ag.algebraic geometry – When does a (higher) Deligne-Mumford stack ‘have enough geometric points’?

We take the following as our definition of spectral Deligne-Mumford stack (following Lurie):

  • A pair $(X,O_X),$ where $X$ is an $infty$-topos and $O_X$ is a strictly Henselian sheaf of $E_infty$-rings.
  • Such that there exists a family of objects ${U_alpha}$ in $X$ such that the associated locally ringed $infty$-topos $$(X_{/U_alpha}, O_Xrvert_{U_alpha})$$ is equivalent to $operatorname{Spacute{e}t}(R)$ for some connective $E_infty$-ring $R$.

Say that a spectral DM stack has enough geometric points if for every topos-theoretic point $$gamma_ast:mathcal{S} to X$$ there exists a separably closed field $kappa$ and a map $$operatorname{Spacute{e}t}(kappa)to X$$ of spectral DM stacks inducing $gamma_ast$.

Lurie proves that when $X$ is a quasiseparated spectral algebraic space, $X$ has enough geometric points, but this seems like an extremely strong condition. Is there a weaker condition for which the claim still holds (preferably one that doesn’t demand that $X$ is a $0$-stack)?

dnd 5e – Implications of liberally handing out higher quality healing potions

The term healing potions in this context refers to the ensemble of {Potion of Healing; Potion of Greater Healing; Potion of Superior Healing; Potion of Supreme Healing}, not to the item Potion of Healing.

Potion of Healing appears on the table of purchasable goods (PHB p. 150). Furthermore, the descriptions of Potion of Healing and the Herbalism Kit (pp. 153/154) suggest that the latter can be used in the production of the former, which was discussed in this question: How does one craft Potions of Healing?

Xanathar’s Guide to Everything (p. 128) provides rules for crafting magic items, involving the acquisition of a formula. Healing potions, however have simpler rules, without the need for a formula (p. 130).
XGtE further has a suggested magic item progression detailing the number of magic items that should be gained on average on different levels.

The fact that Potions of Healing can be purchased at 50 gp a piece, suggests that they are in fact exempt from the progression. Similarly, the fact that magic items can be crafted in downtime questions the magic item progression. For most magic items, I see no problem, since opportunities to acquire formulae can be restricted in the same manner as opportunities to obtain complete magic items. Healing Potions do not require any formulae, however, and are much easier to craft, which bears the question if they are, in fact, generally exempt from the magic item progression.

In know that I can ignore or modify the magic item progression, that magic items are completely optional, and that I could hand out e.g Potions of Greater Healing within the boundaries of the progression.

I liberally hand out potions of healing, through loot and quest givers. I’m wondering, however, what would happen if extended this to other healing potions. For example, the loot in one of the more dangerous dungeons would contain four potions of greater healing instead of four potions of healing, or I replace half of all potions of healing in loot with potions of greater healing, or I increase the quality of the healing potions given out with tier of play.

My question therefore is: Given the crafting rules on XGtE p. 130 (and potentially other rules that I overlooked), are healing potions exempt from magic item progression? If not, what would be the balance implication of changing that?

dnd 5e – How are warlocks viable at higher levels?

Warlock is as viable as other classes

So, let us start by your assumptions:

it seems like the easiest way to play one is as an Eldritch Blast spammer. I understand the damage output can be quite high.

This is right. While it is not the only way to play it, it is arguably the easiest and, as this answer shows, a very viable choice.

Now, let us go to the problem I see you are having:

However, at mid to high levels don’t enemies start having resistance to spells below a certain level, making EB unviable?

No. In 5e, the way monsters get magic resistance is the following:

Magic Resistance. The creature has advantage on saving throws against spells and other magical effects.

Note that Eldritch Blast is not a saving throw inducing spell, it is a Ranged Spell Attack, and is, therefore, not affected by it.

The other way monsters get to resist to effects are:

  • Resistance or immunity to a type of damage. Eldritch Blast does Force damage, which is the least resisted type of damage.

  • Legendary Resistance. Some enemies can automatically succeed a saving throw. Again, this does not matter, as EB is not a saving throw inducing spell.

Overall, EB will be as effective in high levels as in low levels, and you will not, in any moment, become useless in combat.

Edge Cases

There are some encounters which will make EB useless. From my head, I can think on 3, all of them quite unlikely.

The Helmed Horror is immune to force damage, and, to the best of my knowledge, there is no way to bypass this immunity. From the 3 monsters, this is possibly the one more likely to be met in a regular adventure – even appearing in the published Princes of the Apocalypse. You should avoid this encounter, if possible.

The Rakshasa is immune to spells of 6th level or lower. This is a CR 13 monster, so it is pretty much a screw all spellcasters, not specific to Warlock, but Warlock will be the one most screwed by it. But then again, it is one specific creature among hundreds of others.

And finally, the Tarrasque. It has the Reflective Carapace feature, which states

Reflective Carapace. Any time the tarrasque is targeted by a magic missile spell, a line spell, or a spell that requires a ranged attack roll, roll a d6. On a 1 to 5, the tarrasque is unaffected. On a 6, the tarrasque is unaffected, and the effect is reflected back at the caster as though it originated from the tarrasque, turning the caster into the target.

So, yeah, Eldritch Blast is useless against a Tarrasque. But hey, so is resisting. Just let it devour you, your friends and the world. Jokes aside, while it is an extremely iconic monster, it is also unlikely your choice of class and build should be made around it, because you are not likely to encounter it any time soon.

unity – Unity3D – Despite using Time.deltaTime, my camera and player move faster at higher framerates

I’m pretty new to c# and Unity so I wouldn’t be surprised if I had made some stupid mistakes, but as far as I know, I don’t think I misused Time.deltaTime. Any help would be greatly appreciated.

Movement Script:

using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class PlayerMovement : MonoBehaviour
    public CharacterController controller;

    public float speed = 12f;
    public float gravity = -9.81f;
    public float jumpHeight = 3f;

    public Transform groundCheck;
    public float groundDistance = 0.4f;
    public LayerMask groundMask;
    private float crouchMultiplier = 1f;

    private Vector3 velocity;
    private bool isGrounded;
    private bool isCrouched;
    private float velocityDecay = 1f;

    // Update is called once per frame
    void Update()
        isGrounded = Physics.CheckSphere(groundCheck.position, groundDistance, groundMask);

        if (isGrounded && velocity.y < 0)
            velocity.y = -2f;

        float x = Input.GetAxis("Horizontal");
        float z = Input.GetAxis("Vertical");

        Vector3 move = transform.right * x + transform.forward * z;

        controller.Move(move * speed * crouchMultiplier * velocityDecay * Time.deltaTime);
        if (Input.GetButtonDown("Jump") && isGrounded)
            velocity.y = Mathf.Sqrt(jumpHeight * -2f * gravity);

        if (Input.GetKeyDown(KeyCode.LeftControl))
            isCrouched = true;
            gameObject.transform.localScale = new Vector3(1, 1f, 1);

        if (Input.GetKeyUp(KeyCode.LeftControl))
            crouchMultiplier = 1f;
            gameObject.transform.localScale = new Vector3(1, 2f, 1);
            velocityDecay = 1;
            isCrouched = false;

        if (isCrouched)
            if (velocityDecay > 0.2f && isGrounded)
                crouchMultiplier = 2.5f;
                velocityDecay *= 0.995f;

        velocity.y += gravity * Time.deltaTime;

        controller.Move(velocity * Time.deltaTime);

Camera Script:

using System;
using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class MouseLook : MonoBehaviour

    public Transform playerBody;
    public float mouseSensitivity = 100f;
    float xRotation = 0f;
    // Start is called before the first frame update
    void Start()
        Cursor.lockState = CursorLockMode.Locked;
        Cursor.visible = false;

    // Update is called once per frame
    void Update()
        float mouseX = Input.GetAxis("Mouse X") * mouseSensitivity * Time.deltaTime;
        float mouseY = Input.GetAxis("Mouse Y") * mouseSensitivity * Time.deltaTime;
        xRotation -= mouseY;
        xRotation = Mathf.Clamp(xRotation, -90f, 90);
        transform.localRotation = Quaternion.Euler(xRotation, 0f, 0f);
        playerBody.Rotate(Vector3.up * mouseX);

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algorithms – Generalized Josephus problem and solutions for higher cases

So from the Wikipedia page about this problem, there is a variant of this problem where for $n$ people in a circle, instead of killing the next person, kills the $k$th person instead. (Thus the original problem will have $k=2$.)

Apparently there are solutions for the original $k=2$ which is $f(n)=2(n-2^{lfloorlog_2nrfloor})+1$ and for $k=3$ which is
$f(n)=3(n-round(alphacdot(3/2)^m))+2$ with $m$ being the biggest number such that $round(alphacdot(3/2)^m)le n$ and this constant $alphaapprox0.8111$.

So I was wondering how is the solution for $k=3$ case found (lack of citation) and is there any way to search for solutions for higher $k$?
Any research/paper links will be appreciated.

Does 2-dimensional Gauss-Bonnet theorem applicable in higher dimensions?

This is a cross-post of this MSE post that users commented that it is appropriate for MO.

I want to know

Question: Does 2-dimensional Gauss-Bonnet theorem applicable (any topological or geometrical obstruction) in higher dimensions?

My idea is that one can consider 2-dimensional embedded sub-manifolds of $(M^n,g)$ and then applying Gauss-Bonnet theorem to all of such sub-manifolds then gluing them together somehow and finding a topological or geometrical property. Is that possible at all?

“Additional content download” requested on some higher spec android phones after reinstalling/deleting game app data- what criteria would there be?

I had two phones both fine and then something changed so that on the higher spec it has to download additional content which is a real pain as it takes at upto and over 10mins depending on network and cannot use mobile data. The older spec has no extras requested, the curious thing is the game started offering a low Res version but my low spec by default uses the standard res version.

Im wondering what it is about the low spec that tells either Google play or the app developer what the phone can handle. I need to replace the low spec due to a broken port but would like to get a faster phone which still avoids this additional content.

The game is large, CSR2 and the low spec has 2GB ram, 16GB storage (no media is stored, no extra apps besides the game so space isnt an issue) running only android 6 with no more updates only security.

Ive asked game support and all theyve said is they recommend 2GB ram or more, android4.4 or better and 3GB or more available space to play the game (nothing about this additional content).

adb – TWRP 3.1.0 or higher for Sony Xperia C3 Dual (D2502)

Sorry for bad English.
I have a Sony Xperia C3 Dual (D2502) and want to backup of entire Android directly to pc using TWRP and adb. I know we should use TWRP 3.1.0 or higher on the phone, but I could not find this versions compatible for my devices. My current TWRP version is I need it urgently.

Thanks in advance.