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Bach
  • Member for 4 years, 8 months
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8 votes
3 answers
586 views

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

7 votes
1 answer
318 views

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

6 votes
2 answers
357 views

What is the intuition behind the law of quadratic reciprocity?

6 votes
2 answers
122 views

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

5 votes
1 answer
83 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]

5 votes
1 answer
203 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.

5 votes
2 answers
244 views

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

5 votes
1 answer
60 views

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

5 votes
2 answers
101 views

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

5 votes
1 answer
161 views

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

5 votes
1 answer
291 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.

4 votes
2 answers
338 views

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

4 votes
1 answer
99 views

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

4 votes
1 answer
162 views

Ask for example: A simple ring with zero divisors

4 votes
2 answers
366 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, $\psi \in Aut(H)$

3 votes
1 answer
158 views

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

3 votes
1 answer
90 views

Product of commutative nilpotent linear transformations is zero

3 votes
1 answer
74 views

Differentiability of $G_k(w)$

3 votes
3 answers
253 views

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

3 votes
2 answers
35 views

Solving a system of ordinary differential equations

3 votes
3 answers
187 views

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

3 votes
3 answers
465 views

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

3 votes
1 answer
120 views

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

3 votes
1 answer
98 views

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

3 votes
1 answer
64 views

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

3 votes
1 answer
151 views

If $ \int_0^{2\pi}|f(re^{i\theta})|d\theta\le Ar^k $ for $f$ holomorphic and every $r>0$, then $f(z)=Cz^k$ for some constant $C$.

3 votes
1 answer
141 views

Non-negative integrable functions converging in measure on $[0,1]$ with $ \lim_{k\to\infty}\int_0^1 f_k(x)dx=\int_0^1 f(x)dx .$

3 votes
1 answer
237 views

Matrices commuting with a given $3\times 3$ complex matrix.

3 votes
1 answer
151 views

On analytic function with $f(e^z)=z$

3 votes
1 answer
199 views

Is there a continuous transformation that does not preserve zero measure?

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