## How do you get motivated / inspired every day?

I often need blood pumping or motivation to make me think at the beginning of the day to start. I listen to some podcast / music or watch videos like this:

How do you ensure you maintain a growth mindset during the day?

Do you have any special tricks or tips that you are willing to share?

## dnd 5e – How can I create a character inspired by WoW's Illidan Stormrage?

I want to make a D&D 5e character inspired by Illidan Stormrage, a character from World of Warcraft.

I love Illidan's appearance, but it seems impossible to copy. His skills are amazing. I have a little idea about how to build it, but my construction never fits well, so I need help to build the character.

For those who do not know:

1. He is a double-handled night elf, with demon horns and wings.

I have 2 ideas on how to recreate this. One way is to use a sorcerer who can launch Alter Self at will to create horns. The other method is to make a Draconic Bloodline sorcerer, who gets wings (draconic but similar). Both methods are mutually exclusive.

2. It has a high attack speed, deals a lot of damage and does not use armor.

My idea for attacks / damages is to use a class with additional attack or barbaric reprisals. To deal with the lack of armor, there are things like wizard's armor, Draconic Resistance, or Disarmed Defense.

3. Abilities:

• His best ability is metamorphosis; He can become a demon.
For this, my idea is to launch true polymorph on myself.

• Fire Aura (DH in WoW, as well as certain talents in HotS and a main basic skill in WC3, a DoT that treats aura for fire damage)
My idea: fire shield

• Evasion
My idea: rogue and monk classes have evasion, but I think that just having a high AC is enough, like 18-20; wizard's armor It seems enough to achieve this.

If anyone knows how to create a character like Illidan, especially how it looks, I would like to play with a drow with wings and horns. Please help me build this character.

## A Quori is not destroyed when its host is

When we look at the statistics blocks for Hashalaq, Kalaraq and Tsucora Quori, which are quori that currently do not have a body, we see an ability called "Possession", which does exactly that, and includes the text

Possession lasts until the body falls to 0 hit points, the quori ends as a bonus action, or the quori is expelled for an effect such as dispelling the spell from evil and good. When the possession ends, the quori reappears in an unoccupied space less than 5 feet from the body.

(emphasis mine)

As this text states that when a possessed body falls to 0 hp, the quori reappears outside it, we can infer that this is also true for Inspired. Also, if you assume that they act exactly like these, then the Quori would not even necessarily be sent back to the darkness of dreams, simply expelled from their body.

## nt. number theory: a problem inspired by the definition of tau numbers and a divisibility relationship related to the powers of two

It is (I suppose this easy fact is well known) obvious that an integer $$n> 1$$ it's a power of two $$n = 2α$$, where $$alpha geq 1$$ it's whole, if only yes $$n$$ satisfies the divisibility ratio $$( text {rad} (n) cdot varphi (n)) mid n tag {1}$$
where $$text {rad} (n) = prod_ {p mid n} p$$ denotes the product of the division of distinct prime numbers $$n$$ Y $$varphi (n)$$ It is Euler's totient function.

(A draft of the test is that for integers $$n> 1$$ one can write $$text {rad} (n) varphi (n) = n prod_ {p mid n} (p-1)$$ so our whole $$n$$ satisfied $$(1)$$ have $$omega (n) = 1$$ different prime factor in its factorization, this arithmetic function $$omega (x)$$ count the number of different prime factors, and one has $$p-1 = 1$$ implies that our whole has the form $$n = 2α$$, where $$alpha geq 1$$ it's whole)

Observations A) One knows that the following series is convergent $$sum_ {n = 1} ^ infty frac {1} {2 ^ n}$$, therefore, the sum of the reciprocals of the solutions of $$(1)$$ it's convergent (really $$n = 1$$ It will also be a solution and therefore the previous series has value $$2$$ when we consider that $$1$$ it is also a power of two $$1 = 2 0$$ and solution of $$(1)$$) B) We note that Euler's totient function is a counting function, because it counts positive integers $$k$$ until $$n$$ that satisfy $$(n, k) = 1$$.

This week I met the beautiful article (1) in which problems related to the one known as tau numbers (See the introductory section of (1), this magazine is an open-access electronic magazine, in the first paragraph the author remembers the definition, or see the linked Wikipedia). From the definition of tau numbers, (see if you want Wikipedia Refeasible Number) I was inspired to define the following variation of tau numbers, or if you prefer a variation of the divisibility ratio $$(1)$$ as $$tau (n) = sigma_0 (n) = sum_ {1 leq d mid n} d$$ It is also a counting function.

Definition. For integers $$n> 1$$, I say that $$n$$ is a rad-refactorable number yes and only if it satisfies $$( text {rad} (n) cdot tau (n)) mid n, tag {2}$$
where $$tau (n)$$ denotes the function of number of divisors.

The first terms of this sequence are the following where I have considered adding like this $$n = 1$$ as a solution $$(2)$$

$$a_1 = 1, a_2 = 8, a_3 = 9, a_4 = 72, a_5 = 128, a_6 = 384, a_7 = 625, ldots$$

Therefore, we have redefined solutions as those positive integers $$n geq 1$$ satisfactory $$(2)$$ in order to include $$a_1 = 1$$ as a legitimate retractable radial number.

Question. Can you prove or disprove that? $$sum _ { substack { text {positive integers} a_m \ text {which are rad-refactorible}}} frac {1} {a_m}$$
is it convergent? Thank you.

I have calculated more terms in the sequence, I think it is not easy to link the terms $$frac {1} {a_m}$$ using a geometric series, and I don't know if there are infinite refactorizable radial numbers.

## References

(1) Joshua Zelinsky, Tau numbers: a partial proof of a conjecture and other results, Journal of Integer Sequences, vol. 5 (2002), Article 02.2.8.

## Universal property: why should Microtek Greenburg, a residential paradise inspired by nature, invest?

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## Find solutions and get a first statement for these diofanthine equations inspired by certain figurative numbers

Yesterday I was thinking about speculative relationships between certain figurative numbers, please see if you need the tables and references of the article
MathWorld Encyclopedia Figurative number.

I was wondering about speculative relationships of the type: first I take prism base the hexadecimal number and $$z$$ as height, we represent the hexadecimal number of this base with the integer $$x$$. Secondly we do the same, we multiply $$z$$ by a centered pentagonal number using $$and$$ in his representation to build a second prism. We finally tested the comparison

$$z cdot left ( text {hexadecimal number} right) + z cdot left ( text {centered pentagonal number} right) = text {Octahedral number of Haüy}$$
where the variable for the representation of our octahedral number of Haüy in RHS is precisely the height $$z$$. Such an equation for positive integers $$x, y, z geq 1$$ It is equivalent to
$$18x ^ 2z + 18xz + 15y ^ 2z + 15yz-8z ^ 3 + 12z ^ 2-4z + 6 = 0 tag {1}.$$

Another equation that I have considered was

$$z cdot left ( text {hexadecimal number} right) -z cdot left ( text {centered pentagonal number} right) = text {Octahedral number of Haüy}$$
that is equivalent to

$$18 x ^ 2 z + 18 x z – 15 y ^ 2 z – 15 and z – 8 z ^ 3 + 12 z ^ 2 – 16 z + 6 = 0 tag {2}.$$

Question. Can you find solutions for $$(1)$$ or $$(2)$$? Please see the computational facts below. What can be a first step / professional statement that can be deduced in the study of these equations? $$(1)$$ or $$(2)$$? I ask about how to obtain a first professional proposition (which is an interesting first statement) for some of these equations, when the study of these equations is evoked. Thank you.

Fact. If we divide by $$2$$ Genuine equations, if I'm not mistaken, it's easy to show that we just need to verify the equations for odd integers $$z geq 1$$ and integers $$and geq 1$$ such the corresponding centered pentagonal number is an even integer, since the other cases are absurd. I cannot deduce a similar statement for the congregation $$equiv text {mod} 3$$ (when we divide the equation by $$3$$) For each equation $$(1)$$ Y $$(2)$$ The Wolfram Alpha online calculator gives me partial leads and the polynomial discriminant, but I don't know if it is useful to solve the problem in my question.

Computational facts

1) I can't find solutions $$(x, y, z)$$ of the equation $$(1)$$ for integers $$x, y, z geq 1$$ when our integers travel the segments $$1 leq x leq 300$$ Y $$1 leq and leq 300$$ Y $$1 leq z leq 300$$.

2) Similarly for the second equation and the same integer segments, our difference of figurative numbers $$(2)$$, I can only find the solutions $$(x, y, z) = (1,1,1)$$ Y $$(x, y, z) = (31,34,1)$$

## dnd 5e – After the game tests: Planescape Inspired Lady of Pain Patron, is it balanced?

A couple of weeks ago I tried a subclass inspired by Planescape for the sorcerer.
Since then, I have been able to incorporate the comments offered by the community, as well as play the tests with the player who asked me to make this character possible for her. For a bit of context, our campaign has become a kind of adventure for planar rangers. We have an evocation assistant, a forester from Horizon Walker and now the sorcerer from LoP. We started the campaign at 5th level and since then we have progressed to 9th. The initial goal of the subclass was to provide another front-line option other than Hexblade that is more defensive when the Hexblade is offensive.

For this we have simplified the central function, The shadow of serenity to offer the same CA bonus while exchanging the ability to make creatures vulnerable to damage by a few Hellish Rebuke castings.

I still worry about the actual wording of certain features to maintain consistency with the published materials, as well as the overall balance of the Subclass at the highest levels.

You made a silent pact with the Lady of Pain and her cage. As a shadow of His serenity, you have been accused of maintaining the balance between the planes and preserving the privacy of the Lady. As part of this agreement, you have the following restrictions and benefits:

• You should never disclose the nature of your covenant, even under penalty of death.

• You must protect Dabu and the city of Sigil, and never cause harm.

• Alcohol and other intoxicants do not have a diminished effect on the pain you feel. On the contrary, their tolerance to pain has increased twice that of most mortal creatures.

• You can understand the strange visual language of Dabu, but you can not communicate using it.

Expanded spell list

1st: Shield, inflict Wounds.

2nd: Cloud of daggers, silence.

4th: stone skin, death room

5th: binding planar, Geas

Features

Starting at the first level, you can like an action project a sinister shadow that hides your enemy form within a 10-foot radius centered on you for 8 hours, which gives you several benefits:

• You gain a bonus to your CA equal to half of your Charisma modifier,

• In reaction to being hit by a creature within your aura, you can
launches a version of the Hellish Rebuke spell. The level of the spell is
equivalent to the amount of Serenity & # 39; s Shadow charges that
spend. This spell can not exceed the 5th level (6d10) and you must have
The ability to cast the spell to the corresponding corresponding.
level. The damage by this reaction is considered a magical touch.
and use your spell to save DC.

• You can spend 1 charge of Serenity & # 39; s Shadow to take the Hide or
Undock the action as a bonus action.

Dexterity (stealth controls).

The Shadow of Serenity has a number of charges equal to your wizard level +1. Once the Duration of the Shadow ends, you end it soon as an action, or if you use all the Serenity Shadow charges, you must finish a long rest to use the Shadow again.

At level 10, the aura of the shadow grants CA bonus to the allies within the range equal to half of your Charisma modifier rounded down (minimum of +1). The radius of the Serenity Shadow expands to 30 feet.

At level 14, the shadow aura gives you twice the Dexterity skill bonus (stealth controls) and allows allies on your radio to take the Unlink or Hide action as a bonus action.

Breaking off

On the sixth level, your body has adapted to recover from frightful injuries. You have an advantage in saving death shots. Also, if you have hit points equal to or less than half of your maximum hit points, you can use an additional action to throw a Hit Die to recover hit points, the number of hit points retrieved is equal to the Hit die roll. given more your Charism. modifier

Planar goalkeeper

At level 14, you can cast the plane change spell once without using a spell slot or material components as long as you address Sigil. You can use this function to travel to another place if you own the cost of the material component and the spell slot does it normally. In reaction to the plane change spell cast by another creature, you can counteract that spell immediately by using a level 7 spell slot. Once you use this function, you must finish a long rest before using it again.

Lady of Pain Patron Eldritch Invocations

The radius of the Shadow feature of your Serenity expands 30 feet. Also, while the shadow is active, you can cast the Dark spell by spending 3 charges from the Shadow of Serenity.

Berk & # 39; s Bain

Prerequisites: Level 5, Lady of Pain Patron

You are becoming more cruel to your enemies, since you now have a critical hit with an attack roll of 19 or 20, while the Serenity Shadow feature is active, in addition, if you are not, you acquire the ability of Intimidation .

Tormentor

Prerequisites: Level 10, Patron of the Lady of Pain, Covenant of the Leaf

You have learned to foster a fear of abnormal and persistent pain in your enemies.
Once per turn, you can inflict additional psychic damage of 5d6 on a creature that you hit with an attack if you have an advantage on the attack roll. The attack must use your covenant weapon.

You do not need an advantage in the attack roll if another enemy of the target is within 5 feet of it, that enemy is not incapacitated and you have no disadvantage in the attack roll.

I thought you were dead for sure!

Prerequisites: Level 12, Lady of Sorrow Patron

Injuries are a daily occurrence in your line of work, allowing you to obtain more information about anatomy and medicine. You gain skill in medicine ability, in addition, when you use the Separation function, you can now add twice the Charisma Modifier and you can reattach a cut limb by holding a cut body part (other than your head) in the place where it fell. Once you use this function to reposition a limb, you can not use it again until you have finished a long rest.

Prerequisites: Level 18, Lady of Pain Patron

You gain mastery of the specific dimensions of the pocket in which the Lady of Pain fills Berks who dislike it. You can cast the maze spell once without using a spell slot. The labyrinth seems to be a circular series of platforms, roads and portals. The labyrinth can also be cast on a group of willing creatures, just like your Charisma modifier. The objects and inanimate objects that remain in the labyrinth will remain there. If the maze is used in this way, you must complete a DC 20 Intelligence test to finish the spell before. Once this Invocation has been used for any of the functions, you must take a long break to regain access to the maze.

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## A graph theory question inspired by a question about Morse functions in \$ mathbb {S} ^ 2 \$

This question arose from my attempt to solve the problem posed in this fascinating question about the gradient of a soft function in $$mathbb {S} ^ 2$$.

Fix a finite set, $$P$$, of an even number of points $$p_1, ldots, p_N in mathbb {S} ^ 2$$. Leave $$C (p_i) = {q in mathbb {S} ^ 2: q cdot p_i = 0 }$$ Be set of large circles with axis of $$p_i$$.

We make the following assumptions

• $$C (p_i) neq C (p_j)$$ for $$p neq j$$
• $$C (p_i) cap C (p_j) cap C (p_k) = emptyset$$ for different $$p_i, p_j, p_k$$.

Now let $$V = {q in mathbb {S} ^ 2: q in C (p_i) cap C (p_j), mbox {some} p_i neq p_j }$$. This is the set of {vertices}.
Leave
$$Gamma = bigcup_ {p in P} C (p)$$
and let $$E$$ be the set of components of $$Gamma backslash V$$. This is the set of edges. Clearly, $$(V, E)$$ It is a flat graph. Finally, let's go $$F$$ be the set of the components of $$mathbb {S} ^ 2 backslash Gamma$$. This is the set of faces. We can think of $$(F, E)$$ as the dual chart for $$(V, E)$$.

Keep in mind that the antipodal map $$A: mathbb {S} ^ 2 to mathbb {S} ^ 2$$ Induces an involution in the sets. $$V$$, $$E$$ Y $$F$$ which we will also denote by $$A$$.

Now suppose we have a map $$sigma: F to {0, ldots, N }$$ with the property that

• $$sigma (f) + sigma (A (f)) = N$$
• Yes $$f_1$$ Y $$f_2$$ they are adjacent faces, then $$| sigma (f_1) – sigma (f_2) | = 1$$ (that is to say., $$sigma$$ Jump through one through an edge.

Look at the construction ensuring that each vertex. $$v$$ It limits four edges and thus one has four faces. $$f_1$$, $$f_2$$, $$f_3$$ Y $$f_4$$ meet in $$v$$, then, until the re-labeling $$sigma (f_1) = sigma (f_2)$$, $$sigma (f_3) = sigma (f_2) -1$$ Y $$sigma (f_4) = sigma (f_2) + 1 = sigma (f_3) + 2$$.

Now let $$F_e = {f in F: sigma (f) mbox {even} }$$ and y $$F_o = {f in F: sigma (f) mbox {odd} }.$$
Leave
$$F_o & # 39; = {f in F_o: sigma (f) geq 3 mbox {y} sigma (A (f)) geq 3 }.$$
As $$N$$ it's even, we have $$A (F_e) = F_e$$ Y $$A (F_o) = F_o$$.

by $$F_o & # 39; & # 39;$$ some unspecified subset of $$F_o & # 39;$$, we are interested in the following problem:

Issue: Determine if $$F_o & # 39; & # 39; cup F_e$$ contains a connected path (in the dual graph $$(F, E)$$) connecting antipodal faces.

When $$N = 2$$ or $$N = 4$$, $$F_o & # 39;$$ it is empty and it is easy to see any path between the antipodal points in $$F_e$$ must go through $$F_o$$ – That is, there is no route as requested by the problem. I have no idea what can happen when $$N = 6$$. My suspicion is that for $$F_o & # 39; & # 39;$$ big enough one can have such a way.