Square in a finite field I need to prove: $a$ is a square in $\mathbb{Z}$ iff $a\mod p$ is a square in $\mathbb{F_p}$ for every $p$ prime. First side ($\rightarrow$) is trivial and is derived directly from multiplicativity of modulo. Now I need to prove the other side ($\leftarrow$). I tried taking $p&gt;a$ so $a\equiv a\mod p$, and I know that $a$ is a square so there exists $b&lt;p$ s.t $\space$$b^2\mod p \equiv a$. But, $b$ is not necessarily the root of $a$ in $\mathbb{Z}$, so I'm stuck. I also know that for each $p$, $(\frac{a}{p})=1$ but this doesn't get me anywhere. Any hint would help.
An integer is a $n$'th power if that holds true for all moduli. I have not been able to solve this problem. Any insights would be appreciated!   Let $x, n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_{k}$ such that $x − a_k^n$ is divisible by $k$. Prove that $x = A^{n}$ for some integer $A$.
In natural language parsing, what is the feature function? I'm working in the area of natural language processing, to be specific I'm reviewing the parsers that take advantage of data-mining techniques.   I've read an introduction to natural language processing, specifically the  by Della Pietra and others. In that paper, the authors explain that features are indicator functions that take into account certain aspects of the text. The problem is that this aspects affect the maximum entropy function.  In the case of parsing, I have problems understanding the concept of features. For parsing one must have:   Two sets, $X$ for the sentences and $Y$ for the parses For each sentence $x \in X$, one must have the N-best list of candidate parses; denoted $GEN(x) \subset Y$. Training samples $(x_j,y_j)$. The $x_j$ are sentences and the $y_j$ are parses. And a feature mapping $\Phi(X,Y)$ that maps each pair (x,y) to a vector of feature values.   The problem of parsing will consist of finding the weight vector such that the following expression is minimized  $$ -\sum log P(y_j|x_j,w) + \lambda ||w||_1 $$  where   $$ P(y_j|x_j,w) = \frac{exp \{w^T \Phi(x_j,y_j)\}   }{ \sum_{u \in GEN(x)} exp\{ w^T\Phi(x_j,u) \}  } $$  My question is, what are the features in this case? Some codified version of the parsing tree? Are the features a statistical concept?  I will also appreciate any bibliography that can be useful in tackling this issue.  UPDATE: I've talked to a professor and he had just told me that the features are obtained after doing a classification algorithm based on the parse-tree. Do you know what kind of classification is done with the parse tree? In Python, how can I parse a numeric string like "545.2222" to its corresponding float value, 545.2222? Or parse the string "31" to an integer, 31?  I just want to know how to parse a float str to a float, and (separately) an int str to an int. Change the Fill Color of a cell if the value of that cell equals the value of another specific cell I want to change the fill color of cell C5 if it has the same value as cell A5.   Can this be done? I know how to format cell C5 if it has a certain value - but not when it is to be compared to the value of another cell. Avoiding the second leg of an international flight - different rates for a return ticket? Possible Duplicate:          I am in the process of booking a flight from LAX - ZUR. I found a flight for approx. 1100 USD. However, if I book that flight and then continue, after a 5 hour layover, onward to SVO, The flight is approx. half of the initial price, and the first leg of the second flight is the exact same flight as the first booking. Assuming I travel light and do not check any baggage, are there any potential issues with this? Seems like a hell of a way to save a good chunk of change. I'm just starting to learn LaTeX. I tried typing \begin{equation*}  (math goes here)  \end{equation*}  and I got a whole bunch of errors. It seems LaTeX interpreted the empty line as the equation: when it saw nothing there, I guess it just ended math mode? It then said my other commands needed $'s so I assume that's what happened. When I typed \begin{equation*} (math goes here) \end{equation*}  everything was fine. So I'm confused. If LaTeX sees an empty line in an environment, does it just interpret this as me changing my mind, and deciding not to use the declared environment? CreateService get error: ERROR_INVALID_ADDRESS (0x000001e7) Specific system of differential equations I have the following system of equations:  \begin{eqnarray}\frac{dx}{dt} = x(1 - x^2 - y^2) \\ \frac{dy}{dt} = y(4 - x^2 - y^2) \end{eqnarray}  I want to prove that if a solutions starts (at time $t = 0$) in the region   $$ G = \{(x, y) \in \mathbb{R}^2 : x &gt; 0, y &gt; 0 \text{ and } 1 &lt; x^2 + y^2  &lt; 4\} $$  that it remains there for all $t \geq 0$. I have already proven that, from $G$, a solution cannot cross the $x$-axis where $0 &lt; x &lt; 2$ or the $y$-axis where $0 &lt; y &lt; 2$ (since those line segments are orbits), but the pieces of the circles with radius 1 and 2 (where $x &gt; 0, y &gt; 0$) are the lines I haven't done yet. In this question we can focus on just the circle with radius 1 since the proof for the other circle is completely analogous to it.  In my book, there is a similar problem where they show that if a solution crosses a certain line, that solution attains its maximum or minimum value for $x$ and $y$, and then they derive a contradiction. However, in this case, for $n \geq 1$ the $n$-th derivative of $x$ equals 0 on the circle with radius 1. (If the second derivative were negative, $x$ would attain its maximum on the circle, a contradiction since $\frac{dx}{dt} &lt; 0$ in $G$, so $x$ is decreasing.)  I don't see why a solution can't cross the circle at all. On the circle, $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} &gt; 0$, so after crossing the circle, the solution would go back up to $G$.  Is the circle itself an orbit?  EDIT I think I have found the answer. Using polar coordinates, I found $$ \frac{dr}{dt} = r - r^3 + 3r \sin^2(\theta) $$ Now, $$ \frac{dr}{dt} &gt; 0 \Leftrightarrow 0 &lt; r - r^3 + 3r \sin^2(\theta) \leq r - r^3 + 3r = 4r - r^3 $$ since $0 \leq \sin^2(\theta) \leq 1$. Note that $0 &lt; r \leq 1$ satisfies $4r - r^3 &gt; 0$, and thus $\frac{dr}{dt} &gt; 0$. Therefore, if a solution goes from $G$ to the edge of the unit circle, $r$ must, on the edge, either be decreasing or constant (i.e., its derivative is less than or equal to 0), contradicting the fact that the derivative is positive.  Can someone tell me if this is correct? Calculate posterior distribution (gamma-prior, poisson-likelihood) I want to calculate the posterior distribution given a gamma-prior and a poisson likelihood. The task is from a textbook and I just have the solutions (without a walkthrough). Please find all given information in the following:  Given information:  $m$ countries each with a population of $n_i$; $y_i$ is the number of people having a disease; $y_i$ is a poisson random variable with mean $\frac{n_i*\lambda}{100}$  The prior distribution for $\lambda$: $\lambda \sim Gamma(a_0,b_0)$  Task:  Show that the posterior distribution is also a gamma distribution with $a_m = a_0 + \sum_{i = 1}^{m}y_i$ and $b_m = b_0 + \sum_{i = 1}^{m} \frac{n_i}{100}$  My solution:  $Posterior \propto Likelihood \times Prior$  $p(\theta|y) \propto \prod_{i=1}^{m}(\theta^{y_i}e^{-\theta})\times \theta^{a_0-1}e^{-b_0\theta}$  $p(\theta|y) \propto \theta^{\sum_{i = 1}^{m}y_i}e^{-m\theta}\times \theta^{a_0-1}e^{-b_0\theta}$  $p(\theta|y) \propto \theta^{\sum_{i = 1}^{m}y_i+a_0-1}e^{-\theta(m+b_0)}$  $a_m = a_0 + \sum_{i = 1}^{m}y_i$  $b_m = b_0 + m \neq b_0 + \sum_{i = 1}^{m} \frac{n_i}{100}$  I think I am doing a mistake, because I don't know how to incorporate the information that the mean of $y_i$ is given as $\frac{n_i*\lambda}{100}$. Or do I just have to transform $b_m$ somehow? Can someone may give me a hint?   Thank you already in advance!  Edit 1: Considering first advices  $Likelihood: p(y | \theta) \propto \prod_{i=1}^{m}(\frac{n_i\lambda}{100})^{y_i}e^{-\frac{n_i\lambda}{100}}$  $\propto \prod_{i = 1}^{m} ((\frac{n_i}{100})^{y_i})\lambda^{\sum_{i=1}^{m}y_i}e^{-\lambda\sum_{i=1}^{m}\frac{n_i}{100}}$  $\propto \lambda^{\sum_{i=1}^{m}y_i}e^{-\lambda\sum_{i=1}^{m}\frac{n_i}{100}}$  Is the transformation of $\prod_{i=1}^{m}$ I did from the first to the second step correct? I'd like to emulate HTML's infinite page height using LaTeX/XeTeX, i.e. have the PDF pages grow higher until a manual \newpage is issued.  This could be emulated by setting a very large page height and using pdfcrop on the output (or just setting the page height manually), but I'd like a pure TeX variant and if possible support for footnotes etc at the bottom of the cropped page. NSWindow Mac App Store like Title Bar How to make a grid-to-sphere animation like this? Proving that two vector spaces of equal dimension are isomorphic Given the following problem     Let $U$ and $V$ be finite dimensional vector spaces, such that $\dim(U) = \dim(V)$. Prove that $U$ and $V$ are isomorphic.   I was just wondering if somebody could critique and validate my proof.  Proof  Two vector spaces $U$ and $V$ are isomorphic if and only if there exists a bijection $\phi: U \to V$. That is, an invertible linear map between the two. Let $\dim(U) = \dim(V) = n$ and let $\left\{u_1, \dots, u_n\right\}$ and $\left\{v_1, \dots, v_n\right\}$ be bases for $U$ and $V$, respectively. It follows that any vector $u \in U$ can be written as  $$ u = c_1u_1 + \cdots + c_nu_n $$  We define $\phi: U \to V$ as  $$ \phi(c_1u_1 + \cdots + c_nu_n) = c_1v_1 + \cdots + c_nv_n $$  Observe that $\phi$ is invertible. That is, given a vector  $$ v = d_1v_1 + \cdots + d_nv_n $$  in $V$, we define $\phi^{-1}: V \to U$ as  $$ \phi^{-1}(d_1v_1 + \cdots + d_nv_n) = d_1u_1 + \cdots + d_nu_n $$  Since there exists a bijection between $U$ and $V$, they are isomorphic. I have already bought the PC version on minecraft for my account. However, when attempting to change my skin, the game simply states:"You have to buy the PC edition of Minecraft."   Which is odd because I already have.  Why is this happening? How can I fix it? Is there any scientific experiment that can lead us to conclude we live in 3 spatial dimensions without the premise of the conception of limited dimensions?  Thank you all who helped in the improvement of this question (which was not clear at first).   EDIT:  I know that this can be a little philosophical, but it is also a scientific question.  Let's consider the scenario where the mankind was not ever able to see.  Let's also consider that this limitation could be surpassed thus not limiting us to reach a scientific and technological knowledge "similar" to what we have today.  Would this civilization of blind people reach the conclusion that they are living in a 3D spatial world?  Is the sense of touch enough to reach that conclusion? Is there any scientific experiment that can lead us to that conclusion without the premise of the conception of limited dimensions?  Would it be easier, harder, or just different to reach a conclusion predicted by the M-Theory? (please do not focus only on this last question) Sending `time` command into text file I want to get the result of time command into a text file but it's not working it only put blank space in the text file.  I already tried this commands,  A-  $ x=`time` $ echo $x &gt; log.txt $ cat log.txt  $   B-   $ time &gt; log.txt  real    0m0.000s user    0m0.000s sys     0m0.000s $ cat log.txt  $   C-  $ time &gt; log.txt 2&gt;&amp;1  real    0m0.000s user    0m0.000s sys     0m0.000s  $ cat log.txt  $    What I really want is this.   $ time &gt; log.txt $ cat log.txt  real    0m0.000s user    0m0.000s sys     0m0.000s Sole minimal element: Why not also the minimum? A minimal element (any number thereof) of a partially ordered set $S$ is an element that is not greater than any other element in $S$.  The minimum (at most one) of a partially ordered set $S$ is an element that is less than or equal to any other element of $S$.    Let's consider the power set $\mathcal P (\{x,y,z\})$ together with the binary relation $\subseteq$.  The Hasse diagram shows what element(s) we're looking for:    It's easy to see that:   $\emptyset$ is a minimal element $\emptyset$ is the minimum   Now if we remove $\emptyset$ and consider $\mathcal P (\{x,y,z\})\setminus \emptyset$ instead, we get the following:   $\{x\}$, $\{y\}$ and $\{z\}$ are minimal elements there is no minimum   (1) We know that a minimum is unique and it is always the only minimal element.  (2) And from the example above, it seems that, if a sole minimal element exists, it is always the minimum.  But I read that (2) is false. Why? Flows/Processes, Version Control and CI I've started using Processes and Flows and am really excited about these features -- but am still wrapping my mind about how to manage them in Version Control (which I use) and Continuous Integration (which I don't yet... but am trying to work in that direction). The underlying challenge is that each time a process or flow is edited, the metadata file is renamed to reflect its update version, e.g. MyFlow-1.flow, MyFlow-2.flow, etc. The Challenge for Version Control If each version of the flow has its own file name, this makes it more complicated to track changes to flows.  It seems like there could be two strategies here:  Rename the files to remove the version number.  I'd need a script to run through my src folder pre-commit, and remove the number so that my file name is always MyFlow.flow.  This is an annoying extra step, but makes version control simpler.  On the other hand, it probably makes deployment much harder.  Track each file in version control with its original name.  We'd need to be diligent about deleting MyFlow-1 in the commit where we add MyFlow-2 so that git would recognize this as a rename/move and show helpful diffs.  Otherwise, the whole file appears to change with every change, which makes version control rather less useful.   The Challenge for Deployment/Continuous Integration From my experiments so far, once a flow has been activated in a given environment, the metadata file can't be deployed again (either it is currently active, or has been active in the past and superseded). That means that I need to prevent my previous flow versions from being included in a deployment package (either by not tracking them at all, or by using a script to delete earlier versions).  I also need some way to determine whether the current version (in my repo) should be included in the deployment payload for a given environment or not. I could imagine a script that (1) downloads flows from the target environment, and removes *.flow files from the payload if they already exist in the target environment, but that feels complicated.  And it's even harder if I've removed the version numbers from my flow files. Are you using flows in VS or CI?  How are you managing these issues? Note:  I'm not sure whether VC and CI should be one question or two - depends on how integrated the strategy for dealing with these two challenges.  If you think these should be separate questions, comment and I'll re-post. How do I specify a pointer to an overloaded function? Inverse of $f(x)=\sin(x)+x$ What is the inverse of   $$f(x)=\sin(x)+x.$$  I thought about it for a while but I couldn't figure it out and I couldn't find the answer on the internet.  What about  $$f(x)=\sin(a \cdot x)+x$$  where $a$ is a known real constant.  Thank you for taking the time to read this question! Sorry if this has been asked before... Is  (i.e. repulsive gravity) possible in string theory?  I have read some articles about scientists making assumptions about the existence of anti-gravity, but is it possible in string theory?