How to show $15 \mid 2^{4n}-1$ by induction
Prove $5\mid2^{4n}-1$ by induction
Modern ice-age/plans for weapon hidden in genome Hide some fields in the NewForm.aspx for a custom list I have 5 columns in the custom list. Out of which only 3 have to displayed in the NewForm.aspx page.  All of the 5 columns have to be displayed on the EditForm.aspx page. While googling, I have come across the following  :  When I implemented the code that was given on this page, I get an error which says that parentNode is not found. Please guide me how can I implement this requirement.  Any other alternatives other than this are also appreciated. Cauchy-Schwarz inequality proof (but not the usual one) Before you downvote/vote-to-close, I am not asking for a proof of: $$\sum^n_{i=1}a_ib_i\le\sqrt{\sum_{i=1}^na_i^2}\sqrt{\sum^n_{i=1}b_i^2 } $$ Which is what EVERY link I've found assumes is the inequality (for a proof of that see:  )    I am reading a book that claims the Cauchy-Schwarz inequality is actually: $$\vert\langle x,y\rangle\vert\le\Vert x\Vert\Vert y\Vert$$ where $\Vert x\Vert :=\sqrt{\langle x,x\rangle}$  with the additional claim: equality holds $\iff\ x,y$ are linearly dependent  I cannot find a proof of this claim (only proofs for the dot product inner product). My question is: what is the simplest way to prove this (I define simplest to mean "using as few definitions outside of the inner product itself as possible" - so that's not including Gramm-Schmitt orthonormalisation, for example.)  I've just found some possible answers in the links on the right (annoyingly) but fortunately I have a second, albeit softer, question:  What inequalities that are common for say $\mathbb{R}^n$ are actually based on the inner product?   (In future I will check this site first, I was targeting lecture notes and such in my search)  Addendum: I want to use this on not-finite dimensional vector spaces. I've found one proof that relies on finite-ness. There are so many questions on this website. There is so many questions on this website.  The former &quot;sounds right,&quot; but the contracted form of the latter does as well:  There's so many questions on this website.  Which is correct? How to create svg symbols that have modifiable fill color, stroke color and stroke width? I want to create some svg symbols for Qgis using Inkscape, those symbols must have attributes as described in the question. For the last two days, I have been experimenting as instructed by  and mimicked some symbols came bundled with qgis that have desired attributes without any success.  Finally I experimented with the simplest form of symbol: I created an svg that contains only a circle in Inkscape and tried to modify it.  The original file () have this line:  &lt;path    sodipodi:type="arc"    style="fill:#ffffff;fill-opacity:1;stroke:#000000;stroke-width:1.01988637;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:none"   I modified it into:  &lt;path    sodipodi:type="arc"    style="fill:param(fill) #ffffff;fill-opacity:1;stroke:param(outline) #000000;stroke-width:param(stroke-width) 1.01988637;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:none"   Saved as  and selected it as point symbol, but Qgis couldn't change all three parameters at the same time.  What did I do wrong or should do differently?  My system: Qgis 1.8.0 on Ubuntu 12.04 64 bit, Inkscape 0.48, Gedit 3.4.1 text editor. Sometimes, others just inform us about things without us asking them, other times they do not do that even if we ask them. I want a verb for when others do not give us information — whether because they do not like to or they just don't give it, but we cannot directly ask them to — but we actively extract the information from them sometimes without them being aware of it.   So the word I am looking for shouldn't be only for cases where we coerce or force others to give us information, and it shouldn't be only for cases where others do not like to give us information.  Example:     I needed that information. But I couldn't ask her for it, or most   probably she would not give to me even if I asked her. So, I ....   the information I wanted.   It could be a phrasal verb and so not just a single word, but I couldn't devise a  blank suitable for all possibilities!   Two verbs I've already had but I think are not suitable:  to give someone the third degree   To prize something from/out of somebody Splicing a Javascript array from within the callback passed to forEach Which programs can I use to edit PDF files? I know that there is a very similar question, but I'm not interested in highlighting. What I'm looking for is some tool in Linux to edit pdf files. I've tried pdfedit which did the work for me, but if freezes after a few edits. I've tried this tool under Ubuntu 9.10 and under KDE Fedora 14 and the same thing happens. I've tried Okular but the notations are too big to fill out exercises from pdf workbooks.    Is there any tool that suits this job? I don't care whether it is open source or I have to pay a reasonable price for it. A diagram which is not the torus At page 17 of Munkres' Elements of Algebraic topology it says, referring to fig. 3.6, that "the diagram does not determine [the torus]. It does more than paste opposite edges together":   Is it so? I tried to glue them together, but end up with a normal torus. Why is a (simple) mask not recommended for asymptomatic COVID-19 patients? Information about (simple) mask wearing (concerning covid19) is a complete mystery to me. On several official places it is not recommended for a person without symptoms to wear a mask. For example:    (time 0:20)  or here:    However no explanation is given - which makes me wonder (and even angry). I think everybody should wear a mask because:   Many persons feel ashamed wearing it, even those with symptoms. If everybody wears it, they do not feel ashamed and wear it. As far as I know, one is (wrt covid19) contagious before developing symptoms. So if everybody wears the mask also these people do.   The two above points act as a collective protection.   Simple mask certainly will lower "getting / spreading illness" probability. Of course the "holes" (in a simple mask) are too big to prevent completely virus to enter (or get out when coughing), but it lowers probability! Even if it lowers the transmission probability by e.g. 5%, it might be a lot. As a hypothetical example: it might reduce R0 from 1.02 to 0.97 and so stop the virus spread so fast.   The videos are full of self-contradiction. In the same video    time 0:50 the lady says I should wear a mask if I am in contact with an infected person... why if it is inefficient?  Question: What are reasons for not recommending wearing a mask? I computed the homology groups of the torus, can someone tell me if this is correct? The calculation, not the result that is. Thanks!  The cells of $T^2$ are $e^0, e^1_a, e^1_b, e^2$  The chain groups are  $$ C_0(T^2) = \{ k e^0 |&#160;k \in \mathbb{Z} \} = \mathbb{Z}$$  $$ C_1(T^2) = \{&#160;k_1 e^1_a + k_2 e^1_b |&#160;k_1 , k_2 \in \mathbb{Z} \} = \mathbb{Z} \oplus \mathbb{Z}$$  $$ C_2(T^2) = \mathbb{Z}$$  $$ C_k(T^2) = \{ 0 \} , k &gt; 2$$  Now the homology groups:  $$ H_0(T^2) = \ker \partial_0 / im \partial_1 = \mathbb{Z} / 0 = \mathbb{Z}$$  where $im \partial_1 = 0$ because there is no chain in $C_1(T^2)$ whose boundary is a zero-chain in $C_0(T^2)$. (Is this reasoning correct?)  $$ H_1(T^2) = \ker \partial_1 / im \partial_2 = \mathbb{Z} \oplus \mathbb{Z}$$  where $\ker \partial_1 =  \mathbb{Z} \oplus \mathbb{Z} $, i.e. again everything maps to zero because there is no element in $C_1$ whose boundary maps to an element in $C_0$.  $im \partial_2 = 0$ again because there is no element in $C_2$ whose boundary is an element of $C_1$.  I don't want to use Hurewicz to do $H_1$.  $$ H_2(T^2) = \mathbb{Z}$$ using similar arguments as above.  Thanks for your help.  Edit  I posted the computations as an answer below. I got two up votes but I don't know by who so I'm not yet sure I can trust them... Why does this not work with an imaginary exponent? Could someone please tell me why this exponentiation rule does not work when the exponent is imaginary?  The following seems to be true but is only true if $x$ is an integer.  $$e^{2\pi ix} = \left( e^{2\pi i} \right)^x = 1^x = 1$$ complex number question involving modulus Let $z, w$ be distinct complex numbers. Show that if $|z| = 1$ or $|w| = 1$, then $$\frac{|z − w|}{|1 − z^*w|} = 1$$ [Hint: Note that $|a|^2 = aa^*$.]  Hey guys, couldn't get my thinking cap on for this question. Any helpful input? Will appreciate it! Dropbox: ext4 isn't ext4 I simply know that a single free quark does not exist. What is the reason that we can not get a free quark?  If we can't get a free quark then what is single-top-quark? tikz package causing error in IEEE access template I have been trying for a while to load the tikz package to the IEEE ACCESS template. Whenever I use the command \usepackage{tikz}, I am getting the error   /usr/local/texlive/2017/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex, line 33 Missing number, treated as zero.   It was working fine with the IEEEtran template and I was able to draw my circuit diagrams in that.   Anyone has any idea how to solve this issue? Can you use twinned spell on the same target? So me and my DM have been arguing on how Twinned spell works. Twinned spell:  When you cast a spell that targets only one creature and  doesn't have a range of self, you can spend a number of  sorcery points equal to the spell's level to target a second  creature in range with the same spell (1 sorcery point if  the spell is a cantrip).  I argued that &quot;a second creature&quot; can be a the same target as it is not worded as &quot;another creature&quot; For this example counter spell with twinned spell.  A wizard casts fireball on me I use CounterSpell#1 with Twinned Spell Wizard casts CounterSpell on my CounterSpell#1 I use him as the same target for my twinned Spell CounterSpell#2  Result countered Is this possible? I’m looking for an adjective/ noun that describes a person who loves life, traveling, great food, always doing things which gets adrenaline kicking and is a always ready for attempting things for pleasure only What is better performance wise with i7 2600k: 1x4GB or 2x2GB RAM? Is it better to use 2x 2GB or 1x 4GB RAM with a i7 2600k Sandy Bridge performance-wise for:   Applications? Games? Both? How could I find $x$ in this equation $x^2-x+6 \equiv 0 \pmod {9}$ This equation is like this: $$x^2-x+6 \equiv 0 \pmod{9}$$  I want to find x by using modular arithmetic. How I could do that?   My solution was:     $$\begin{align*}  x(x-1) &amp;\equiv -6 \pmod{9}\\ x(x-1) &amp;\equiv 3 \pmod{9} \end{align*}$$      So $x$ is either:   $$ x \equiv 3\pmod{9}$$      or       $$x - 1 \equiv 3 \pmod{9}$$      Which is $x \equiv 4 \pmod{9}$   I know what I have done is very dumb. So what is the correct solution? Give me the instructions also please.