## Are there paths of exact forms joining two symplectic structures on open manifolds?

There is a Theorem of Conolly, L$$hat{text{e}}$$ and Ono which states that on a closed simply-connected $$4$$-manifold two cohomologous symplectic forms can be joined by a path of non-degenerate $$2$$-forms.

I was wondering is there any open analogue of this theorem? For example:

Let $$omega_0$$ and $$omega_1$$ be two cohomologous symplectic structures on an open simply-connected $$4$$-manifold $$M$$ which are equal on the complement of some compact set. Are there any results on the existence of a path $${omega_t}_{tin [0,1]}$$ of non-degenerate $$2$$-forms joining $$omega_0$$ and $$omega_1$$ such that all $$omega_t$$ are equal on the complement of some compact set?

I am especially interested in the case $$M=mathbb{R}^4$$.

## Prioritize search result to show matching exact result

Suppose I search "TA" on wordpress query, the results show the posts with words containing "TA" first, like sTAnd, I want to display exact word "TA" first then words containing "TA".

## powershell – How to modify the same attribute of multiple lists with exact schemas in Sharepoint online through coding or ms flow

I have several lists with the same exact schemas in SharePoint Online. How can I do the following?

1- How to add/remove a column to/from all of them?

2- How to add/remove an option to a choice/multichoice column?

I think there should be a way to accomplish this in C#/Python or through Powershell. I prefer coding as it makes life much easier and faster. MS flow is fine.

Please let me know if it would be better to break this question into two seprate questions.

## ct.category theory – Is every additive, left exact functor isomorphic to a hom functor?

One can prove the following result: let $$F colon mathrm{Mod}_R to mathrm{Mod}_S$$ be left-exact and preserve small products (equivalently, a continuous functor). Then $$F$$ is of the form $$mathrm{Hom}(M,-)$$.

Let me first take a step back. The context of your question is (as I think you know) the Eilenberg-Watts theorem, a form of which states that if $$G:{}_Rmathrm{Mod}^{mathrm{op}} to mathrm{Mod}_S$$ is left exact and preserves products, then there is an $$(R,S)$$-bimodule $$M$$ such that $$G simeq mathrm{Hom}(-,M)$$. In fact applying the functor to $$R$$ shows that $$M =G(R)$$, so the candidate module was obvious all along. And in a dual form, if $$H:mathrm{Mod}_R to mathrm{Mod}_S$$ is right exact and preserves coproducts, then there is an $$(R,S)$$-bimodule $$M$$ such that $$H simeq -otimes M$$, and necessarily $$M cong H(R)$$.

But in this case it’s not as immediate what to plug into the functor $$F$$ to get a candidate module. However one can apply the special adjoint functor theorem. Module categories are complete, well powered, and have a cogenerator. So $$F$$ has a left adjoint $$H$$. You can apply Eilenberg-Watts to $$H$$ to deduce that $$H simeq – otimes M$$ for a bimodule $$M$$. Then $$F simeq mathrm{Hom}(M,-)$$.

In your case you have a left-exact additive functor $$mathrm{mod}_R to mathrm{Mod}_S$$, where $$mathrm{mod}_R$$ denotes the compact objects (finitely presented modules). It extends uniquely to a functor $$mathrm{Mod}_R = mathrm{Ind}(mathrm{mod}_R) to mathrm{Mod}_S$$ preserving filtered colimits. If we make the additional assumption that the extended functor preserves all products, then by the above argument it is of the form $$mathrm{Hom}(M,-)$$. Since it commutes with filtered colimits, $$M$$ is moreover compact.

## MySQL freezes every hour at exact time

MySQL freezes every hour at exact time – Database Administrators Stack Exchange

## Is there a way to have my mouse operate on 2 different monitors/computers at the same exact time?

I’d like to know if there’s a way I can perfectly sync (or close enough at least) my mouse on 2 different computers/monitors (preferably computers) at once. Essentially I want to mirror my mouse on 2 different monitors at once. I have 2 computers and would like to be able to just have the mouse be able to operate on both at the same time without needing to use a switch. If worst comes to worst though I can hook up both monitors to one. Thanks!

## gmail – Anyway to search for an exact phrase that contains quotes?

Im trying to search for a phrase containing quotes but as the quotes is what gmail uses to delimit the exact phrase filter, it closes before it can.
For instance

I need to search for the exact phrase:

and then james said “come here”

Issue is when I put that around quotes

``````"and then james said "come here""
``````

It creates two exact phrases search:

1. and then james said
2. come here

Which doesnt work for me

Is there any way to escape the “?
On programming languages we usually have a escape character like that you add before the character you want to escape, so it would be something like:

``````"and then james said "come here""
``````

## ag.algebraic geometry – Looking for the exact and the precise statement of Ogus conjecture

I have been looking for several weeks for the exact and the precise statement of Ogus conjecture, but, I cannot find it.
The only book which made me discover the statement of this conjecture is that of Yves André: Théorie des motifs. See here, http://tomlr.free.fr/Math%E9matiques/Andre,%20Y%20-%20Une%20Introduction%20aux%20Motifs%20%28SMF%202004%29.pdf, pages, $$79$$ and $$80$$.

To understand how this conjecture is formulated, the author of this book directs us to a paper of Ogus, the holder of this conjecture, which is entitled, Hodge Cycles and Crystalline Cohomology.

The paper can be found here, https://www.jmilne.org/math/Books/DMOS.pdf, page, $$359$$, in the introduction.

The statement of Ogus conjecture is not very clear. I formulated it as follows, following my efforts to understand its statement.

Here is the statement that I propose,

Let $$k$$ be a number field.

Let $$R$$ be an étale $$mathbb {Z}$$ -algebra.

Let $$X$$ be a smooth projective $$R$$ -scheme.

Let $$R’ supseteq R$$ be another étale $$mathbb{Z}$$ – algebra.

Let $$s$$ be a closed point of $$mathrm{Spec} R ‘$$, and let $$W$$ be the completion of $$R’$$ in $$s$$.

We have an isomorphism, $$H_ {mathrm{dR}}^{i} (X / R) otimes_k W simeq H_{mathrm{cris} }^{i} (X (s) / W )$$

$$H_{mathrm{cris}}^{i} (X(s) / W)$$ is a $$F_ {displaystyle v}$$ – crystal, therefore, equipped with the Frobenius $$F_{ displaystyle v} : H_{mathrm{cris}}^{i} (X(s) / W) to H_{mathrm{cris}}^{i} (X(s) / W)$$ defined by, $$F_{displaystyle v} (z) = p^r z$$.

So, we can pass this Frobenius $$F_{displaystyle v}$$, to $$H_{mathrm{dR}}^{i} (X / R) otimes_k W$$ by this isomorphism.

Let the integral class cycle map (i.e., on $$mathbb {Z}$$), be defined by,
$$mathrm{cl}_X : mathcal{Z}_{sim}^{i} (X) to displaystyle bigcup_{v in I} displaystyle Big (H_{mathrm{dR}}^{i} (X / R) otimes_k W displaystyle Big)^{textstyle F_{v}},$$ where, $$Big (H_{mathrm{dR}}^{i} (X / R) otimes_k W Big)^{textstyle F_{v}}$$ is the $$F_{displaystyle v}$$ – crystal of $$F_{textstyle v }$$ – invariants.

$$I$$ is the collection of the closed points $$s$$ of $$mathrm{Spec} R’$$.

So, Ogus conjecture asserts that, the rational class cycle map (i.e., over $$mathbb{Q}$$), which is $$mathrm{cl}_X otimes mathbb{Q}$$, as follow, $$mathrm{cl}_X otimes mathbb{Q} : mathcal{Z}_{sim}^{i} (X) otimes_{ mathbb{Z} } mathbb{Q} to displaystyle bigcup_{v in I} displaystyle Big (H_{mathrm{dR}}^{i} (X / R) otimes_k W displaystyle Big)^{textstyle F_{v}} otimes_{ mathbb{Z} } mathbb{Q}$$

is surjective?

So, is that right ?

Can you correct that statement for me to see if I got it right?

How is $$W$$ defined ?

Does $$W$$ vary when the closed point $$s$$ of $$mathrm{Spec} R’$$ varies ?

See here, Berthelot-Ogus comparison isomorphism for others interesting informations.

## How can I measure the exact range of focus of a given fixed focus webcam?

The concept of depth of field is really just an illusion, albeit a rather persistent one. Only a single distance will be at sharpest focus. What we call depth of field are the areas on either side of the sharpest focus that are blurred so insignificantly that we still see them as sharp. Please note that depth-of-field will vary based upon a change to any of the following factors: focal length, aperture, magnification/display size, viewing distance, etc.

There’s only one distance that is in sharpest focus. Everything in front of or behind that distance is blurry. The further we move away from the focus distance, the blurrier things get. The questions become: “How blurry is it? Is that within our acceptable limit? How far from the focus distance do things become unacceptably blurry?”

What we call depth of field (DoF) is the range of distances in front of and behind the point of focus that are acceptably blurry so that to our eyes things still look like they are in focus.

The amount of depth of field depends on two things: total magnification and aperture. Total magnification includes the following factors: focal length, subject/focus distance, enlargement ratio (which is determined by both sensor size and display size), and viewing distance. The visual acuity of the viewer also contributes to what is acceptably sharp enough to appear in focus instead of blurry.

The distribution of the depth of field in front of and behind the focus distance depends on several factors, primarily focal length and focus distance.

The ratio of any given lens changes as the focus distance is changed. Most lenses approach 1:1 at the minimum focus distance. As the focus distance is increased the rear depth of field increases faster than the front depth of field. There is one focus distance at which the ratio will be 1:2, or one-third in front and two-thirds behind the point of focus.

At short focus distances the ratio approaches 1:1. A true macro lens that can project a virtual image on the sensor or film that is the same size as the object for which it is projecting the image achieves a 1:1 ratio. Even lenses that can not achieve macro focus will demonstrate a ratio very near to 1:1 at their minimum focus distance.

At longer focus distances the rear of the depth of field reaches all the way to infinity and thus the ratio between front and rear DoF approaches 1:∞. The shortest focus distance at which the rear DoF reaches infinity is called the hyperfocal distance. The near depth of field will very closely approach one half the focus distance. That is, the nearest edge of the DoF will be halfway between the camera and the focus distance.

For why this is the case, please see:

Why did manufacturers stop including DOF scales on lenses?
Is there a ‘rule of thumb’ that I can use to estimate depth of field while shooting?
How do you determine the acceptable Circle of Confusion for a particular photo?
Find hyperfocal distance for HD (1920×1080) resolution?
Why I am getting different values for depth of field from calculators vs in-camera DoF preview?
As well as this answer to Simple quick DoF estimate method for prime lens

## Find the nearest exact form of an irrational number

Is it possible to find the nearest exact form of an irrational number if the decimal number is given? Suppose i only have $$6$$ digits i.e. $$3.14159$$ the output should tell me one of the possibilities is $$pi$$. I don’t have any attempts since i have no idea how to do this. Hope you can help me.