unity – Baked lightmap: Parts of scene appear too dark

I’m having trouble setting up global illumination in my Unity 2021.1.1 URP project

Parts of my scene are appearing completely dark, I’m only using directional light and area light on the windows, here’s the baked lightmap view: enter image description here

In the corner as you can see it’s completely dark

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Here’s the Shaded view: enter image description here

The corner again

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Generally the whole scene is pretty dark, it’d like to brighten it up a bit, but the corner is just horrendous.

Those are my lighting settings

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Is it better to load all of the scene at once or load small parts of it as the player moves?

I want to create a lot of detail in my game but I don’t want to kill the frame rate tracking all the models and movements (like all the grass). Will loading them in pieces save the frame rate or will constantly loading and unloading models with every step negate any benefits of loading in pieces? I want to only load what the player will see at that moment and nothing else until the player either moves the camera or walks. I also want to know if loading a scene on Unity uses the gpu or the cpu?

pathfinder 1e – What Parts Of Price Are Reduced By Magical Item Restrictions?

My question here is that, for creating magical items in Pathfinder 1e, when applying item price reductions due to restrictions (like only for certain classes, or requiring skill usage), do you apply the price reduction at the very end of calculating up the items cost (thus also reducing the included base item’s price, the masterwork addition and any special material cost components)? Or, do you only apply the reductions to whatever portion of the price is derived from magical enhancements?

So, for example, for a ranger’s flaming longbow +1, would the final price be 5,975 gp (8,000 x 0.7 = 5,600, + 75 + 300), or 5,862 gp and 5 sp (8,375 x 0.7)?

I was applying the reductions at the very end, but on second thought, have realized that may not be right. Does anyone have any thoughts on this?

nt.number theory – Partity of partitions with distinct parts of parts $>1$

This question is motivated by my earlier (unanswered) MO post.

The number of partitions into distinct parts is generated by $sum_{ngeq0}Q(n)x^n=prod_{kgeq1}(1+x^k)$. Focusing on parity of coefficients $Q(n) ,,text{mod},, 2$, the series takes the form $sum_{jinmathbb{Z}}x^{j(3j-1)/2}$.

In light of these matters, consider the modified product $prod_{kgeq2}(1+x^k)$. It might be easy but I still like to ask:

QUESTION. Is this true?
$$prod_{kgeq2}(1+x^k) mod 2=1+(x^2+x^3+x^4)+(x^7+x^8+x^9+x^{10}+x^{11})+cdots$$
to say that consecutive terms appear with block-lengths: $1, 3, 5, 7, 9,dots$; that is, odd numbers.

NOTE. $prod_{kgeq2}(1+x^k)$ generates partitions into distinct parts with parts larger than $1$.

Parts of display glitching, Ubuntu 20.04

Parts of the screen are randomly glitching. An area from one part of the screen may end up in a different place or get mixed up.
The problem occurred after installing the Spotify app via snap. However, after deleting it, it remained.
I’ve tested this problem in Windows and there this problem is not observed.

It’s an Asus x550vb laptop with Intel HD Graphics 4000 / Nvidia GeForce 740M. Ubuntu 20.04.2 LTS

Screenshot #1
Screenshot #2

calculus and analysis – Integration by parts

I know this question is already present in many variations, and it seems that for each one you have to define your own rules, and I am struggling to invent them in this case.

I want to integrate a big list of expressions of the form

{ r^n y^{(p)} y^{(q)}},,qquad n,p,qin mathbb{Z}_{>0}

where $y$ is an unknown function. Namely, by taking integrals by parts, I want to bring the expressions to the form of sum where remaining integrals contain minimal power of $r$ inside.
For example,

int dr,, r^2 y’ y” = frac{1}{2} r^2 y’^2 – int dr,, r^2 y’y”-2int dr,,r y’^2

Of course, straightforward application of some naive rules and then using FixedPoint will yield nothing but an infinite loop.

Probably, this is already implemented either in some package, or in some Mathematica function. I would be glad if you could point it to me.