t)dt $ on $ (0,1) $

asked Jul 20, 2019 at 14:07

6 votes

3 answers

376 views

Suppose $ H\leqslant G $, prove that if $ (H, G')=\langle e \rangle $, then $ (H', G)=\langle e \rangle $.

asked Sep 29, 2018 at 12:12

6 votes

2 answers

236 views

Suppose that $G$ is a group of order $924=2^2\cdot3\cdot7\cdot 11$. Prove that $G$ has an element of order $77$.

asked May 31, 2019 at 9:08

5 votes

1 answer

97 views

A set $E$ such that $E$ is dense in $[0,1]\times [0,1]$, and the intersection of $E$ and any line parallel to the axes has at most one point [duplicate]

asked Apr 24, 2019 at 8:00

5 votes

1 answer

343 views

Let $ R $ be a p.i.d. and $ A\in M_n(R) $. If $ \det(A)=1 $, prove or disprove that $ A $ can be expressed as products of elementary matrices.

asked Jan 23, 2019 at 13:12

5 votes

1 answer

63 views

$ af'(\xi)+bf'(\eta)=0 $ for some $ 0<\xi<\eta<1 $

asked Mar 24, 2019 at 11:41

5 votes

2 answers

121 views

Find the maximal value for $ \left| \int_{0}^{1}(f(x))^2-f(x)dx \right| .$

asked Sep 23, 2018 at 9:03

5 votes

1 answer

206 views

Calculate the integral $ \int_{0}^{1}\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} (x^2+x^2y^3)\, dy dx .$

asked Sep 25, 2018 at 13:05

5 votes

1 answer

715 views

$f(z)=z^n+a_{n-1}z^{n-1}+\cdots+ a_0\in\mathbb Z[z]$ has all its roots on the unit circle. Prove that any root of $f(z)=0$ is a root of unity.

asked Aug 14, 2019 at 15:45

4 votes

2 answers

401 views

Real roots of $ 1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots +\frac{x^n}{n!} $ [duplicate]

asked Aug 15, 2019 at 23:07

4 votes

2 answers

2k views

On entire functions of finite order

asked Jul 12, 2019 at 7:57

4 votes

1 answer

69 views

Is $\left\{f\in L^1(\mu)\colon\int_{\mathbb R} |f(x)|^2d\mu\ge 1\right\}$ a closed set of $L^1(\mu)$?

asked Jun 5, 2019 at 12:37

4 votes

1 answer

216 views

Proving that a certain space of compact sets is a complete metric space

asked Jun 14, 2019 at 12:39

4 votes

3 answers

267 views

How to understand the notation $ \frac{\partial f}{\partial \overline{z}} $

asked Jul 19, 2018 at 11:31

4 votes

3 answers

623 views

Does there exist a formula for product of the primitive $ n $th roots of unity.

asked Apr 28, 2018 at 17:33

4 votes

1 answer

253 views

Ask for example: A simple ring with zero divisors

asked Jan 10, 2019 at 5:20

4 votes

1 answer

137 views

Any cyclic submodule $ Dz $ such that $ \operatorname{ann} z\subset \operatorname{ann} x $ for every $ x \in M $, is a pure submodule.

asked Jan 26, 2019 at 12:45

4 votes

2 answers

527 views

$N\rtimes_{\phi}H\cong N\rtimes_{\phi\circ\psi}H$ for $N,H$ be groups, $\phi \colon H\rightarrow Aut(N)$ be a homomorphism, $\psi \in Aut(H)$

asked May 14, 2019 at 4:06

3 votes

1 answer

214 views

Prove that $g(x):=\int_0^1f(x,y)dy$ is Borel measurable.

asked Apr 24, 2019 at 14:59

3 votes

1 answer

115 views

Product of commutative nilpotent linear transformations is zero

asked Jun 1, 2019 at 11:03

3 votes

1 answer

87 views

Differentiability of $G_k(w)$

asked May 19, 2019 at 7:26

3 votes

3 answers

307 views

Map $ \mathbb C\setminus\gamma $ conformally to a punctured disk, where $ \gamma = \{ z\in S:Re(z)\le 0 \} $

asked Mar 27, 2019 at 7:18

3 votes

2 answers

38 views

Solving a system of ordinary differential equations

asked Oct 23, 2018 at 1:49

3 votes

1 answer

192 views

Why the projective space $ \mathbb{P}^{1} $ is homeomorphic to $ S^{3}/S^{1} $?

asked Jul 17, 2018 at 14:45

3 votes

1 answer

143 views

Showing the existence of invariant $2$-dimensional subspaces for some certain endomorphisms of $V$

asked Jun 17, 2019 at 16:16

3 votes

2 answers

74 views

Show that for every $v\in\mathbb R^6$ there is a $0\ne w\in\mathbb R^6$ such that $B(v,w)=B(w,w)=0$.

asked Jun 6, 2019 at 15:43

3 votes

1 answer

128 views

Prove that $\sum_{n=1}^\infty \int_{-\infty}^\infty\cos(n^2x)I(x)dx$ converges absolutely.

asked Jun 23, 2019 at 7:21

Stack Exchange Network

156 Questions

What is the intuition behind the law of quadratic reciprocity?

Prove $ \sum_{i=0}^{n}(-1)^{i}q^{(i+1)i/2}{n\choose i}_{q}{n+m-i\choose n}_{q}=1 $

Suppose $ \alpha, \beta>0 $. Compute: $ \int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx $

Differentiability of $ G(x)=\int_{\mathbb R} e^{tx}f(t)dt $ on $ (0,1) $

Suppose $ H\leqslant G $, prove that if $ (H, G')=\langle e \rangle $, then $ (H', G)=\langle e \rangle $.

Suppose that $G$ is a group of order $924=2^2\cdot3\cdot7\cdot 11$. Prove that $G$ has an element of order $77$.

A set $E$ such that $E$ is dense in $[0,1]\times [0,1]$, and the intersection of $E$ and any line parallel to the axes has at most one point [duplicate]

Let $ R $ be a p.i.d. and $ A\in M_n(R) $. If $ \det(A)=1 $, prove or disprove that $ A $ can be expressed as products of elementary matrices.

$ af'(\xi)+bf'(\eta)=0 $ for some $ 0<\xi<\eta<1 $

Find the maximal value for $ \left| \int_{0}^{1}(f(x))^2-f(x)dx \right| .$

Calculate the integral $ \int_{0}^{1}\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} (x^2+x^2y^3)\, dy dx .$

$f(z)=z^n+a_{n-1}z^{n-1}+\cdots+ a_0\in\mathbb Z[z]$ has all its roots on the unit circle. Prove that any root of $f(z)=0$ is a root of unity.

Real roots of $ 1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots +\frac{x^n}{n!} $ [duplicate]

On entire functions of finite order

Is $\left\{f\in L^1(\mu)\colon\int_{\mathbb R} |f(x)|^2d\mu\ge 1\right\}$ a closed set of $L^1(\mu)$?

Proving that a certain space of compact sets is a complete metric space

How to understand the notation $ \frac{\partial f}{\partial \overline{z}} $

Does there exist a formula for product of the primitive $ n $th roots of unity.

Ask for example: A simple ring with zero divisors

Any cyclic submodule $ Dz $ such that $ \operatorname{ann} z\subset \operatorname{ann} x $ for every $ x \in M $, is a pure submodule.

$N\rtimes_{\phi}H\cong N\rtimes_{\phi\circ\psi}H$ for $N,H$ be groups, $\phi \colon H\rightarrow Aut(N)$ be a homomorphism, $\psi \in Aut(H)$

Prove that $g(x):=\int_0^1f(x,y)dy$ is Borel measurable.

Product of commutative nilpotent linear transformations is zero

Differentiability of $G_k(w)$

Map $ \mathbb C\setminus\gamma $ conformally to a punctured disk, where $ \gamma = \{ z\in S:Re(z)\le 0 \} $

Solving a system of ordinary differential equations

Why the projective space $ \mathbb{P}^{1} $ is homeomorphic to $ S^{3}/S^{1} $?

Showing the existence of invariant $2$-dimensional subspaces for some certain endomorphisms of $V$

Show that for every $v\in\mathbb R^6$ there is a $0\ne w\in\mathbb R^6$ such that $B(v,w)=B(w,w)=0$.

Prove that $\sum_{n=1}^\infty \int_{-\infty}^\infty\cos(n^2x)I(x)dx$ converges absolutely.