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I'm a student of mathematics in Germany.


Mar
11
comment How can $\frac12 \log(x) = \log(\sqrt{x})$?
If $\mathrm{e}^y = √x$, then $\mathrm{e}^{2y} = (\mathrm{e}^y)^2 = x$ and so $2\log √x = \log x$. This implictly uses $\exp (a + b) = \exp a · \exp b$ which is equivalent to saying $\log x + \log y = \log xy$ if one defines $\log = \exp^{-1}$, setting $x = \exp a$ and $y = \exp b$. But by this, you directly get $\log x^z = z \log x$ for $z ∈ ℤ$, which algebraically extends to $z ∈ ℚ$ and by continuity extends to $z ∈ ℝ$.
Mar
10
revised If the following bijective, injective, both, or neither?
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Mar
10
revised Let $f_n:[a,b] \to \mathbb{R}$ be a sequence of piecewise continuous functions.
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Mar
10
revised Let $f_n:[a,b] \to \mathbb{R}$ be a sequence of piecewise continuous functions.
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Mar
10
answered Let $f_n:[a,b] \to \mathbb{R}$ be a sequence of piecewise continuous functions.
Mar
10
comment Let $f_n:[a,b] \to \mathbb{R}$ be a sequence of piecewise continuous functions.
Strange things can happen. Wait a second.
Mar
10
comment Let $f_n:[a,b] \to \mathbb{R}$ be a sequence of piecewise continuous functions.
What are your thoughts?
Mar
10
comment Notation to refer to all the n element subsets of a set?
Some people use ${S}\choose{n}$, confer an answer on mathoverflow on suggestions for good notation.
Mar
10
comment conjugate function prove derivative
You mean $g(z) = \overline{f(\overline{z}))}$, right?
Mar
10
comment Completeness & Closedness in Metric Spaces
@copper.hat Just made the same mistake: $(-1..0]$ is closed in $(-1..2)$.
Mar
10
revised Prove that $\frac{x^2+1}{x^2(1-x)}>8$ for $x\in(0,1)$
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Mar
10
comment Prove that $\frac{x^2+1}{x^2(1-x)}>8$ for $x\in(0,1)$
@Gregor I expanded this idea into a full answer.
Mar
10
revised Prove that $\frac{x^2+1}{x^2(1-x)}>8$ for $x\in(0,1)$
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Mar
10
answered Prove that $\frac{x^2+1}{x^2(1-x)}>8$ for $x\in(0,1)$
Mar
10
comment Prove that $\frac{x^2+1}{x^2(1-x)}>8$ for $x\in(0,1)$
$x^2(1-x) > 0$ for $x ∈ (0..1)$, so instead show $x^2+1 > 8x^2(1-x) = 8x^2 - 8x^3$.
Mar
10
revised $M \rightarrow M^T M$ is a continuous mapping.
added 61 characters in body
Mar
10
answered $M \rightarrow M^T M$ is a continuous mapping.
Mar
10
comment Let $A \in \operatorname{Mat}_n(\mathbb R), A(i,j) := \langle v_i,v_j\rangle$ where $v_1,\dotsc,v_n$ is a basis. Show $A$ is invertible.
@NicolasLykkeIversen Oh, yeah. You can.
Mar
10
answered Let $A \in \operatorname{Mat}_n(\mathbb R), A(i,j) := \langle v_i,v_j\rangle$ where $v_1,\dotsc,v_n$ is a basis. Show $A$ is invertible.
Mar
10
comment Let $A \in \operatorname{Mat}_n(\mathbb R), A(i,j) := \langle v_i,v_j\rangle$ where $v_1,\dotsc,v_n$ is a basis. Show $A$ is invertible.
Maybe this is yet to prove!