I take photos with my phone (a Galaxy S9) frequently and this problem seems to plague me every time I try taking photos indoors near a window during daytime. The light from the window which is above the table seems to be leaking into the image and it messes with the white balance very badly, along with producing a photo that looks like it was taken with a smudged lens. The first image below is the photo taken on auto mode normally. The second one is similar but I used my finger to cover the top of the lens where the light is leaking in. I put my finger as close as possible to the lens without having it show up in the image. As you can see, the second photo came out much better, without any sunlight leaking in from the top. As a result, the colors are so much better in the second image compared to the first. Is this a problem with my lens, my photography skills, or is it something else?
Someone mentioned that "veiling glare" or "veiling flare" was to blame for reduced contrast in a photo. What does this mean exactly?  What causes veiling glare, and how can it be avoided? How does it relate to those floating polygonal lens flares one often sees in movies?
Calculate sums of inverses of binomial coefficients How to calculate the sum of sequence $$\frac{1}{\binom{n}{1}}+\frac{1}{\binom{n}{2}}+\frac{1}{\binom{n}{3}}+\cdots+\frac{1}{\binom{n}{n}}=？$$ How about its limit? Proof that the relation $5 \mid (a + 4b)$ is symmetric and transitive Take the relation $R$ to be defined on the set of integers:  $$aRb \iff 5 \mid (a + 4b)$$  As part of a larger proof, I have to show that $R$ is both symmetric and transitive. I'm lost.  I see the first steps, but I can't find how to progress further. Here's what I have at this point:    Proof of Symmetry  We have to prove that if $5 \mid (a + 4b)$, then $5 \mid (b + 4a)$. Clearly, this is true if $a = b$, but apart from that, it's unclear in my mind.  Proof of Transitivity  We have to prove that if $5 \mid (a + 4x)$ and $5 \mid (x + 4b)$, then $5 \mid (a + 4b)$.    I've fiddled around with sample values, but I still don't see it. I'm pretty lost here. Thoughts? Executing multiple commands from a Windows cmd script I'm trying to write a Windows cmd script to perform several tasks in series. However, it always stops after the first command in the script.  The command it stops after is a maven build (not sure if that's relevant).  How do I make it carry on and run each task in turn please?  Installing any software or configuring the registry etc is completely out of the question - it has to work on a vanilla Windows XP installation I'm afraid.  Ideally I'd like the script to abort if any of the commands failed, but that's a "nice to have", not essential.  Thanks. Ubuntu 19.10 change desktop environment How can I always run the command prompt as administrator? Should Indicate which Positions I've Already Applied For Pattern in Isomorphisms between $\mathbb Z_m \times \mathbb Z_n$ and $\mathbb Z_{mn}$ In general, the group $\mathbb Z_{mn}$ under addition $\bmod mn$ is isomorphic to the direct product $\mathbb Z_m \times \mathbb Z_n$ if and only if $\gcd(m,n)=1.$ This is easy to see since $\mathbb Z_{mn}$ is generated by the element $(1,1)$, and if $\gcd(m,n)=1$, then this will only be zero if it is multiplied by itself $mn$ times (resulting in the cyclic group).   However I'm interested in constructing an isomorphism between the two groups as an explicit expression. I've been playing around a bit with some examples, one of which is the function $\phi : \mathbb Z_2 \times \mathbb Z_3 \to \mathbb Z_6$ where $\phi(a,b) = 3a + 4b \pmod 6$, which gives us the desired isomorphism. I'm interested here in the relationship between the integers $2$, $3$, and the coefficients $3, 4$ of $a$ and $b$. After playing around in Wolfram|Mathematica, I managed to obtain the following isomorphisms $\phi_{m,n} : \mathbb Z_m \times \mathbb Z_n \to \mathbb Z_{mn}$ for coprime $m,n$ in $\{1, 2, \dots, 10\}$.   First of all, the following results seem to hold in general.  $$\phi_{1, n}(a, b) = a + b \qquad \text{for all $n$}$$  $$\phi_{m, 1}(a, b) = a + b \qquad \text{for all $m$}$$  $$\phi_{2, n}(a, b) = na + (n + 1)b \qquad \text{for all $n$}$$ $$\phi_{m, n}(a, b) = K a + L b \iff \phi_{n,m}(a, b) = L a + K b \qquad \text{for all $m, n$}$$  Beyond this point, I could not spot a general pattern. The isomorphisms I obtained are:  $$\phi_{3, 4}(a, b) = 4a + 9b$$ $$\phi_{3, 5}(a, b) = 10a + 6b$$ $$\phi_{3, 7}(a, b) = 7a + 15b$$ $$\phi_{3, 8}(a, b) = 16a + 9b$$ $$\phi_{3, 10}(a, b) = 10a + 21b$$ $$\phi_{4, 5}(a, b) = 5a + 16b$$ $$\phi_{4, 7}(a, b) = 21a + 8b$$ $$\phi_{4, 9}(a, b) = 9a + 28b$$ $$\phi_{5, 6}(a, b) = 6a + 25b$$ $$\phi_{5, 7}(a, b) = 21a + 15b$$ $$\phi_{5, 8}(a, b) = 16a + 25b$$ $$\phi_{5, 9}(a, b) = 36a + 10b$$ $$\phi_{6, 7}(a, b) = 7a + 36b$$ $$\phi_{7, 8}(a, b) = 8a + 49b$$ $$\phi_{7, 9}(a, b) = 36a + 28b$$ $$\phi_{7, 10}(a, b) = 50a + 21b$$ $$\phi_{8, 9}(a, b) = 9a + 64b$$ $$\phi_{9, 10}(a, b) = 10a + 81b$$ If you want to try and generate a few yourself, here is the Mathematica function I implemented. I'm sure it can be optimised, but for the first 10 coprime integers it sufficed.  isomorphism[{m_, n_}] :=       Module[{x = Null},       For[i = 1, i &lt; m*n,           i++, (If[#[[2]] == Table[i, {i, 0, m*n - 1}],           x = ToString[#[[1]][[1]]] &lt;&gt; "a+" &lt;&gt; ToString[#[[1]][[2]]] &lt;&gt;          "b"; Break[]] &amp;) /@       Table[{{i, j},       Mod[i #[[1]] + j #[[2]], m*n] &amp; /@       Table[{Mod[i, m], Mod[i, n]}, {i, 0, m*n - 1}]}, {i, 1,      m*n}, {j, 1, m*n}][[i]]]; x];   I'd appreciate if anyone can help me find the pattern to determine $\phi_{m, n}$ in general. error: file '/grub/i386-pc/normal.mod' not found error: file '/grub/i386-pc/normal.mod' not found. grub rescue&gt;   What can I do? I just sit and stare at it.   I found my old netbook (Dell Inspiron 1010) which I have not used for about four years. I replaced Windows XP with Ubuntu 12.10. I used my bootable USB drive. I installed and rebooted. I got the message that normal.mod is not found.   What should I do? Type exit, reboot, or quit? Should I re-install? Understanding the length extension attack I have been trying to understand exactly how a length extension attack works on SHA-1. I'll detail below what I've understood so far, so that I can convey my understanding of the same and hopefully get advice on where I'm going wrong.  Lets assume that for a message $m$, a secret $s$ is appended to it and the SHA-1 hash is calculated. If $\textrm{len}(m) + \textrm{len}(s)$ is not a multiple of the block size, a certain amount of padding is added to it to make it a multiple, and the hash is generated using what is essentially $s || m || \textrm{padding}$.  Now SHA-1 internally uses 5 registers — it takes in an block, generates new register values, turns the crank, takes in another block, generates new register values, turns the crank ... and so on till the end. When all blocks are exhausted, the register values are concatenated and spit out.  Assume that an attacker knows a certain value of $m$, for which he also knows the hashed key. In such a case, he knows the state of the 5 registers when $(s || m || \textrm{padding}$ was processed. Lets also assume that the attacker knows the length of $s$, so that he can predict how much padding was added.  Here is where I get confused. From what I've read so far, the attack involves taking the known hash, adding a certain amount of padding and then the value that we want. However, isn't the padding already used up in creating the hash? Why are we using padding in our attack? Is voltage the speed of electrons? When I try to connect to servers or join up on lan worlds in Minecraft it says "Failed to Login : Invalid session (Try restarting your game)". Multiplayer used to work fine but now keeps coming up with this error. Can anybody help? For about 2 days I'm trying to configure Ubuntu so that I can send an email, but when I type in:  echo "Subject: sendmail test" | sendmail -v my@email.com   I'll get:  postdrop: warning: unable to look up public/pickup: No such file or directory Mail Delivery Status Report will be mailed to &lt;andreigbr&gt;.   Even if I change my@email with mine, the inbox will be empty. Can somebody help me, please? I watched about 5 tutorial about how to send an email fom the command line but still doesn't work. A field that is conservative must have a curl of zero everywhere. However, I was wondering whether the opposite holds for functions continuous everywhere: if the curl is zero, is the field conservative? Can someone please give me an intuitive explanation and insight into this, and if it is true, why? Also, please try to not be too rigorous (only in grade 9). Create bash file to open minecraft launcher I use this command in the terminal to run the minecraft launcher with java 8:  /home/max/Java/jre1.8.0_25/bin/java -jar Minecraft.jar   I included the full java path because I installed it on /home as a normal user and i want to include it.   That command works, it opens the launcher with Java 8 but I want to make a bash file so I don't have to open the terminal. And, most important, I don't want the terminal to remain open when I click play in the Minecraft launcher and the game starts.  Now if i close manually the terminal when the game have started, the game is closed too. Icon missing when application is launched I've created an application launcher in /usr/share/applications that launches rdesktop. The file specifies an icon name and I've added the icons to:   /usr/share/icons/HighContrast/48x48/apps /usr/share/icons/HighContrast/22x22/apps /usr/share/icons/HighContrast/256x256/apps /usr/share/icons/HighContrast/32x32/apps /usr/share/icons/HighContrast/scalable/apps-extra /usr/share/icons/HighContrast/24x24/apps /usr/share/icons/HighContrast/16x16/apps /usr/share/icons/hicolor/scalable/apps /usr/share/icons/hicolor/128x128/apps /usr/share/icons/hicolor/16x16/apps   I then drug the launcher to the taskbar and the launcher with the correct logo is visible in the taskbar. However, when I start the task, a new icon is created in the taskbar, a square with a question mark in the center.  What icons do I need and where, so that the taskbar displays the correct icon for the application when it is launched?  The .desktop file contents:   [Desktop Entry] Version=1.0 Terminal=false Icon=mswindows Type=Application Categories=Network; Exec=rdesktop win7shared1 -g 1680x1000  Name=win7shared1 GenericName=rdesktop Comment=Open windows desktop  Keywords=Windows StartupNotify=false What is the JavaScript >>> operator and how do you use it? When I open the file in Vim, I see strange ^M characters.  Unfortunately, the world's favorite search engine does not do well with special characters in queries, so I'm asking here:   What is this ^M character? How could it have got there? How do I get rid of it? How are Anonymous inner classes used in Java? Does anyone know a proof from the first principles that a nilpotent matrix has zero trace. No eigenvalues, no characteristic polynomials, just definition and basic facts about bases and matrices. A composition of a periodic function that is somehow not periodic.