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Using Craft 3.4.9 and Commerce Pro 3.0.11 What’s the best way to get more information onto a Paypal invoice so that I can match Paypal data with a Craft Commerce order? Does anyone have some success doing this? At the very least, I would love to be able to pass the item SKU over into […]

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## Is there a way to have Shadow Hand save DCs based on Charisma?

I am playing a Spellthief/Rogue and just picked up the Martial Study feat to gain access to Shadow Hand maneuvers. Shadow Hand is a Swordsage-exclusive school and its maneuvers key off Wisdom for their save DCs, but as a Spellthief, I boosted my Charisma and my Wisdom is pretty low. Is there any way to […]

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## Solution for logging into PC with macbook, using PC speakers to play music

We have a Windows 10 PC that is connected to our sound system. I would like to be able to log into that PC from my macbook (High Sierra) and play music, either off the (PC) hard drive or from an online service (Deezer). Essentially, i would be using the macbook only for it’s keyboard […]

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## Magento 2: Add VIP pricing that’s always 5% off normal pricing?

I’m trying to add a new line of pricing info to my Magento 2 website, for example, if regular price of an item is $10, it will show “VIP price:$9” under the regular pricing, the new price will always be a fixed % off the retail price. And I need this new price to […]

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## Triangular prism, centroid of a triangle

Let $ABCA_1B_1C_1$ be a triangular prism. There is a point $T\in\Delta ABC$. Let $T_1$ be a centroid of $\Delta A_1C_1T$. If the following holds: $$\overrightarrow{A_1T_1}=\frac{1}{3}\overrightarrow{A_1A}+\frac{1}{9}\overrightarrow{A_1B_1}+\frac{4}{9}\overrightarrow{A_1C_1}$$ prove that $T$ is the centorid of $\Delta ABC$. My attempt: I wanted to use the following fact: Let $\overline{AT}$ be a median to $\overline{BC}$. Then: $$\overrightarrow{AT}=\overrightarrow{AB}+\overrightarrow{AC}=\overrightarrow{AB}+\frac{\overrightarrow{BC}}{2}=\overrightarrow{AC}-\frac{\overrightarrow{BC}}{2}$$ because a median […]

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## Triangular prism, centroid of a triangle

Let $ABCA_1B_1C_1$ be a triangular prism. There is a point $T\in\Delta ABC$. Let $T_1$ be a centroid of $\Delta A_1C_1T$. If the following holds: $$\overrightarrow{A_1T_1}=\frac{1}{3}\overrightarrow{A_1A}+\frac{1}{9}\overrightarrow{A_1B_1}+\frac{4}{9}\overrightarrow{A_1C_1}$$ prove that $T$ is the centorid of $\Delta ABC$. My attempt: I wanted to use the following fact: Let $\overline{AT}$ be a median to $\overline{BC}$. Then: $$\overrightarrow{AT}=\overrightarrow{AB}+\overrightarrow{AC}=\overrightarrow{AB}+\frac{\overrightarrow{BC}}{2}=\overrightarrow{AC}-\frac{\overrightarrow{BC}}{2}$$ because a median […]

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## nilpotent uniform pro-$p$ groups of dimension 2

I read in a paper that all nilpotent uniform pro-$p$ groups of dimension $\leq 2$ are Abelian (Prime Decomposition and the Iwasawa $\mu$-Invariant by Hajir-Maire, Math. Proc. of the Camb. Phil. Soc. (2019) Section 5.1). This is probably obvious, but I am unable to prove this nor have I seen a reference. This would mean […]

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## $\lim_{r\to 0}\int_C \log(1-e^{2i\pi z})dz=0$ for quarter circle $C$ from $ri$ to $r$

Consider the quarter circle from $z=ri$ to $z=r$ for $r>0$. Denote this quarter circle by $C$. I would like to show that $\lim_{r\to 0}\int_C \log(1-e^{2i\pi z})dz=0.$ How can I evaluate this limit? My rough idea is that $\log(1-e^{2i\pi z}) \sim \log r$ and the length of $C \sim r$, so we use \$r\log r […]

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## Two projectiles at an initial height with a missing angle of projection

David and Goliath are testing what they learned in school, that the distance travelled by launching a projectile is dependent on a range of variables; the launch angle, height and velocity of projection. As Goliath is significantly taller, David decides to throw a small projectile while standing on a ladder so that he is throwing […]

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## What does “across” in “get across the river” mean?

Cambridge Dictionary gives this definition about across from one side to the other of something with clear limits, such as an area of land, a road, or a river and this example They’re building a new bridge across the river apparently, a bridge is one of the possible ways to across the river. simply in […]