## Is this a valid proof of Hall’s Theorem on System of Distinct Representatives?

Hall’s Theorem states that for sets $$A_1,…,A_n$$ if and only for all $$J\subseteq \{1,…,n\}$$ we find $$|\bigcup_{j\in J}A_j|\geq|J|,$$ then we can choose $$x_i\in A_i$$ for all $$i$$ such that $$\{x_1,…,x_n\}$$ is a system of distinct representatives for $$A_1,…,A_n.$$ To me it seems easiest to some inductive argument (or a contradiction that is similar to induction), however, in my graph theory class we’re working with Flow Algorithm’s, so the goal is to solve it using the Max-Flow-Min-Cut Theorem. I’ve seen one proof using that theorem, but I think I’ve came up with another, which to me seems more natural. I was wondering if my proof is correct. Here’s the proof.

$$\textbf{Proof:}$$

Throughout $$x$$ shall refer to an element of some $$A_i.$$ We set up a capicated network as follows $$\overrightarrow{E}=\{(s,A_i):\forall A_i\}\cup\{(A_i,x):\forall A_i\text{ and }x\in A_i\}\cup\{(x,t):\forall x\in \cup_{i=1}^n A_i\}$$ with capacities for $$(s,A_i)$$ and $$(x,t)$$ all $$1$$ and all other capacities infinite. This is ‘depicted’ in the following link, where any unwritten capacity is infinite.

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Now we suppose that we have a minimal network cut $$U.$$ So the sum of capacities of edges leaving $$U$$ is minimal. If $$A_i\in U,$$ and $$x\not\in U$$ for sum $$x\in A_i,$$ then $$(A_i,x)$$ is an edge leaving $$U$$ with infinite capacity, and $$\{s\}$$ has smaller capacity than $$U,$$ impossible. So $$x\in U$$ for all $$x\in A_i$$. We compute $$\text{capacity}(U)=\sum_{A_i\not\in U}\text{capacity}(s,A_i)+\sum_{x\in U}\text{capacity}(x,t)=|\{A_i\}\backslash U|+|\{x:x\in U\}|$$ $$\text{capacity}(U)\geq|\{A_i\}\backslash U|+|\{x\in A_i: A_i\in U\}|\geq|\{A_i\}\backslash U|+|\{A_i\in U\}|=n.$$ Thus since $$\text{capacity}\{s\}=n$$ we find that $$\{s\}$$ is a min cut, and we can find a system of distinct representatives. $$\blacksquare$$

## Help! I’ve renamed a system apk, and the phone is on bootloop now!

I have an xiaomi redmi note 5 rooted: to get rid of the notification system about new updates I’ve renamed the ” Updater.apk ” file on root /system/data/Updater , but now the phone in in bootloop. I’ve tried to rename it back booting from twrp, but after mounted the system partition ( mount /system ) I cannot find the same directories I saw when I was browsing using a root file manager. How can I locate the ” Update.apk ” file and rename it back on its original name? Thank you

## iPhone hook up to Car audio system for music

Was trying to hook up music from my iPhone to car audio – failed and then couldn’t hear anything from the iPhone except when I put on earbuds. Why?

## Unable to modify file permissions.xml, get error Read-only file system

I have a rooted Galaxy S5 SM-G900V, lineage_klte, ARMv7 Processor rev 1 (v71), Kernel: 3.4.113-lineageos #1 SMP, Oreo 8.1 that a friend asked me to modify.

I need to modify the platform.xml file to allow the system to write data to the external sd card. I need the ability to modify this file. The current permissions are: rw-r–r– 1 root root, however I am unable to save any modifications.

I am unable to modify the file permissions and would like to know how to do so. Note, I am looking how to change the file permissions and not how to or what syntax to use in the xml file, or if it will allow me to have the system recognize the external sd card for saving data. I am perplexed on why I am unable to modify this file.

> adb devices  <-- verify I have the device > adb shell    <-- enter shell > su           <-- super user # ls -l drwxr-xr-x  17 root   root      4096 2019-03-20 10:26     system # cd system #ls -l drwxr-xr-x 20 root root     4096 2019-03-20 10:25 etc # cd etc  # ls -l  > drwxr-xr-x  2 root root     560 2013-12-31 21:23 permissions  # cd permissions  # ls -l  -rw-r--r-- 1 root root  9116 2008-12-31 18:00 platform.xml  # chmod 755 platform.xml  >> chmod: chmod 'platform.xml' to 100755: Read-only file system 

So, how do I change the file attributes for platform.xml allowing it to be modified? I thought that there was a hidden immutable attribute on the file but there is not(at least that I can determine).

Related question, is there any other website to post this question? Any help is most appreciated.

## how to make a volume with a system image on it the Windows 7 boot volume

I’ve created a Windows 7 x64 system partition image on an empty volume on another drive because the original boot drive was throwing errors and SMART BIOS thinks it is “bad” even though the manufacturer’s utility reports it as in good health.

The image is here:

S:\img\imagename

Is there a simple way to make S: the boot volume, change its drive letter to C:, and take the original drive offline, without having to copy the image somewhere else? I don’t have any free space anywhere to put it, and all the SATA cables are in use.

## debug.log found in my mac system

I want to know if this is a bitcoin transaction i found in my computer wallet.

019-03-16T21:22:38Z UpdateTip: new best=00000000000000000e5a76e7fe7b72319617f515a4d7c76243f8062e64fb311b height=344194 version=0x00000002 log2_work=82.26815 tx=60096802 date=’2015-02-19T10:25:08Z’ progress=0.158562 cache=304.0MiB(2066812txo) 2019-03-16T21:22:38Z UpdateTip: new best=000000000000000001ddd9e738e1edd65bff59b77ae7ef407bcee2ce518e4a6a height=344195 version=0x00000002 log2_work=82.268197 tx=60098439 date=’2015-02-19T10:56:54Z’ progress=0.158567 cache=304.1MiB(2067260txo) 2019-03-16T21:22:38Z UpdateTip: new best=000000000000000002984fc795c48138c8699a746f1057dd8ceb490b079bb9fd height=344196 version=0x00000002 log2_work=82.268245 tx=60098646 date=’2015-02-19T10:59:57Z’ progress=0.158567 cache=304.1MiB(2067024txo) 2019-03-16T21:22:38Z UpdateTip: new best=00000000000000001697b0bbd30989718e304a1a369a60962788ed5a5eead4b2 height=344197 version=0x00000002 log2_work=82.268292 tx=60098741 date=’2015-02-19T11:00:57Z’ progress=0.158567 cache=304.0MiB(2066270txo) 2019-03-16T21:22:38Z UpdateTip: new best=0000000000000000006ae9fa940419e7956c7d0f3020996f7a7978f9550138d4 height=344198 version=0x00000002 log2_work=82.268339 tx=60099493 date=’2015-02-19T11:05:55Z’ progress=0.158569 cache=304.0MiB(2066604txo) 2019-03-16T21:22:38Z UpdateTip: new best=0000000000000000177c917e30518c47e71b330094eaaced604b045619d8582a height=344199 version=0x00000002 log2_work=82.268387 tx=60099865 date=’2015-02-19T11:13:51Z’ progress=0.158570 cache=304.0MiB(2066641txo) 2019-03-16T21:22:52Z Leaving block file 232: CBlockFileInfo(blocks=363, size=133839871, heights=343644…344280, time=2015-02-15…2015-02-19) 2019-03-16T21:22:52Z Pre-allocating up to position 0x1000000 in blk00233.dat 2019-03-16T21:22:59Z Pre-allocating up to position 0x100000 in rev00233.dat 2019-03-16T21:22:59Z UpdateTip: new best=0000000000000000024497df68e3d8d38557a5af53b737ce509257d59fbdace3 height=344200 version=0x00000002 log2_work=82.268434 tx=60100126 date=’2015-02-19T11:18:23Z’ progress=0.158571 cache=304.1MiB(2067320txo) 2019-03-16T21:22:59Z UpdateTip: new best=0000000000000000043fde47bc57a0446dc2b8d498c219e1a7f0f02300b24bf9 height=344201 version=0x00000002 log2_work=82.268481 tx=60100141 date=’2015-02-19T11:18:46Z’ progress=0.158571 cache=304.1MiB(2067324txo) 2019-03-16T21:22:59Z Pre-allocating up to position 0x1100000 in rev00232.dat

## What would be a good radiation homebrew system?

I am creating a campaign with a nuclear power plant. As the players get closer to it, it will deal more radiation damage. But I don’t have a balanced system for radiation damage. What I’ve got so far is CON checks every so often, becoming more often and increasing DCs. But this doesn’t account for rests and increases in challenge rating for encounters (to make it harder, the monsters will have evolved to not be affected). I’ve created a new type of damage to deal with it (radiation damage). This is on account of the characters will receive radiation suits. I’m working on a by distance system, with the DCs and timing of checks heavily decreasing when blocked by something.

## Where does the responsibility for clearing higher-level caches lie in a pyrimidal calculation system with caching at multiple levels?

I’m working on system that analyses non-trivial amounts of data. The analysis is pyramidal, in that various combine to make intermediate values, which in turn combine with each other and more inputs to make higher-order intermediate values, and so on, until the value is produced. As a simple example: inputs A & B combine to make intermediate product X, inputs A & C combine to make intermediate product Y, inputs C & D combine to make intermediate product Z, and then intermediate products X, Y, and Z are combined with input E to make the final result. See below for a crude diagram representing this example relationship.

Nearly all of these steps are expensive enough to have user-noticeable performance costs, both retrieving the inputs and calculating the intermediate products. On the plus side, all of the inputs change only when requested by the user, so their values can be cached and reused, and the system can be specifically notified when one of the inputs has changed. Thus worrying about when to invalidate caches isn’t a problem.

The problem I’m struggling with is where the responsibility lies for clearing the caches for the intermediate products; after all, if input A changes in the example above, we need to recalculate X and Y, but we can keep the cached values of Z, as well as B, C, D, and E.

I’m using the Event Aggregator pattern, but I can’t decide what event the cache for, say, Y should use to trigger its invalidation. I can see a couple options:

1. The cache for Y can subscribe to both the A-data-changed and the C-data-changed events and invalidate when it receives either of them. This requires Y to know that it depends on A & C, and means that higher-level caches have to subscribe to more and more data-changed events to cover all their dependencies (and their dependencies’ dependencies, and so on).
2. The caches for A & C could send a Y-data-changed event whenever they receive their own data-change events. This would require them to know that Y depends on them, and would also require them to send data-changed events for all the downstream products that depend on them. Additionally, it would generate unnecessary duplicate change events for higher-level products (as multiple dependencies all try to pass on the message).
3. A separate dependency manager / event translator could keep track of all the dependencies and know that, e.g. if it sees a C-data-changed event it needs to send Y-data-changed and Z-data-changed events, as well as events for anything that depends on Y or Z, all the way up the pyramid.

I’m leaning towards going with #3 and putting the new manager with/near the DI root. Are there advantages/disadvantages to these approaches I’ve overlooked? Or alternate approaches, at that?

## Blacked out GUIs after logging in, can’t use the system properly

After logging in, many elements of the GUI go black.

enter image description here

In the above image you’ll see that the terminal is just a black box on the right. The black bar just above the unity launcher is the onscreen keyboard which is still responsive, even though it’s blackened, which is true of all apps; I can close the terminal (which in the picture is the black box on the right) by randomly clicking in the upper-left corner. The same happens even for drop-downs; clicking on the wifi or the power logo on the menu bar on the top just gives me a black box.

This problem used to happen before and it would be fixed with just a simple restart. Now a restart wont’t work. As you can see, I can’t use the terminal, and I cant access a lot of things since the GUI will just be blacked out most of the time. I can’t use my system like this. Please help.