Is there a standard expression for  $$\sum_{r=0}^{n}\binom nr^2$$
I would like to prove inductively that $${2n\choose n}=\sum_{i=0}^n{n\choose i}^2.$$  I know a couple of non-inductive proofs, but I can't do it this way. The inductive step eludes me. I tried naively things like $${2n+2\choose n+1}={2n+2\over n+1}{2n+1\choose n}=2\cdot {2n+1\over n+1}{2n\choose n},$$  But I don't think it can lead me anywhere. I would like the proof to be as simple as possible.
How to set Windows bootloader as default bootloader? I have a Win 7/11.10 dual boot system with Grub 2 as the default bootloader. However, I want to make Windows bootloader the default bootloader (grub 'dos' like interface scrares the hell out of my sister) so when I start the PC I will have the following boot options: 1. Windows 7 2. Ubuntu  NB: A process to reverse this operation so Grub can be the default bootloader again will also be appreciated On the photos where the lens flare is present, my Nikon D7000 is drawing strange labyrinth-looking artifacts. Here's an example.  The first image is a screenshot of a little part of the original photo zoomed 2:1 in Lightroom.  The second image is the same photo with Lightroom's contrast, clarity and sharpening set to the maximum to make the labyrinth pattern more visible.      This affects only some of the photos presenting lens flare. Here, for instance, the lens flare is present and light conditions are very close to the first photo, but there is no labyrinth pattern, only usual noise (screenshot of a part of a photo at 1:1 with contrast, clarity and sharpening set to the maximum):    Notes:   I can see this pattern only on some of the photos when the sun is in the frame or nearly in the frame. It is present with any lens I tested. The actual photo was shot at f/8, ISO 200, 1/640 s. I've seen this pattern in photos shot at ISO 100 as well. The photo is in RAW format, so this is not a JPEG artifact.   What is that? How do I avoid it? Let $M$ be a smooth manifold and let $N$ be a manifold with boundary. If $dF_{p}$ is an isomorphism, then $F\left(p\right)\in\mbox{int}M $. Let $M$ be a smooth manifold and let $N$ be a manifold with boundary, both with the same dimension $n$. If $dF_{p}$ is an isomorphism, then $F\left(p\right)\in\mbox{int}N  $.  I am trying to prove this theorem to prove a result about smooth embeddings.  Here is how I am thinking the problem could be solved. Assume thet $F(p)$ is a boundary point of $N$. Then there exists a chart $(V,\psi)$ at $F(p)$ such that $\psi(V) $ is an open subset of the upper half space $\mathbb{H}^{n}$. I guess we have to use some fact about $M$ having no boundary and the the fact that $dF_p$ is an isomorphism to show that there is a contradiction, but I cannot figure that out. I am having trouble loading csv files into layers using QgsVectorLayer().  I know this looks like a repeat, but I have carefully read     and    but still cannot get it work.  An example of my data file, test.csv:  Index,y,x 0,37.7646618,-122.4960858 1,37.777494,-122.416311 2,37.717707,-122.3997056 3,37.732284,-122.498118 4,37.749998,-122.392333   My code:  import os import urllib  import pathlib from qgis.core import QgsVectorLayer  local_path = 'C:\dev\FCMS_conflation\data\TMC_nodes.csv' node_layer = 'test'  abspath = os.path.abspath(local_path) params = {'delimiter': ',',           'decimalPoint': '.',           'xField': longitude,           'yField': latitude}  uri = "%s?%s" %(pathlib.Path(abspath).as_uri(), urllib.unquote(urllib.urlencode(params)))  layer = QgsVectorLayer(uri, node_layer, "delimitedtext")   The formatted URI looks like:  [] uri &gt;&gt;&gt; 'file:///C:/dev/FCMS_conflation/data/TMC_nodes.csv?decimalPoint=.&amp;delimiter=,&amp;xField=longitude&amp;yField=latitude'   I have tried many variations of the parameters, including the crs and type. But no matter what I try, I cannot create a valid layer.  [] layer.isValid() &gt;&gt;&gt; False   I have verified that I can add the file using the QGIS gui. I even tried using the uri provided by layer.metadata from within the Python plugin.   I am using QGIS 2.10.1 on Windows 7 64-bit machine. Tiny lifeforms on planetoid that live at accelerated time rate A book I once owned and read back in late-80's or early-90's that I have not been able to identify title &amp; author for:  It was a SF story concerning tiny native lifeforms, living on either a planetoid or possibly a dead/neutron star, that evolve at a seemingly accelerated time rate. Most of their story is told in the few days of observation (from orbit) by a human expedition to the planet, during which time-frame the race below progresses from stone-age tech level to being more advanced than the human observers, and eventually flying up to meet the humans in small advanced craft powered by micro-singularities... one of which accidentally 'disintegrates' the tip of a human scientist's nose in the process. I am a super newbie for Ubuntu and basically need Ubuntu for a Python package.  I would like to run a Python file in say e:/username/folder1/folder2/python.py. I tried every method online and the only reply is bash: cd./e: No such file or directory. I am not sure whether it is because of the bash windows or virtualenv.  When I use bash on Ubuntu on Windows, ls, it says VIRTUALENV_DIR. cd, it says /home/username. The cd .. etc changes the directory a little, but not another drive, nor the folder I would like it to find. Minimizing Application to system tray using WPF ( Not using NotifyIcon ) Browsing the internet from the command line How can I browse the Internet from the command line in Ubuntu? I would like to know where gravity gets its energy to attract physical bodies?  I know that the law of conservation states that total energy of an isolated system cannot change. So gravity has to be getting its energy from somewhere, or else things like hydropower plants wouldn't be able to turn the power of the falling water into a spinning rotor.  Just to be clear, Lets create an example:  Lets say we have two objects with equal mass close to each other. So gravity does its job and it pulls each other closer, this gets turned into kinetic energy. This is where I'm lost. According to the law of conservation energy can't be created or destroyed and the kinetic energy comes from the gravitational pull so where does the gravitational pull gets its energy.  If that energy isn't being recycled from some where else then that means you have just created energy, therefore breaking the law of conservation. How to disable X server autostart in Debian Jessie? We have this large database (>1TB) that we intend to "shrink".  The database revolves around one main entity, let's call it "Visit".  For discussion, let's say it is a database for a medical practice.  There are a total of 30 visit "types", such as procedure, annual, follow-up, immunisation etc, each of which is a subsidary table to "Visit", e.g. "visit_immuno".  The database has accummulated some 12 years of data since 2000.  Someone has proposed that we keep about 3 years of data in the "live" version and have the rest live in an "old_data" database.  The date is ONLY stored in the "Visit" table since it is normalised.  The Visit table also contains a ROWVERSION column and a BIGINT pseudo-identity (clustered) column.  For all intents and purposes, let's say the clustering key is populated by a SEQUENCE (SQL Server 2012 Enterprise) - we shall name it cid.  The visit.date is not always in the same order as the clustering key, for example when a doctor goes on extended visitations and returns with his "briefcase" of data, it gets merged into the main table.  There are also some updates to the "visit" table that will cause the ROWVERSION column to be out of sync with both the cid and date columns - to put it simply, neither ROWVERSION nor cid would make suitable partition keys for this reason.  The business rule for removing data from the "live" is that the visit.date must be greater than 36 months and a child visit_payment record must exist.  Also, the "old_data" database does not contain any of the base tables except visit%.  So we end up with:  Live DB (daily use) - All tables Old-Data DB - older data for the visit% tables  The proposal calls for a Combined DB that is a shell containing Synonyms to ALL the base tables in the Live DB (except visit%) plus Views that UNION ALL across the visit% tables in the two databases.  Assuming the same indexes are created in the Old-Data DB, will the queries perform well on the UNION-ALL Views?  What type of query patterns might trip up the execution plan for the UNION-ALL Views? Ubuntu 19.04 problems install upgrades How do I create an 8 sided die from one object? Simply put, is there a way to make a cube balance solely on one point/vertex?  I'm struggling. I do not want to have to join two objects, nor have to extrude an object. I just want to have a simple eight-sided die created with one mesh. Can only search social network messages in the Dash after 13.10 upgrade Missing SOAP action header Im trying to get the session ID of another user in the same org by sending the following request by using the data from my enterprise wsdl and its an developer edition.  &lt;?xml version="1.0" encoding="UTF-8"?&gt; &lt;soapenv:Envelope xmlns:soapenv="http://schemas.xmlsoap.org/soap/envelope/" xmlns:urn="urn:enterprise.soap.sforce.com"&gt;    &lt;soapenv:Header&gt;       &lt;urn:organizationId&gt;xxxxxxx&lt;/urn:organizationId&gt;    &lt;/soapenv:Header&gt;    &lt;soapenv:Body&gt;       &lt;urn:login&gt;          &lt;urn:username&gt;xxxxxxxxx&lt;/urn:username&gt;          &lt;urn:password&gt;xxxxxxxxx&lt;/urn:password&gt;       &lt;/urn:login&gt;    &lt;/soapenv:Body&gt; &lt;/soapenv:Envelope&gt;   But when i send this request i get      Missing SOAP ACTION header  but in my wsdl i dont have the action ,its blank    how to overcome this problem ?I got totaly struck over here since there is no action though its asking for action header.Following method handles the http request.   public string   processrrequest(string ep,string message,string method) {      HttpRequest req = new HttpRequest();     req.setEndpoint(ep);     req.setMethod(method);     req.setHeader('Content-Type', 'text/xml');     req.setHeader('SOAPAction','');     req.setBody(message);     system.debug('Request::'+req.getbody());     Http http = new Http();     HttpResponse res =http.send(req);     //Dom.Document doc  = res.getbodyDocument();     system.debug('Response::'+res.getbody());     system.debug('Response2::'+res.getStatus());     Dom.Document doc= new Dom.Document();     doc=res.getBodyDocument();     system.debug('doc::'+doc);     return 'hai'; } How to prove that the radial basis function is a kernel? How to prove that the radial basis function $k(x, y) = \exp(-\frac{||x-y||^2)}{2\sigma^2})$ is a kernel? As far as I understand, in order to prove this we have to prove either of the following:   For any set of vectors $x_1, x_2, ..., x_n$ matrix $K(x_1, x_2, ..., x_n)$ = $(k(x_i, x_j))_{n \times n}$ is positive semidefinite. A mapping $\Phi$ can be presented such as $k(x, y)$ = $\langle\Phi(x), \Phi(y)\rangle$.   Any help? Node for setting render colour, regardless of lighting I'm trying to make a material node setup to set final render colors, regardless of lighting conditions. Is such a setup possible? What is a deterministic method for evaluating a sensible buffer pool size? I'm trying to come up with a sane way to understand if the max server memory (mb) setting is appropriate (either should be lower, or higher, or stay the way it is).  I am aware that max server memory (mb) should always be low enough to leave room for the operating system itself, etc.    The environment I'm looking at has several hundred servers; I need a reliable formula I can use to determine if the present size of the buffer pool is appropriate since RAM is costed per GB allocated to each server.  The entire environment is virtualized, and "physical" RAM allocated to a VM can easily be changed up or down.  I have a particular SQL Server Instance I'm looking at now with a PLE of 1,100,052 seconds, which equates to 12.7 days (the amount of time the server has been up).  The server has a max server memory setting of 2560MB (2.5GB), of which only 1380MB (1.3GB) is actually committed.    I've read several items including one by Jonathan Keheyias () and another by Paul Randal (), and several others.  Jonathan advocates monitoring for a PLE below 300 per 4GB of buffer pool as being too low.  For the SQL Server Instance above, 300 * (2.5 / 4) = 187 results in a really really low target PLE below 300.  This instance has 290GB of SQL Server data (not including log files), and is only used for integration testing.  Assuming the last 12 days is representative of typical usage for this server, I'd say the max server memory (mb) setting could be lowered.  At the other end of the scale, I have another integration-test server with a PLE of 294, that has a max server memory (mb) setting of only 1GB.  This server has only 224MB of SQL Server data not including logs, and is running some BizFlow databases.  This server might benefit from a higher max server memory (mb) setting.  I'm thinking a good starting place for targets that might be assigned too much memory could include looking at:  SELECT      RamMB = physical_memory_in_bytes / 1048576     , BufferPoolCommittedMB = bpool_committed * 8192E0 / 1048576     , BufferPoolCommitTargetMB = bpool_commit_target * 8192E0 / 1048576     , PercentOfDesiredSizeMB = CONVERT(INT,(CONVERT(DECIMAL(18,2),bpool_committed)                              / bpool_commit_target) * 100) FROM sys.dm_os_sys_info;   If BufferPoolCommitTargetMB / BufferPoolCommittedMB is greater than 1, the server is not using the entire buffer pool.  If the machine in question also has a PLE of greater than "x" then it might be a good candidate for a decrease in max server memory (mb).  Since the Buffer Manager:Lazy writes/sec performance counter tracks the number of times SQLOS has written pages out to disk between checkpoints due to memory pressure, this might be another good thing to look at.  DECLARE @WaitTime DATETIME; SET @WaitTime = '00:00:15'; DECLARE @NumSeconds INT; SET @NumSeconds = DATEDIFF(SECOND, 0, @WaitTime); DECLARE @LazyWrites1 BIGINT; DECLARE @LazyWrites2 BIGINT;  SELECT @LazyWrites1 = cntr_value  FROM sys.dm_os_performance_counters dopc WHERE (         dopc.counter_name LIKE 'Lazy writes/sec%' COLLATE SQL_Latin1_General_CP1_CI_AS     )     AND dopc.object_name = 'MSSQL$' + CONVERT(VARCHAR(255),                SERVERPROPERTY('InstanceName')) + ':Buffer Manager';  WAITFOR DELAY @WaitTime;  SELECT @LazyWrites2 = cntr_value  FROM sys.dm_os_performance_counters dopc WHERE (         dopc.counter_name LIKE 'Lazy writes/sec%' COLLATE SQL_Latin1_General_CP1_CI_AS     )     AND dopc.object_name = 'MSSQL$' + CONVERT(VARCHAR(255),                SERVERPROPERTY('InstanceName')) + ':Buffer Manager';  SELECT LazyWritesPerSecond = (@LazyWrites2 - @LazyWrites1) / @NumSeconds;   The above code assumes the server is under load during the 15 seconds it takes to run, otherwise it will report 0; which might be a misleading false-negative.  Should I also look at PAGELATCHIO_* wait stats or some other wait type as an indicator of memory pressure, or lack thereof?   My question is, how can I reliably determine a "good" target value for PLE and max server memory (mb)? In the past, both Philosophy and Science were one. However, because of the vastness of Science, it was cut off. I am inclined to go along the same line of thinking, but, is there a way to merge both disciplines once again? biblatex: separator for citing multiple sources, some of them with page, some not I'm using biblatex for my references.  Normally, if I use \cite{source1, source2, source3}, I get a citation like  [3, 45, 98]   If I now want to give a page number for one of them, it looks confusing imho, as I should get (did not try it out up to now, honestly)   [3, 45, 78, p. 23f, 98]   So imho it would make sense to change one of the separator from a comma to e. g. a semicolon like:  [3; 45; 78, p. 23f; 98]   Question 1: Does that make sense, or are there other conventions for citing sources with and without page numbers together?  (Up to now I collected the citations without page number in one cite command and added singular \cite[][p. xy]{} commands for those with page numbers., so I'd get:   [3, 45, 98][78, p. 23f]   )  the \cites[]{}[]{}[]{} could also do that, but it also uses the same separators for entries and postnotes.  Question 2: How could I change the separator globally for those number-citations from comma to semicolon in biblatex?