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{\displaystyle \mathbf {d} \mathbf {c} ^{\mathrm {T} }-\mathbf {c} \mathbf {d} ^{\mathrm {T} }={\begin{bmatrix}0&c_{2}d_{1}-c_{1}d_{2}&c_{3}d_{1}-c_{1}d_{3}\\c_{1}d_{2}-c_{2}d_{1}&0&c_{3}d_{2}-c_{2}d_{3}\\c_{1}d_{3}-c_{3}d_{1}&c_{2}d_{3}-c_{3}d_{2}&0\end{bmatrix}}}
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{\displaystyle \mathbf {d} \mathbf {c} ^{\mathr...
{\displaystyle [\mathbf {a} ]_{\times }={\begin{bmatrix}0&c_{2}d_{1}-c_{1}d_{2}&c_{3}d_{1}-c_{1}d_{3}\\c_{1}d_{2}-c_{2}d_{1}&0&c_{3}d_{2}-c_{2}d_{3}\\c_{1}d_{3}-c_{3}d_{1}&c_{2}d_{3}-c_{3}d_{2}&0\end{bmatrix}}}
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{\displaystyle [\mathbf {a} ]_{\times }={\begin...
{\displaystyle \mathbf {d} \mathbf {c} ^{\mathrm {T} }={\begin{bmatrix}c_{1}d_{1}&c_{2}d_{1}&c_{3}d_{1}\\c_{1}d_{2}&c_{2}d_{2}&c_{3}d_{2}\\c_{1}d_{3}&c_{2}d_{3}&c_{3}d_{3}\end{bmatrix}}}
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{\displaystyle \mathbf {d} \mathbf {c} ^{\mathr...
{\displaystyle \mathbf {c} \mathbf {d} ^{\mathrm {T} }={\begin{bmatrix}c_{1}d_{1}&c_{1}d_{2}&c_{1}d_{3}\\c_{2}d_{1}&c_{2}d_{2}&c_{2}d_{3}\\c_{3}d_{1}&c_{3}d_{2}&c_{3}d_{3}\end{bmatrix}}}
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{\displaystyle \mathbf {c} \mathbf {d} ^{\mathr...
{\displaystyle {\begin{aligned}\|f\|_{\mathcal {H}}^{2}&=\left\langle \sum _{i=1}^{n}\gamma ((x_{i},t_{i}),\cdot )c_{i}^{t_{i}},\sum _{j=1}^{n}\gamma ((x_{j},t_{j}),\cdot )c_{j}^{t_{j}}\right\rangle _{\mathcal {H}}\\&=\sum _{i,j=1}^{n}c_{i}^{t_{i}}c_{j}^{t_{j}}\gamma ((x_{i},t_{i}),(x_{j},t_{j}))\\&=\sum _{i,j=1}^{n}\sum _{s,t=1}^{T}c_{i}^{t}c_{j}^{s}k(x_{i},x_{j})A_{s,t}\\&=\sum _{i,j=1}^{n}k(x_{i},x_{j})\langle c_{i},Ac_{j}\rangle _{\mathbb {R} ^{T}}\\&=\sum _{i,j=1}^{n}k(x_{i},x_{j})\langle c_{i},AA^{\dagger }Ac_{j}\rangle _{\mathbb {R} ^{T}}\\&=\sum _{i,j=1}^{n}k(x_{i},x_{j})\langle Ac_{i},A^{\dagger }Ac_{j}\rangle _{\mathbb {R} ^{T}}\\&=\sum _{i,j=1}^{n}\sum _{s,t=1}^{T}(Ac_{i})^{t}(Ac_{j})^{s}k(x_{i},x_{j})A_{s,t}^{\dagger }\\&=\sum _{s,t=1}^{T}A_{s,t}^{\dagger }\langle \sum _{i=1}^{n}k(x_{i},\cdot )(Ac_{i})^{t},\sum _{j=1}^{n}k(x_{j},\cdot )(Ac_{j})^{s}\rangle _{{\mathcal {H}}_{k}}\\&=\sum _{s,t=1}^{T}A_{s,t}^{\dagger }\langle f_{t},f_{s}\rangle _{{\mathcal {H}}_{k}}\end{aligned
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Multi-task learning - Wikipedia
Should we move from a 1.5°C-2°C target to a 5°C or even a 10°C target? (I'm not crazy, it would be more radical, given the right probability demand)
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Should we move from a 1.5°C-2°C target to a 5°C...
{\displaystyle \mathbf {a} =\mathbf {c} \times \mathbf {d} ={\begin{pmatrix}c_{2}d_{3}-c_{3}d_{2}\\c_{3}d_{1}-c_{1}d_{3}\\c_{1}d_{2}-c_{2}d_{1}\end{pmatrix}}}
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{\displaystyle \mathbf {a} =\mathbf {c} \times ...
{\displaystyle {\begin{aligned}(\mathbf {a} \times \mathbf {b} )\times (\mathbf {a} \times \mathbf {c} )&=(\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} ))\mathbf {a} \\(\mathbf {a} \times \mathbf {b} )\cdot (\mathbf {c} \times \mathbf {d} )&=\mathbf {b} ^{\mathrm {T} }\left(\left(\mathbf {c} ^{\mathrm {T} }\mathbf {a} \right)I-\mathbf {c} \mathbf {a} ^{\mathrm {T} }\right)\mathbf {d} \\&=(\mathbf {a} \cdot \mathbf {c} )(\mathbf {b} \cdot \mathbf {d} )-(\mathbf {a} \cdot \mathbf {d} )(\mathbf {b} \cdot \mathbf {c} )\end{aligned}}}
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{\displaystyle {\begin{aligned}(\mathbf {a} \ti...
{\displaystyle \mathbb {C} \otimes _{\mathbb {R} }\mathbb {C} =\mathbb {C} [x]/(x^{2}+1)=\mathbb {C} [x]/(x+i)\times \mathbb {C} [x]/(x-i)=\mathbb {C} ^{2}.}
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{\displaystyle \mathbb {C} \otimes _{\mathbb {R...
{\displaystyle {\begin{aligned}\mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=\mathbf {b} (\mathbf {a} \cdot \mathbf {c} )-\mathbf {c} (\mathbf {a} \cdot \mathbf {b} )\\(\mathbf {a} \times \mathbf {b} )\times \mathbf {c} =\mathbf {b} (\mathbf {c} \cdot \mathbf {a} )-\mathbf {a} (\mathbf {b} \cdot \mathbf {c} )\end{aligned}}}
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{\displaystyle {\begin{aligned}\mathbf {a} \tim...
Big C or little c – The C in GRC, Quantifying Risk Compliance
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Big C or little c – The C in GRC, Quantifying R...
{\displaystyle \operatorname {Var} (c)=\mathbb {E} [(c-\mathbb {E} [c])^{2}]=\mathbb {E} [(c-c)^{2}]=0}
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Varianz (Stochastik) – Wikipedia
Graph showing projected greenhouse gas emissions and future warming levels depending on actions taken. It compares the 1.5C target against the temperature rise predicted if current policies continue (2.5C-2.9C). In the 1.5C scenario, emissions drop very s
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What is climate change? A really simple guide
{\displaystyle \cos(A,B)\geq \cos(A,C)\cdot \cos(C,B)-{\sqrt {\left(1-\cos(A,C)^{2}\right)\cdot \left(1-\cos(C,B)^{2}\right)}}.}
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{\displaystyle \cos(A,B)\geq \cos(A,C)\cdot \co...
{\displaystyle \cos(A,C)\cdot \cos(C,B)+{\sqrt {\left(1-\cos(A,C)^{2}\right)\cdot \left(1-\cos(C,B)^{2}\right)}}\geq \cos(A,B),}
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{\displaystyle \cos(A,C)\cdot \cos(C,B)+{\sqrt ...
{\displaystyle B_{i}C^{i}=B_{1}C^{1}+B_{2}C^{2}+\cdots +B_{n}C^{n}}
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{\displaystyle B_{i}C^{i}=B_{1}C^{1}+B_{2}C^{2}...
{\displaystyle {\frac {\mathbb {C} [x,y,z]}{(f)}}\otimes _{\mathbb {C} [x,y,z]}{\frac {\mathbb {C} [x,y,z]}{(g)}}\cong {\frac {\mathbb {C} [x,y,z]}{(f,g)}}}
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{\displaystyle {\frac {\mathbb {C} [x,y,z]}{(f)...
{\displaystyle a_{y}={\begin{bmatrix}b_{z}\\b_{x}\end{bmatrix}}\times {\begin{bmatrix}c_{z}\\c_{x}\end{bmatrix}},\ a_{z}={\begin{bmatrix}b_{x}\\b_{y}\end{bmatrix}}\times {\begin{bmatrix}c_{x}\\c_{y}\end{bmatrix}}}
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{\displaystyle a_{y}={\begin{bmatrix}b_{z}\\b_{...
{\displaystyle \mathbf {a} ={\begin{bmatrix}a_{x}\\a_{y}\\a_{z}\end{bmatrix}},\ \mathbf {b} ={\begin{bmatrix}b_{x}\\b_{y}\\b_{z}\end{bmatrix}},\ \mathbf {c} ={\begin{bmatrix}c_{x}\\c_{y}\\c_{z}\end{bmatrix}}}
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{\displaystyle \mathbf {a} ={\begin{bmatrix}a_{...
{\displaystyle C(x_{1},\ldots ,x_{k})=C(x_{1})C(x_{2})\cdots C(x_{k})}
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{\displaystyle C(x_{1},\ldots ,x_{k})=C(x_{1})C...