If I have two vectors from $\mathbb{R}^q$ then their inner product gives the length of the projection of one on them onto the other multiplied by the other's length. I have searched but couldn't find an intuitive analoge interpretation for the complex inner product. And why do we need complex inner products?  (I'm learning from the book Linear Algebra Done Right of Axler, and they don't really give a motivation for it either, only for the real case..)
To be clear on this, I know what is the definition of an inner product space and some properties and theorems about them. What I am asking for is an intuition for this definition in the complex case. In the real case, the intuition (or at least one of them) is  geometric: The inner product of two vectors is the length of the projection of the first to the second scaled by the norms of both vectors so that it is symmetric (modulo some details). In particular I (and everybody else) think of "inner product zero" as geometric orthogonality and of orthonormal bases as, well, orthonormal bases and so on. The question is, what should I think about when working with complex (or should I say hermitian?) inner product spaces? what is the "meaning" of the complex number associated to two vectors called their inner product?  I will be happy to hear all kinds of answers. For example, what physical phenomena does it model or in what mathematical situations does in "naturally" appear. Answers that stress the "nice structure" resulting are also welcome, yet I feel that by itself  it is a bit unsatisfying.
Fantasy book where a young prince becomes a mage It is perhaps a bit embarrassing that while doing higher-level math, I have forgot some more fundamental concepts. I would like to ask whether the square root of a number includes both the positive and the negative square roots.  I know that for an equation $x^2 = 9$, the solution is $x = \pm 3$. But if simply given $\sqrt{9}$, does one assume that to mean only the positive root? And when simply talking about the square root of a number in general, would one be referring to both roots or just the positive one, when neither is specified? Get total duration of video files in a directory I have a list of .ts files:   out1.ts ... out749.ts   out8159.ts  out8818.ts   How can I get the total duration (running time) of all these files? I need to insert 800000 records into an MS Access table. I am using Delphi 2007 and the TAdoXxxx components. The table contains some integer fields, one float field and one text field with only one character. There is a primary key on one of the integer fields (which is not autoinc) and two indexes on another integer and the float field.  Inserting the data using AdoTable.AppendRecord(...) takes > 10 Minutes which is not acceptable since this is done every time the user starts using a new database with the program. I cannot prefill the table because the data comes from another database (which is not accessible through ADO).  I managed to get down to around 1 minute by writing the records to a tab separated text file and using a tAdoCommand object to execute  insert into table (...) select * from [filename.txt] in "c:\somedir" "Text;HDR=Yes"   But I don't like the overhead of this.  There must be a better way, I think.  EDIT:  Some additional information:   MS Access was chosen because it does not need any additional installation on the target machine(s) and the whole database is contained in one file which can be easily copied. This is a single user application. The data will be inserted only once and will not change for the lifetime of the database. Though, the table contains one additional field that is used as a flag to indicate that the corresponding record in another database has been processed by the user. One minute is acceptable (up to 3 minutes would be too) and my solution works, but it seems too complicated to me, so I thought there should be an easier way to do this. Once the data has been inserted, the performance of the table is quite good. When I started planning/implementing the feature of the program working with the Access database the table was not required. It only became necessary later on, when another feature was requested by the customer. (Isn't that always the case?)   EDIT:  From all the answers I got so far, it seems that I already got the fastest method for inserting that much data into an Access table. Thanks to everybody, I appreciate your help. The canonical way to return multiple values in languages that support it is often . Option: Using a tuple Consider this trivial example: def f(x):   y0 = x + 1   y1 = x * 3   y2 = y0 ** y3   return (y0, y1, y2)  However, this quickly gets problematic as the number of values returned increases. What if you want to return four or five values? Sure, you could keep tupling them, but it gets easy to forget which value is where. It's also rather ugly to unpack them wherever you want to receive them. Option: Using a dictionary The next logical step seems to be to introduce some sort of 'record notation'. In Python, the obvious way to do this is by means of a dict. Consider the following: def g(x):   y0 = x + 1   y1 = x * 3   y2 = y0 ** y3   return {'y0': y0, 'y1': y1 ,'y2': y2}  (Just to be clear, y0, y1, and y2 are just meant as abstract identifiers. As pointed out, in practice you'd use meaningful identifiers.) Now, we have a mechanism whereby we can project out a particular member of the returned object. For example, result['y0']  Option: Using a class However, there is another option. We could instead return a specialized structure. I've framed this in the context of Python, but I'm sure it applies to other languages as well. Indeed, if you were working in C this might very well be your only option. Here goes: class ReturnValue:   def __init__(self, y0, y1, y2):      self.y0 = y0      self.y1 = y1      self.y2 = y2  def g(x):   y0 = x + 1   y1 = x * 3   y2 = y0 ** y3   return ReturnValue(y0, y1, y2)  In Python the previous two are perhaps very similar in terms of plumbing - after all { y0, y1, y2 } just end up being entries in the internal __dict__ of the ReturnValue. There is one additional feature provided by Python though for tiny objects, the __slots__ attribute. The class could be expressed as: class ReturnValue(object):   __slots__ = [&quot;y0&quot;, &quot;y1&quot;, &quot;y2&quot;]   def __init__(self, y0, y1, y2):      self.y0 = y0      self.y1 = y1      self.y2 = y2  From the :  The __slots__ declaration takes a sequence of instance variables and reserves just enough space in each instance to hold a value for each variable. Space is saved because __dict__ is not created for each instance.  Option: Using a  (Python 3.7+) Using Python 3.7's new dataclasses, return a class with automatically added special methods, typing and other useful tools: @dataclass class Returnvalue:     y0: int     y1: float     y3: int  def total_cost(x):     y0 = x + 1     y1 = x * 3     y2 = y0 ** y3     return ReturnValue(y0, y1, y2)  Option: Using a list Another suggestion which I'd overlooked comes from Bill the Lizard: def h(x):   result = [x + 1]   result.append(x * 3)   result.append(y0 ** y3)   return result  This is my least favorite method though. I suppose I'm tainted by exposure to Haskell, but the idea of mixed-type lists has always felt uncomfortable to me. In this particular example the list is -not- mixed type, but it conceivably could be. A list used in this way really doesn't gain anything with respect to the tuple as far as I can tell. The only real difference between lists and tuples in Python is that lists are , whereas tuples are not. I personally tend to carry over the conventions from functional programming: use lists for any number of elements of the same type, and tuples for a fixed number of elements of predetermined types. Question After the lengthy preamble, comes the inevitable question. Which method (do you think) is best? What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity? Review option on mobile phone not available? How do I update jQuery to the latest version I can download? `time echo` got no output I'm trying to use the eVis plugin in QGIS 3.16.3 to display geotagged photos (as I have in the past without difficulty), but the plugin has totally disappeared. Not only can't I find it as an icon or under the Database menu, but it doesn't appear in the list of plugins (and yes, I am looking in the complete list of all plugins!). Has it been moved somewhere I haven't come across or is it a bug? Prevent questions on Hot List from being upvoted by casual visitors (only rep is from association bonus) Running tcpdump starts ssh flood I have runtime Linux running in a PLC. My development machine is running Ubuntu 14.04. The PLC and the development machine are connected through a five port switch. I ssh to PLC from my development machine to transfer executable of a networking application that I develop on my development machine.   I run tcpdump on PLC to debug the kind of packets my network application is receiving BUT as soon as I start tcpdump, there starts a flood of ssh sync and acks back and forth from my ubuntu machine to the PLC. The LED lights of ethernet controller of PLC as well on the switch start blinking rapidly. The flood overwhelms the PLC and it almost dies down while all other packets get dropped. It feels like a DOS attack whenever I run tcpdump but as soon as I stop tcpdump, everything calms down and gets back to the normal.   I would like know remedy on this problem and why it happens?     Any help would be appreciated. Proof explanation $A\setminus B = A \setminus B \cap A$ $A\setminus B = A \setminus B \cap A$  Proof $A\setminus B \subseteq A \setminus B \cap A$:  $x\in A\setminus B \to x\in A \land x \notin B$  $x\in A \land (x \notin B \color{red}{\lor x\notin A})$  $x \in A \land (x\notin B \cap A)$  $x \in A\setminus B\cap A$  Proof $A \setminus B \cap A \subseteq A\setminus B$:  $x \in A \setminus B \cap A \to x\in A \land x\notin B \cap A$  $x \in A \land (x\notin B \color{red}{\lor x\notin A})$  $x \in A \land x \notin B$  $x \in A \setminus B $  Therefore $A\setminus B = A \setminus B \cap A$  I am confused how the stuff in the red is formed. Can we add an or statement to anything and ignore if its true or not if the first part of the or is true? On the importance of the i.i.d. assumption in statistical learning Windows Starter Edition, Home and Home Premium do not include gpedit, how do I install it? I want to make some changes to Windows that require using the Group Policy Editor (gpedit.msc).  Unfortunately the Group Policy Editor is not included with the Starter Edition, Home and Home Premium editions of Windows.  How can I install it? How do you rename root? Not that it's a very good idea to change it, but for fun. According to  post, there are still some problems even after changing entries in /etc/passwd, /etc/shadow, and /etc/sudoers. Any suggestions? Reference label with included file and relaxed clearpage I am running into a rather weird error that apparently originates from the combination of \include, \let\clearpage\relax and a referenced label after a page break.  In this particular example, the label mylabel defined in the listing cannot be referenced anymore when the reference and the label are close enough to a page break.  The MWE consists of two files, the main-file and an included file. I am using the \let\clearpage\relax command to . The error does not occur if the file is \input instead of \included.  \documentclass[11pt,twoside]{book} %  \usepackage[utf8]{inputenc} % Enoding \usepackage[english]{babel} % Language \usepackage{listings} \usepackage{forloop}  \begin{document} \begingroup \let\clearpage\relax  \include{includedfile} % changing include for input would work  \endgroup \end{document}   Included file:   \newcounter{ct} \forloop{ct}{1}{\value{ct} &lt; 40}% {%     .\\ }  Listing \ref{mylabel}.  \begin{lstlisting}[caption=somecaption, label=mylabel] some code; \end{lstlisting}   Running this example will have an output where the label cannot be referenced (even after multiple runs):    Now, one solution is to use \input as mentioned earlier but I am rather interested in why this happens. Does the \clearpage command fulfil an important function for references to work? And can the error be avoided without removing \let\clearpage\relax while using \include?  EDIT: I just found  with the same issue (though it does not have a definite answer). Ignore folders/files when Directory.GetFiles() is denied access I am trying to display a list of all files found in the selected directory (and optionally any subdirectories). The problem I am having is that when the GetFiles() method comes across a folder that it cannot access, it throws an exception and the process stops.  How do I ignore this exception (and ignore the protected folder/file) and continue adding accessible files to the list?  try {     if (cbSubFolders.Checked == false)     {         string[] files = Directory.GetFiles(folderBrowserDialog1.SelectedPath);         foreach (string fileName in files)             ProcessFile(fileName);     }     else     {         string[] files = Directory.GetFiles(folderBrowserDialog1.SelectedPath, "*.*", SearchOption.AllDirectories);         foreach (string fileName in files)             ProcessFile(fileName);     }     lblNumberOfFilesDisplay.Enabled = true; } catch (UnauthorizedAccessException) { } finally {} How to run grep on a single column? What is the difference between "I am done" and "I am through"?