I am asking  if 4 is a mathematical  expression  and  it is a monomial.  Monomials are polynomials with single term.  so 4 is a polynomial  expression .  which is a constant  polynomial  expression  because it can be written  in this form 4x^0.  all polynomial  expressions  are algebraic  expressions .  so 4 is an algebraic  expression  also. But 4 is a numerical  expression  rather than an algebraic  expression  because there must have a variable  to be called an algebraic  expression.  Its a paradox please solve.  one more question  what algebraic  expressions with  one  term is called if it is not a polynomial  eg.  root (x).
I mean to ask  that $7$ is an arithmetic  expression  but it can also  be  written  as $7x^0$.  which is a constant polynomial  expression.  Every polynomial  expression  is an algebraic  expression  so with  this logic is $7$ an algebraic  expression  or an arithmetic  expression.  Further $7=5+2$, is it an arithmetic  equation  or algebraic  equation  because  it can also be written  in  $7x^0 = 5x^0 + 2x^0$ form. please help.  Thank you.
Want to copy the file name quickly in Windows. I always use rename, copy , but feel it is quite slow. Any better way? I'm not sure which of the following sentences is correct. Is the comma mandatory, optional or unnecessary? I've googled a bit about this topic, but I got confused.     Our mapping contains 2000 words, which map to more than one lemma.   or     Our mapping contains 2000 words which map to more than one lemma.   Context: In total, there are more than 2000 words.  I've read  . If I understand the answers correctly, my "which map to more than one lemma" is a defining clause and no comma is required. But is it wrong to add one? I gzip directories very often at work. What I normally do is  tar -zcvf file.tar.gz /path/to/directory   Is there a way to specify the compression level here? I want to use the best compression possible even if it takes more time to compress. Custom fields not useable in Approve/Reject Approval Request I'm trying to get through my first SalesForce mod and it's definitely been a rocky path. I have one task I'm trying to do, but this section is holding me back.  I have two custom fields.  One a pick list and the other a text box. I have added them to the Approve/Reject Approval Request section in the Opportunity page. I added the fields through the Select Fields to Display on Approval Page Layout and they DO show up on the Approve/Reject Approval Request section. I added them in like ().  However, just the label for the fields show, the field itself is nowhere to be found and can't be used.    You can see and even use the fields when you open the Opportunity page itself, but when you go to use the Approve/Reject, you can't do anything with them or see any values.  I was never a SalesForce user and got this tossed in my lap and really don't know my way around yet. I appreciate any help on this!!  Thanks!!  Kurt Diplomacy Attacking a Territory as they move How can I reduce the delay before window previews are shown in the Windows 7 taskbar? I really love the preview feature of the Windows 7 taskbar. There's only one reason I don't use it more often -- it takes a relatively long time to display the actual window.  There is a registry trick () to reduce the delay used by general menus, and make the submenus open instantly when hovered upon. I'm wondering if there's a similar trick that can be used for the Windows 7 taskbar's previewing. $\mathbb{Q}(\sqrt{n}) \cong \mathbb{Q}(\sqrt{m})$ iff $n=m$ Page 105 of D. Burton's A First Course in Rings and Ideals reads     It is not difficult to show that if (I'll call them $n$ and $m$ instead of $n_1$ and $n_2$) $n$, $m$ are square free integers, then $\mathbb{Q}(\sqrt{n}) \cong \mathbb{Q}(\sqrt{m})$ if and only if $n=m$.   Well, it is getting difficult for me, any help or would be appreciated.    My progress so far: Suppose $\phi :\mathbb{Q}(\sqrt{n})\to\mathbb{Q}(\sqrt{m})$ is an isomorphism. Then $\phi(u)=u$ for every $u \in \mathbb{Q}$, so that $\phi(\sqrt{n})^2=\phi(n)=n$. If $\phi(\sqrt{n})=a+b\sqrt{m}$ for some $a,b\in\mathbb{Q}$, then $(a+b\sqrt{m})^2=a^2+b^2m + 2ab\sqrt{m}=n$ implies $$a^2+b^2m = n$$ and $$2ab=0.$$  Then $b=0$ cannot happen since that would imply that $\sqrt{n}\in\mathbb{Q}$, where $n$ is a square-free integer. Also $a=b=0$ cannot happen. Hence $a=0$ and we have $$b^2m=n.$$ If $\psi:\mathbb{Q}(\sqrt{n})\to\mathbb{Q}(\sqrt{m})$ were an isomorphism then there would exist some $s\in\mathbb{Q}$ such that $$s^2m=n,$$ then $s=b$ or $s=-b$. Therefore the only isomorphisms from $\mathbb{Q}(\sqrt{n})$ to $\mathbb{Q}(\sqrt{n})$ are $\phi(u+v\sqrt{n})=u+vs\sqrt{m}$ and $\psi(u+v\sqrt{n})=u-vs\sqrt{m}$.    I don't know if there's an easier way and I don't know if I'm in the correct way here. Thanks beforehand :) How to read high voltages on microcontroller? Add a binary to my path I have an executable. I want to execute the executable in terminal with name only like other commands.  I can put my executable in /usr/local/bin or I could add its PATH to ~/.bashrc. Both will work.  What is better?  Is there  any difference? How to read ANSI encoded files in the right way? I have some files that Ubuntu can't read it ( ANSI encoding ) but Windows can read it well.  When I open it in gedit or notepad++ it seems like this :     Êã ÇáÊÍæíá áÜÜ   How can I make Ubuntu read ANSI encoded files well? This question isn't so much about how to achieve the effect of making a scene look like it's modelled in miniature, but why tilting the focal plane creates the effect. What is it about photographing miniatures that makes them look like miniatures, and how is this replicated by tilting the focal plane?  I can understand how when photographing a model of a scene, you have a much higher magnification, which leads to a shallower depth of field compared to when capturing the real-sized scene, but how does tilting the focal plane help? Is it just that you have the focal plane tilted relative to the plane that most of your subject is on, so that you can achieve a narrower depth of field for the focal length and aperture, or is there more to it than that? Drush from GitHub stopped working I have a  that has been working since it moved there.  I did a git pull this morning, and am now getting errors:     $ drush      Warning: require(/home/mpd/drush/vendor/autoload.php): failed to open   stream: No such file or directory in   /home/mpd/drush/includes/bootstrap.inc on line 1007      Fatal error: require(): Failed opening required   '/home/mpd/drush/vendor/autoload.php' (include_path='.:') in   /home/mpd/drush/includes/bootstrap.inc on line 1007   How can I fix this? solution :-  $21= 3 \times 7$  there is only one Sylow $3$ and Sylow $7$ subgroup  so, Sylow $3$ and Sylow $7$ subgroup are normal in group $G$  so $G$ is cyclic group of order $21$.  Am I right ?  somebody told me that group of order $21$ is not cyclic.  he gave me this .  If group of order 21 is not cyclic, then can we understand it by Sylow method ? In statistical learning, implicitly or explicitly, one always assumes that the training set $\mathcal{D} = \{ \bf {X}, \bf{y} \}$ is composed of $N$ input/response tuples $({\bf{X}}_i,y_i)$ that are independently drawn from the same joint distribution $\mathbb{P}({\bf{X}},y)$ with  $$ p({\bf{X}},y) = p( y \vert {\bf{X}}) p({\bf{X}}) $$  and $p( y \vert {\bf{X}})$ the relationship we are trying to capture through a particular learning algorithm. Mathematically, this i.i.d. assumption writes:  \begin{gather} ({\bf{X}}_i,y_i) \sim \mathbb{P}({\bf{X}},y), \forall i=1,...,N \\ ({\bf{X}}_i,y_i) \text{ independent of } ({\bf{X}}_j,y_j), \forall i \ne j \in \{1,...,N\} \end{gather}  I think we can all agree that this assumption is rarely satisfied in practice, see this related  and the wise comments of @Glen_b and @Luca.  My question is therefore:      Where exactly does the i.i.d. assumption becomes critical in practice?    [Context]   I'm asking this because I can think of many situations where such a stringent assumption is not needed to train a certain model (e.g. linear regression methods), or at least one can work around the i.i.d. assumption and obtain robust results. Actually the results will usually stay the same, it is rather the inferences that one can draw that will change (e.g. heteroskedasticity and autocorrelation consistent HAC estimators in linear regression: the idea is to re-use the good old OLS regression weights but to adapt the finite-sample behaviour of the OLS estimator to account for the violation of the Gauss-Markov assumptions).   My guess is therefore that the i.i.d. assumption is required not to be able to train a particular learning algorithm, but rather to guarantee that techniques such as cross-validation can indeed be used to infer a reliable measure of the model's capability of generalising well, which is the only thing we are interested in at the end of the day in statistical learning because it shows that we can indeed learn from the data. Intuitively, I can indeed understand that using cross-validation on dependent data could be optimistically biased (as illustrated/explained in ).   For me i.i.d. has thus nothing to do with training a particular model but everything to do with that model's generalisability. This seems to agree with a paper I found by Huan Xu et al, see "Robustness and Generalizability for Markovian Samples" .   Would you agree with that?  [Example]   If this can help the discussion, consider the problem of using the LASSO algorithm to perform a smart selection amongst $P$ features given $N$ training samples $({\bf{X}}_i,y_i)$ with $\forall i=1,...,N$ $$ {\bf{X}}_i=[X_{i1},...,X_{iP}] $$ We can further assume that:   The inputs ${\bf{X}}_i$ are dependent hence leading to a violation of the i.i.d. assumption (e.g. for each feature $j=1,..,P$ we observe a $N$ point time series, hence introducing temporal auto-correlation) The conditional responses $y_i \vert {\bf{X}}_i$ are independent. We have $P \gg N$.    In what way(s) does the violation of the i.i.d. assumption can pose problem in that case assuming we plan to determine the LASSO penalisation coefficient $\lambda$ using a cross-validation approach (on the full data set) + use a nested cross-validation to get a feel for the generalisation error of this learning strategy (we can leave the discussion concerning the inherent pros/cons of the LASSO aside, except if it is useful). Can binary data be ordinal? Binary data is often mentioned as a nominal sub-category, especially in such examples as female/male, smoker/non-smoker, etc. However, binary data with such values as pass/fail, correct/incorrect, absent/present, etc, seems to give some weight to its values. It's not like in the example of the gender, where both values are equal and differ primarily by the nominal and other context-related traits. Instead, this type of binary data clearly indicates that one value means something and the other means nothing.   In case of such distinction, can binary be considered ordinal? If yes, what are statistical tests that are usually used for such data? Also, are there any interesting books or papers on this case? "Open in terminal" not working on nautilus as root When using natilus as root, open in terminal used to work as using with any user. However, when I press open in terminal when using nautilus as root now, it does nothing.  How to solve it?  The error:  GLib-GObject-CRITICAL **: g_object_set_data: assertion 'G_IS_OBJECT (object)' failed Where can I ask questions regarding Firefox? I can't seem to find info on installing a Swedish language package on my machine. The language is not listed  and the hits on Google I got led nowhere (or, rather, to the said page).  Where can I ask such a question (within the realm of Stack Exchange)? The maximum energy stored by a capacitor is when it’s (theoretically) fully charged, or just before it reaches dielectric breakdown, if you will. It seems, then, that the maximum energy ($E$) $=QV$, where $Q$ is the maximum charge attained and $V$ is the maximum ‘voltage’. However, the correct equation for the maximum energy is $E=\frac{1}{2}QV$. Why? I’m looking for a simple, intuitive explanation because my reasoning that $E=QV$ seems sound to me. I know about , but I’m looking for a reason to disprove my specific argument that $E=QV$ and not the “Oh, it comes from integration.” response. I need to make my (BibTeX) references section appear in the table of contents of my LaTeX document (documentclass: article), with section numbering too.  My approach until now has been making a new section and including the bibliography (references.bib) at that point:  \section{References} \bibliography{references}   However, the final document shows both the section title that I have written and the section title that BibTeX writes, which is quite redundant and I definitely dislike.  How can I either remove BibTeX's section title, or make the BibTeX bibliography appear in the table of contents without making a new section?  If I were to make the BibTeX bibliography appear in the table of contents without making a new section, how could I assure that the section title that BibTeX writes looks exactly like sections typeset with \section? Indiscernible to create descending chain of elementary models