combinatorics – Number of ways a committe can be formed

In how many ways can a committee of four be formed from 10 men (including Richard)
and 12 women (including Isabel and Kathleen) if it is to have two men and two women

a) Isabel refuses to serve with Richard,

b) Isabel will serve only if Kathleen does, too

My Thoughts :
a) Total number of ways to select 4 people = 22C4 now for part a) I just need to deduct the
pairs where Isabel and Richard are both in the committe which = 20C2

I am not sure how to proceed with part (b) and if part (a) is entirely correct

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reference request – Different ways of defining the Chern character of a complex

Consider a finite complex $E$ of (holomorphic) vector bundles on a (complex) manifold $X$, i.e, the complex is of the form
0 to E_N to E_{N-1} to dots to {E_0} to 0,

where the bundles are equipped with connections $D_i$. By K-theory, one may consider the complex $E$ as the alternating sum $sum_i (-1)^i (E_i)$, and it is then natural to define the Chern character (as a form) as $ch(E,D) := sum_{i=0}^N (-1)^i ch(E_i,D_i)$, and the Chern form as $c(E,D) := prod_{i=0}^N c(E_i,D_i)^{(-1)^i}$, where $ch(E_i,D)$ and $c(E_i,D)$ denote the Chern character and Chern form of $(E_i,D_i)$.

Alternatively, for a fixed $k$, one may express the Chern character as a polynomial in the Chern forms, $ch_k = S_k(c_1,dots,c_k)/k!$, where $S_k$ is the polynomial which expresses the Newton polynomials in terms of the elementary symmetric polynomials, i.e., what is sometimes called the Hirzebruch-Newton polynomial. For example, $S_1(t_1)=t_1$, $S_2(t_1,t_2)=t_1^2-2t_2$ etc. With the help of the polynomials $S_k$ above, one could alternatively define a Chern character of $E$ by

I have only found mentioned in passing or implicitly that these definitions coincide, i.e., $ch_k(E,D) = widetilde{ch}_k(E,D)$, but not any precise argument. Does anyone know of a convenient reference or proof of this fact?

probability – Ways of solving the following recurrence relation system.

Consider the following system of linear recurrence relations.

$$begin{aligned} p_n &= a cdot p_{n-1} – c_{n-1}\ c_{n-1} &= p_{n-1} – b cdot p_{n-2} + c_{n-2}end{aligned}$$

with $p_0 = 1$ and $c_1 = 1$.

I’ve tried to represent $p_n$ as a finite linear combination of $p_{k}, k < n$, but this doesn’t work for me. Maybe there is any chance to represent the final solution?

Any ideas?

dnd 5e – Are there any ways to reinforce Paper Birds?

A Paper Bird is sufficiently sturdy; don’t punish players for using their limited resources.

A Paper Bird is a magic item. Magic items have Magic Item Resilience which makes them

objects of extraordinary artisanship, […] Thanks to this
combination of careful crafting and magical reinforcement, a magic
item is at least as durable as a regular item of its kind. Most magic
items, other than potions and scrolls, have resistance to all damage. […]

The Paper Bird is neither a scroll nor a potion.

The bird is an object that has 1 hit point, an Armor Class of 13, a
flying speed of 60 feet,

Regular weather should be no obstacle for the Paper Bird

It travels to within 5 feet of its intended recipient by the most
direct route, whereupon it turns into a nonmagical and inanimate sheet
of parchment that can be unfolded only by the intended recipient. If
the bird’s hit points or speed is reduced to 0 or if it is otherwise
immobilized, it turns into ash.

So the issues arise when something threatens to attack or immobilise the Paper Bird while the party that has no better way to contact an NPC via spells like sending – with its unlimited range – is unlikely to have access to any methods that add further protection to the Paper Bird. Nor is it likely that they could keep up with its flying speed of 60 feet that is not subject to exhaustion and doesn’t need resting, so chasing and protecting the Paper Bird is also not an option.

But as you stated, they have access to multiple of them – initially, one would assume 1d6 + 3. So if they are willing to spend them as resources, it is likely that some or one of them will eventually reach the NPC.

If I were the DM, I would collaborate with their idea and acknowledge their effort, unless there is a more interesting development along the path. Taking away limited resources like magical items without them paying off is frustrating for players, so I wouldn’t recommend doing that. Moving the story towards what is compelling for the players creates a cooperative and more invested experience for all of you.

peripherals – Logitech non-unifying receivers compatibility (and ways to get a spare)

I have bought an MK235 keyboard-mouse set and a M220 mouse which I use for different computers. Recently I lost the receiver of the MK235 set, but, doing some testing and the Logitech app, I have found that the M220 receiver is also compatible.

Both of these receivers are P/N C-U0010. After searching a bit in ebay, it seems that this specific P/N is used for variety of products, while it seems that some mouses and keyboards are also compatible with other non-unifying receivers. I have an older one (C-U0003) but it is not compatible, but this does not seem to be the rule. As expected, I have found nothing from official sources. The questions are:

  1. Is there any compatibility list for devices and receivers?
  2. The standard cost of such a receiver in ebay is 10+ euros. I can get a mouse with the same receiver for a similar price. Is there an official Logitech e-shop to get one at a normal price?
  3. Has anyone the wireless mouse B170 or M170 to tell me the P/N of the receiver that comes with the product?


seo – What are the best ways to improve a site’s position in Google?

There is no definitive answer and all answers will be completely opinion based.

Keyword meta tags are NOT used in search engine ranking. If you would like a more definitive answer see the following post here.

The best thing you can do is:

1.] Keep adding fresh content, comments and blog posts and create RSS feeds for Google and submit these as sitemaps to Google.

2.] Keep all content unique and don’t duplicate content. Google is currently fighting to keep all content on there search engine unique and non-duplicated. So make sure you add canonical links to your page. See here to learn how to do so.

3.] Make sure your website is responsive. Google also wants sites to display fast and correctly on all devices and you may be penalised if your website is too slow on mobile device.

4.] H1 tags, Meta descriptions, titles, and Meta Tags. This said, with meta tags not being relevant, it won’t hurt to make them relevant to your content. However have one and only one H1 tag, and make sure it is relevant to your content. Same goes for your meta description and title tags.

5.] Website speed is an important factor. Google recommends that all websites should load in less then 2 seconds. In this case make sure all images are optimised, don’t load unnecessary resources, set a cache in your .htaccess, enable persistent TCP connections, add gzip to your resources, and combine all off your CSS files into one and minify them, same with you JS files.

All my suggestions are of my own opinion and they are all onsite optimisation. You will be required to gather relevant backlinks but make sure you don’t participate in a linking scheme. Please also note that bad links and spam content can potentially harm your website causing much bigger issues.

Social Media can also be a good source of traffic and can be used as a very powerful SEO tool if used properly.

The main point would be: Don’t expect results overnight, this process will take a very long time and it may take a lot of work and time before you start seeing results. Make sure you target the right keywords for your site doing research on your target keywords, analyse your competition, what are they doing that you aren’t doing, how many backlinks do they have. None of this however will guarantee you a top spot. If it was that easy then everyone would be doing it.

The best place to start is by looking here, as this is written by Google and will advise you on the best practices.

Galaxy S10+ – Gallery broken in weird ways – can’t see pictures, can’t save camera photos and can’t take screenshots

I’m having a very strange problem today with my Galaxy S10+.

I had an update yesterday evening, and all of today it’s been broken in a weird way. This’ll be a longish post because I want to be really clear about the problems I’m having and explain what I’ve done so far to try fix it. Right off the bat I want to note that these problems are not exclusive to my SD card. My SD card works just fine and my pictures are stored across both my internal storage and the SD card. I’ve tried most of my troubleshooting with and without it in.

These problems occurred immediately after I finished an update. The Software Update Information says “G975FXXU9FUBD / G975FOTF9FUBD / G975FXXU9DUBD”

Basically, when I open the stock Gallery app, it shows absolutely nothing. None of my pictures or albums show at all. However, the automatically generated Stories work just fine. I can’t select anything, I can’t highlight anything, I can’t change anything. All I see is a blank screen where I can swipe endlessly between Pictures and Albums and see a whole lot of nothing.

My files are perfectly fine and safe from what I can see. I’ve hooked it up to my computer and nothing is missing, damaged, broken or corrupted. They are all there in the My Files app and show all the thumbnails for the pictures. I’ve downloaded another Gallery app from the Play Store and all the pictures are there in the right order and the albums are there too.

But, if I say open a picture in My Files and click Gallery as the app to open the image with, it’ll hang for a while then open a black screen with no buttons on it except the home bar, and all I can do is back out of it. Strangely, when I open the app tray after this, it’ll only show the My Files app being open and not Gallery.

There are two very strange aspects to my problem too – I can’t take photos with the camera or screenshots.

The Camera app won’t save photos no matter whether I choose to point them to the internal storage or the SD card. It’ll show the little preview box showing a picture was taken and do the little animation, but after a while that just turns back into the generic Gallery icon, says “Warning, camera failed” and closes itself.

If I’m quick and click on the icon that shows the picture I took, it’ll just do the same thing. If I try click there before taking a picture, it’ll just open the broken Gallery app after a delay of a few seconds.

I’d love to show some screenshots of my problems but I also can’t take screenshots. I can press the button for it and get the overlay, but if I try to crop them or let it save, it just hangs on the overlay screen and I can’t interact with anything and can only back out.

I can still save photos from online or from apps like Twitter, and apps like Messenger or Line which I use very often allow me to open and browse media and send pictures just fine. They also let me save pictures taken with the camera as long as it’s through those apps and it saves those photos as I’ve set it in the settings of those apps, and I can see them in the My Files app right after.

After spending the whole day googling as best I can, so far I’ve tried the following;

  • Booting in Safe Mode (both with and without the SD card in) – The same problem exists in Safe Mode
  • Clearing the Cache/Data for various apps like Gallery, Media and Devices, My Files, Camera and all sorts
  • Looked for files called “.nomedia” on both the phone itself and through my computer and found nothing.
  • I was going to try deleting the “” folder as suggested in a post which sounded a lot like my problem, but turns out I don’t have that folder in the directory the post suggested.
  • Using a “media rescan” app.
  • Forcing an update through the Galaxy Store but it couldn’t find an update for Gallery and wouldn’t let me do anything with it.
  • Pressing the “uninstall update” button on the Gallery app settings page and it remained the same. Updated it through the Galaxy Store and no prizes for guessing the outcome there.
  • Literally just turning my device off for an hour and leaving it, then back on.
  • Literally just leaving the device on, locked for an hour.
  • Literally just leaving the device on with the Gallery app open in case it needed to refresh or something.

I’ve tried most of these both with and without the SD card in too.

I feel the problem has to be centred around Gallery because it’s only ever when an app needs to use it as a medium for saving, viewing or interacting with a picture where it has issues.

The only thing I can really see that might be causing a problem is that I’ve got a LOT of pictures and videos. We’re talking well over ten thousand pictures spread across the internal storage and the SD card. But a day ago when it worked perfectly fine that didn’t seem to be an issue.

I’ve been using it all day and besides the gallery thing it’s worked just fine. No other apps seem to be broken or have had any weird issues. My notifications are as they normally are. It’s literally just any time I need to use the Gallery app.

So far all my troubleshooting has done is bother me by deleting the wallpapers and theme/icons I had on after coming out of Safe Mode. I’m completely stumped as to what to try short of a full factory restore, and I really don’t want to have to take that option.

Any help would be really appreciated.

algebra precalculus – Looking for alternative ways to solve $3^{2^x}=2^{3^x}$

If $log2=m$ and $ log3=n$, What is the answer of
$large3^{2^x}=large2^{3^x}$? $1)log
mnquadquadquadquadquadquad2)frac{m-n}{log m-log
n}quadquadquadquadquadquad3)frac{log frac
mn}{m-n}quadquadquadquadquadquad4)frac{m+n}{log mn}$

Here is my approach:

To solve the equation I took logarithm in base $3$ of both sides of the equation:

So we have $(frac23)^x=frac mn$. by taking logarithm in base $frac23$ we have:

$$x=log_{largefrac23}(frac mn)=frac{log(frac mn)}{log(frac23)}=frac{log frac mn}{m-n}$$

My questin: Is there other approach to solve this problem?

Please help me think of ways to improve my network diagram

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