Prove that the additive group $ℚ$ is not isomorphic with the multiplicative group $ℚ^*$. Prove that $ℚ^*_{&gt;0}$ is not isomorphic with $ℚ$.
A question that may sound very trivial, apologies beforehand. I am wondering why $( \mathbb{Q}_{&gt;0} , \times )$ is not isomorphic to $( \mathbb{Q} , + )$. I can see for the case when $( \mathbb{Q} , \times )$, not required to be positive, one can argue the group contains elements with order 2 (namely all negatives). In the case of the requirement for all rationals to be positive this argument does not fly. What trivial fact am I missing here?
Proof that $\sum\limits_i \frac{(p-1)!}i$ is divisible by $p$ I'm stuck on the following phrases. Is it correct to use preposition "of" in the sentences bellow?    It's not a little unpleasant of being her so fussy. It's awfull of him having done this.   Thank you. What makes a bread either close or open crumbed/textured? What causes the texture of bread to be open, ie with lots of large holes, or close, ie a regular, uniform tender crumb with no large holes? How can I achieve either? How can I merge properties of two JavaScript objects dynamically? I need to be able to merge two (very simple) JavaScript objects at runtime.  For example I'd like to:  var obj1 = { food: 'pizza', car: 'ford' } var obj2 = { animal: 'dog' }  obj1.merge(obj2);  //obj1 now has three properties: food, car, and animal   Does anyone have a script for this or know of a built in way to do this?  I do not need recursion, and I do not need to merge functions, just methods on flat objects. Script at Startup A great Python IDE for Ubuntu i'm surprised there is so little documentation on this issue. But i'm trying to duck the audio on all the mix buses in Unity so that you can clearly hear the VO of the game.  Now it works great except that I have some reverb zones in the game and any sound that plays in these reverb zones, the actual reverb itself is NOT DUCKED.  So the VO is accompanied with a huge wall of wishy washy muddy noise.  Anyone deal with this before? How can we deal with this? Is God either immoral or not omnipotent? The usual (Christian) justification for suffering/evil in the world created by a benevolent God is freedom of the will. However, the more interesting question is not about the source of evil (which the free will may very well be) but why God chose to create the world such as it is. Why not create a world with freedom of the will, but without suffering? Even assuming that God is bound by logic such combination doesn't seem to be logically impossible: creatures certainly can do good of their own free will. There are two alternatives.  1) God is powerful enough for all practical intents and purposes, enough to be called God, but not omnipotent. He created only the best of all possible worlds (Leibniz). This position is perhaps the most defensible philosophically, it also avoids logical problems with , etc., but it does diminish deity's status and invites unpleasant questions like what limits God (logic only, inert matter, outer chaos, other beings), is there someone/thing more powerful, etc.    Interestingly enough, the original Christian theodicy wasn't free will. St.Augustine says poetically "the canvas of creation requires both the black and the white paint". "Requires" here can be interpreted differently,  is to explain the black by the "just punishment" for the original sin, which brings us to Adam and Eve, and back to the free will. But if God is still omnipotent and hence omniscient creating creatures he knew would sin, and then punishing them for it is hardly an improvement.   2) However, Augustine can be read more straightforwardly. Evil is "required" for the perfection of the world just like black paint is required for the perfection of a painting (evil can be interpreted as lack of good rather than something onto itself but that is beside the point). God is immoral, and he is beyond our conception of morality, his notion of perfection is his own. This is much bolder, and similar to Plotinus's position on the existence of the One, the One can not exist because it is "prior to all existence". But it does evoke the unpleasant taste of "ends justify the means". It is one thing to use "God is beyond our logic and comprehension" to get rid of unliftable stones, it is quite another to convince people to worship an immoral perfectionist.  Formally, there is another logical possibility of asserting both morality and omnipotence, but defending the morality of creating a world full of suffering, for a higher purpose perhaps. For example, the suffering only appears bad to us, not to an absolute being in its wisdom. But this simply redefines morality and is only different in phraseology from the Augustine's position. The question is asked in human terms and has to be answered in kind. One is free to argue that God is beyond human logic and/or morality, but it still leaves him immoral as commonly understood.   There does not appear to be a way to square this circle, one has to accept Leibniz's or Augustine's alternative (or both) and deal with the downsides. Option 2) looks more consistent, but it is rarely embraced, at least not openly. Perhaps, for "public relations" reasons it is more attractive to dismiss the question with something like "this is beyond us" or change the subject to the free will.      What are the modern theological positions on the morality/omnipotence in catholisism, protestantism, etc.? Is the question relevant to other religions?   EDIT: I am not sure that it came through but by immoral I did not mean morally deficient or evil. Rather at the God's plane of being morality becomes meaningless, like tastes or smells are meaningless when applied to elementary particles. Broken image link on the new Stack Exchange blog Before I jump on to the problem, let me first congratulate  and his team for the newly re-designed Stack Exchange blog. I like it very much, great job! While searching on Google, I stumbled upon  blog post of Jeff Atwood and found few broken images. There are currently four images are broken on this post, Here is one of them:    Upon doing an inspect element on one of the broken image (same goes for all other broken images on the post), I found out that it is currently pointing to: https://blog.stackoverflow.com/wp-content/uploads/stackexchange-genuine-example11.png The newly re-designed Stack Exchange blog is now moved to a new location:  and the old blog URL i.e.  has the correct mapping of URL redirection (also called URL forwarding) and works perfectly. But, the problem lies in the latter part i.e. /wp-content/uploads/ directory structure does not exist anymore at the new blog location. I read the Jon Chan's  post (nice writeup! btw) where he talks about his custom Python script. Here is the relevant excerpt taken from that post:  I wrote a Python script to go through all the imported posts, looked for references to WordPress hosted content, download all the static assets into images/wordpress in the repo, and changed all the links to relative paths on Jekyll. Worked surprisingly well.  In other words, all of the old /wp-content/uploads/ static assets is now resides under the /images/wordpress/ directory at the new blog location. So, Jon's find and replace script should have worked (which it did for the most of the part) as intended to be but apparently it did not catch few of them. At least on  blog post and few other like ,  or . We could possibly have more but these are the ones I have found so far. I believe it's a minor bug somewhere in the script and could be fixed with few lines of code tweaking. Also, the static content link mapping problem would not have occurred at all if the directory name was unchanged as the URL redirection (as stated above) would have taken care of it. But, there must a reason for this change. Upon investigating further, I found out that the issue with broken images does not end here. They will still appear as broken even if Jon's script replaces the path to the correct one. Because the images have not been uploaded (or copied) to the blog's new /images/wordpress/ directory at all. They are missing! For example, try clicking on the following link: ## I replaced old dir. path /wp-content/uploads/ to the new /images/wordpress/  So, in a nutshell, in order to fix this broken images link problem, we need to make sure that not only they are pointing to new path i.e. /images/wordpress/ but also do exists (i.e. uploaded) at the blog's new location. Please fix this at your earliest. Newbie Q: Is "Field Level Security" the same as "Field Accessibility"? There are two buttons that seem very similar when viewing the detial page for an object: Set Field Level Security and View Field Accessibility. They both seem to allow you to set the field level security.  Are these ("Field Level Security" and "Field Accessibility") basically the same thing? How do I enable the "Universe" repository? Where does Google Chrome automatically save pages? I opened the following page:   When disconnecting WiFi and even after rebooting the mobile phone, the page can still be read due to being saved automatically.  Left to the URL bar, an indicator looks like this:  This also shows up when opening manually saved pages, which are saved as MHTML files in the default download folder.  However, an automatically saved page can not be re-downloaded manually to get to the download folder, because the arrow is greyed out.I wish to copy some automatically saved pages into the download folder.  Where are the files of automatically saved pages stored? Let $p$ be prime number ($p\gt2$) and $a,b\in\mathbb Z$ ,$a+b\neq0$ ,$\gcd(a,b)=1$ how to  prove that $$\gcd\left(a+b,\frac{a^p+b^p}{a+b}\right)=1~~\text{or}~~ p$$  Thanks in advance . How to type en-dash and em-dash in Windows 7/8? I write blog posts, and I often need to type en-dash (–) and sometimes em-dash (—) in my blog posts. Currently I keep these characters saved in a text file, and copy-paste them whenever I need them. Unnecessary to say, it is a headache to move between keyboard and mouse when you type large texts.  It would be better for me if there ware a keyboard shortcuts available for these two characters. Is there a way to type en-dash and em-dash in Windows? I use Windows 7 and Windows 8 in different computers.    Here's an aligned comparison of dashes, if you need to see the difference for some reason.   Here's-dash Here's–en-dash Here's—em-dash How do I disable search-as-you-type text in Ubuntu 17.10 and later? I cannot find enable-interactive-search in    gsettings set org.gnome.nautilus.preferences enable-interactive-search true.     I searched using    gsettings list-recursively | grep search     and    gsettings list-recursively | grep interactive     but no such option.  Can you help me disable this in Ubuntu 17.10 and later? I am using scikit learn  to build a Random Forest classifier.  I have heard that it might be possible to build a single decision tree from a Random  Forest. The suggestion  is that although the decision tree may not be as good a classifier as the Random Forest it may be better than the decision tree you would get by using the standard method.  However I haven't been able to find this method online. Does it exist?     My question is not about extracting one  of the decision trees from a Random Forest. It is asking about a method to construct a new decision tree from the whole Random Forest, perhaps by combining the trees in the Random Forest somehow. I saw this bug mentioned in image below, where in I was about to mark the question as duplicate..    Has you can see there is no radio buttons to select and go further.. And I am stuck..  Is this a KNOWN bug..  I am using Mozilla Firefox Browser..  Edit:-  As guided by @AzizShaikh, This kind of behaviour is observed when you have already flagged a post. In my opinion, I think the page as I mentioned above should not have come in first place, I mean don't allow the user to enter it doesn't belong here, or it is a duplicate section if his vote has been posted successfully for that section rather than showing go nowhere Im lost Screen ..  Can we do something about this...?? Induction proof: n lines in a plane Assume that there are $n$ infinitely long straight lines lying on a plane in such a way that no two lines are parallel, and no three lines intersect at a single point. Prove that these lines divide the plane into $\frac{n^2+n+2}{2}$ regions. (Hint: Any two non-parallel straight lines on a plane must intersect at exactly one point.)  Can someone tell me where to start here? I know to prove the base where $n=0$, but don't know how to proceed.  Any help is greatly appreciated! How do you turn off developer mode on an s5? I turned developer options 'off', but is there anything else that needs to be done? Not my phone so don't want to mess anything up :-) Kernel of injective group homomorphism If $\phi$ is a homomorphism from the group $G$ with identity $e$ to the group $G'$ with identity $e'$, then $\phi$ is injective if and only if $\ker\phi=$?  I think the $Ker\phi = \{x\in G:\phi(x)=1\}$  because you are looking at an identity then it is all reals $0$ does not work a because if $x=0, e'x=0$  but then i realize it is injective how does that affect the kernel?