How fast would a Tenser’s Floating Disk descend if I pulled it over a long drop?

So I’m designing a variant human warlock with the wizard ritual caster feat and while considering which rituals to start with I read the description for Tenser’s floating disk and looking through the eldritch invocations I saw the Ascendant step invocation allows levitation on myself at will so if I was to make a floating disk, have a party member or some equipment placed on it and then go down a chasm or hole or off the side of a flying ship/island etc would the disk follow at my levitate speed (20 feet descent or ascent per turn) or my movement speed (30 feet per turn) or would it drop like a rock? I’m picturing using it like a down elevator. Additionally would I be able to hold a wooden tabletop under the disk and levitate up and have it ascend to stay 3 feet above the surface?

For ease of reference here is the description of the relevant spells (quoted from D&D Beyond).

Tenser’s floating disk:

This spell creates a circular, horizontal plane of force, 3 feet in diameter and 1 inch thick, that floats 3 feet above the ground in an unoccupied space of your choice that you can see within range. The disk remains for the duration, and can hold up to 500 pounds. If more weight is placed on it, the spell ends, and everything on the disk falls to the ground.
The disk is immobile while you are within 20 feet of it. If you move more than 20 feet away from it, the disk follows you so that it remains within 20 feet of you. It can move across uneven terrain, up or down stairs, slopes and the like, but it can’t cross an elevation change of 10 feet or more. For example, the disk can’t move across a 10-foot-deep pit, nor could it leave such a pit if it was created at the bottom.
If you move more than 100 feet from the disk (typically because it can’t move around an obstacle to follow you), the spell ends.

Levitate:

One creature or loose object of your choice that you can see within range rises vertically, up to 20 feet, and remains suspended there for the duration. The spell can levitate a target that weighs up to 500 pounds. An unwilling creature that succeeds on a Constitution saving throw is unaffected.
The target can move only by pushing or pulling against a fixed object or surface within reach (such as a wall or a ceiling), which allows it to move as if it were climbing. You can change the target’s altitude by up to 20 feet in either direction on your turn. If you are the target, you can move up or down as part of your move. Otherwise, you can use your action to move the target, which must remain within the spell’s range.
When the spell ends, the target floats gently to the ground if it is still aloft.

To be clear I am not asking about whether I can move the disk over a hole, I am aware of that limitation and can easily put a plank over the hole and move the disk over the void, I am only asking about the vertical movement speed of the disk.

Factoring long trigonometric expressions separating specific variables

I have a very long expression with trigonometric functions:

n (2 a - h - 2 z) (-Sin[p (β - δ + t ω)] + Sin[p (β + δ + t ω)]) 

I would like to simplify the term (-Sin[p (β - δ + t ω)] + Sin[p (β + δ + t ω) to 2 Sinδ Cos (ωt + β).

Any suggestions? Just using Simplify does not work on this case. Thank you for your help.

If you cast a spell with a focus component and a long duration, does it end if the focus component goes missing?

I can’t seem to find a good example outside of 3rd party materials right now and pending GM or Dev input I figured I’d ask here. The example I’m specifically looking at is a spell with a 24 hour duration that uses a focus component that is an armlet worth at least 25g. If that armlet were removed for some reason (stolen, confiscated, handed to someone else, etc) would the spell duration end at that point or is the focus component only needed for the initial casting? If it were a Concentration spell I could see it needing to stick around, but IDK otherwise. Anyone have any official rulings on this?

Machine has 64 bit architecture and two word long instruction

A machine has a 64-bit architecture, with 2-word long instructions. It has 128 registers, each of which is 32 bits long. It needs to support 49 instructions, which have an immediate operand in addition to two register operands. Assuming that the immediate operand is a signed integer, the maximum value of the immediate operand is that can be stored is?

edit: would 1 word long instruction for this case be of 64 bit and 2 word long instruction be of 128 bits? Also do we have to add an extra bit while calculating the total bits required for 49 instructions?

I have password reset link with a long string of characters. What do those characters mean? [closed]

I have password reset link with login/reset_password?h=f7f7935cf3f63b3c01fc6987fb80f05c what does this h=32 characters mean?

I am testing a password reset functionality and found out that there is an URL parameter h with 32 characters in the password reset link. What is the purpose of these 32 characters?

Does casting a spell with a long casting time require the target to stay in range and line of sight the whole time?

When casting a spell with a casting time greater than an action, do range and line of sight need to be maintained for the entire time?

For instance, does a cleric casting Prayer of Healing need the targets to be within 30 ft. and in line of sight for just the start of the spell, just the end or for the entire casting time.

Certain spells, such as Identify or Planar Binding, explicitly mention needing to stay within range for the casting time. Other spells like Magic Circle only imply you stay in range since the casting time represent drawing out the circle.

I’m fairly confident that one does indeed, need to stay in range at least. This line suggests that before a spell is cast, it remains limited by its range:

Once a spell is cast, its effects aren’t limited by its range, unless the spell’s description says otherwise.

I haven’t found a rule requiring that one maintains line of sight.

The situation that this question arose from had an uncontrolled celestial being summoned into a magic circle. The wizard began to cast Planar Binding on it and realized that they could simply step into another room while they completed the spell, which seemed to trivialize what might otherwise be a very challenging task.

In Eberron: Rising from the Last War, can Warforged become exhausted if they decide not to take a long rest?

In “Eberron: Rising from the Last War” under “Constructed Resilience”, it states the following as a bonus:

You don’t need to sleep, and magic can’t put you to sleep.

and it goes on to say

Sentry’s Rest. When you take a Long Rest, you must spend at least six hours in an inactive, motionless state, rather than sleeping. In this state, you appear inert, but it doesn’t render you Unconscious, and you can see and hear as normal.

Now, what I’ve noticed that has changed from “Wayfinder’s Guide to Eberron” it read

Warforged Resilience … You don’t need to sleep and don’t suffer the effects of exhaustion due to lack of rest, and magic can’t put you to sleep.”

So, in this approved version of warforged, if this living construct refused to take a long rest, would it suffer exhaustion that a normal humanoid of the flesh would likely take, or would it be safe from this kind of treatment?

What is the long term interaction between grapple and Armor of Agathys?

Armor of Agathys (PHB 215, emphasis mine):

A protective magical force surrounds you, manifesting as a spectral frost that covers you and your gear. You gain 5 temporary hit points for the duration. If a creature hits you with a melee attack while you have these hit points, the creature takes 5 cold damage.

Grappling (PHB 195, emphasis mine):

When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple. If you’re able to make multiple attacks with the Attack action, this attack replaces one of them.The target of your grapple must be no more than one size larger than you and must be within your reach. Using at least one free hand, you try to seize the target by making a grapple check instead of an attack roll: a Strength (Athletics) check contested by the target’s Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). You succeed automatically if the target is incapacitated. If you succeed, you subject the target to the grappled condition. The condition specifies the things that end it, and you can release the target whenever you like (no action required).

I believe that grapple counts as a melee attack that activates Armor of Agathys. (Evidence in Bold)

If so, what happens on consecutive rounds of grappling? Assuming the attempt to escape has failed, and Armor of Agathys can still cause damage to opponents who hit it with melee attacks, does Armor of Agathys inflict more damage to the grappler on consecutive rounds of being grappled? Or is it just the initial grapple attempt that activates the spell and causes damage?

How likely is for a pseudorandom number generator to generate a long sequence of similar numbers?

How likely is for a pseudorandom number generator to generate a long sequence of similar numbers? “Similar numbers” could be same numbers, or numbers from a given range.

For example, if we consider PRNG algorithm being a simple counter counting from 0 tom MAX, the distribution is uniform and there’s a guarantee of not repeating numbers in a sequence. So, not repeating numbers does not break uniformness. But probably it breaks randomness, does it? To what extent? If so, does it mean, that the better the algorithm, the less guarantee we have to not generate similar numbers in sequence?

I’m particularly interested in the answers regarding Mersenne Twister as a most popular PRNG in programming languages implementations. It’d also be great to know how things are in operating systems’ crypto-secure PRNGs – Yarrow (macOS), Fortuna (FreeBSD) or ChaCha20 (Linux).