Calculate the following integral when $a^2+b^2=1&lt;c$  : $\int_{0}^{2\pi}\frac{dt}{acos(t)+bsin(t)+c}$ I thought maybe setting $\cos(t)=\frac{z+z^{-1}}{2},\sin(t)=\frac{z-z^{-1}}{2i}$ and then try to calculate but not sure about if it's the right way to do ti
Let $a,b,c\in\mathbb{R}$ such that $\sqrt{a^2+b^2}=1&lt;c$.  Find $I:=\int_{0}^{2\pi} \frac{1}{a\cos(t)+b\sin(t)+c}dt$.  So, I set $z=e^{it}$ and then $dz=ie^{it}dt$ and so $\frac{dz}{iz}=dt$. Therefore,  $\begin{align*} I &amp; =\frac{1}{i}\int_{|z|=1} \frac{dz/z}{a\cdot\frac{z+z^{-1}}{2}+b\cdot\frac{z-z^{-1}}{2i}+c} \newline &amp;=\frac{2}{i}\int_{|z|=1} \frac{dz}{a\cdot(z^2+1)-ib\cdot(z^2-1)+2cz} \newline &amp;=\frac{2}{i}\int_{|z|=1} \frac{dz}{(a-ib)z^2+2cz+a+ib}\newline &amp;= \frac{2}{i}\int_{|z|=1} \frac{dz}{(z-z_+)(z-z_-)}\end{align*}$  Where $z_{\pm}=\frac{-2c\ \pm\ \sqrt{4c^2-4(a+ib)(a-ib)}}{2(a-ib)}=\frac{-c\ \pm\ \sqrt{c^2-(a^2+b^2)}}{(a-ib)}=\frac{-c\ \pm\ \sqrt{c^2-1}}{(a-ib)}$.  Therefore, if we define $f(z)=\frac{1}{z-z_-}$, we get that $f$ is analytic in $\{z:|z|\leq 1\}$ since  $$|z_-|=\Bigg|\frac{-c-\sqrt{c^2-1}}{a-ib}\Bigg|=\bigg|c+\sqrt{c^2-1}\bigg|&gt;|c|&gt;1$$  Finally from Cauchy's Integral Formula:  $$I=\frac{2}{i}\cdot 2\pi i\cdot f(z_+)=4\pi \frac{1}{z_+-z_-}=\frac{2\pi(a-ib)}{\sqrt{c^2-1}}$$ Which is not always real... So my question is what is the problem.  Moreover, sometimes it is correct, e.g. $a=1,b=0,c=2$ gives the correct answer $\frac{2\pi}{\sqrt{3}}$.
Explain proof by contradiction. As our users regularly upload ~6MB images to use on the website (and aren't too familiar with how to resize them first), WordPress stores the originals as well as resizing them to several different sizes.  I'd like a function or plugin that takes the uploaded image, resizes it down to something more manageable and then replaces the original.  I've seen some functions that delete the original but do not replace it, meaning it is impossible to regenerate the thumbnails at a later date. I need this to be replaced so the user can upload a large image and it is automatically resized down and stored for future resizing if needed. I have an unlocked North American (Canada) iPhone 5s, where can I buy a data only SIM card in Japan ? I'm not interested in renting nor in a voice plan, I'll be staying for 4 weeks.  I'll be landing in Tokyo, so preferably around there. Can I get them from the airport somewhere ? I'm assuming I won't have problems with the network/frequencies, am I correct ? Uncaught SyntaxError: Unexpected token with JSON.parse The following MWE produces a PDF whose TOC has two entries: “Foo” and “Bar” with correct page numbers. So far, so good. However, upon clicking on the link to “Bar”, I am taken to the page for “Foo” instead.  \documentclass{report} \usepackage{hyperref} \newcommand*\backmatter{\setcounter{chapter}{0}}  \begin{document} \tableofcontents \chapter{Foo} \backmatter \chapter{Bar} \end{document}   (Produced with pdflatex, run twice.)  This MWE is whittled down from a (much!) more complex document but the above does seem to reproduce the error reliably.  Re-setting the chapter count seems to confuse hyperref, even though the page references are still accurate. Notice that in my real document, \backmatter does other things, including setting the numbering style to alphabetical so there’s no chance of confusion even though both chapters are ostensibly labelled as “Chapter 1”.  How do I fix this? Is this a bug? Are there workarounds? Find html element which id starts with I heard Andrew Ng (in a video I unfortunately can't find anymore) talk about how the understanding of local minima in deep learning problems has changed in the sense that they are now regarded as less problematic because in high-dimensional spaces (encountered in deep learning) critical points are more likely to be saddle points or plateaus rather than local minima.   I've seen papers (e.g. ) that discuss assumptions under which "every local minimum is a global minimum". These assumptions are all rather technical, but from what I understand they tend to impose a structure on the neural network that make it somewhat linear.  Is it a valid claim that, in deep learning (incl. nonlinear architectures), plateaus are more likely than local minima? And if so, is there a (possibly mathematical) intuition behind it?   Is there anything particular about deep learning and saddle points? How can I define the different properties of an Cycles/Eevee material? Such things as glossy, glass, diffuse, RGB-color, emissive via python scripting? RSA, DES, AES, etc., all use (relatively) complicated mathematics to encrypt some message with some key. For each of these methods, there have been several documented vulnerabilities found over the years. Some even believe that there is a vulnerability in AES known only to the NSA.  On the other hand, the one-time pad is not only ridiculously simple, but also has been shown to be impossible to crack (if used appropriately).  Say you have some message to be encrypted with length n. Simply generate a random string with length m where m is at least as large as n, and perform modular addition of the random string onto the message.  As long as the random string is generated with a high-enough-quality random number generator, and as long as the same one-time pad isn't reused, it should be impossible to crack.  If this is true, all we would need to perfect are fast, cryptographically secure deterministic random number generators. Use the key as a starting seed for the RNG and you have a powerful uncrackable encryption scheme which is simple to understand and implement.  Blum-Blum-Shub is a fast, cryptographically secure PRNG, and there are others too.  I have written a C++ program which uses two 2048-bit primes (which match the necessary criteria) for Blum-Blum-Shub and uses a password which the user enters - converted into a starting seed - to run byte-wise modular addition. The process is fast, deterministic, and if the literature I have read is correct then it should be very secure.  So why is so much time and effort invested in coming up with convoluted encryption cyphers when mathematics shows that the simplest is often the best? How do I compute the normalisation of $A=k[X,Y]/(Y^3 - X^5)$? I'm trying to solve exercise 4.7 in Reid's UCA:  "Find the normalisation of $A=k[X,Y]/(Y^3 - X^5)$."  I can easily show $A$ is not normal: let $x$ and $y$ denote the images of $X$ and $Y$ in $A$. Thus $y^3 = x^5$. So there exists $t=y/x\in\operatorname{Frac} (A)$ which is a root of the monic polynomial $T^3 - x^2 \in A[T]$. But since $t\notin A$, $A$ is not normal.  However, I don't know how to compute the normalisation of $A$; I have not seen dimension theory developed, so  to a similar question doesn't really help me. Similarly I cannot try and apply something like  to a related question, because in this case the map $k[X,Y]\rightarrow k[t]$ sending $X\mapsto t^3, Y\mapsto t^5$ is injective so the image isn't isomorphic to $A$  UCA never really gives a good explanation on how to actually compute normalisations, so I'd be grateful if anyone could walk me through it. Geometric interpretation of the multiplication of complex numbers? I've always been taught that one way to look at complex numbers is as a Cartesian space, where the real part is the $x$ component and the imaginary part is the $y$ component.  In this sense, these complex numbers are like vectors, and they can be added geometrically like normal vectors can.  However, is there a geometric interpretation for the multiplication of two complex numbers?  I tried out two test ones, $3+i$ and $-2+3i$, which multiply to $-9+7i$.  But no geometrical significance seems to be found.  Is there a geometric significance for the multiplication of complex numbers? Does white whole wheat flour need to be refrigerated? I know that regular whole wheat flour needs to be refrigerated, but does white whole wheat flour also need refrigeration? I hate to use up the space in the refrigerator if I don't have to.  I have 5 lb. (2.3 kg), and it would probably take me a month or so to use it all. How to calculate $\int_0^{\pi/2} \sin^a x \cos^b x \,\mathrm{d} x$ How can I use a variable in the replacement side of the Perl substitution operator? Upgrade Debian Jessie (8) to Buster (10) possible or need to go sequential in versions upgrades? Just started working with a Debian Jessie (8) in a production environnent. While presently deployed it is not seeing any heavy use yet. I was looking at the OS , and I was thinking about how convenient it would be to just upgrade it before is start seeing some more steady use in our environment and possibly making things more complicated with a later upgrade.  I was looking for a procedure to upgrade from Debian v.8 to v.10 but most documented  are sequential in order (e.g. 8 -> 9, or 9 -> 10).  My question: should I upgrade sequentially, or there is some proper documented way to skip a main version while upgrading? Highlight some words of a beamer Microsoft.ACE.OLEDB.12.0 provider is not registered I have a Visual Studio 2008 solution with two projects (a Word-Template project and a VB.Net console application for testing).  Both projects reference a database project which opens a connection to an MS-Access 2007 database file and have references to System.Data.OleDb.  In the database project I have a function which retrieves a data table as follows   private class AdminDatabase    ' stores the connection string which is set in the New() method    dim strAdminConnection as string     public sub New()    ...    adminName = dlgopen.FileName    conAdminDB = New OleDbConnection    conAdminDB.ConnectionString = "Data Source='" + adminName + "';" + _        "Provider=Microsoft.ACE.OLEDB.12.0"     ' store the connection string in strAdminConnection    strAdminConnection = conAdminDB.ConnectionString.ToString()    My.Settings.SetUserOverride("AdminConnectionString", strAdminConnection)    ...    End Sub     ' retrieves data from the database    Public Function getDataTable(ByVal sqlStatement As String) As DataTable         Dim ds As New DataSet         Dim dt As New DataTable         Dim da As New OleDbDataAdapter         Dim localCon As New OleDbConnection           localCon.ConnectionString = strAdminConnection          Using localCon             Dim command As OleDbCommand = localCon.CreateCommand()             command.CommandText = sqlStatement             localCon.Open()             da.SelectCommand = command             da.Fill(dt)             getDataTable = dt         End Using      End Function End Class   When I call this function from my Word 2007 Template project everything works fine; no errors. But when I run it from the console application it throws the following exception     ex = {"The 'Microsoft.ACE.OLEDB.12.0'   provider is not registered on the   local machine."}   Both projects have the same reference and the console application did work when I first wrote it (a while ago) but now it has stopped work.  I must be missing something but I don't know what.  Any ideas? Disable ohead on a single page What is the lowest level of mathematics required in order to understand how encryption algorithms work? If $p,q$ are primes $(pq,a)=1$ and $p-1|q-1$ then prove $a^{q-1}\equiv 1\pmod{pq}$ $p,q$  primes, $(pq,a)=1$ and $p-1|q-1$ then prove $a^{q-1}\equiv 1\pmod{pq}$   Got this in an exam today and had no idea how to do it.