inequality problem tough - putnam level
Interesting property of finite integer sequence with sum 1
Prove that if a function $f$ has a jump at an interior point of the interval $[a,b]$ then it cannot be the derivative of any function. Prove that if a function $f$ has a jump at an interior point of the interval $[a,b]$ then it cannot be the derivative of any function.  I know that for $f$ is differentiable in $(a,b)$ and that it has one-sided derivative $f_+' (a)≠f_-' (b)$ at the endpoints. If $C$ is a real number between $f_+' (a)$  and $f_-' (b)$, then there exists $c∈(a,b)$ such that $f' (c)=C $. How can I use this to prove the above? How to install qt 4.8 from repository? Why don't custom upstart job names tab complete in the service command? Does the regularization of a divergent infinite sum yield a unique value?  I.e. do different regularization schemes acting on the same infinite sum produce the same exact value independent of the regulator?  What, exactly, do these values mean?  Or what are they?  My understanding is that they are not "convergent" values.  (Sorry in advance for being a physicist. What do I need to consider to choose between dSLR, mirrorless, or a compact as my first "serious" camera? How does a spinning object "know" that it is spinning? I am constructing a thought experiment about a spinning object that is floating in intergalactic space.  I assume that this object is about the size of a planet so that it will have enough gravity so that a Foucault pendulum will work, although I'm not sure that this is necessary for the thought experiment.    I can easily determine that this object is spinning if I stand on the object and observe the galaxies around me rise and set.  Similarly, I can determine the axis of rotation.    For simplicity, I go to one of the poles of my object, and I set up my Foucault pendulum.  What will I see?  And why will I see it?  I assume that in intergalactic space, the gravity is very small (galaxies are very far away, and the mass of my object is very small compared to the mass of a galaxy) so that there will be little coupling between the gravitational field of my object and the gravitational field of the galaxies around me. Detecting start bit in software UART I'm experimenting with writing a software UART on my microcontroller using GPIO pins. This is to temporarily add a UART channel on a project until we get the new design implemented that uses a uC with more UART ports.  What I'm having difficulty with is correctly detecting a start bit in a serial stream. The source of the stream is external and doesn't care when my device powers up. So it's very likely that my device will power on and start seeing data bits in the middle of a byte transmission. Undoubtedly, that will cause my software UART to read erroneous values, as it won't be able to tell the difference between a start bit and any other high-to-low transition.  Is this an inevitable issue with a UART channel? Or is there some clever trick that the uC manufacturers use in their hardware UARTs? I'm confused. Running Fedora Linux, lscpu yields:  Architecture:            i686 CPU op-mode(s):          32-bit, 64-bit ...   But when I try to install a 64-bit program (Chrome) I get error like:     Package /....x86_64.rpm has incompatible architecture x86_64. Valid architectures are ['i686', 'i586', 'i486', i386']   I'm less interested in being able to install Chrome and more interested in why lscpu says that my CPU can run in 64-bit mode; clearly this can't mean I can run 64-bit programs. Can anyone clarify? How to install Unity 8? I'm using Ubuntu 14.04 and found out that we can install Unity 8 on it. Will Unity 8 replace Unity 7 once I install it? And if it does can I switch back to Unity 7 if I want? How would one go about it? Thousands of 404 errors in Google Webmaster Tools Because of a former error in our ASP.Net application, created by my predecessor and undiscovered for a long time, thousands of wrong URLs where created dynamically. The normal user did not notice it, but Google followed these links and crawled itself through these incorrect URLs, creating more and more wrong links.  To make it clearer, consider the url     example.com/folder   should create the link     example.com/folder/subfolder   but was creating     example.com/subfolder   instead. Because of bad url rewriting, this was accepted and by default showed the index page for any unknown url, creating more and more links like this.     example.com/subfolder/subfolder/....   The problem is resolved by now, but now I have thousands of 404 errors listed in the Google Webmaster Tools, which got discovered 1 or 2 years ago, and more keep coming up.  Unfortunately the links do not follow a common pattern that I could deny for crawling in the robots.txt.  Is there anything I can do to stop google from trying out those very old links and remove the already listed 404s from Webmaster Tools? RESTful Authentication What does RESTful Authentication mean and how does it work?  I can't find a good overview on Google.  My only understanding is that you pass the session key (remeberal) in the URL, but this could be horribly wrong. Sending and Parsing JSON Objects in Android I have a canon 760D. It has a EF-S lens mount. Am I able to use a EF lens with my camera? Why "a user" instead of "an user"? I am getting confused while using an article an. Because an is used while referring to vowels (a,e,i,o,u).  But today I read an article where for user they used a user.  My question is, why was an not used?  Could someone explain this with an example? It is quite easy to show that for every prime $p$ and $0&lt;i&lt;p$ we have that $p$ divides the binomial coefficient $\large p\choose i$; one simply notes that in $\large \frac{p!}{i!(p-i)!}$ the numerator is divisible by $p$ whereas the denominator is not (since it is a product of numbers smaller than $p$ and $p$ is prime).  My problem is with generalizing this argument for $q=p^n$. I'm looking for the most elegant and simple way to prove that $p$ still divides $\large q\choose i$. How did Harry become a Horcrux? The function $f(z)=z+\frac{1}{z}$ maps $\Omega$ biholomorphically on the upper plane The question states:     Let $\Omega$ be the upper half plane minus the unit disk. Show that the map $f: z \mapsto z+1/z$ maps $\Omega$ biholomorphically on the upper plane. Describe the preimages of the lines $\mathrm{Im}(z)=\text{a constant}&gt;0$.   So how exactly does one show the mapping is biholomorphic. I'm not sure where I should start, so unfortunately I haven't made any progress I was trying to concatenate text files in sub-folders and tried:  cat ./{mainfolder1,mainfolder2,mainfolder3}/{subfolder1}/book.txt &gt; out$var   However this did not return anything. So, tried adding a non existing 'subfolder2'  cat ./{mainfolder1,mainfolder2,mainfolder3}/{subfolder1,subfolder2}/book.txt &gt; out$var   And this time it did work out, concatenating the files successfully. Why does this happens? I have close to two hundred images where the subject looks like he just walked out of a sauna. It was 36 degrees with humidex, but he was not that red.   For comparison, attaching another picture, same camera, same lens, auto white balance, next day.       So, two questions, 1) how did that happen?, and 2) how do I fix it? I can bring down the red, but then I have to manually bring back the lip color in 200 pictures.   And next time when I deal with an older guy at 36 degrees C I will be shooting indoors.   *Edit: The camera is Olympus EPM 2. I am now guessing that he may have been THAT red, but my eyes tricked me. The last picture is from the beginning of the shoot before he got very hot and had two glasses of wine AND the background worked better for his skin and clothes. I do not have colour management setup, but I will get on it. I created a test MVC sharepoint app succesfully in a solution. Now I need to integrate this into a main solution project.   But when I added a SharePoint MVC App into the solution and hit F5, I am getting following error:  Error occurred in deployment step 'Install app for SharePoint':  Failed to install app for SharePoint. Please see the output window for details.   Detailed stack trace:  CorrelationId: 50583239-71cb-4b07-8bcb-4bbc51a22631      ErrorDetail: There was a problem with activating the app web definition.      ErrorType: App      ErrorTypeName: App Related      ExceptionMessage: Microsoft.SharePoint.SPException: Exception from HRESULT: 0x81070964 ---&gt; System.Runtime.InteropServices.COMException: Exception from HRESULT: 0x81070964 at Microsoft.SharePoint.Library.SPRequestInternalClass.EnableModuleFromXml(String bstrSetupDirectory, String bstrFeatureDirectory, String bstrUrl, String bstrXML, Boolean fForceUnghost, ISPEnableModuleCallback pModuleContext) at Microsoft.SharePoint.Library.SPRequest.EnableModuleFromXml(String bstrSetupDirectory, String bstrFeatureDirectory, String bstrUrl, String bstrXML, Boolean fForceUnghost, ISPEnableModuleCallback pModuleContext) --- End of inner exception stack trace --- at Microsoft.SharePoint.Administration.SPElementDefinitionCollection.ProvisionModules(SPFeaturePropertyCollection props, SPSite site, SPWeb web, SPFeatureActivateFlags activateFlags, Boolean fForce) at Microsoft.SharePoint.Administration.SPElementDefinitionCollection.ProvisionElements(SPFeaturePropertyCollection props, SPWebApplication webapp, SPSite site, SPWeb web, SPFeatureActivateFlags activateFlags, Boolean fForce) at Microsoft.SharePoint.SPFeature.Activate(SPSite siteParent, SPWeb webParent, SPFeaturePropertyCollection props, SPFeatureActivateFlags activateFlags, Boolean fForce) at Microsoft.SharePoint.SPFeatureCollection.AddInternal(SPFeatureDefinition featdef, Version version, SPFeaturePropertyCollection properties, SPFeatureActivateFlags activateFlags, Boolean force, Boolean fMarkOnly) at Microsoft.SharePoint.SPFeatureCollection.AddInternalWithName(Guid featureId, Int32 compatibilityLevel, String featureName, Version version, SPFeaturePropertyCollection properties, SPFeatureActivateFlags activateFlags, Boolean force, Boolean fMarkOnly, SPFeatureDefinitionScope featdefScope) at Microsoft.SharePoint.SPFeatureCollection.AddInternal(Guid featureId, Version version, SPFeaturePropertyCollection properties, Boolean force, Boolean fMarkOnly, SPFeatureDefinitionScope featdefScope) at Microsoft.SharePoint.SPFeatureCollection.Add(Guid featureId, Boolean force, SPFeatureDefinitionScope featdefScope) at Microsoft.SharePoint.SPUserSolutionCollection.EnsureWebFeaturesActivated(SPUserSolution solution) at Microsoft.SharePoint.Packaging.SPUserCodeSolutionDeploymentGroup.ActivateSolution(SPWeb web, SPUserSolution solution) at Microsoft.SharePoint.Packaging.SPUserCodeSolutionDeploymentGroup.Deploy() at Microsoft.SharePoint.Administration.SPAppTask.DeployOperation() at Microsoft.SharePoint.Lifecycle.MonitoredTaskExecution.DoTask()      Source: AppWeb      SourceName: App Web Deployment   Can somebody shed a light on this issue? :)