The question is that simple, Why isomorphisms are important ? I know how we define mathematically isomorphisms, and I know what it means, if two things are isomorphic they are essentially the same thing. I want you to give some examples where they answer the question &quot;Why isomorphisms are important ?&quot; because I am an undergraduate I don't have that much experience to give my self that example. I think isomorphism like the analog of matrix similarity, meaning if we want to find a big power of a matrix we don't calculate all the matrix multiplications we just find the Jordan form or the diagonal matrix, and we take it from here. So we make problems easier and faster to calculate by connecting them with isomorphisms. Do you have in mind a fundamental example? Or one you think condenses the beauty of mathematics. Thank you.
In many courses, say Linear Algebra, Group Theory, Topology, instructors often say that knowing that certaing objects are isomorphic, sometimes makes life easier because we can work with more friendly objects.  Is there any example of using isomorphism concept to solve a non trivial concrete problem?   I must say, I'm not looking on important or famous isomorphic objects, but on non trivial application of the concept of isomorphism used in order to solve another kind of problem.
I'm looking to simultaneously send the same email to multiple recipients.  Let's say I want to email John and Bill the same email, I want them to be in the "To:" field, but I don't want them to see each other in the address list.  I don't want to bcc them both because then they see they are bcc'd.  I want them to each get their own copy of the identical email and to have 2 emails in my sent folder - one for each of them.  Essentially I want the same result as if I separately sent them each the identical email, but I'd like to do it in fewer steps.  Can I accomplish this either natively or with a plug-in, or some pre-existing code?  NOTE: I am looking to do this interactively (in real time), not in batch. In attempting to wrap my mind around the basic vocabulary, concepts, and methods of philosophy, I find myself wondering what the difference is between a philosopher and a theologian.   can have two meanings: 1. Theology is a rational study of the existence of God/gods and the nature of religious ideas. 2. Theology is simply a study of a particular religion (or all religions), really more the practice than the theory, but maybe a mixture.  It seems from an immediate reading that definition 1 would lead to the conclusion that a theologian is a philosopher (perhaps that theology is a branch or subset of philosophy), but definition 2 appears to be rather unphilosophical in nature.    So, how does philosophy answer this question?   Is theology considered a proper subset/branch of philosophy overall? Does it depend entirely on which of the above 2 definitions of theology fits a particular theologian as to whether he is a philosopher or not?   I think this answer is simple and fairly self-evident, but somewhere in my gut I feel like I'm missing something.  Is there a treatment of this question in the literature?    Edit: the answers so far seem to want to redefine theology or other terms - that is fine, but please be clear that you are redfining them, and provide, if possible, sources for why the term should be different than given (support from historical works, other sites, etc.). I have 5 tables and a section after them. After creation of an output file I have two tables on one page, then a section and last 3 tables on next page.  What I want to achieve is to put tables and section below them.  \begin{table}[ht]\footnotesize \centering \begin{tabular}{|ll|} ...... \end{tabular} \label{tab:result} \end{table}  \section{Section below tables} What does 5' and 3' mean in DNA and RNA strands? What are 5' and 3' in DNA and RNA strands? Please clarify with some images and please use simple English. From time to time, I need to typeset Welsh. Sometimes, I need to typeset citations and bibliographies in Welsh. I think that last time I did this, I may still have been using BibTeX although I am honestly not sure. In any case, my entire work-flow now uses biber and biblatex...  I found  on changing the names of things, including the names of things used by biblatex. However, that cannot deal with the case of an unsupported language.  Suppose that I want to typeset a document in Welsh:  \documentclass[a4paper,welsh]{article}  \usepackage{babel} \usepackage[utf8]{inputenc}% welsh needs utf8x but biblatex doesn't like it \usepackage[T1]{fontenc}% not great for Welsh but the best pdfTeX can do \usepackage{csquotes}% doesn't like Welsh though biblatex likes csquotes on account of babel... \usepackage[backend=biber]{biblatex} \bibliography{biblatex-examples}  \title{Fy Nogfen} \author{Rhywun}  \begin{document}   \maketitle   \tableofcontents   \section{Cyntaf}   \autocite{westfahl:space}    \printbibliography  \end{document}   There are multiple problems in such a case. (One is that I need utf8x but can't use it on account of biblatex. Another is that I need utf8x at all. Another is that biblatex likes csquotes but csquotes doesn't do Welsh. But these are irrelevant to this question.)  The output is as follows:    Note that the use of the English citation is just so this is easy to reproduce. Obviously, the content of the reference might well be in Welsh. (Though it might be in English, too.)  As can be seen, while elements of the document controlled by babel are handled correctly, those under the control of biblatex are not. However, I cannot simply modify the solution offered by lockstep for changing strings to customised English variants. That is, I can't just use a Welsh version of  \DefineBibliographyStrings{english}{%   bibliography = {Works Cited},   references = {Works Cited}, }   because biblatex has never heard of it.  What is the most appropriate way to configure biblatex for use with an unsupported language? Matching a space in regex I need to match a space character in a PHP regular expression. Anyone got any ideas?  I mean like "gavin schulz", the space in between the two words. I am using a regular expression to make sure that I only allow letters, number and a space. But I'm not sure how to find the space. This is what I have right now:  $newtag = preg_replace("/[^a-zA-Z0-9s|]/", "", $tag); I'm writing a technical documentation and I want to explain step-by-step how to go through various menus.  For example I used to write Open with &gt; VLC media player.  That's how it looks in Windows 8:     What I want in Latex should looks similar to this: light grey box, dark grey border and black windows-like font.    I've seen , suggesting the package  - but it looks very old fashioned - are there any more modern approaches? How to remove \n from a list element? Alternate Row Coloring in Django Template with More Than One Set of Rows Django templates offer the builtin tag cycle for alternating between several values at different points in a template (or for loop in a template) but this tag does not reset when it is accessed in a scope outside of the cycles definition.  I.e., if you have two or more lists in your template, the rows of all of which you'd like to use some css definitions odd and even, the first row of a list will pick up where the last left off, not with a fresh iteration from the choices (odd and even)  E.g., in the following code, if the first blog has an odd number of entries, then the first entry in a second blog will start as even, when I want it to start at odd.  {% for blog in blogs %}   {% for entry in blog.entries %}     &lt;div class="{% cycle 'odd' 'even' %}" id="{{entry.id}}"&gt;       {{entry.text}}     &lt;/div&gt;   {% endfor %} {% endfor %}   I've tried obviating this by patching with the resetcycle tag offered here:    to no avail.  (The code didn't work for me.)  I've also tried moving my inner loop into a custom tag, but this also did not work, perhaps because the compile/render cycle moves the loop back into the outer loop?  (Regardless of why, it didn't work for me.)  How can I accomplish this simple task!?  I'd prefer not to create a data structure in my view with this information pre-compiled; that seems unnecessary.  Thanks in advance. How to prevent updating of a specific package? Help me understand Support Vector Machines I understand the basics of what a Support Vector Machines' aim is in terms of classifying an input set into several different classes, but what I don't understand is some of the nitty-gritty details. For starters, I'm a bit confused by the use of Slack Variables. What is their purpose?  I'm doing a classification problem where I've captured pressure readings from sensors I've placed on the insole of a shoe. A subject will sit, stand, and walk for a couple of minutes while pressure data is recorded. I want to train a classifier to be able to determine whether a person is sitting, standing or walking and be able to do that for any future test data. What classifier type do I need to try? What is the best way for me to train a classifier from the data I've captured? I have 1000 entries for sitting, standing and walking (3x1000=3000 total), and they all have the following feature vector form. (pressurefromsensor1, pressurefromsensor2, pressurefromsensor3, pressurefromsensor4) If you open any of localized meta site in browser's private mode, e.g.  you will see untranslated text on the top right corner:        This message is not found in Transifex, hence unable to localize.  Related bug report on ruSO.meta: I am trying to create a stacked bar chart above a table, basically identical to the one below.    I have found similar posts such as:   However, my experience is very limited to simple tables and I am struggling to modify this to suit my needs.  Any ideas on how I can achieve such a graph?  Thanks! Explain why $\mathbb{Z \times Z}$ and $\mathbb{R \times R}$ is not a field Google Analytics: How can I track between http:// and https:// profiles for a domain? I'm making a call:  myResult = MakeMyCall(inputParams, out messages);   but I don't actually care about the messages.  If it was an input parameter I didn't care about I'd just pass in a null.  If it was the return I didn't care about I'd just leave it off.    Is there a way to do something similar with an out, or do I need to declare a variable that I will then ignore? How to Combine two Normal Maps? I made a high poly model and then baked its textures onto a low poly model. Now I want to give another normal map's details to the model. How can this be done? How do I iterate over the properties of an anonymous object in C#? I think there's an unintended motivation for users to post questions to higher-traffic Stack Exchange sites rather than their more specific or appropriate lower-traffic ones.  I think I have a solution.  What motivated this suggestion for me was that I am confident that if users were to post Android design questions to  beta and/or , that they would likely not be noticed by the appropriate audience compared to the vast army of Android experts here on Stack Overflow.    There are plenty of Android questions on SO that really should be on one or both of the aforementioned sites.  Presently there is little motivation for users to start questions like this on the appropriate site.  Is there something we could do to cross-pollinate related sites?  For instance, what if we let certain tags span certain sites?  Obviously the selection would be controlled by high-reputation users.   There could also be a feature to show (or explicitly hide) certain questions from other sites regardless of the tags.  This could be controlled by lower-reputation users than the global tag since it wouldn't have much statistical impact.  For instance, the android-layout tag could by default span sites, but OPs and other users could eliminate from the Graphic Design site questions that are of a purely technical nature.  Users interested in Android tagged questions would see related questions on design and UE sites--whether they are searching, posting, or viewing a question through the related items.  This would help eliminate cross-site duplicates, and encourage people to post on the correct site in the first place.   For related question lists and search results, there could even be a specific cross-site icon  next to an off-site question to show that it's from UX, for example. Explicit solution for linear regression with two predictors I have some samples of data of the form $x,y$ and $z=f(x,y)$. I wish to fit a plane $$z = Ax + By + C$$ to the data with the smallest mean square errors.  I have found an "answer" in section 3 of  document, but it is left in the form of some equations left to be solved. I just about have the ability to solve these equations, but the process gets so messy that the likelihood of me making a mistake is quite high. Surely someone has written out the full solution longhand somewhere (it might be called "closed form") anyway, in the form $$A=\ldots,$$ $$B=\ldots,$$ $$C=\ldots.$$  EDIT: Maybe "closed form" is the wrong phrase. So let me be clear. I want an explicit solution for $A$, $B$, and $C$, and not a solution that ends with "So now if you can solve these equations then you can discover the values of $A$, $B$ and $C$".