Assume $f$ is entire and $\int \int |f(x+iy)|^2 dx dy &lt; \infty$ , then show that $f \equiv 0$.  My approach was to notice that $|f|^2$ is a subharmonic function and then we can use the mean value inequality. Note that $$\int \int |f(x+iy)|^2  dxdy =  \int_0^{\infty} \int_{0} ^{2\pi} |f(re^{i\theta})|^2 rd\theta dr$$Now we know from subharmonicity $|f(0)|^2 \leq \frac{1}{2\pi} \int_0^{2\pi} |f(Re^{i\theta})|^2d\theta = \frac {1}{\pi R^2}\int_0^{R} \int_{0} ^{2\pi} |f(re^{i\theta})|^2 rd\theta dr$   As $R \rightarrow \infty$ , the rightmost term tends to  $0$ as we know $\int_0^{\infty} \int_{0} ^{2\pi} |f(re^{i\theta})|^2 rd\theta dr &lt; \infty$, thus we have |f(0)| = 0. We could have chosen any $z$ in place of 0 and the argument would follow the same and thus we see $f \equiv 0$.   I was wondering if there is any way to get this result without the subharmonicity, I was trying to apply Liouville's but couldn't, if there is a simpler solution please let me know.
Suppose $f$ is entire and    $$\iint_\mathbb{C}|f(z)|^2dxdy &lt; \infty$$Prove that $f\equiv 0.$   So far I have:  Suppose $f$ is bounded.  Then $f$ is constant by virtue of Liouville and so the conclusion is obvious.  Thus, assume $f$ non-bounded.  Then  $$\Big|\iint_\mathbb{C}f^2(z)dA\Big| \leq \iint_\mathbb{C}|f(z)|^2dxdy &lt; \infty$$ We can parameterize \begin{align*} \iint_\mathbb{C}f^2(z)dA &amp;= \int_0^{\infty}\int_0^{2\pi}f^2(re^{i\theta})\;d\theta\;rdr \\ &amp;= \int_0^{\infty}\int_0^{2\pi}f^2(re^{i\theta})\;ire^{i\theta}[-i\frac{1}{r}e^{-i\theta}]d\theta\;rdr \\ &amp;= \int_0^{\infty}\oint_{C_R}f^2(z)\;[-i\frac{1}{z}]dz\;rdr \\ &amp;= \int_0^{\infty}2\pi r dr\frac{1}{2\pi i}\oint_{C_R}\frac{f^2(z)}{z}dz \\ \\ &amp;= 2\pi f^2(0)\int_0^\infty rdr  \end{align*}  Whence f(0) = 0.  And I have no idea where to go from there.  The "entire" bit seems to suggest Liouville but I have already dealt with the bounded case.  Please advise ...
Difference estimate & confidence intervals for $\chi^2$ test between 2 proportions I am using a chi-sq test between two proportions (). I am using this over the z-test for proportions because I do not think my data is normal.   My question is how do I get the difference (shift) estimate &amp; confidence intervals for this test.   For example,     Data1: 193/252=.77   Data2: 154/227=.68   P-value=.032 (I used the formula in the link above. I will also add my exact code below)  Shift Estimate: I assume its the difference between proportions, ie .77-.68=.09. Is this correct?   Confidence Interval: ??????????????????  Code for generating P-value (matlab):   % Pooled estimate of proportion  p0 = (n1+n2) / (N1+N2); % Expected counts under H0 (null hypothesis)  n10 = N1 * p0;  n20 = N2 * p0; % Chi-square test, by hand  observed = [n1 N1-n1 n2 N2-n2];  expected = [n10 N1-n10 n20 N2-n20];  chi2stat = sum((observed-expected).^2 ./ expected);  p = 1 - chi2cdf(chi2stat,1);  H=0; if(p&lt;.05), H=1; end Bug in editor for numbered lists? I noticed what seemed to be a bug in the way the stackoverflow editor handles numbered lists. I did a search on meta for this and it didn't come up, but if it is a duplicate, I'll delete the question.  Consider the following code:  Here are some points:  1. One 2. Two 3. Three  Here are some more points which continue the theme:  4. Four 5. Five   When you actually type this into the text editor, this is the result:  Here are some points:   One Two Three   Here are some more points which continue the theme:   Four Five   In this case, the ordered list resets after the break. This is probably just an edge case (and not necessarily worth fixing), but I ran into the problem today and figured this would be a good place to raise the topic! My workaround was to use back-ticks around the numeral, but then you lose the hanging indent. (ie, 1. One)  Might this be worth fixing? Is it expected behavior? Or am I doing it wrong? Can a parallel computer simulate a quantum computer? Is BQP inside NP? Unable to log in to Stack Exchange I don't know why but now, when i go in material preview or rendered (in solid mode it works fine), my object is entirely gray, or invisible idk, because it has the same color of the background. I've been working for days on this project without having this issue, it suddenly showed up. I tried opening other old projects and suddenly they all do the same so i guess it is some blender program setting that applies for all the projects, but idk how it happened cuz i didn't change anything :/, anyone got any tips? Suppose that $S_0$ and A are positive numbers, let $$S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right)$$ with $n \geq 0 $.      (a)Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$       (b)Show that $S_{n+1} \leq S_n $ , if $n \geq 1$      (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists      (d) find s   (a) Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$  Given $$P_n:  S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \geq \sqrt{A}$$  $$P_0: S_{1} = \frac{1}{2}\left(S_0 +\frac{A}{S_0}\right) \geq \sqrt{A}  $$  We assume that $P_n$ is true  $$P_{n+1}: S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right)$$ by assumption $$S_{n+2}= \frac{1}{2}\left(S_{n+1}\left(1 +\frac{A}{(S_{n+1})^2}\right)\right) \geq \frac{1}{2}\left(\sqrt{A}\left(1 +\frac{A}{(\sqrt{A})^2}\right)\right)$$ $$ S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right) \geq \sqrt{A}  $$  It follows that $S_{n+1} \geq \sqrt{A} $    (b) Show that $S_{n+1} \leq S_n $ , if $n \geq 1$  $$S_{n+1} \leq S_n$$ $$\frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \leq S_n $$ Dividing by $S_n$ $$\frac{1}{2}\left(1 +\frac{A}{S_n^2}\right) \leq 1 $$ $$\frac{A}{2S_n^2} \leq \frac{1}{2}$$ $$A \leq S_n^2$$ $$S_n \geq \sqrt{A}$$ As  $S_{n+1} \leq S_n$ yields a true statement, it follows $S_{n+1} \leq S_n$ is true.    (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists  Since $S_{n+1} \leq S_n$, the sequence is non-increasing,   using the non-increasing theorem stating that      if $\{S_n\}$ is non-increasing then $$\lim\limits_{n \rightarrow &gt; \infty} S_n = \inf\{S_n\} $$     (d) find s    Is the argumentation in (a) and (b) appropriate? Also, I have to admit I m getting less confident in my argumentation (c) and (d). How to proceed in (c) and (d)?  Much appreciated for your input or help. I've heard a lot about the Undying Lands but I don't fully understand how they're different from Middle-earth and why only certain people/races are allowed to go there.  Can someone explain? What is the full "for" loop syntax in C? What security systems and processes are in place to prevent malicious third parties from hacking/compromising the security of code in the Debian mirrors, or to verify that the packages we get are in fact the ones the maintainers think they are? Given a real, skew-symmetric matrix $\mathbf{A}\in\mathbb{R}^{m\times m}$, and a nonzero vector $X\in\mathbb{R}^{m}$, classify the scalar $\lambda$ (real, complex, imaginary, etc.) in the eigenvalue equation $$ \mathbf{A}X = \lambda X $$  The matrix $\mathbf{A}$ is skew symmetric if $\mathbf{A}^{T}=-\mathbf{A}$.  My try : I am new to matrices , so can not get a idea how to deal with it , Any hint will help Improving upon ArcGIS Buffer/Dissolve performance? Check if all elements in a list are identical How can I remove Amazon search results from the dash or disable the feature? How can I disable Amazon search results in the Dash? What test can I use to compare slopes from two or more regression models? I would like to test the difference in response of two variables to one predictor. Here is a minimal reproducible example.   library(nlme)  ## gls is used in the application; lm would suffice for this example m.set &lt;- gls(Sepal.Length ~ Petal.Width, data = iris,                 subset = Species == "setosa") m.vir &lt;- gls(Sepal.Length ~ Petal.Width, data = iris,                 subset = Species == "virginica") m.ver &lt;- gls(Sepal.Length ~ Petal.Width, data = iris,                 subset = Species == "versicolor")   I can see that the slope coefficients are different:  m.set$coefficients (Intercept) Petal.Width    4.7771775   0.9301727 m.vir$coefficients (Intercept) Petal.Width    5.2694172   0.6508306  m.ver$coefficients (Intercept) Petal.Width     4.044640    1.426365    I have three questions:   How can I test the difference between slopes? How can I test the difference between residual variances? What is a simple, effective way to present these comparisons?    A related question, , suggests re-running the model with a dummy variable to differentiate the slopes, are there options that would allow the use of independent data sets? Ok... so this is embarrassing.  I typed in "boobs" on EncyclopediaDramatica's search because, well, it's a really funny page.  But now I've got a problem... EVERY TIME I click a search field in Chrome now, it shows "boobs" as the first suggestion in the drop down!  This is not really socially acceptable as this laptop goes with me everywhere and Chrome is my primary browser.  Is there a way I can isolate and remove just that one entry? Payment from Apple for developer Actually snapping to pixel grid Does anyone know if there's a way to have Illustrator CC (2014) constrain control points to the pixel grid? I understand that "Align to pixel grid" makes it so that a shape will try to match the pixel grid, but the actual control points don't line up to said grid... The image I've attached illustrates my problem. USB WiFi dongle device that can emulate a virtual mass storage device for a home video player Is there a USB WiFi dongle device that can:   Emulate a virtual mass storage device for a home video player But in fact shows the files from a network-shared folder? I need to test this code, but when writing a trigger, I get an invalid type I need help. I am currently trying to test this code before uploading  public class CoreValuesExt { ApexPages.StandardController sc; public boolean readonly {get; set;} public Core_Values__c cv {get; set;} public integer countEdit = 1; public string buttonValue {get; set;} public integer recordScan;  public CoreValuesExt(ApexPages.StandardController sc) {     this.sc = sc;     cv = new Core_Values__c();     cv.Account__c = sc.getid();      readonly = true;     buttonValue = 'Edit';     recordScan = [SELECT COUNT() FROM Core_Values__c WHERE Account__c = :cv.Account__c];    if (recordScan != 0){    cv = [SELECT id, HSE__c, Professionalism_Integrity__c, Quality_Accuracy__c, Technology_Innovation__c, Timeliness__c, Value_for_money__c FROM Core_Values__c WHERE Account__c = :cv.Account__c];     }  }  public ApexPages.PageReference Save() {     return sc.Save(); }  public void ApplyAction() {     countEdit++;     Integer modolus = Math.mod(countEdit, 2);     String total = cv.HSE__c + cv.Professionalism_Integrity__c + cv.Quality_Accuracy__c + cv.Technology_Innovation__c + cv.Value_for_Money__c + cv.Timeliness__c;     if(modolus == 0)     {         readonly = false;         buttonValue = 'Save';     }     else     {         if(total != 'nullnullnullnullnullnull')            {         if (recordScan == 0){         insert cv;         readonly = true;         buttonValue = 'Edit';         }         else         {         update cv;         readonly = true;         buttonValue = 'Edit';         }         }     }    }   }  With this trigger.  trigger CoreValuesTrigger on Core_Values__c (before insert) {  Core_Values__c cvs = Trigger.new; CoreValuesExt.ApplyAction(cvs);  }   But when I try to save, I get a message saying:      Illegal assignment from List to   CoreValuesExt__Core_Values__c at line 3 column 1   "  How do I go through with this? I need help. Thanks how to use these hash functions in python? In this article , specifically in the section 4.1 of the initialization phase,the following cryptographic hash functions have been chosen: \begin{align} h_1&amp;:   \mathbb G\to\mathbb Z_q^*\\ h_2&amp;:  \{0,1\}^*\times\{0,1\}^*\times\mathbb G\to\mathbb Z_q^*\\ h_3&amp;:  \{0,1\}^*\to\mathbb Z_q^* \end{align} where G is an additive group with order q. Can anyone explain to me what's the meaning of their notations and how to use them for example in SHA algortihm? Or maybe give me a simple example with real inputs?