Show that $\|T\|=\|T^*\|$
Norm of adjoint operator in Hilbert space
Existence of a Pivot Point in Euclidean Geometry In geometry there is a term called a pivot point. There exists a pivot point 'P' in the interior of a convex polyhedron in a euclidean space if and only if every single line through 'P' contains exactly zero or two vertices of the polyhedron.  I asked one of my professors, but he gave me a very narrow example of a cube, but I want to know the maximum number of pivot points that exist in a polyhedron. Can someone please help me prove how many exist? Read Value from URL in PHP What are some alternatives to "hover" on touch-based devices? Came up recently in a 37signals podcast:    The straightforward, "brute-force" solution is to just use the first touch on a UI object as the hover.    It works. But are there other, more elegant solutions? For instance, could the hardware ever be made to support actual finger hovering? biblatex: Tighter integration of \textcite "in the flow of text" This question led to a new feature in a package:      Many biblatex citation styles feature the \textcite command which is "intended for use in the flow of text, replacing the subject of a sentence" (manual, section 3.6.2). At the moment, this works well for citations of single works -- e.g. (assuming the authoryear style), one may write in the source  The reference work is \textcite{knuth:ct:a}.   which will be typeset as     The reference work is Knuth (1984).   Citing multiple works, on the other hand, is somewhat tedious. If one simply writes (1):  Important works are \textcite{knuth:ct:a,knuth:ct:b,markey}.   the output is rather a list than a correct sentence (2):     Important works are Knuth (1984); Knuth (1986); Markey (2005).   One has to write (3):  Important works are \textcite{knuth:ct:a}, \textcite{knuth:ct:b}, and \textcite{markey}.   to get (4):     Important works are Knuth (1984), Knuth (1986), and Markey (2005).   Can \textcite be modified so that input (1) will yield output (4), hereby achieving a tighter integration "in the flow of text"?  \documentclass{article}  \usepackage[style=authoryear]{biblatex}  \addbibresource{biblatex-examples.bib}  \begin{document}  This is how \texttt{\textbackslash textcite} currently works:\\ Important works are \textcite{knuth:ct:a,knuth:ct:b,markey}.  \bigskip  But I'd like \texttt{\textbackslash textcite} to output the following:\\ Important works are \textcite{knuth:ct:a}, \textcite{knuth:ct:b}, and \textcite{markey}.  \printbibliography  \end{document} Finding the conjugate hyperbola Assume we are given a general proper hyperbola $  a_{11}x^2 + 2 a_{12} x y + a_{22} y^2 + 2 a_{13}x +2 a_{23}y + a_{33} = 0$ with  $D =\det \begin{pmatrix}   a_{11} &amp; a_{12} \\   a_{12} &amp; a_{22} \end{pmatrix} &lt; 0$ and $ A = \det \begin{pmatrix}   a_{11} &amp; a_{12} &amp; a_{13} \\   a_{12} &amp; a_{22} &amp; a_{23} \\   a_{13} &amp; a_{23} &amp; a_{33} \end{pmatrix} \neq 0$.  Is there an easy way to find the implicit description of the conjugate hyperbola?  I could translate and rotate the hyperbola and obtain a canonical representation of the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = \pm 1$, find the conjugate of the canonical one and the transform back. However, I'm looking for a more straightforward way to achieve this. Distribution of orbital velocities in a disk galaxy for N-body simulation? Please add a system to allow file uploads attached to questions and answers Maintain the aspect ratio of a div with CSS How do I select all the columns in a table that only contain NULL values for all the rows?  I'm using MS SQL Server 2005.  I'm trying to find out which columns are not used in the table so I can delete them. Unicode Processing in C++ What is the best practice of Unicode processing in C++? How to make HTML code inactive with comments Producing PDF/X compliant document with XeLaTeX If $id:(X,d_1)\to (X,d_2)$ is continuous then what will be $X$? How do I disable updating a form in Windows Forms? During a complicated update I might prefer to display all the changes at once. I know there is a method that allows me to do this, but what is it? Negative values for AICc (corrected Akaike Information Criterion) I have calculated AIC and AICc to compare two general linear mixed models; The AICs are positive with model 1 having a lower AIC than model 2.  However, the values for AICc are both negative (model 1 is still &lt; model 2).  Is it valid to use and compare negative AICc values? Multivariable limit - perhaps a trickier problem I am stuck on. I am trying to solve the following limit:  $\lim_{(x,y) \to (0,0)} \frac{x^4y^4}{(x^2 + y^4)^3}$  (This is a more challenging problem from Folland Calculus, it seems).  I am pretty sure this limit does not exist (however, this is just a guess, and I am not 100% sure.)  I am trying to approach via the paths y = mx and x = my but... they don't seem to work. I have tried simpler cases such as x = 0 and y = 0.   Any help appreciate. How can Cooper communicate with his daughter before the time he went to space? According to Ohm's Law, if the resistance is zero, the power dissipated by a resistor is unmeaning.  $$P_d = R \times I^2$$  $$I = \sqrt{ \frac {P_d}{R} }$$  Can one calculate the resistance given that power is defined? Only by knowing power, how can one calculate the maximum current of a zero-Ohm resistor?   I'm a bit confused as I've seen there is zero-Ohm resistors with specified power in the market. When I have these facts,   French coins were not found in the ruin. German coins were not found in the ruin.   which of these sentences are grammatically correct?  1.     No French and German coins were found in the ruin.   2.     No French or German coins were found in the ruin.   3.     No french nor German coins were found in the ruin.   I think 1 is correct but I'm not sure.  By the way, I can also say     Neither German nor French coins were found in the ruin.   Also, when you have an additional fact   Italian coins were not found in the ruin   which of these statements are grammatically correct?  1.     No French, German and Italian coins were found in the ruin.   2.     No French, German or Italian coins were found in the ruin.   3.     None of French, German and Italian coins were found in the ruin.   4.     None of French, German or Italian coins were found in the ruin.   Thanks, What are the differences in implementing interfaces implicitly and explicitly in C#?  When should you use implicit and when should you use explicit?  Are there any pros and/or cons to one or the other?    Microsoft's official guidelines (from first edition ) states that using explicit implementations are not recommended, since it gives the code unexpected behaviour.  I think this guideline is very valid in a pre-IoC-time, when you don't pass things around as interfaces.  Could anyone touch on that aspect as well?