expression manipulation – Arranging a function in powers of a variable

My function defined as

myfun = ( 
g^2 (2 + 4 x + a1 + 2 y a1 + a2 + 
    2 y a2) + ((1 + 2 x) (1 + 2 g + 8 g^2) + a1 + 
    2 y a1) (Theta)^2 + (1 + 
    2 x) (Theta)^4)/((Theta)^4 (1 + (1 + 2 y) (1 + 
      2 g (1 + g)) a2 + 
   2 (Theta)^2 + (1 + 2 y) a2 (Theta)^2 + (Theta)^4 + 
   4 x^2 (1 + (Theta)^2)^2 + (1 + 2 y) a1 (1 + 2 (-1 + g) g + 
      a2 + 2 y a2 + (Theta)^2) + 
   2 x (2 (1 + (Theta)^2)^2 + (1 + 2 y) a1 (1 + 
         2 (-1 + g) g + (Theta)^2) + (1 + 2 y) a2 (1 + 
         2 g (1 + g) + (Theta)^2))));

has the numerator arranged in ascending powers of $theta$. However, denominator is looking messy! How can I arrange the denominator also in powers of $theta$?

dnd 5e – What class, race or spell; damage dealing personal “Aura” powers exist in D&D 5e?

Racial Traits

Scourge Aasimar

Radiant Consumption Starting at 3rd level, you can use your action to unleash the divine energy within yourself…… at the end of each of your turns, you and each creature within 10 feet of you take radiant damage equal to half your level (rounded up).

Class Features

Ranger-Hunter Level 11

Whirlwind Attack
You can use your action to make a melee attack against any number of creatures within 5 feet of you, with a separate attack roll for each target.

Druid-Circle of Spores Level 2

Halo of Spores When a creature you can see moves into a space within 10 feet of you or starts its turn there, you can use your reaction to deal 1d4 necrotic damage to that creature unless it succeeds on a Constitution saving throw against your spell save DC.

Symbiotic Entity When you deal your Halo of Spores damage, roll the damage die a second time and add it to the total. Your melee weapon attacks deal an extra 1d6 necrotic damage to any target they hit.
These benefits last for 10 minutes

Warlock Invocation XGtE

Cloak of Flies As a bonus action, you can surround yourself with a
magical aura that looks like buzzing flies. The aura extends 5 feet from you in every direction, but not through total cover. It lasts until you’re incapacitated or you dismiss it as a bonus action. The aura grants you advantage on Charisma (Intimidation) checks but disadvantage on all other Charisma checks. Any other creature that starts its turn in the
aura takes poison damage equal to your Charisma modifier (minimum of 0 damage). Once you use this invocation, you can’t use it again until you finish a short or long rest.

Spells

Spirit Guardians

You call forth spirits to protect you. They flit around you to a distance of 15 feet for the duration…..An affected creature’s speed is halved in the area, and when the creature enters the area for the first time on a turn or starts its turn there, it must make a Wisdom saving throw. On a failed save, the creature takes 3d8 radiant damage (if you are good or neutral) or 3d8 necrotic damage (if you are evil). On a successful save, the creature takes half as much damage.

Sword Burst

Casting Time: 1 action
Range: Self (5-foot radius)
Components: V
Duration: Instantaneous
You create a momentary circle of spectral blades that sweep around you. All other creatures within 5 feet of you must each succeed on a Dexterity saving throw or take 1d6 force damage. This spell’s damage increases by 1d6 when you reach 5th level (2d6), 1 1th level (3d6), and 17th level (4d6).

Thunderclap

You create a burst of thunderous sound, which can be heard 100 feet
away. Each creature other than you within 5 feet of you must make a
Constitution saving throw. On a failed save, the creature takes 1d6
thunder damage.

Arms of Hadar

You invoke the power of Hadar, the Dark Hunger. Tendrils of dark
energy erupt from you and batter all creatures within 10 feet of you.
Each creature in that area must make a Strength saving throw. On a
failed save, a target takes 2d6 necrotic damage and can’t take
reactions until its next turn. On a successful save, the creature
takes half damage, but suffers no other effect.

Thunderwave

A wave of thunderous force sweeps out from you. Each creature in a
15-foot cube originating from you must make a Constitution saving
throw. On a failed save, a creature takes 2d8 thunder damage and is
pushed 10 feet away from you. On a successful save, the creature takes
half as much damage and isn’t pushed.

Darkness

Misc.

CULT OF ITRANTH RAXUS THE FLAMED ONE

Radiant Flames (1/Day).

Multihued flame surrounds the cultist for 1 minute, until the cultist is incapacitated or dies, or until the cultist extinguishes the flame {no action required). While inflamed, the cultist has telepathy with a range of 30 feet, and it can teleport as a bonus action up to 30 feet to an unoccupied space it can see. In addition, every creature that starts its turn within 5 feet of the cultist must make a DC 15 Dexterity saving throw, taking 16 (3dl0) radiant damage on a failed save, or half as much damage on a successful one.

functions – Can we determine higher powers of a matrix in terms of lower powered matrices?

Consider a n-ordered square matrix A. Using Cayley-Hamilton Theorem, I can represent the matrix $A^n$ as a matrix polynomial P(A) of degree n-1.

Further any matrix $A^k$ where $k>n$ can also be represented as follows:

$$A^k= a_{k,n-1} A^{n-1} + a_{k,n-2} A^{n-2}+a_{k,n-3} A^{n-3} … + a_{k,2} A^{2}+a_{k,1} A^{1} + a_{k,0}I$$

What I want to know that is there any way to determine the coefficients as a function of k?

reference request – Terminology for a ring satisfying the DCC on chains of principal right ideals generated by the powers of an element

Question. Is there any standard name for a (commutative or non-commutative) unital ring $R$ with the property that, for every $a in R$, the (descending) chain $R, aR, a^2 R, ldots,$ is eventually constant?

Let me refer to this condition as the DCCPRP, that is, the “DCC on chains of principal right ideals generated by the powers of an element”.

The DCCPRP is satisfied by left perfect rings, which, by a famous theorem of Bass (Theorem 28.4 in Anderson & Fuller’s book), can be equivalently characterized as the rings satisfying the DCC on principal right ideals (DCCPR). However, the DCCPRP is much weaker than the DCCPR.

E.g., let $R$ be the commutative, boolean ring with identity obtained by endowing the power set $P(X)$ of a set $X$ with the operations of symmetric difference (as addition) and intersection (as multiplication). The DCCPRP is trivially verified in this case (as it would be in any boolean ring), while the DCCPR holds if and only if $X$ is finite: For, suppose $X$ is infinite. Accordingly, let $x_1, x_2, ldots$ be an infinite sequence of pairwise distinct elements from $X$, and set $X_0 := X$ and $X_k := X setminus {x_1,ldots, x_k}$ for each $k in mathbf N^+$. Then $X_0 P(X), X_1 P(X), ldots$ is a strictly decreasing sequence of principal right ideals of $R$, because $X_k P(X) = P(X_k)$ for every $k in mathbf N$ and, by construction, $X_{k+1} subsetneq X_k$.

Collect all terms containing the same powers of a function in a polynomial

I have a polynomial including powers of (CapitalPhi)((Xi))
enter image description here

  • I want to collect all terms containing the same powers of (CapitalPhi)((Xi))

  • Then, I want to equate the coefficients of the terms to $0.$ (It yields to the system of algebraic equations.)

  • Finally, I want to solve the equation system including the unknowns Subscript(a, 0), Subscript(a, 1), Subscript(b, 1), r. (Other terms such as Subscript((Lambda), 1), Subscript((Lambda), 2) are parameters. They can be assigned to $1$)


POLY:=(r - p Subscript((Lambda), 1) +p^2 Subscript((Lambda), 2)) (Subscript(a, 0) + Subscript(b, 
                    1)/(CapitalPhi)((Xi)) + Subscript(a, 1) (CapitalPhi)((Xi))) + 
                 Subscript((Lambda), 
                  3) (Subscript(a, 0) + Subscript(b, 1)/(CapitalPhi)((Xi)) + 
                    Subscript(a, 1) (CapitalPhi)((Xi)))^3 - 
                 Subscript((Lambda), 
                  2) (2 Subscript(a, 
                     1) (CapitalPhi)((Xi)) (2 + (CapitalPhi)((Xi))^2) + 
                    Subscript(b, 
                     1) (-((2 (2 + (CapitalPhi)((Xi))^2))/(CapitalPhi)((Xi))) + (
                       2 (2 + (CapitalPhi)((Xi))^2)^2)/(CapitalPhi)((Xi))^3));
        
CoefficientList(Expand(POLY), (CapitalPhi)((Xi))^_)

representation theory – Irreducibility of exterior powers of the symmetric group over finite fields

Let $V$ be the $(n-1)$-dimensional standard representation of the symmetric group $S_n$ over the finite field of order $p.$

For which triples $(n,p,k)$ is $bigwedge^k V$ irreducible? Or for a weaker version, for which pairs $(n,p)$ is $bigwedge^k V$ irreducible for all $0leq k leq n-1$?

I know that Fulton-Harris answers this in characteristic 0, but I haven’t found a reference for other fields.

mutants and masterminds 3e – Are Grab-Based powers limited by Campaign PL, or as a tradeoff with the Grab attack bonus?

Semi-related to Does the target of a grab-based power remain grabbed after the attack?, there are a few different ways in which the Grab-based Flaw can work. One is that you have to make the attack roll for grabbing the person, they make their resistance roll against the grab, and then they save against the DC of the Grab-based power. One is that it works as an option to replace the Strength Damage as a Standard action against someone who you’re grabbing. And, honestly, both cases might apply depending on how you interpret the rules text.

If it’s the first case, it’s got a clear trade-off, much like the Grab itself. The attack bonus to Grab, and the rank of either Strength or the grabbing effect, can’t exceed PL*2 just like any attack, so similarly, a power that just adds an additional resistance check would be similarly PL-limited.

If it’s the second case, then we’re in more ambiguous waters, since the Standard action to inflict damage in a Grab doesn’t mention an attack roll, and in prior editions explicitly didn’t require a Strength check, basically modeling that someone who has a hold on you just squeezes it tighter. Thus, it could be argued that this should be limited to campaign PL, just like other attacks that lack an attack roll.

So, is a Grab-based power limited by the attack bonus or the Campaign PL?

dnd 5e – Does the Gulthias Staff have inherent powers that function regardless of attunement?

In Curse of Strahd, there is a magic item known as the Gulthias Staff, found on page 221. It says it requires attunement; however, two features of the staff seem more inherent.

Both are in the main section of the magic item. One reads: “If the staff is broken or burned to ashes, its wood releases a terrible, inhuman scream that can be heard out to a range of 300 feet. All blights that can hear the scream immediately wither and die.” And the other reads: “Its evil makes beasts visibly uncomfortable while within 30 feet of it.”

In the Dungeon Master’s Guide, on page 138, about attuning: “This focus can take the form of weapon practice (for a weapon), meditation (for a wondrous item), or some other appropriate activity.” It then reads, “Otherwise, at the end of the short rest, the creature gains an intuitive understanding of how to activate any magical properties of the item, including any necessary command words.”

Beasts being visibly uncomfortable while near the staff doesn’t seem like the wielder is doing anything in particular. It looks like this is an inherent aspect of it.

And as for if one destroys the staff, I wouldn’t think this would fall under attunement either because it’s not like the wielder needs to learn how to break the thing. The effect it causes appears to be inherent, sort of like someone destroying the One Ring from The Lord of the Rings.

Do I understand correctly that these two aspects don’t require attunement to function?

galois theory – Polynomial equation with perfect squares powers

I am currently trying to solve these kinds of equations, such as:

$$x^{16}-4x^9+9x^4-12x+1=0$$
$$3x^9+7x^4-2x+6=0$$

How can I make an approach to these kinds of questions?
What have I tried:

Solving $x^9+3x^4-5x+7=0$
Assume that the equation is
$$(x^5+a)(x^3+b)(x+c)=0$$
Or something like that, then the $x^9$ will cancel but the problem is to solve the remaining $a, b, c$

If there’s is a quicker way feel free to answer me. I have heard of the Galois theory and if that helps, please write a detaided solution.

18.04 powers off during boot or soon thereafter

I have a Ubuntu 18.04 system that I created to provide remote access to our office March 2020, when the COVID-19 lockdowns started. It has been running great since then … until an update a couple days ago. Now it runs for no more than a few minutes and often powers itself down during the boot sequence without ever reaching the login prompt! That makes it extremely difficult to work on, to say the least.

I used my phone to video the boot sequence (because it disappears too fast to read) and found that when it powers down during a power on sequence, the last message logged to the screen is this:

(  3.144966) fb: switching to nouveaufb from VESA VGA

After that, the screen goes black and the power light on the computer goes out.

Is there any way to recover from this? FYI: I’m no linux/ubuntu expert, so I may need a few more details than are sometimes provided in answers here… (smile) I will try to discover any information requested, but I need to be able to collect it FAST during my very limited time when I can login (and those are few.)