dg.differential geometry – Does higher integrability of Jacobians hold between manifolds when the Jacobians are concentrated?

$$newcommand{M}{mathcal{M}}$$
$$newcommand{N}{mathcal{N}}$$

Let $$M,N$$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds.

Let $$f_n rightharpoonup f$$ in $$W^{1,2}(M,N)$$ with $$Jf_n > 0$$ a.e., and suppose that the volume $$V({x in M , | , Jf_n le r}) to 0$$ when $$n to infty$$, for some $$0. Is it true that $$Jf_n rightharpoonup Jf$$ in $$L^1(M)$$?

I am fine with assuming that $$f_n$$ are Lipschits and injective and that $$V(f_n(M)) to V(N)$$.

The “higher integrability property of determinants” implies that if $$M,N$$ are open Euclidean domains, then $$Jf_n rightharpoonup Jf$$ in $$L^1(K)$$ for any compact $$K subset subset M$$.

Without the assumption $$V(Jf_n le r) to 0$$, this clearly doesn’t hold, even when $$f_n$$ are conformal diffeomorphisms:

Take $$M=N=mathbb{S}^2$$. Let $$s: mathbb{S}^2 to mathbb{R}^2 cup {infty}$$ be the stereographic projection, and let $$g_k(x) = k x$$ for $$x in R^2$$ (and $$g_n(infty) = infty$$.).

Set $$f_n = s^{-1} circ g_n circ s$$. $$f_k$$ are conformal, orientation preserving, smooth diffeomorphisms
and thus $$int_{mathbb{S}^2 }Jf_n=V(mathbb{S}^2 )$$. By conformality $$int_{mathbb{S}^2 } |Df_n|^2 =2int_{mathbb{S}^2 }Jf_n$$ is uniformly bounded, so $$f_n$$ is bounded in $$W^{1,2}$$, and converges to a constant function. (asymptotically we squeeze bigger and bigger parts of the sphere to a small region around the pole).

So, we do not have weak convergence of $$Jf_n$$ to $$Jf=0$$. (the
Jacobians converge as measures to a Dirac mass at the pole.) The question is if by adding the non-degeneracy constraint $$V(Jf_n le r) to 0$$ we recover this ‘Jacobian Rigidity’ under weak convergence.

*(In my case of application $$r=frac{1}{4}$$ but I don’t think it matters).

mining pools – Higher Hashrate VS Incorrect Shares

I recently started mining eth on my 5700 bios flashed to XT, mining on Nanopool. Getting around 48MH/s but i saw a method by Red Panda Mining which can increase my MH/s by around 10 giving me 60MH/s but a high ratio of incorrect shares using a config edit on Phoenix Miner. Im wondering if this is worth the increase in hashrate but ultimately still leads me to incorrect shares. My incorrect share rate is 2:1 using this edit. Thoughts?

Explain How Can You Align Picture So That One May Be Higher Or Lower Than The Other?

Explain how can you align picture so that one may be higher or lower than the other?

dnd 5e – Can a 4th level or higher Dispel Magic dispel the effects of a Duration: Instantaneous spell?

You cannot dispel the effects of an instantaneous spell

The Sage Advice Compendium states (emphasis mine):

Q. Can you use dispel magic on the creations of a spell like animate dead or affect those creations with antimagic field?

A. Whenever you wonder whether a spell’s effects can be dispelled or suspended, you need to answer one question: is the spell’s duration instantaneous? If the answer is yes, there is nothing to dispel or suspend.

There’s really no ambiguity in that statement. A spell that has an instantaneous duration simply cannot be dispelled regardless of what level dispel magic is being cast at.

Furthermore, the definition of Instantaneous spells states (emphasis mine):

Many spells are instantaneous. The spell harms, heals, creates, or alters a creature or an object in a way that can’t be dispelled, because its magic exists only for an instant.

This is just a fact of instantaneous spells and dispel magic is not making some unusual exception to this fact.

The dispel magic spell should be better worded

Technically, the rule on instantaneous spells only says the spell cannot be dispelled, which may or may not imply anything about the dispel-ability of its effects. Meanwhile, the dispel magic spell says first that:

(…) Any spell of 3rd level or lower on the target ends. (…)

And then, when upcast, that:

(…) you automatically end the effects of a spell on the target (…)

And these use different wording… Whether a spell being unable to be dispelled (because it is instantaneous) applies also to its effects is going to be up to the GM because the rules don’t give an explicit answer to the question.

I would hope (and the Sage Advice Compendium does agree here) that if a spell cannot be dispelled, neither can its effects be dispelled. Though this is not explicitly laid out within the rules, which is why I fell back upon the Sage Advice Compendium ruling at the beginning.

seo – How can I use the backlink strategy to make my website get higher?

I have a new website and my competition is so hard among over than 300 websites on only one keyword. My competitors are using backlink strategy through posting their links in blog comments. I heard that it is not legal, but they are dominating the organic search by the this strategy. My question is: How can I use the backlink strategy to make my website get higher?

nt.number theory – Modularity of higher genus curves

The modularity conjecture for smooth projective algebraic curves of genus 1, aka elliptic curves, over number fields is well known, and indeed, is a theorem for all elliptic curves over $$mathbb{Q}$$, and at least potentially, over any CM field.

What is the precise statement of the conjecture for higher genus curves? What are the modular/automorphic forms we expect to correspond to Galois representations realized in the $$l$$-adic cohomology of a smooth projective algebraic curve of genus $$g$$ > 1 over a number field $$F$$ via equality of $$L$$ functions? What is the state of progress towards the conjecture? References would be very welcome.

Unity installed version higher than game project version – can’t open the game project anymore

Long story short – I had to uninstall Unity and Unity Hub and reinstall them again.
I ended up installing Unity version 2020.2.1f1.
Some of my game projects were made using version 2020.1.6f1
Now I cannot open them – there is a yellow warning triangle next to those games in Unity Hub.

I am installing version 2020.1.17f1 as I type these.

So my questions:

1. Is there no backwards compatibility between newer versions of Unity and older games?
2. How do people deal with this?
I’ve heard people transition their games during development into newer versions of Unity but I wasn’t even able to open the games made with older version.
3. How would I go around this if I had spent 2 years or more of development time and was suddenly locked out from my game project – how do I prevent situations like these?

Here are my Unity versions installed:

And here are my games (tutorials mostly):

I may have answered my own question, but still need your input guys. I just noticed that there is a ‘Select Version’ dropdown on the right that allows me to open a game project in Unity version I have installed on my system. Is this how you deal with those issues?

dedicated hosting – WordPress /wp-content in a higher internet speed/higher latency datacenter

Can I speed up WordPress site loading by moving the /wp-folder (containing themes, plugin files, images) to another server with higher latency but also a much faster connection?

Example: host website on low latency server (slow connection) and download files larger than 1mb from the other server (whose both connection speed and latency are 10x as much)

L_inf norm equivalent of L_2 norm. in higher dimensions

I am working on a research subject where I need to implement different adversarial attack types on MNIST data.

I use max perturbation limit “epsilon” under L_infinity norm. This L_inf norm is limiting the max amount of allowed perturbation to each pixel of 28×28 image using a chosen epsilon parameter.

For example: I use epsilon value of 0.15 with L_inf norm metric, use PGD attack type and get an attack success rate around %60 for all MNIST test data.

But I am not sure which epsilon I need to chose when I need to implement the same attack under L2_norm metric. ( to simulate the same volume of perturbation)

I know that in higher dimensions the volume of L2 norm ball is smaller than the L_inf norm cube. And for the same amount of perturbation ( to make the volumes of L_2 ball equal to L_inf cube), I need to use a bigger epsilon value for L_2 norm.

For example, if I chose an epsilon of 0.1 for L_inf norm, then may be I need to chose an epsilon value of around 1.3 for L_2 norm.

But I don’t know the exact conversion formula for the correct epsilon from L_inf norm to L2_norm.

I would be very happy if somebody could help me on this.