What type of function would be suieted to describe the curve of figure?

I need a function to describe the curve of the figure, preferred an exponential function.
I have looked for some functions such as sign(x), rectangular, and but none of them can be useful.
enter image description here

ag.algebraic geometry – Describe $mathcal{N}_{G(mathbb{P}^1,mathbb{P}^k)mid G(mathbb{P}^1,mathbb{P}^n)}$ [from MSE]

(This question came from MSE, but since I’ve received some useful observations I was wondering if anyone in this forum could be mehere the link: https://math.stackexchange.com/questions/3844944/describe-mathcaln-g-mathbbp1-mathbbpk-mid-g-mathbbp1-mathbbp/3845228#3845228 )

Consider $1 leq k < n$ positive integers, and denote by $G(mathbb{P}^k,mathbb{P}^n)$ the Grassmannian of $mathbb{P}^k$‘s in $mathbb{P}^n$. It is well known $G(mathbb{P}^k,mathbb{P}^n)$ admits a structure of projective variety in $mathbb{P}^N$, where $N=binom{n+k}{2}-1$.

It is moreover known that, denoting by $mathcal{N}_{mathbb{P}^kmid mathbb{P}^n}$ the normal bundle of $mathbb{P}^k$ in $mathbb{P}^n$, it holds the following

$$mathcal{N}_{mathbb{P}^kmid mathbb{P}^n} =mathcal{O}_{mathbb{P}^k}(1)^{oplus n-k}.$$

I was wondering if there exists a similar results if we consider a Grassmannian instead of the projective space, that is

$$?? text{ } mathcal{N}_{G(mathbb{P}^1,mathbb{P}^k)mid G(mathbb{P}^1,mathbb{P}^n)}=mathcal{O}_{G(mathbb{P}^1,mathbb{P}^k)}(1)^{2(n-k)} text{ }??$$

I’ve put the double question mark to stress this may be completely nonsense, but it’s my guess (nothing rigorous).


  • The $2(n-k)$ came from a comment on MSE and it is based ona simple computation of dimension of Grassmannian.

  • In order to talk about $mathcal{O}_{G(mathbb{P}^1,mathbb{P}^k$}$ I need a projective embedding, but there’s the natural Plucker embedding we can consider.

I was wondering if there exists such a similar description, I’ve thought about it for a while but didn’t managed to find anything. Thanks in advance!

dnd 5e – How to describe Symbiotic entity wildshape from the spore druid?

Symbiotic Entity isn’t Wild Shape, it’s something you do instead of Wild Shape.

From the ability description (dndbeyond link, emphasis mine):

As an action, you can expend a use of your Wild Shape feature to awaken those spores, rather than transforming into a beast form, and you gain 4 temporary hit points for each level you have in this class.

The Wild Shape ability is indeed metamorphosis, but the ability being described here isn’t a form of Wild Shape, it’s an alternate ability that expends uses of Wild Shape as a resource. As a separate ability, there’s no strong reason to suppose it does something like turn you into a giant mushroom just because Wild Shape is all about transformation.

For insight into what it does do, we’ll turn to the Halo of Spores ability described immediately before Symbiotic Entity (emphasis mine):

Starting at 2nd level, you are surrounded by invisible, necrotic spores that are harmless until you unleash them on a creature nearby.

And immediately after that, in Symbiotic Entity:

At 2nd level, you gain the ability to channel magic into your spores.

So Symbiotic is you channeling your magic into the spores that make up your Halo of Spores ability, empowering them and yourself.

As for what that looks like, it’s largely up to you and your imagination. Maybe you’re surrounded by a cloud of glowing spores? Maybe they embed themselves into your flesh and you do grow mushrooms all over yourself? The rules don’t specify what abilities look like at this level of detail, leaving stuff like that up to player creativity.

You should feel free to describe Symbiotic Entity in whatever way seems coolest to you, or best fits your character, within the constraints described in the text of the ability.

mechanism – What is a good way to describe all kinds of basic equations in elasticity by tensor

I see here that we can use tensors to represent the fundamental equations in elasticity.

enter image description here

But I don’t know how to use the tensor operation in MMA to reproduce these equations. If you can, please provide some useful related resources, please.

I hope these tensors can be involved in operations, not just signs.

first order logic – How to describe Deterministic Transitive Closure in FOL

In “Finite Model Theory and Its Applications”, page 152, it is said that Deterministic Transitive Closure, on ordered finite structures, captures LOGSPACE.

Hence, taking into account that FOL captures LOGSPACE, it should be possible to:

  • Given a functional relation R(X,Y) (e.g. “motherOf(X,Y)”)
  • It should be possible to define, in FOL, a predicate T(X,Y) that captures its transitive closure (e.g. “femaleAncestor(X, Y)”)

I would like to know how to define such predicate. This would be quite easy in Datalog, using recursion, but it should be possible to define it without recursion, which is the question that puzzles me.

How (and when) do you describe “affordance” in your Interaction Design?

Take the following example of an animation in a webstore.

Selecting a whine will make it float and it will fly to your "bag" when clicked "add to cart"

Say you have to write an Interaction Design document about this. I would imagine it would look something like this (VERY simplified of course):


“As a customer, I can add a wine bottle to my cart, so I can buy it”


  1. Move the mouse over the wine bottle
  2. Click the “Add to Cart” button
  3. Result: Wine bottle is now in the “My bag” section


Basic mock-up

But where/when do I describe the fact that the bottle will float when the mouse hovers over it, and visually moves to the cart box when “add to cart” is clicked? I don’t see it as part of the Story/Case/Mock-up at all.

history – Did Polaroid’s patents on integral photography describe a different technology than what Polaroid used?

I had a colleague at Polaroid in the 1980s who told me that the patents Kodak was accused of violating did not describe the system that Polaroid was actually using in its SX-70 and Spectra film, but instead described a chemistry that Polaroid had decided not to use. Polaroid kept the better system as a trade secret.

I’ve looked through coverage of the Polaroid v. Kodak case for several years and have been unable to find any substantiation to this claim. I’ve also read in some cases the same plants were producing both Polaroid and Kodak film. So I’m not sure what’s really going on.

My colleague also told me that during discovery a memo from Kodak was found which stated that Kodak knew that it was in violation of the Polaroid patents, but that Polaroid would probably never sue because of the expense. Likewise, I’ve been unable to find reporting of this memo. You would think that it would be in the court record, but I haven’t been able to find it (and I assume it’s huge).

So I’m looking for an answer to this question — Did Polaroid’s patents on integral photography describe a different technology than what Polaroid used? I don’t know how to even research this question, other than lots of Google searches…

describe website – Social Media Marketing (SMM)

the earnx site is very easy and earn the money 

and very simple task given the site . and the task is watched videos and simply asked own answers of surveys , then youtube videos also watched in 40 sec minimum 


with compare all the website this one is good  


development process – Who should describe the testing procedure of a bug?

The answer to your question depends entirely on the way each company runs their business, how much money they’re willing to devote to QA efforts, the kind of software being written, and other factors. There is no “correct way.”

Your method adds substantial overhead, requiring close coordination between the QA and development teams. We don’t do it this way where I work, and I daresay we are doing just fine, but we work very closely with our stakeholders to make sure the software behaves in ways they expect it to work.

The way that software testing works should be tailored to the nature of the software itself. Not every testing procedure for every software project is going to be the same. The testing and Q&A process for a line-of-business application is going to be very different than the testing and Q&A process for spacecraft software, financial calculations or medical devices.

As the person who understands the actual code our team has written, I’m probably in a better position to craft tests for a specific bug than the QA team is. Teams that are concerned with Quality Assurance and stakeholder acceptance generally focus more on “is the software meeting its stated requirements” than “has this specific bug been fixed.”

But if I were part of a Q&A team that reported a bug, I would certainly make it my business to see that the bug I reported was fixed, whether the developer had written tests for it or not.

automata – Describe this Turing machines

I need to describe a Turing machine that it can be multitape.

$M = (Q,Σ,Γ,δ,q_0,q_a,q_r)$.

Who decide that language.

$L={#uqv#u’q’v’#:u,v,u’v’∈(Γsetminus{sqcup}^*,q,q’∈ Q$ and$ (u,q,v) Rightarrow (u’,q’,v’)}$ according to M.

on alphabet.


But I not sure to understand the language and the alphabet $Sigma’$, do you have exemple that could help me to understand. Thanks