One definition of continuity is that the pre-image of an open set is open. Can I define Continuity as that the image of an open set is open AND the image of close set is close? If this is not correct, are there any counterexamples?
I wanted to discuss my intuition of why we formulate the concept of continuity in terms of pre-image of open set is open instead of images for example if we consider $f(x) = c$ where $c$ is some constant, then that should be continuous, but if we formulated continuity in terms of images it won't be continuous. Is the reason we do it that way, because pre-images will always guarantee that function in the domain can only have $1$ value?
Show that $z^5-1=(z-1)(z^2+2z\cos{\pi \over 5}+1)(z^2-2z\cos{2\pi \over 5}+1)$ UMVUE of $e^{-\lambda}$ from poisson distribution Let $X_1,\ldots,X_n \sim$ Poisson$(\lambda)$. I wish to find UMVUE of $e^{-\lambda}$. Here $\overline{X}_n$ is complete and sufficient for $\lambda$ (hence for $e^{-\lambda}$??). Define $Y_i=1$ if $X_i=0,$ $Y_i=0$ otherwise. Clearly (I checked!) each $Y_i$ and hence $\overline{Y}_n$ is unbiased for $e^{-\lambda}$. Then I should be able to use the Lehmann Scheffe theorem to deduce the UMVUE. The estimator I get looks like $$\frac{1}{n} E\left\{\#X_i:X_i=0 \mid \sum_{i=1}^n X_i\right\}$$ whereas the correct answer is $$\left(\frac{n-1}{n}\right)^{\sum_{i=1}^n X_i}$$ I'd appreciate if anyone lets me know where I'm going wrong and guides me into the right path. Thank you. I recently upgraded to Snow Leopard, from Leopard and I was using an awesome app called  that let you remap your keys, set scripts/macros and even had repeat functionality.   However, this app doesn't seem to be in active development anymore, and 10.6 broke the ability to bind certain modifier keys (Shift) with other keys.   I totally miss things like Shift+Insert for paste (Linux/Windows), setting Ctrl+Shift+Esc for Activity Monitor, and Ctrl+Alt+Del for the login window.   So, how can I achieve this instead? How have Superman comics dealt with the Man of Steel, Woman of Kleenex problem? When I was a teen and Superman II came out, when Superman realized he and Lois were in love, he de-supered himself in the Fortress of Solitude so he could be with Lois.  Right after he emerged from the crystal cage that drained his powers, the first thing he does is take Lois' hand and escort her to a conveniently placed bed that we had never seen there before.  This implies it was necessary for him to lose his powers so he could have sex with Lois.  The problems involved in this situation are well covered in the wonderful essay by Larry Niven, .  But I also know that Superman has been in the comics for decades.  A friend of mine once said, "Oh, he tells her who he is a lot, but he always makes her forget it again later."  So how have the comic books (since this is Superman's primary medium) handled this?  Were there different ways it was handled, or was it always handled the same way?  Have they literally kept Superman and Lois from having sex for decades?  Or have there been times when they've ended up in bed together?  How do they deal with the issues of the Man of Steel having sex with, and not killing, a woman of Kleenex? This is a canonical question   How do I jailbreak an iPhone or iPad (to install a package manager like Cydia or Sileo for tweaks) on:   iOS/iPadOS 13 iOS 12 iOS 11 iOS 10 iOS 9 iOS 8 Which Pokémon can speak with humans? Inspired by Detective Pikachu, I was wondering which Pokémon can "speak" with humans.  I would like to include  Pokémon like Meowth in the anime that speak English/Japanese with their mouths, Pokémon like Mewtwo in the anime/movies that speak telepathically, or other Pokémon we've seen like Lugia/Entei that seem to speak without moving their mouths but still out loud for all to hear (presumably).  The format doesn't matter (manga, anime, video games). And while this is inspired by Detective Pikachu, it's not about it, as the movie is not out yet      And it seems like the human character has the special ability, not the Pikachu 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? Suppose the mean noon-time temperature for September days in San Diego is 24∘ and the standard deviation is 4.9. (Temperature in this problem is measured in degrees celsius)  On September 26, 1963, the all-time record of noon-time temperature in San Diego of 44∘ was hit. Assume the temperature distribution is symmetric around the mean, what is the Chebyshev bound for the probability of breaking (or tieing) this record?  I am having a hard time understanding this. Could someone explain to me how to do this? VMware Linux Server -- how can you tell if you are a vm or real hardware? An interesting question.  I have logged into a Linux (most likely SuSE) host.  Is there some way that I can tell programmatically that I am a VM host or not?    Also assume that the vmtools are not installed. Show that the sequence $a_0 = 1$, $a_{n+1 }= \sqrt{2+a_n}$ is monotonically increasing Given the sequence $a_0 = 1$, $a_{n+1}= \sqrt{2+a_n}$, how can I show that it is monotonically increasing?  I need it to show, that the sequence converges. I already proved boundedness but I can't figure out this last part. While browsing on Safari and listening to iTunes I come across a video and then press the "play" button on the keyboard (to stop the music while I watch the video) it starts/stops the video instead of the music (I have no control over the music).  How can I quickly play/pause my music without affecting the video? My professor sometimes asks me to referee papers for him.  Should I list the corresponding journals in the professionals activity section of my CV? Question: If $c\neq 1$ is an $n^{th}$ root of unity then, $1+c+...+c^{n-1} = 0$  Attempt: So I have established that I need to show that $$\sum^{n-1}_{k=0} e^{\frac{i2k\pi}{n}}=\frac{1-e^{\frac{ik2\pi}{n}}}{1-e^{\frac{i2\pi}{n}}} =0$$ by use of the sum of geometric series'. My issue is proceeding further to show that this is indeed true. 1.In classical mechanics, using Newton's laws, the ellipticity of orbits is derived. It is also said that the center of mass is at one of the foci.   2.Each body will orbit the center of the mass of the system.  My question is : Are the assumptions in 1 and 2 correct?   Follow up question : Assuming the distance from the centre of the mass to each body remains the same, do we have two bodies orbiting the centre of the mass of the system in an elliptical or circular orbit?   Finally : With elliptical orbits, if the heavier mass is supposed to be in one of the foci, if there is any significance to second focus, what is it? Is it a Lagrange point by any chance or does it have some other mathematical property? I came across this formula in a list:  The number of ways of distributing $n$ identical objects among $r$ groups such that each group can have $0$ or more $(\le n)$ objects  I know that standard way of doing this is to solve the problem of distributing n identical objects and $(r-1)$ partitions among themselves which can be done in $C(n+r-1,r-1)$ ways.   But I am unable to prove to myself why it is not $(r+1)^n$.  Because each of the n objects has r+1 choices, either group1, group2,... group r or none at all. "I cannot seem to find it" Journey SMS Fails - Object reference not set? I have a Journey which sends the Subscriber SMS messages, 2/4 journey entries (all unique subscribers) don't receive the SMS.  The error displayed in the SMS activity in my Journey History is the following:      Failed - Object reference not set to an instance of an object.   Can anyone advise what this means? If it was configuration surely all subscribers wouldn't receive the message? I can confirm all subscribers have a mobile number and they use the same format and country code. Not sure if it helps but the entry source is Salesforce Data Entry.  Edit:  There can be entries where the Subscriber record does not exist in the AllSubscribers list as they do not have email addresses attached (only mobile numbers) but my journey only contains SMS message activities, no email. I have checked and even on some Subscribers that do have AllSubscribers records suffer the same activity error. After swapping the positions of the hour and the minute hand, when will a clock still give a valid time? Why is my document pdf generated by pdflatex looks different from the one generated by latex+dvips+ps2pdf ? From the code below, i have tow different papers switch to the tow cited commands above :  \documentclass[10pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[francais]{babel} \usepackage[T1]{fontenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{graphicx} \usepackage{lipsum} \author{M ROUAN SERIK} \title{Article with \LaTeXe} \fontfamily{cmr} \begin{document} \maketitle \section{Titled section} \lipsum \end{document}   Here's my pdffonts for the tow pdfs : Pdflatex:    name                   type     encoding  emb sub uni object ID   ------------------------------- ---------------- --- --- --- ---------   AMTRHA+SFRM1728        Type 1    Custom   yes yes no       4  0   DPTCLU+SFRM1200        Type 1    Custom   yes yes no       5  0   JVZRKA+CMMI12          Type 1    Builtin  yes yes no       6  0   UEWNID+SFBX1440        Type 1    Custom   yes yes no       7  0   FOVERL+SFRM1000        Type 1    Custom   yes yes no       8  0     latex+dvips+ps2pdf      name                                 type              encoding         emb sub uni object ID   ------------------------------------ ----------------- ---------------- --- --- --- ---------   WVMUEH+SFRM1000                      Type 1C           WinAnsi          yes yes no        16  0   ZHRBNN+SFBX1440                      Type 1C           WinAnsi          yes yes no      14  0   IMJWCD+CMMI12                        Type 1C           Custom           yes yes no      12  0   MDMSVR+SFRM1200                      Type 1C           WinAnsi          yes yes no      10  0   TWHNMT+SFRM1728                      Type 1C           WinAnsi          yes yes no       8  0 Simulating a Brownian Excursion using a Brownian Bridge?