How do I self-study String Theory? I am an absolute beginner, with little to no knowledge on the subject but I have somehow developed a strong interest in theories like String theory and M-theory, by reading about them on other forums. So can you   recommend me any farily accessible books or other materials that could help me understand these theories in-depth? Or do I have to extensively study other areas in physics before I am able to comprehend them, if so, what are those other topics?
Introduction to string theory I am in the last year of MSc. and would like to read string theory. I have the Zwiebach Book, but along with it what other advanced book can be followed, which can be a complimentary to Zwiebach. I would like a more mathematically rigorous book or lecture notes along with Zwiebach.   Specifically, mention whether the book discusses string theory   Rigorously? Intuitively?   What's the scope of the book? Does it cover the advanced materials, e.g. Matrix string theory, F-theory, string field theory, etc. Maybe even String Phenomenology?
An unfair three-sided die. An unfair three-sided die. Side $1$ will be rolled with probability $p_1$ , Side $2$ with probability $p_ 2$ and Side $3$ with probability $p_ 3$ . Find a formula for the probability that the $8^{th}$ $3$ will be rolled on the $10^{th}$ roll.  Attempt:  First 9 rolls there are 7 three's.  $$[\binom{9}{7}(p_3)^7(p_1+p_2)^2]\cdot p_3$$ $$=[84\cdot(p_3)^7(p_1+p_2)^2]\cdot p_3$$  any thoughts of this solution? How to transform negative values to logarithms? I would like to know how to transform negative values to Log(), since I have heteroskedastic data. I read that it works with the formula Log(x+1) but this doesn't work with my database and I continue getting NaNs as result. E.g. I get this Warning message (I didn't put my complete database because I think with one of my negative values is enough to show an example):   &gt; log(-1.27+1) [1] NaN Warning message: In log(-1.27 + 1) : NaNs produced &gt;    Thanks in advance  UPDATE:  Here is an histogram of my data. I'm working with palaeontological time series of chemical measurements, E.g the difference between variables like Ca and Zn is too big, then I need some type of data standardization, that is why I'm testing the log() function.   This is my I did some mathematical induction problems on divisibility   $9^n$ $-$ $2^n$ is divisible by 7. $4^n$ $-$ $1$ is divisible by 3. $9^n$ $-$ $4^n$ is divisible by 5.   Can these be generalized as $a^n$ $-$ $b^n$$ = (a-b)N$, where N is an integer? But why is $a^n$ $-$ $b^n$$ = (a-b)N$ ?  I also see that $6^n$ $- 5n + 4$ is divisible by $5$ which is $6-5+4$ and $7^n$$+3n + 8$ is divisible by $9$ which is $7+3+8=18=9\cdot2$.  Are they just a coincidence or is there a theory behind?  Is it about modular arithmetic? Selection of logical connectives {¬,∨,∧,⇒,⇔} in set theory? Operator precedence in a find command? See below example:  $ ls -1p appledir/ applefile orangedir/ orangefile $ find . -type f -name apple\* ./applefile $ find . -type f -name orange\* ./orangefile $ find . -type f -name apple\* -o -name orange\* ./applefile ./orangedir ./orangefile $ find . -type f \( -name apple\* -o -name orange\* \) ./applefile ./orangefile $    I was surprised to discover I needed the parentheses for this to work as expected; apparently I haven't internalized the rule of precedence by which find evaluates its arguments.  How can I easily predict when I will and when I will not need to use parentheses to explicitly group find primaries?  Put another way, what are the rules by which I can imagine find inserting parentheses into the commands I give it, which will allow me to accurately predict how it will evaluate ungrouped expressions? Im trying to understand the solution about this, and all the other solutions are.. a bit weird for me and dont answer all my questions!  Reference :   Im using the answer there as reference, but my understanding is a bit different!     Find the error in the following proof that all horses are the same   color. CLAIM: In any set of h horses, all horses are the same color.      PROOF: By induction on h.      Basis: For h = 1. In any set containing just one horse, all horses   clearly are the same color. Induction step: For k > I assume that the   claim is true for h = k and prove that it is true for h = k + 1. Take   any set H of k + 1 horses. We show that all the horses in this set are   the same color. Remove one horse from this set to obtain the set H1   with just k horses. By the induction hypothesis, all the horses in H,   are the same color. Now replace the removed horse and remove a   different one to obtain the set H2 . By the same argument, all the   horses in H2 are the same color. Therefore all the horses in H must be   the same color, and the proof is complete.   Here's what i have come up with :  Suppose a set of 1 horse. $H_a = { horse_{1} }$  Obviously, all the horses in set $H_a$ are the same, since theres just 1 horse.  So using that as the Basis of the induction we can proceed and try to prove its true for $N$ horses.  -- I dont fully understand the part below   As an example : set $H_0$ :   $H_0 = { horse_{1}, horse_{2}, horse_{3}... horse_{n} }$  In my understanding, its because, I can make a subset  formed by 1 horse :  $K_1  = {horse_{1}}$ $K_2  = {horse_{2}}$ ...  I don't get it. I clearly know that this proof is a lie.  And for me, its because you cant claim theres a relation on a set with size $&gt;= 2$ , since :  $H_1 = { horse_{1}, horse_{2}}$  Theres nothing that says $horse_{1}$ equals $horse_{2}$. The basis was for one horse. But that alone is not a proof.  So what should I do ? Is there flair for Careers.SO? Are there any image badges like those on  for Careers.SO?         I would like to add a nice image to my website profile. I was trying to dual boot elementary os with windows on my laptop (Asus Zenbook UX305). To do that I shrank the windows partition using the windows disk manager, and then used the Elementary installer's "install alongside windows" option to set up the linux partitions automatically. When I rebooted after installing I was put into the grub console. I was able to boot linux manually with the commands  grub&gt; set root=(hd0,gpt4) grub&gt; linux /boot/vmlinuz-3.19.0-26-generic root=/dev/sda4 grub&gt; initrd /boot/initrd.img-3.19.0-26-generic grub&gt; boot     and I found that running the command  grub&gt; configfile (hd0,gpt4)/boot/grub/grub.cfg   would load the normal grub menu as is supposed to happen - so there don't seem to be any problems with the config file itself. I also found that the root and prefix are on (hd0,gpt1), so running  grub&gt; set root=(hd0,gpt4) grub&gt; set prefix=(hd0,gpt4)/boot/grub grub&gt; insmod normal grub&gt; normal   also loaded the normal grub menu. So possibly the problem is that grub is looking on the wrong partition?  In my BIOS, I have a choice between using grub and the windows bootloader to boot. Selecting the windows bootloader boots me directly into windows without any problems.  However, I can't find any way to get any fixes to stick after a reboot. I've tried rerunning grub-install /dev/sda, recreating the config file with grub-mkconfig, and using the Ubuntu boot repair gui tool - none of these have worked. Is there any way to fix this short of reinstalling either or both OSs? I'd rather not have to reinstall windows if possible. Can anyone tell me how Google decides which image to use when it shows an image with the search results (as is the case with mobile search, sometimes)?  If you look at these mobile search results …    … and take the AliExpress.com result as an example, by inspecting  you can see that  used by Google seems to have no special declaration, etc. It is just the first JPG that appears in that page's HTML - so I was thinking maybe that's why it's the image that is used in the Google search results.  However, when you look at these mobile search results …    … and consider the Backpack Billboards result,  reveals that  being used is actually the seventh image declared in the HTML.  So can anyone shed any light on how Google determines which image to be displayed? I would like to know so I can then have control over which image is displayed for my site.  Update / Solution  Thanks to the answers, below, this is the code I have now added to my head HTML in order to get the thumbnail shown in Google when searching for backpack billboards:  &lt;script type="application/ld+json"&gt; {   "@context": "http://schema.org/",   "@type": "Product",   "name": "Regular Backpack Billboard",   "image": [     "https://backpackbillboards.com/wp-content/uploads/2017/12/backpack-billboards-1x1.jpg",     "https://backpackbillboards.com/wp-content/uploads/2017/12/backpack-billboards-4x3.jpg",     "https://backpackbillboards.com/wp-content/uploads/2017/12/backpack-billboards-16x9.jpg"    ],   "description": "Backpack Billboards feature a rechargeable battery which powers the internal LED lights providing a bright illuminated display for up to 8 hours.",   "sku": "BPBB8",   "brand": {     "@type": "Thing",     "name": "Backpack Billboards"   },   "offers": {     "@type": "Offer",     "priceCurrency": "USD",     "price": "299.00",     "priceValidUntil": "2020-11-11",     "itemCondition": "http://schema.org/NewCondition",     "availability": "http://schema.org/InStock",     "seller": {       "@type": "Organization",       "name": "Backpack Billboards"     }   } } &lt;/script&gt; css class definition with multiple identifiers Factorial of 0 - a convenience? If I am correct in stating that a factorial of a number ( of entities ) is the number of ways in which those entities can be arranged, then my question is as simple as asking - how do you conceive the idea of arranging nothing ?   Its easy to conceive of a null element in the context of arrays, for example - so you say that there is only one way to present a null element.   But, in layman terms - if there are three humans $h1, h2, h3$ that need to be arranged to sit on three chairs $c1, c2, c3$ - then how do you conceive of a) a null human, and b) to arrange those ( that ? ) null humans ( human ? ) on the three chairs ?   Please note that referral to humans is just for easy explanation - not trying to be pseudo-philosophical. Three balls to be arranged on three corners of a triangle works just fine. So basically, how do you conceive of an object that doesn't exist, and then conceive of arranging that object ?  So, in essence ... is $0! = 1$, a convenience for mathematicians ? Not that its the only convenience, but just asking. Of course, there are many.  If yes, then its a pity that I can't find it stated like so anywhere.  If not, then can anybody suggest resources to read actual, good proofs ? New profile pages are out - bugs and feedback master list How to treat English questions which have code in a foreign language? What does the symbol $\subset\subset$ mean? In some texts (mainly complex analysis or harmonic analysis) I sometimes see the following double subset symbol $\subset\subset$ for inclusion relation of two regions, e.g., $\Omega$ and $\Omega'$ are two regions in $\mathbb{C}$ such that $\Omega \subset\subset \Omega'$. I never figured out what it means exactly; I always interpreted it as the closure $\overline{\Omega}$ is contained in $\Omega'$ (so that some nasty boundary effects can be avoided). Is that right? Or does $\subset\subset$ mean some other kind of inclusion? Thanks. Schengen Visa Refusal: Justification for the purpose and conditions of the intended stay was not provided Schengen refusal formulae can be difficult to understand sometimes.    When an applicant receives a refusal with "Justification for the purpose and conditions of the intended stay was not provided", what does it mean? Propagate all arguments in a bash shell script Increase the highlighted button bezel thickness (GNOME) Prove that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$ Assume $f:A\to A'$ is a function, $B\subset A'$, $C\subset A'$, and $f^{-1}(B\cap C)=\emptyset$  How can we see that $f^{-1}(B)\cap f^{-1}(C)=\emptyset$? Full Disk Encryption using Ubuntu Core\Snappy I'm not sure if this is the right place to ask , but anyway :   I'm running ubuntu-core (snappy) on Raspberry Pi 3 and I'm willing  to perform partition\full disk encryption using cli\code. is it possible?  thanks