\textit{x_{min}} have error?
Superscript outside math mode
Which is faster in Python: x**.5 or math.sqrt(x)? I've been wondering this for some time. As the title say, which is faster, the actual function or simply raising to the half power?  UPDATE  This is not a matter of premature optimization. This is simply a question of how the underlying code actually works. What is the theory of how Python code works?  I sent Guido van Rossum an email cause I really wanted to know the differences in these methods.  My email:     There are at least 3 ways to do a square root in Python: math.sqrt, the   '**' operator and pow(x,.5).  I'm just curious as to the differences in   the implementation of each of these.  When it comes to efficiency which   is better?   His response:     pow and ** are equivalent; math.sqrt doesn't work for complex numbers,   and links to the C  sqrt() function. As to which one is   faster, I have no idea... My case here is that I was writing something for school, and my teacher explained an error which I really can't see. She referred to articles about &quot;dangling participles&quot;, which was previously unknown to me. I find it hard to relate the examples I've found about it to my situation. So this is a text I've written and the bold part is where the problem arises: (You could read only the bold part if you please)  Wandering, and lost in his thoughts, there is a deep despair occupying the mind of this man. He is barely aware of his surroundings, except for those few moments when the loud blissful voices break through the protective barriers, developed by his immense sadness and grave frustration, into his mind. ”How can they all possibly be so happy?” he thinks to himself. Staring blindly, with his head low, as he walks in a slow pace, whispering ”Mary, Mary, my Mary,” over and over again. Yesterday was her funeral. She had passed away from a severe type of cancer, at a relatively young age. It was his wife. They had been together since high school and were greatly attached to each other. Having spent nearly half his life by her side, we cannot but imagine his grief. Indeed, he could hardly picture a life without her. So, alone he roams in the midst of all the seemingly incessant cheering and noises, drowned by his sorrow.  Her comments on that part was:  we is slightly out of place here – see my comment below These few sentences are about him and his emotions, so ‘we’ seems out of place. If the sentence starts: ‘Having spent nearly half his life by her side’, the reader expects it to continue: ‘he’. After all we have not spent half our lives by her side.  Is she right? If so, could anyone try to explain why? I'm trouble with exercise 1.8 of Carroll's Space-Time and Geometry:     If $\partial_\nu T^{\mu \nu} = Q^\mu$, what physically does the spatial vector $Q^i$ represent? Use the dust energy momentum tensor to make your case.    First of all, since the momentum energy tensor of dust $T^{\mu \nu}=\rho U^\mu U^\nu$ is symmetric, one has the energy-momentum conservation law $$ \partial_\nu T^{\mu \nu} = \partial_\nu T^{\nu \mu}  =0.$$ A vector $Q$ is constant in any frame related by a Lorentz transformation, while this isn't the case for the components $Q^\mu$. But here $Q=0$ and hence $Q^\mu=0$ also for every component $\mu =0,\cdots,4$?   Even if a straightforwardly apply the equation: $$ Q^i = \partial_\nu T^{i \nu} = (\partial_\nu \rho) U^i U^\nu + \rho(\partial_\nu U^i)  U^\nu + \rho U^i(\partial_\nu  U^\nu ),$$ doesn't seems to yield something...   Is my first reasoning right ? Otherwise: can someone give me a hint ?  Thanks Find the order of $8/9, 14/5,48/28$ in the additive group of $\mathbb{Q}/\mathbb{Z}$ Find the order of $8/9, 14/5,48/28$ in the additive group of $\mathbb{Q}/\mathbb{Z}$    JUNK  Guessing asking for $8/9 * k = 0 $ (being a whole integer) for smallest integer so for this one is 9 . another guess would be the smallest element of $Q/Z$ being something like $9/8$  $14/5$ would be 5  $48/28=12/7$ would be 7? What Does it Really Mean to Have Different Kinds of Infinities? E: Unable to correct problems, you have held broken packages Old-style/Antique typesetting in LaTeX/TeX I'm  having some trouble editing the content of a Managed Email Template, I can quite easily modify the content of the Email Template, but when I wish to refer to an unmanaged component, like this:  &lt;messaging:attachment renderAs="PDF" filename="invoice.pdf"&gt;     &lt;c:MyComponent/&gt; &lt;/messaging:attachment&gt;   I get the error message:     Error: Unknown component c:mycomponent   Does anyone know if its possible to embed an unmanaged component in a managed Email Template? Scale Image Using PHP and Maintaining Aspect Ratio If a subsequence of a Cauchy sequence converges, then the whole sequence converges. Let $(X,d)$ be a metric space, and say $(x_n)$ is a Cauchy sequence such that it has a convergent subsequence $(x_{n_k})$ that converges to $x$. We show $x_n \to x$. Let $\epsilon &gt; 0$. Take $N &gt;0$ such that for all $n,m &gt; N$, we have   $$d(x_n,x_m) &lt; \frac{\epsilon}{2}.$$  By hypothesis, we can take also $K &gt;0$ such that for all $n_k &gt; K$, we have  $$ d(x_{n_k},x) &lt; \frac{\epsilon}{2}.$$  Put $M = \max \{N,K\}$. Therefore, for all $n,m,n_k &gt; M$, we have  $$ d(x_n,x) \leq d(x_n, x_{n_k}) + d( x_{n_k},x) &lt; \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.$$  Hence, $x_n \to x$ as desired.  Is this a correct approach? Thank you very much in advance. Print out elements from an array with a comma between elements except the last word If ${n \choose 3} + {n+3-1 \choose 3} = (n)_3$, compute $n$.  Just making sure I am on the right track...  Do I expand each piece out and solve algebraically? I'm reading Galois Theory by Steven H. Weintraub (second edition), and finding that I'm at least somewhat short on the prerequisites. However the following proof looks wrong to me - am I misunderstanding something, or is it actually an incorrect proof?  Lemma 2.2.3. Let $F$ be a field and $R$ an integral domain that is a finite-dimensional $F$-vector space. Then $R$ is a field. Proof. We need to show that any nonzero $r \in R$ has an inverse. Consider $\{1, r, r^2, \cdots\}$. This is an infinite set of elements of $R$, and by hypothesis $R$ is finite dimensional as an $F$-vector space, so this set is linearly dependent. Hence $\sum_{i=0}^n{c_i r^i} = 0$ for some $n$ and some $c_i \in F$ not all zero.  It then goes on to show, given the above, that we can derive an inverse for $r$. However, if I consider examples like $r = 2 \in Q[\sqrt{2}]$, $r = \sqrt{2} \in Q[\sqrt{2}]$ or $r = 2 \in Q[X]/{&lt;X^2&gt;}$, the set $\{1, r, r^2, ...\}$ doesn't look linearly dependent to me. I do believe the lemma is true (and might even be able to prove it), but this does not look like a correct proof to me. Am I missing something? [Edit] Well yes, I am. Somehow I had managed to discount the possibility of any $c_i$ being negative, despite repeatedly looking at each fragment of the quoted text in an attempt to find what I might be misunderstanding. $$\int^2_{-2}\frac{x^2}{1+5^x}$$  How do I start to integrate this?  I know the basics and tried substituting $5^x$ by $u$ where by changing the base of logarithm I get $\frac{\ln(u)}{\ln 5}=x$, but I got stuck.  Any hints would suffice preferably in the original question and not after my substitution.  (And also using the basic definite integrals property.)  Now I know only basic integration, that is restricted to high school, so would prefer answer in terms of that level. YA dystopia novel of battle school with unusual military control scheme Some sort of battle school (no, not Ender's Game), where teens are being trained as military commanders.   The control system for their army hinges around some sort of system that requires extremely fine motor control -- being able to "push" around digital representations of military units using a special interface that interprets minute finger movements.  The boy protagonist eventually meets a girl in the battle school, and is astonished to learn that most people do not live as well as he does -- for instance, he gets a fresh jumpsuit outfit every day and disposes of the old one; this is a luxury.  One scene I recall is them kissing for a moment in some sort of underground passageway; the boy protagonist notes the exact length of the kiss, as he has been trained to have heightened time perception.  I don't recall much about other commander characters; I think they have different special abilities, and one other major character might have physical deformities, possibly because of postwar-radiation-related birth defects.  They have been training for an upcoming, prearranged final battle between all the existing national governments; the military units are very possibly robotic or entirely digital in nature, so as to avoid wasting materiel / human life.    They ultimately win the battle, just barely; I think the girl protagonist gives them an edge in the competition by hacking the timing / control system in favor of their team, causing a discrepancy which only the boy protagonist notices due to his heightened perceptions.  It was agreed that the winning government gets dominion over one of the few islands of land left that was not irradiated in some previous war; the protagonists get to live there as a reward for their victory.  I believe the boy protagonist later realizes he's gradually losing his ability to exactly quantify the passage of time.  Published no later than 2004, likely somewhere between 1980-2000, probably sometime in the '90s. Proof by induction that $ 169 \mid 3^{3n+3}-26n-27$ $ 169$ | $3^{3n+3}-26n-27$ ?  Fulfilled for $n=0$. Induction to $n+1$: An integer $x$ exists so that  $ 169x= 3^{3n+6}-26n-27-26$ $ 169x= 27*3^{3n+3}-26n-27-26$ $ 169x= 26*3^{3n+3}+3^{3n+3}-26n-27-26$  An integer $m$ exists so that   $ 169x= 26*3^{3n+3}+169m-26$ ($ 13x= 2*(3^{3n+3}-1)+13m$)  Now I'm stuck although it looks simple. Thanks for any input in advance. How do you join on the same table, twice, in mysql? I have 2 tables. One (domains) has domain ids, and domain names (dom_id, dom_url).  the other contains actual data, 2 of which columns require a TO and FROM domain names. So I have 2 columns rev_dom_from and rev_dom_for, both of which store the domain name id, from the domains table.  Simple.  Now I need to actually display both domain names on the webpage. I know how to display one or the other, via the LEFT JOIN domains ON reviews.rev_dom_for = domains.dom_url query, and then you echo out the dom_url, which would echo out the domain name in the rev_dom_for column.  But how would I make it echo out the 2nd domain name, in the dom_rev_from column? List the files accessed by a program Installing Guest Additions causing problems xcode4 doesn't support Perforce?