I was reading the Proof of Hölder's Inequality in Measure Theory, and i saw, that our professor used an inequality and didn't explain, why it is true, and the inequality is;  For $\alpha\in(0,1)$ and $x\in\mathbb R_+$ $x^\alpha-\alpha x\le 1-\alpha$  $\textbf{I tried ;}$ $x^\alpha-\alpha x\le 1-\alpha$ $\Leftrightarrow x^\alpha\le 1-\alpha+\alpha x=1+\alpha(x-1)$ and $x^\alpha\le 1+\alpha(x-1)$ looks similar to bernoulli inequality, if you take $x'=\alpha(x-1)$ and $r=\frac{1}{\alpha}$ $(1+x')^r\ge x$ Is my justification correct, and does Bernoulli hold also for rational exponents ?
Prove: $$x^\alpha-\alpha x \le 1- \alpha  \\ \forall x\ge 0 \ ,  \ 0&lt;\alpha &lt;1$$   It does resamble Lagrange's MVT, in order to get to the RHS, say $\alpha\in[0,1]\Rightarrow \frac{1-\alpha-0^\alpha-\alpha 0}{1-0}=1-\alpha$.  But I'm not sure what to do about the LHS. I always get a negative value in the LHS and we know that the RHS will always be postive but that doesn't really help with proving it...  Edit: maybe define a function: $g(x) = x^\alpha-\alpha x - 1+ \alpha$ Derive it: $g'(x) = \alpha x^{\alpha-1}-\alpha $ but then I get to $x^{\alpha-1}=0$    Also, I had two questions when trying to figure this out:   Can we use deriviation of both sides of an inequality to prove it ? Can we use limit of both sides of an inequaltiy to prove it ?   Thanks.
I would like to have multiple bibliographies (reference lists) as described in   ().   However, I'd like to use with a single .tex file, which would contain the bibliography database embedded, as in  ...   Now, I want to use letter documentclass, for which the bibliography needs to be defined ()   So what I end up is with this example (code below, compiled with just pdflatex, no bibtex(which I'd like to avoid in this case)):     ... obviously, something works (multibib recognized primary and secondary cites, assigning them number "1" as start) -- however I don't get two, separate primary and secondary list of references; would that be possible in this context? (doesn't have to be with multibib, but I'd like to avoid a separate .bib file and a bibtex call)  EDIT: I'd also like separate cite keys (i.e. rendered as [a1] and [b1]; not yet sure if multibib can do that; edit: no it cant,  says "You can also edit the .bbl files created by BibTEX and add 'a', 'b',. . . to the labels, but your changes will be overwritten by subsequent BibTEX runs." and recommends \cite[a]{key}) - how possible would that be?  &nbsp;  The code is this:   \documentclass{letter}  % Environment thebibliography undefined when using letter: http://www.latex-community.org/forum/viewtopic.php?f=4&amp;t=3359 \makeatletter \setlength \labelsep {.5em} \newcommand\newblock{\hskip   .11em\@plus.33em\@minus.07em} \let\@openbib@code\@empty \newcommand\refname{References} \newcommand\section{\@startsection   {section}{1}{\z@}%   {-3.5ex \@plus -1ex \@minus -.2ex}%   {2.3ex \@plus.2ex}%   {\normalfont\Large\bfseries}} \newenvironment{thebibliography}[1] {\section*{\refname}%   \@mkboth{\MakeUppercase\refname}{\MakeUppercase\refname}%   \list{\@biblabel{\@arabic\c@enumiv}}%   {\settowidth\labelwidth{\@biblabel{#1}}%     \leftmargin\labelwidth \advance\leftmargin\labelsep \@openbib@code     \usecounter{enumiv}%     \let\p@enumiv\@empty \renewcommand\theenumiv{\@arabic\c@enumiv}}%   \sloppy \clubpenalty4000 \@clubpenalty \clubpenalty   \widowpenalty4000%   \sfcode`\.\@m} {\def\@noitemerr {\@latex@warning{Empty       `thebibliography' environment}}%       \endlist} \makeatother  \usepackage{multibib} \newcites{pri}{Primary sources} \newcites{sec}{Secondary sources}  \begin{document}   \begin{letter}{Addressee} Blah, blah - content...  Primary cite: \citepri{lamport94} ...  Secondary cite: \citesec{lamport94} ...   \begin{thebibliography}{9}  \bibitem{lamport94}   Leslie Lamport,   \emph{\LaTeX: A Document Preparation System}.   Addison Wesley, Massachusetts,   2nd Edition,   1994.  \bibitem{lamport95}   Leslie Lamport,   \emph{\LaTeX: A Document Preparation System}.   Addison Wesley, Massachusetts,   2nd Edition,   1995.  \end{thebibliography}  \end{letter}   \end{document} I have this problem as part of a school assignment:  NOTE: In case it's hard to read in the image above, here's the directory tree: $ tree . `-- sample_dir1     `-- sample_dir         |-- admin         |-- cambridge         |   |-- cafeteria         |   |-- library         |   `-- security         |       |-- annex         |       |-- building         |       `-- parking         |-- faculty         |-- history.exe         |-- markham         |   |-- annex         |   |-- building1         |   `-- parking         |-- oxford         |   |-- outline.doc         |   |-- programming         |   |   `-- report.pdf         |   `-- security         `-- stenton             |-- gen_ed             `-- lib_arts                 |-- english.txt                 `-- match.doc  15 directories, 11 files  I tried doing the command: $ cp ../cambridge/security/parking ./parking2  ...but it isn't working. The question in the image above states that the current directory is stenton, and that we need to make a copy of the file named parking (from the security directory), and to name the new file parking2, and place it in the current directory, stenton, using relative pathnames. What am I doing wrong? My cp command seems to be correct. What does increased caster level affect? Let's consider two statements: Zorn's lemma and theorem about existence of maximal ideals in commutative ring with $1$.  It's easy to prove that Zorn's lemma implies existence of maximal ideals.  I wonder if the converse proposition is true, i. e., does existence of maximal ideals implies Zorn's lemma? My approach was to construct a ring structure on an ordered set which satisfies Zorn's lemma condition such that existence of maximal ideal implies existence of maximal element in the set, but I didn't succeed. How do I get Gmail to ask me to be a mailto handler again? Found an error in my thesis I've found a error in my thesis recently, after its submission. I made a mistake that wrongly accepted the hypothesis which should be rejected. What should I do?? Prove that $\sum_{k=0}^n a_k x^k = 0$ has at least $1$ real root if $\sum_{k=0}^n \frac{a_k}{k+1} = 0$ Knowing that $$ \frac{a_0}{1} + \frac{a_1}{2} + \frac{a_2}{3} +\cdots + \frac{a_n}{n+1} =0$$ Prove that  $$ a_0 + a_1x + a_2x^2 + \cdots + a_nx^n = 0$$ has at least one real solution.  I suspect that it's proven with the intermediate value theorem. But can't find two numbers that satisfy it. I am currently using 19.04 and I got "No new release found for 19.10" Boundary Question in $\mathbb{R}^{2}$ (Manifolds) I don't always answer myself, but when I do, I fail to ask a good question Sometimes I'll encounter problems while coding, or sometimes out of simple interest for a question, I'll google for an answer, and most of the time I find that answer is here on stackoverflow.com.   If I can't find an answer, or if it's not enough to satisfy my curiosity, sometimes I expend my own effort to solve those questions. And after I've solved it, I'm happy to make my contribution to the community, and I post them as the Q&amp;A style posts. For example:     It seems relatively easier to ask a question if I didn't already have a solution, and I can put what I have so far in the question body, such as:     The problem to me is, if I have my own solution to my own question, then most of what I have so far becomes part of the solution. It's likely I have failed in making my contribution by asking a question to which I have an answer, for example:     So I'm wondering if there are some guidelines or good practices for Q&amp;A style posts?   How do I do it better? My non-native English speaking friend just asked me: "Data is..." or "Data are..."?  I said both but that's because I've been desensitized from reading/writing both (especially from writing code and adding quick comments).  My question: Is it acceptable to utilize either for a university paper? Or is one safer than the other (when confronted with stickler professors)?    Related questions: HP PSC 1410 all scanner's images are black I have a HP PSC 1410 printer with scanner inside.  I've been scanning documents and images with it for the last couple of years.  Recently my scanner has stopped performing as excepted and i can't scan or copy images\documents.  My problem is that every images\document i'm trying to scan - the result from the printer is a black page.  I've been googling about this issue and couldn't find any helpful stuff beside replacing the whole printer - which is something i'm trying to avoid.  Any ideas ? I've read something about the shipping lock of the scanner but couldn't find any information about how to check it.. Where to put the periods when using a parenthetical sentence? I have two complete sentences; the second is in parentheses, essentially a comment on the first.  Where do I place the periods?  I would use this:     They played opera. (I covered my ears.)   Is this punctuation correct? Prove that the dihedral group $D_4$ can not be written as a direct product of two groups I like to know why the dihedral group $D_4$ can't be written as a direct product of two groups. It is a school assignment that I've been trying to solve all day and now I'm more confused then ever, even thinking that the teacher might have missed writing out that he means normal subgroups.  On another thread it was stated (as the answer to this question) that the direct product of two abelian groups is again abelian. If we consider the direct product of abelian subgroups $H$,$K\in G$ where $HK=G$ (for all $g \in G$, $g=hk$ $h \in H$, $k \in K$.) I can't understand why this would imply $g=kh$? It is not stated anywhere that $H$,$K$ has to be normal! But if this implication is correct I do understand why $D_4$ (that is non-abelian) can't be written as a direct product of two groups. But if it's not, as I suspect, I need some help!  We know that all groups of order 4 and 2 are abelian, (since $4=p^2$), but only 4 of the subgroups of $D_4$ are normal:  Therefor I can easily show that $D_4$ can't be a direct product of normal subgroups: The only normal subgroups of $D_4$ is the three subgroups of order 4, (index 2 theorem): $\{e,a^2,b,a^2b\}$, $\langle a\rangle$, $\{e,a^2,ab,a^3b\}$ and the center of $D_4=\{e,a^2\}$ We can see that these are not disjoint. So $D_4$ can't be a direct product of normal subgroups. The reason for this being that the center is non-trivial. But why can't $D_4$ be a direct product of any two groups?  If we write the elements of $D_4$ as generated by $a$ and $b$, $a^4=e$, $b^2=e$, $ba=a^3b$ why isn't $D_4=\langle b\rangle\langle a\rangle$ ? I calculated the products of the elements of these two groups according to the rule given above and ended up with $D_4$, and also $\langle a\rangle$, $\langle b\rangle$ is disjoint...? Why is this wrong? Very thankful for an answer! The new ask page is now live on the network! Is it possible to spawn mobs in random places? bread that does not get moldy How can a bread have an expiration date of one year into the future?? The bread in questions is Deutsche Küche rye bread (from Aldi).  Yes, it is a dense bread, but it is not dry like a cracker, so there seems to be enough moisture for mold to grow.  There are no preservatives.  Edit (pictures added): How are the file metadata stored in Windows? (I'm using Windows XP but I guess it's similar in all the recent versions of Windows.)  When you create for example a new empty text document, you'll find in its properties that it has size of 0 bytes. Zero bytes means no information. No data. But still, the file has some name, it can still carry the dates of last access, modification and creation. It carries the information whether it's hidden file or not, whether it's read-only or not...  So where are all the metadata stored? Using Leibniz Integral Rule on infinite region I am trying to take the derivative with respect to $a$ of some function $I(a)=\int_{0}^{\infty}f(a,x)dx$.  I would like to make sure that I am using the  correctly.  Various web sources indicate a set of conditions that must hold for $f(x,a)$ and $\frac{\partial f(x,a)}{\partial a}$ when integration is done over infinite region.  From reading  (see Theorem 10.3 on page 13) the conditions that $f(x,a)$ and $\frac{\partial f(x,a)}{\partial a}$ must obey are:   $f(x,a)$ and $\frac{\partial f(x,a)}{\partial a}$ are continuous over $x\in[0,\infty)$ and around $a$ that we are interested in. There exists an integrable function (over $x$) $g(x)$ such that $|\frac{\partial f(x,a)}{\partial a}|\leq g(x)$. There exists an integrable function (over $x$) $h(x)$ such that $|f(x,a)|\leq h(x)$.   Integrable here means $\int_{-\infty}^{\infty}g(x)dx&lt;\infty$.  However,  seems to omit condition 3 above.  I am wondering which source is correct.  If there are "both correct", when is condition 3 necessary? How to create old looking/ cracked glass I am looking for old looking cracked glass for a window in a shack scene. I have looked everywhere but can find a tutorial or material or model on this please can someone help me on this thanks