Suppose that the functions $\ f_1, f_2, g_1, g_2: \mathbb{N}\to\mathbb{R}$ (set of real numbers greater or equal to 0) are such that $\ f_1 \in \Theta \ (g_1)$ and $\ f_2 \in \Theta \ (g_2)$.  Prove that $ \ f_1 + f_2 \in \Theta (\max\{\ g_1, g_2\})$  I have no idea how to go about this proof. Could someone please explain?  Thank you!
I need to prove that     $f_1 \in \Theta(g_1) \land f_2 \in \Theta(g_2) \implies f_1 + f_2 \in \Theta(\max\{g_1, g_2\})$    is relevant, but I have a slightly different case, so I don't know how to translate it into this one, because I need to turn the $f$'s into $g$'s.  Am I allowed to say $\max\{f_1, f_2\} \in \Theta(\max\{g_1, g_2\})$?  Or even $\Theta(\max\{f_1, f_2\}) = \Theta(\max\{g_1, g_2\})$?
"There are no shortage of applications" I've been having an argument with a colleague about this sentence, could you please let me know which one of us is correct:     There are no shortage of applications for our product in this   space.   She is convinced that are should be replaced by is, and I think it should stand as it is. Thanks for your help! I followed the official step-by-step "Visual Studio Code and Unity" guide, but it seems I messed up something, because Intellisense doesn't see any Unity related stuff.  Like the MonoBehaviour class and its members (like .transform), or the Unity callbacks, like Start or Update. As generally acknowledged TeX offers superb formatting capabilities for math in an automated fashion. Nevertheless there are some areas with clear deficiencies  that one either has to accept or manually improve on a case by case basis.  One of the biggies in that respect is TeX's handling of sub-formulas (i.e., material inside a brace group, e.g., and index, but also any group on top-level). TeX sets such sub-formulas always in natural width even if the whole formula is subject to severe stretching or shrinking otherwise. As an example, consider the following (not very sensible) example:  \documentclass{article}  \newlength\x  \begin{document}  \newcommand\formula{$ a+\mathbf{b+c}+d+e+f = \sum_{i=1}^{n-x-y-z} x_{i+j}$}  \settowidth\x{\formula}  \hbox to \x{\formula}    % natural width  \addtolength\x{-20pt}  \hbox to \x{\formula}   % now shrink it a lot  \addtolength\x{100pt}  \hbox to \x{\formula}   % now stretch a lot in the opposite direction  \end{document}   The formula here has a number of obvious sub-formulas in the subscripts and superscript, but for illustration I also added a sub-formula via \mathbf. Of course the usage of \mathbf in this way is wrong!!! (it should not be applied to several symbols), but I'm sure you would find this in documents. In any case just {b+c} would have had the same effect here. Now what do you think happens if we run this?  This:    As one can see the naturally "boxed" sub-formulas are very wrong the moment the rest of the formula is subject to stretching or shrinking.  Until recently no TeX engine successor addressed this issue. With LuaTeX opening up a lot of the internals of TeX I had some hope that this would be different. However, upon studying the manual my conclusion is that this area (sor far it least) has not been addressed (or considered).  As far as I can see the only way something could be done in LuaTeX about this issue would be to  use the mlist_to_hlist callback. However, this would really mean replacing the full math typesetting algorithm, which of course could be a way to solve the problem but ... but what is needed is not that (as 99% of this algorithm is next to perfect) but to add support to not simply box sub-formulas at their natural width.  So long text ... here is the short question:   Is this analysis correct, or did I overlook something? And in case anybody knows: are there plans to look into the sub-formula issue and provide support for it eventually? (it is not listed on the "to-dos" for math)    In case somebody wonders that I talk about "this sub-formula issue" as if it is something like a known thing ... it is, it was already raised way back in  which at the time a some thorough (if not say heated) discussions. A comparison between Grade, Mark, Score and Point Let's suppose a teacher wants to punish a student and wants to decrease his / her achieved points on a test paper; he would say something to the student; I would like to know which one of the following self-made examples may work here:        I have to take two marks of you.   I have to take two grades of you.   I have to take two scores of you.   I have to take two points of you.      For me all the first three mean the same and though the last one is not the same, but works here either. I need to get your feedback. At this time, Wubi does not work with Windows 8 default boot-loader. Thus at this point Wubi would not work on a new Windows 8 machine.      Citation fom    Question  Recently I've learned of , a way to install Ubuntu right from Windows.  Besides installing from Windows, I would like to know key differences between a regular "alongside Windows" or Dual-boot Ubuntu installation and one done with Wubi. Are there any disadvantages (for example a performance penalty) or incompatibilities I should look out for when using Wubi? How to resolve a Java Rounding Double issue Create a virtual file that is actually a command Is there any way to create a virtual file, such that reading from the file actually reads from the stdout of a command; writing to the file is acually writing to the stdin of a command?  So far I have kludged this with an inotifywait on a file, which calls a command when the file is modified, taking it's input from the file and writing back to it.  I don't like that the inotifywait has to be constantly restarted though (and I have to ensure that it is always running). I only use this file perhaps twice a week. How to eliminate Cycles artifacts with many overlapping volumes? I have encountered an issue with Cycles Volumetrics in a recent scene. I am using Blender 2.72, and both CPU and GPU render options. In both I get a solid black artifact when I have two volumes overlapping, as in this campfire smoke test:    I have a solid cube, with a volume scatter and volume absorption material, duplicated by a particle system. As you can see, when the particles get higher up and farther apart, and are no longer overlapping, the artifacts disappear. When are "because", "since","for" and "as" interchangeable? I am not a native speaker. On a previous question of mine, I thanked for an answer by saying:     So the phrase is not an idiom, since it is applied in its literal sense.   I consciously chose since over because, because — well, I like that usage of since, even though I know that some (native speakers) don't like it. Nonetheless, that choice got me thinking for a while, and it occurred to me that I even could have used more alternatives, namely as or for, without changing the meaning.   So, on a general note, here are the alternatives:        Statement A (is true), because statement B (is true).   Statement A (is true), since statement B (is true).   Statement A (is true), as statement B (is true).   Statement A (is true), for statement B (is true).      Are these completely interchangeable? Or interchangeable in certain situations? Or can you point out any (not-so-)subtle differences between any of them? Are there even more synonyms? How can I get the HTML source of a given web address in C#? Beamer how to change color of infolines and frame title How could I change the color of color of infolines and frame title in beamer. I am using this now   \documentclass[xcolor=dvipsnames]{beamer} \usetheme{CambridgeUS} \useinnertheme{rectangles} \useoutertheme{infolines} \usecolortheme[named=Brown]{structure}    I need to change the color from red to brown. [named=Brown] doesn't work for \useoutertheme{infolines}. This is my current output. I also need to change the background of frame title (mouse pointer) "Its" as a Possessive Pronoun How to downgrade a package via apt-get? How can I downgrade a package to an older version via apt-get?  Other tools are also acceptable but apt-get is preferred. How does Cantor's diagonal argument work? I haven't used regular expressions at all, so I'm having difficulty troubleshooting. I want the regex to match only when the contained string is all numbers; but with the two examples below it is matching a string that contains all numbers plus an equals sign like "1234=4321". I'm sure there's a way to change this behavior, but as I said, I've never really done much with regular expressions.  string compare = "1234=4321"; Regex regex = new Regex(@"[\d]");  if (regex.IsMatch(compare)) {      //true }  regex = new Regex("[0-9]");  if (regex.IsMatch(compare)) {      //true }   In case it matters, I'm using C# and .NET2.0. 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This command works:  find dir1 -name '*.png' -exec cp {} dir2 \;   but this command doesn't:  find dir1 -name '*.png' -exec cp {} dir2 + find: missing argument to `-exec'   I also tried:  find dir1 -name '*.png' -exec cp {} -t dir2 + find: missing argument to `-exec'   and:  find dir1 -name '*.png' -exec cp {} dir2 \+ find: missing argument to `-exec'   After looking at , I even tried:  find dir1 -name '*.png' -exec cp {} dir2 {} + find: Only one instance of {} is supported with -exec ... +    says that:     -exec {} + was added in [version] 4.2.12 in 2005   My version of find is 4.4.2.  What am I doing wrong?