## complexity theory – Are minimal boolean circuit sizes known for small problem sizes of an NP-complete problem?

I think a table with the following numeric values ​​would be very interesting, but I have not found any table online displaying them:

Choose any NP-complete problem (say, click, but a problem with well-defined instances of $$n$$– bit size for each $$n$$ is probably the best for this question). For each value of $$n$$ (input size), there is a boolean circuit of minimum size which solves the problem for instances of size $$n$$. Obviously it is difficult to calculate these minimum sizes, but they can be calculated by an exhaustive search for small values ​​of $$n$$.

Is there an array of these values ​​for the little ones $$n$$ known?

## woocommerce – Change the layout of [products] small code

I would like to display additional products on my simple product pages. I use the shortcode as such in the short description of the simple products.

``````[products ids="2113"]
``````

The problem is the layout of the resulting loop. Is there a way to customize the layout to just provide the Name – Price – "Add to Cart" button on one line. Almost as a variation option. Currently, I don't think it's possible with only CSS.

Thank you

## dg.differential geometry – Characterization of the isometric incorporation of Euclidean spaces from small to high dimensions

P.S. I asked the question on MSE over a week ago, but I didn't get the answer I wanted, so ask here.

Let $$m . Let us equip $$mathbb {R} ^ m, mathbb {R} ^ n$$ with their canonical Euclidean (Riemannian) metrics. How to characterize the isometric embeddings of $$mathbb {R} ^ m$$ in $$mathbb {R} ^ n$$? To be more precise, I am looking for transformations that are sufficiently regular, injective and preserve the distance $$Phi: mathbb {R} ^ m to mathbb {R} ^ n$$, so that: $$d_m (x, y) = d_n ( Phi (x), Phi (y))$$, or $$d_m, d_n$$ represents the distances in $$m, n$$ dimensional Euclidean spaces respectively, so e.g. $$d_m (x, y): = || x-y || _ { mathbb {R} ^ m}$$.

$$x cards to A (Bx, O (Bx))$$ or $$A$$ is a Euclidean isometry of $$mathbb {R} ^ n$$ (i.e. rigid movement), $$B$$ is a Euclidean isometry of $$mathbb {R} ^ m$$ (i.e. a rigid movement), and $$O: mathbb {R} ^ m to mathbb {R} ^ n$$ is "zero padding" $$n-m$$"times, namely: $$O (x) = (x, 0, points 0)$$.

If the above is correct / incorrect, how should I prove it or characterize Euclidean embeddings?

## The context

i use macOS Catalina (10.15.3), but this problem exists for at least 5 years in the integrated messaging application (Mail.app).

## The problem

Mail.app in macOS seems to display some incoming mail & # 39; much smaller fonts that they are only supposed to be shown. In any other mail client on Windows, these messages appear completely normal and the font size is clearly readable. Usually this applies to personal emails in RichText or HTML format, which have no particularly stylish text other than bold or colored text. It does not apply to newsletters and most emails generated by online stores, etc.

## Workarounds

1. A way of screed with this problem is to press repeatedly `CTRL``+`to increase the font size. But, as soon as you switch to another email, the size is reset to the default (which in turn is the correct behavior). When you open the same email again, you need to increase the font size again. It is mentioned in Mail from AOL, font to small Fix? (5 year question with no real answer).
2. You can increase the overall font size of emails in settings (Fonts and colors> Font for email). However, this will also increase the subject font size, headers and other text that is displayed in the preview pane for all emails, and it just looks awful.

3. (Not tested) It may be possible to apply an AppleScript which amended the content of the email. It is mentioned in Replace font in incoming mail (mail.app). I can't even think it will succeed. It could be a huge security issue.

## Possible reason

Looking at the source of these problematic emails, I can see a lot of tags, especially parts like this, where the `size="2"` could be the problem:

``````Ich danke dir!
``````

Previously, it was possible (somehow) to solve this problem by modifying `/Applications/Mail.app/Contents/Resources/Message.css`. This file no longer exists.

## Some references to this problem on other sites

This problem seems been around for years and has not been processed properly by Apple. For me it's always most frustrating problem after switching to the Mac from a Windows PC. Any help is appreciated!

## Small deviations

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## unit – The small health bar above the entity is drawn differently when the entity rotates differently

Here is a simple positioning code for my health bar:

``````transform.position = Camera.main.WorldToScreenPoint(target.position - 0.2f * target.up);
``````

Here is the prefab:

• HealthBar is anchored in the center-center and rotates at 0.5, 0.5,
• The background and fill are stretched and rotated by 0.5, 0.5, but the background is positioned left, right, top and bottom.

Here's what it should look like: (all images are enlarged)

And here is a video about the entity that revolves around: https://i.imgur.com/EOzFEvt.mp4

2 examples of rotations, in which the health bar is not drawn correctly:

Why?

Edit: Origin of the entity:

## machine precision – How to make exponentials of small default value to zero?

I have a function `f(t_) := Exp(N(-(t - 30.)*(t - 30)/100))`. When I tried to assess `f(600)` this shows ```Exp(-3294) is too small to represent as a normalized machine number; precision may be lost.``` Is there a global setting to make such a default assessment zero?

Currently, my implementation is: `f(t_) := If(t < 70, Exp(N(-(t - 30.)*(t - 30)/100)), 0)` But I hope to have a way to settle globally.

## performances – Modification of a small delay before loading in full resolution in Lightroom Classic

While browsing the images in Lightroom, it waits a little (maybe a second or two) on each image before loading the full resolution version (when loading toast appears). I think it is for performance reasons preventing a complete rendering because we pass on images without really needing to load them.

I was wondering if there was a way to adjust this delay?

## How do you use range finding devices with small viewfinders while wearing glasses?

I am fairly short-sighted and have been wearing glasses or contact lenses for decades (since the second year). I gave up on contact lenses about ten years ago.

Before I gave up on contacts, I bought a Zeiss Ikon 532/16 (Super Ikonta B), 6×6 with paired rangefinder, but I moved away from photography before returning to full-time glasses. Now, taking the camera back, I find it difficult to use the very small viewfinder (I can pretty much see the RF patch, but I can't see all the edges of the frame both) due to the distance between the eyes and viewfinder reinforced by my glasses.

I just bought a Kiev 4A, a Soviet copy / upgrade of a pre-war 35mm Contax RF camera (interchangeable lenses, broadly similar to an old Leica); I have wanted a camera in this class for a long time, but I see that the viewfinder window (based on a Zeiss-Ikon design, of course) is very similar in size to that of my Super Ikonta, and I suspect that I & # 39; & # 39; He'll have the same problem using the viewfinder.

I know the dioptric eyepieces which can be added to various cameras, but as far as I can see, these viewfinders designed by Zeiss-Ikon have no provision for such a corrective lens (if they are even available in -6.75 diopter). Maybe I could cut out a lens from an old pair of glasses and attach it, but I hate making a permanent change that affects every subsequent user, to a camera that (in the case of the Super Ikonta) is 39; a decade or so old as me.

What else can I do to see through these tiny viewfinder / rangefinder windows?

## rangefinder – How to use RF cameras with small viewfinders while wearing glasses?

I am fairly short-sighted and have been wearing glasses or contact lenses for decades (since the second year). I gave up on contact lenses about ten years ago.

Before I gave up on contacts, I bought a Zeiss Ikon 532/16 (Super Ikonta B), 6×6 with paired rangefinder, but I moved away from photography before returning to full-time glasses. Now, taking the camera back, I find it difficult to use the very small viewfinder (I can pretty much see the RF patch, but I can't see all the edges of the frame both) due to the distance between the eyes and viewfinder reinforced by my glasses.

I just bought a Kiev 4A, a Soviet copy / upgrade of a pre-war 35mm Contax RF camera (interchangeable lenses, broadly similar to an old Leica); I have wanted a camera in this class for a long time, but I see that the viewfinder window (based on a Zeiss-Ikon design, of course) is very similar in size to that of my Super Ikonta, and I suspect that I & # 39; & # 39; He'll have the same problem using the viewfinder.

I know the dioptric eyepieces which can be added to various cameras, but as far as I can see, these viewfinders designed by Zeiss-Ikon have no provision for such a corrective lens (if they are even available in -6.75 diopter). Maybe I could cut out a lens from an old pair of glasses and attach it, but I hate making a permanent change that affects every subsequent user, to a camera that (in the case of the Super Ikonta) is 39; a decade or so old as me.

What else can I do to see through these tiny viewfinder / rangefinder windows?