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Yiorgos S. Smyrlis
  • Member for 9 years, 5 months
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51 votes
Accepted

Which $f$ satisfy the equation $\,\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y\,$?

46 votes

What is the remainder when $1! + 2! + 3! +\cdots+ 1000!$ is divided by $12$?

45 votes

Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

45 votes
Accepted

e is irrational

44 votes
Accepted

The equation $x^3 + y^3 = z^3$ has no integer solutions - A short proof

39 votes
Accepted

Prove that $\int_0^1{\left\lfloor{1\over x}\right\rfloor}^{-1}\!dx={1\over2^2}+{1\over3^2}+{1\over4^2}+\cdots.$

38 votes
Accepted

Equicontinuity on a compact metric space turns pointwise to uniform convergence

35 votes
Accepted

Prove that: $\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$, for $n > 1$

34 votes
Accepted

If $\,x>1$, then $\lim\limits_{n\rightarrow\infty}\frac{\left\lfloor x^{n+1} \right\rfloor}{\left\lfloor x^n \right\rfloor}=x$.

33 votes

Evaluating the nested radical $ \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots}}} $.

33 votes
Accepted

Find the exact value of the infinite sum $\sum_{n=1}^\infty \big\{\mathrm{e}-\big(1+\frac1n\big)^{n}\big\}$

32 votes
Accepted

Determine the convergence of $ \sum_{n=1}^{\infty}\left[1-\cos\left(1 \over n\right)\right] $

31 votes

Is there a function having a limit at every point while being nowhere continuous?

29 votes

Integral of matrix exponential

27 votes
Accepted

Is$\frac{\sqrt{a}}{\sqrt{b}}$ the same as $\sqrt{\frac{a}{b}}$?

25 votes
Accepted

A normal matrix with real eigenvalues is Hermitian

25 votes
Accepted

Convergence of sequence: $ \sqrt{2} \sqrt{2 - \sqrt{2}} \sqrt{2 - \sqrt{2 - \sqrt{2}}} \sqrt{2 - \sqrt{2 - \sqrt{2-\sqrt{2}}}} \cdots $ =?

25 votes

How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$.

24 votes
Accepted

How do you calculate this limit $\lim_{n\to\infty}\sum_{k=1}^{n} \frac{k}{n^2+k^2}$?

24 votes

Does this sequence $\,\sqrt[n]{1+\cos2n}\,$ have a limit?

23 votes

Who came up with the $\varepsilon$-$\delta$ definitions and the axioms in Real Analysis?

23 votes

Finite dimensional subspace of $C([0,1])$

22 votes
Accepted

A uniformly continuous function maps bounded set to bounded sets

21 votes
Accepted

Is there an analytic function satisfying $\,\,f\big(\!\frac 1 n\!\big)=\frac 1 {\sqrt{n}}$ for all $n\in\mathbb N$?

21 votes

If $A^2=2A$, then $A$ is diagonalizable.

21 votes
Accepted

If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$

21 votes
Accepted

Find all bijections $\,\,f:[0,1]\rightarrow[0,1],\,$ which satisfy $\,\,f\big(2x-f(x)\big)=x$.

20 votes
Accepted

Prove that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$

19 votes
Accepted

Show that $({\sqrt{2}\!+\!1})^{1/n} \!+ ({\sqrt{2}\!-\!1})^{1/n}\!\not\in\mathbb Q$

19 votes
Accepted

If $\,A^3-A+I=0,\,$ then $A$ is invertible

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