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Title: Fundamental Solution
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Series: Distributions
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YouTube-Title: Distributions 11 | Fundamental Solution
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Bright video: https://youtu.be/qtFAnhh1Ht4
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Dark video: https://youtu.be/JgGRayxB1lA
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: dt11_sub_eng.srt missing
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Definitions in the video: fundamental solution
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: What is not correct for the map $D^{\alpha}: \mathcal{D}^\prime(\mathbb{R}^n) \rightarrow \mathcal{D}^\prime(\mathbb{R}^n)$ where $\alpha$ is a non-vanishing multi-index?
A1: It’s a linear map.
A2: It’s a continuous map.
A3: For a sequence $(T_k)$ that converges to $T$ in $\mathcal{D}^\prime(\mathbb{R}^n)$, one also has $D^{\alpha} T_k \xrightarrow{n \rightarrow} T$.
A4: $D^{\alpha} (5 T) = 5 \cdot D^{\alpha} (T)$ for all $T \in \mathcal{D}^\prime(\mathbb{R}^n)$
A5: It’s a injective map.
Q2: What does a fundamental solution $f \in \mathcal{L}^1_{\mathrm{loc}}(\mathbb{R}^n)$ for a differential operator $P(D)$ not fulfil?
A1: $ P(D) T_f = \delta $
A2: $ \int_{B_{2}(a)} f(x) ( P(D) \varphi ) , dx = 0$ for every test function and every vector $a \in \mathbb{R}^n$ with $| a | > 2$.
A3: $ \int_{B_{\varepsilon}(a)} f(x) ( P(D) \varphi ) , dx = 0$ for all $\varepsilon > 0$ and test functions $\varphi$.
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Last update: 2024-11
