Let $f$ be analytic in $\mathbb D$ and continuous in $\overline {\mathbb D}$ . Let $A$ $=$ {$ e^{i \theta}| |\theta - \pi |&lt; \epsilon$} , for $\epsilon &gt; 0$ small enough such that $f|_{A} = 0$ . Show that $f$ is identically equal to $0$ on $\mathbb D$.
Suppose I have a function $f$ that is analytic on the unit disk $D = \{ z \in \mathbb{C} : |z| &lt; 1 \}$ that is also continuous up to $\bar{D}$. If $f$ is identically zero on some segment of of the boundary (e.g. $\{ e^{it}, 0 \leq t \leq \pi/2 \}$ ), is it then true that $f$ is identically zero on the entire boundary?   I know that analytic functions that are zero at an accumulation point inside the domain of analyticity are identically zero throughout the entire domain, but I don't know what (if anything) can be said if something similar occurs on the boundary of the domain.
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. Can CBC ciphertext be decrypted if the key is known, but the IV not? Massless charged particles I have long been confused about this notion. I know that for a statement within a single category, forming the dual statement is just reversing every arrows. But what about a statement concerning several categories and functors between them?   Mac Lane suggests reversing all arrows in all categories and leaving the functors invariant in his book "Categories for the working mathematician" (page 32), but it seems that Mac Lane himself doesn't always follow this discipline (when he states "the dual of Yoneda Theorem", he doesn't reverse the arrows in $\mathbf{Set}$ !) Can anyone give me a formal definition of "dualize a statement in category theory"? Is the product of two Gaussian random variables also a Gaussian? Say I have $X \sim \mathcal N(a, b)$ and $Y\sim \mathcal N(c, d)$.  Is $XY$ also normally distributed?  Is the answer any different if we know that $X$ and $Y$ are independent? Problem with \pspicture in a simple plain TeX code Just writing the following very simple program I had some problems. I use to write in plain TeX and in this case I tried to make a picture with PSTricks and first of all sized it with \pspicture. The program is:  \input pstricks \pspicture (0,0)(5,5) \endpspicture \bye   but when I compiled with TeX, I encountered the following error:  Command Line:   tex.exe --src --interaction=errorstopmode "figura.tex" Startup Folder: C:\Users\MT\Documents\TeX  This is TeXk, Version 3.14159265 (TeX Live 2015/W32TeX) (preloaded format=tex)  Source specials enabled. (./figura.tex (c:/texlive/texmf-dist/tex/generic/pstricks/pstricks.tex we are running tex and have to define some LaTeX commands ... (c:/texlive/texmf-dist/tex/generic/xkeyval/pst-xkey.tex 2005/11/25 v1.6 PSTricks specialization of xkeyval (HA) (c:/texlive/texmf-dist/tex/generic/xkeyval/xkeyval.tex (c:/texlive/texmf-dist/tex/generic/xkeyval/xkvutils.tex (c:/texlive/texmf-dist/tex/generic/xkeyval/keyval.tex)) 2014/12/03 v2.7a key=value parser (HA) (c:/texlive/texmf-dist/tex/generic/xkeyval/xkvtxhdr.tex 2005/02/22 v1.1 xkeyval TeX header (HA)))) (c:/texlive/texmf-dist/tex/generic/pstricks/pst-fp.tex `pst-fp' v0.05, 2010/01/17 (hv)) (c:/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex (c:/texlive/texmf-dist/tex/generic/pgf/utilities/pgfutil-common-lists.tex)) (c:/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeys.code.tex (c:/texlive/texmf-dist/tex/generic/pgf/utilities/pgfkeysfiltered.code.tex)) (c:/texlive/texmf-dist/tex/generic/pgf/utilities/pgffor.code.tex (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmath.code.tex (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathcalc.code.tex (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.code.tex (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.basic.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.trigonometric.code .tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.comparison.code.te x) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.base.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.round.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.misc.code.tex) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithmetics .code.tex))) (c:/texlive/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex) )) `PSTricks' v2.64b  &lt;2015/14/11&gt; (tvz) (c:/texlive/texmf-dist/tex/generic/pstricks/pstricks.con)) ! Undefined control sequence. \endpspicture ...z@ \box \pst@hbox \else \fboxsep                                                    =0pt \fbox {\rule [-\pst@s... l.5 \endpspicture  ? x No pages of output. Transcript written on figura.log.  _____________________________________________________________________   TeX Compilation Report (Pages: 0)   Errors: 1   Warnings: 0   Bad Boxes: 0 _____________________________________________________________________   What's the problem? Counting nodes in a tree in Java First of all, I swear this is not homework, it's a question I was asked in an interview. I think I made a mess of it (though I did realise the solution requires recursion). Here is the question:  Implement the count() method which returns the number of nodes in a tree. If a node doesn't have either a left or right child, the relevant getXXChild() method will return null  class Tree {    Tree getRightChild() {     // Assume this is already implemented   }    Tree getLeftChild() {     // Assume this is already implemented   }    int count() {     // Implement me   } }   My reason for asking the question is simply curious to see the correct solution, and thereby measure how bad mine was.  Cheers, Tony I noticed relation between modulo operation and number which is power of two  Example I have to calculate $ 3431242341 \mod 2^5 $, which is $ 5 $ but it is equivalent to  $ ( 3431242341 \mod 2^9 ) \mod 2^5 $   I tried many examples and it seems to be true in general, and I am not sure if it is a coincidence or true in general that I can use first modulo operation ( greater number) and the result will be the same. Coefficient changes sign when adding a variable in logistic regression Showing $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$ How to prove that $\frac{x^2}{yz+2}+\frac{y^2}{zx+2}+\frac{z^2}{xy+2}\geq \frac{x+y+z}{3}$ holds for any $(x,y,z)\in[1,2]^3$ Prove that for $x,y,z\in [1,2]$ the following inequality holds: $$\frac{x^2}{yz+2}+\frac{y^2}{zx+2}+\frac{z^2}{xy+2}\geq \frac{x+y+z}{3}.$$ I tried to apply the Cauchy-Schwarz inequality or the power mean inequality, but my attempts were unsuccessfull, so I am asking for help. What do I need to consider to choose between dSLR, mirrorless, or a compact as my first "serious" camera? I am thinking of buying a camera to learn photography. Most of the time I'll be shooting portraits and landscapes. But I am not sure where I should start. Most of my friends tell me that I should get an entry level DSLR, but I am not sure if I have the persistence and enthusiasm to carry it all day in a backpack. But the camera on my cellphone doesn't seem to be enough to begin with.  So what I want is some expert advice (preferably with professional experience). Can most compact fixed-lens or mirrorless cameras fit the situations I mentioned? Or do I really need to go to a DSLR to get a tool to learn photography? Template Hierarchy: confused with index.php, front-page.php, home.php I'm really confused with index.php, home.php and front-page.php. In many cases even though I had an index.php, I overwrite it with front-page.php. But recently, I became aware about home.php.   What's the difference between the home.php and index.php? What's the ideal condition to use a home.php than an index.php? What's the ideal condition to use a front-page.php? When I'm using a front-page.php then what specific task an index.php is doing for me then?   I've read the  in Codex. Please don't mix my question with Page Templates, I understand 'em, alHamduLILLAH. I am working on a prototype, one function of which is to detect whether an object is "in range" by a few cm. Right now I'm just using an Arduino and a cheap  that operates at 13.56 MHz:    I've successfully gotten it to detect the tag, but only at a distance of about 3-4 cm when the tag is oriented parallel to the antenna.  According to my  , the HF band allows read ranges of between 10 cm and 1 m for passive tags, depending on factors like antenna size, tag size and orientation, and transceiver power.  I realize this inexpensive reader is not up to task as-is, but I'm interested in building something that achieves read ranges of 15-20 cm.  Some criteria:   No need to read multiple tags; only whether one tag is in range or not. No need to write to the tag. Cannot use an active tag.   I'm not experienced with RFID reader construction (yet) but am interested in knowing what I should pursue.  Here are some things I've considered:   Ditch the embedded antenna on the cheap RFID reader and construct my own, larger antenna. I'd need to see if the onboard  is up to the task of providing additional power. Look for a different IC around which to build a reader from scratch. Find a ready-made solution instead of trying to build one (more cost-effective?). Use something other than RFID for simple proximity detection.   I am experienced with microcontrollers and basic electronics, and have made audio and lighting projects, but this is a first with RFID. Since I don't need to read multiple tags, which RFID is suitable for, I've considered maybe some sort of magnet and hall-effect sensor, but the range does not seem adequate. I considered using an ultrasonic proximity detector, but the application calls for detecting the tagged object while ignoring untagged ones (ultrasonic would produce false positives on untagged objects). I've considered some sort of reflective surface placed on the object, and detecting reflected light, but the object orientation is not reliable enough to maintain alignment.  So, in a nutshell, What modifications can I make to my existing reader, or what alternate technology should I pursue to achieve 15-20 cm detection of a passively tagged object?  I plan on connecting whatever proximity detector is used to a microcontroller using SPI. I have read a lot about the SB-800/SB-900 but I am on a low budget (around ~120$). Is there an external flash that I can have to start learning to use it? I do not need TTL compatibility so it can be an older model. I have a Nikon D5000. Dedicated mysql server slow response I want to make the teeth a separate child object from the mouth. I want to know the best way to do this.   I have tried going into edit mode with Tab and trying to select all the vertices but that doesn't move the teeth. Is there an easy way to select just that part and make it a separate child mesh to the rest of the mouth? Why can't a CNAME record be used at the apex (aka root) of a domain? Are data members allocated in the same memory space as their objects in C++? How to escape quotes in shell? I'm having trouble with escaping characters in bash. I'd like to escape single and double quotes while running a command under a different user. For the purposes of this question let's say I want to echo the following on the screen:  'single quote phrase' "double quote phrase"   How can I escape all the special chars, if I also need to switch to a different user:  sudo su USER -c "echo \"'single quote phrase' \"double quote phrase\"\""   Of course, this doesn't produce the right result.