I just can't solve this problem in spite of doing a whole book on logs and inequalities     Where $[\dotsc]$ denotes the greatest integer function, what is the value of the natural number $n$ satisfying the equation   $$   [\log_21]+[\log_22]+[\log_23]+\dotsb+[\log_2n] = 1538  $$   A. 255 B. 256 C. 254 D. 313  The answer is A.  Please give an easy solution as I am a high school student. And I do fully understand the meaning of the question but I can't solve it.  Since stack exchange forces me to provide an attempt here it goes- I wrote the equation as $log_2n!=1538$  after this I feel n! should be $2^{1538}  $  and now i am confused...   I hope you got your attempt for the people who put the question on hold.  nice you pointed out chris.. yeah the question is same but the person who posted it has accepted the answer to it not me!....the answer maybe sufficient to understand for him.. but it may not be for me...  PS-whats the point of giving a fake attempt and not speaking the truth?!!
If $[\cdot]$ denotes greatest integer function, then what is the value of natural number $n$ satisfying the equation $$[\log_21]+[\log_22]+[\log_23]+\dots[\log_2n]=1538 ?$$   My try:  Note that  $$0+1\times2+2\times2^2+3\times2^3+4\times2^4+5\times2^5+6\times2^6+7\times2^7=1538$$  But the answer is $$n=\sum_{i=0}^72^i=255.$$  How is it derived?
Can a dual national visit USA under VWP after visa refusal for other nationality? Recently I was refused an E-2 visa under Section 214b. To be honest I don't really know why because I have strong ties with my home country (house, bank account, family) so my lawyer sent an appeal to the legalnet and I'm  waiting for an answer.  I have a second nationality that I used to travel in USA within the last 15 months under ESTA. A few days after my E-2 refusal I checked my ESTA status  and it is still authorized.  Can I travel to the States right now or not?  I have a purchased business over there and have to take care of my assets. How to get time difference in minutes in PHP How to calculate minute difference between two date-times in PHP? Is the CMB rest frame special? Where does it come from? It seems that we are . This is the speed of our galaxy relative to the cosmic microwave background. Where does this rest frame come from? Is this special in any way (i.e., an absolute frame)? EDIT: I think the more important question is &quot;where does the CMB rest frame come from?&quot;. Is 'efficate' a word in English? I routinely hear the word "efficate" being used.  For example, "The most powerful way to efficate a change in the system is to participate."  I do not find entries for this word in common English dictionaries, but I do not have an unabridged dictionary.  I have checked the OED (I'm not sure if it is considered unabridged), and it has no entry for "efficate".  It does have an entry for "efficiate", which is used in the same way.  Wordnik has an  for "efficate" with over 1800 hits, thus providing some evidence for the frequency of use.  I personally like the word and find the meaning very clear and obvious when others use it.  If it's not currently an "officially documented" word, perhaps its continued use will result in it being better documented. This is a Putnam Problem that I have been trying to solve (on and off) for two years, but I have failed. I am in Calculus BC. This problem comes from the book "Calculus Eighth Edition by Larson, Hostetler, and Edwards". This problem is at the end of the first section of the chapter 8 exercises. Here's the problem:  Evaluate $$\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)} + \sqrt{\ln(x+3)}}.$$  Please. Any help is very much appreciated. So are solutions. Thank you!  Edit: I like the solution given, but I was interested to see if there is any other way of doing the problem? I'm excited to see the results. If $D$ is critical dimension of Bosonic strings, a particular derivation goes like the following, where we arrive finally at  $$ \frac{D-2}{2}\sum_{n=1}^\infty n + 1 = 0. $$ Now mathematically this is clearly a divergent series, but using zeta function regularization here we are taking  $$ \sum_{n=1}^\infty n = \zeta(-1) = -\frac{1}{12}. $$ And obtain $ D = 26 $ where $\zeta $ is the analytic continuation of the zeta function we know. But it makes no sense in putting $ s = -1 $ in the formulae $$ \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}. $$ As the above is only valid for $ Re(s) &gt; 1 $. So what is going on in here? Can anyone give me a reasonable explanation about obtaining $ -1/12 $? I'm writing a script and I would like to pass the results from bc into a variable. I've declared 2 variables (var1 and var2) and have given them values.  In my script I want to pass the results from bc into another variable say var3 so that I can work with var3 for other calculations.  So far I have been able write the result to a file which is not what I'm looking for and also I've been able to echo the result in the terminal but I just want to pass the result to a variable at moment so that I can work with that variable.  echo "scale=2;$var1/var2" | bc Deriving the Lagrangian for a free particle I'm a newbie in physics. Sorry, if the following questions are dumb. I began reading &quot;Mechanics&quot; by Landau and Lifshitz recently and hit a few roadblocks right away.  Proving that a free particle moves with a constant velocity in an inertial frame of reference ($\S$3. Galileo's relativity principle). The proof begins with explaining that the Lagrangian must only depend on the speed of the particle ($v^2={\bf v}^2$): $$L=L(v^2).$$ Hence the Lagrance's equations will be $$\frac{d}{dt}\left(\frac{\partial L}{\partial {\bf v}}\right)=0,$$ so $$\frac{\partial L}{\partial {\bf v}}=\text{constant}.$$ And this is where the authors say   Since $\partial L/\partial \bf v$ is a function of the velocity only, it follows that $${\bf v}=\text{constant}.$$  Why so? I can put $L=\|{\bf v}\|=\sqrt{v^2_x+v^2_y+v^2_z}$. Then $$\frac{\partial L}{\partial {\bf v}}=\frac{2}{\sqrt{v^2_x+v^2_y+v^2_z}}\begin{pmatrix} v_x \\ v_y \\ v_z \end{pmatrix},$$ which will remain a constant vector $\begin{pmatrix} 2 \\ 0 \\ 0 \end{pmatrix}$ as the particle moves with an arbitrary non-constant positive $v_x$ and $v_y=v_z=0$. Where am I wrong here? If I am, how does one prove the quoted statement?  Proving that $L=\frac{m v^2}2$ ($\S$4. The Lagrangian for a free particle). The authors consider an inertial frame of reference $K$ moving with a velocity ${\bf\epsilon}$ relative to another frame of reference $K'$, so ${\bf v'=v+\epsilon}$. Here is what troubles me:   Since the equations of motion must have same form in every frame, the Lagrangian $L(v^2)$ must be converted by this transformation into a function $L'$ which differs from $L(v^2)$, if at all, only by the total time derivative of a function of coordinates and time (see the end of $\S$2).  First of all, what does same form mean? I think the equations should be the same, but if I'm right, why wouldn't the authors write so? Second, it was shown in $\S$2 that adding a total derivative will not change the equations. There was nothing about total derivatives of time and coordinates being the only functions, adding which does not change the equations (or their form, whatever it means). Where am I wrong now? If I'm not, how does one prove the quoted statement and why haven't the authors done it? How do you compute negative numbers to fractional powers? My teachers have gone over rules for dealing with fractional exponents. I was just wondering how someone would compute say: $$(-5)^{2/3}$$ I have tried a couple ways to simplify this and I am not sure if the number stays negative or turns into a positive. I know that if a negative number is raised to an odd power it is negative, but fractional powers are neither odd or even. Is there a general rule for dealing with these types of problems? I'm looking for a symbol for skeptical consequence so that I can write that something, B, is a skeptical consequence of something else, A.  In a book I read it's written like \(A \vdash B\) or \(A \sststile{}{} B\) but the horizontal line is wavy like \sim, has the same width as the vertical line and does not vary in width. So, what I'm looking for is a binary relation symbol that is similar to the combination of the first two and the last relation in the following image:    That is, something similar to the following mockup but the wavy line should be such that it has a constant width and the same width as the vertical line:    I've tried  without success. Is there any way to find out if a file is a directory?  I have the file name in a variable. In Perl I can do this:  if(-d $var) { print "it's a directory\n" } We have been using this result without proof in my class, but I don't know how to prove it. Could someone point me in the right direction?  $$|1-e^{i\theta}|\le|\theta|$$  I believe this is true for all $\theta\in\mathbb{R}$. It is easy to show that the left side is bounded by 2 (triangle inequality), but I'm stuck otherwise. I do not have enough time to bake my bread.  Can I refrigerate it or let it sit on the counter and bake it later in the day? Let $K$ be any field. Is it true that:   If $f_1,\dots, f_k \in K[x_1,\dots,x_n]$ are polynomials of degree $1$, then $I=(f_1,\dots, f_k)$ is all $K[x_1,\dots,x_n]$, or, if it is a proper ideal, then is it prime? In particular, is it true that $K[x_1,\dots,x_n]/I$ is isomorphic to a polynomial ring (in the case that $I$ is proper and hence prime by the statement above)?   If they are true, how can i prove these statements in an elementary but rigorous way?  P.S.: Birth of my question: I found on my book that "affine varieties given by linear equations are irreducibile", so I translated (I don't know if I have made it right) this geometric statement into the analogous algebraic version. What if the universe is rotating as a whole? How to handle relative urls correctly with a nginx reverse proxy Sure I'm not the first one that tried to serve a domain example.com from a example.net/bbb, but I haven't found a solution yet.  My NGINX configuration follows the  and looks something like this:  server {     listen 80;     server_name example.net;     root /path/to/aaa;      location /bbb/ {         proxy_pass http://example.com/;         proxy_set_header Host $host;         proxy_set_header X-Real-IP $remote_addr;     }      location / {         try_files $uri $uri/ /index.html;     }     location ~ \.(svg|ttf|js|css|svgz|eot|otf|woff|jpg|jpeg|gif|png|ico)$ {         access_log off;         log_not_found off;         expires max;     } }   I can manage to render the root of example.com in example.net/bbb but:  ISSUE 1  example.net/bbb/some/path doesn't work as expected and the index.html of example.net is rendered.  ISSUE 2  Any asset in example.com/assets gives 404 because the browser look for example.net/assets. Be great if I could solve this without placing absolute paths everywhere. It would be nice to be able to put emphasis on specific bits of a fragment or of a quote (bold can help for a quote but I find that it makes things less readable sometimes). What do you think? Update: On , you can highlight particular lines by prefixing each line with @@. See also: Name of short sci-fi story involving a musician who is punished for listening to Bach I read a short story a long time ago and I can neither remember the title nor the author.  The main character is a musician.  He is sequestered and allowed to play all day on some kind of a fantastic instrument; however, he has been barred from listening to any music for fear that it will stifle with his innate creativity.  Someone secretes him some Bach, he listens to it, and is then caught and cast out.  He gets a series of jobs and is inevitably punished for breaking some music-related taboo.  Any ideas? Let $R$ be a commutative ring and $A=\{1,a,a^2,\dots\}$ for some $a\in R$. Prove that $A^{-1}R$ is isomorphic to $R[T]/(aT-1)$.    I guess I'm meant to find a surjective homomorphism between $A^{-1}R$ and $R[T]$ and then use first isomorphism theorem. What homomorphism should I use? options for [for /f "usebackq" %%f in (`find "" path`) do] sentence in batch I symplified my problem: @echo off set res=  for /f &quot;usebackq&quot; %%f in (`find &quot;a&quot; list.txt`) do set res=%%f echo %res%  pause&gt;nul and that things.txt file in the code is like:  list.txt file in the code is: apple pitch pear melon mango  As you expect, the result of this code is mango  because the line 'mango' is located below the line 'apple' and 'pear' But the result I want is apple  or pear  Specifically, I want to put each of the other values above mango such as apple and pear into the variable res using option keys provided. is there any way to do that?