I am currently studying this type of Jacobi matrix for all n  Example of the type of matrix $n=4$ $$\mathbf{X}=\left[\begin{array}{*{20}{c}} {0}&amp;{1}&amp;{0}&amp;{0}\\ {1}&amp;{0}&amp;{1}&amp;{0}\\ {0}&amp;{1}&amp; {0} &amp;{1}\\ {0}&amp;{0}&amp;{1}&amp;{0} \end{array}\right]$$  I am struggling to prove that all eigenvalues for all n will be; $ \lambda \leq |2|$
$$A=\begin{array}{ccc} 0 &amp; 1/2 &amp; 0 &amp; \cdots &amp; 0 \\ 1/2 &amp; 0 &amp; 1/2 &amp;\cdots&amp; 0\\ 0 &amp; 1/2 &amp; 0 &amp; \cdots &amp; 0\\ \vdots &amp; \vdots &amp; \vdots &amp;\ddots &amp; 1/2\\ 0 &amp; 0 &amp; 0 &amp; 1/2 &amp; 0 \end{array}$$  I need to know the spectral radius of $A$ in function of the dimension of the matrix $n$. I know only one way to know the spectral radius, using the definition I find all eigenvalues and then find the maximum. but it very hard if $n$ is changing. Some help please!  I did some iteration by   $$det(\lambda I-A)$$  $$n=2 \quad r=\frac{\sqrt{1}}{2}$$ $$n=3 \quad r=\frac{\sqrt{2}}{2}$$  for $n=2$ and $n=3$ are correct.  and if we take this rule, for   $$n=5\quad \quad r=\frac{\sqrt{4}}{2}$$  but this isn´t true since I know radius for this matrix should be $&lt; 1$ and if we take this rule  for $n=5$ we have  $$\frac{\sqrt4}{2}=1$$   Some hint?
Save text area to file Find: $$\lim\limits_{x \to 1} \left( {\frac{x}{{x - 1}} - \frac{1}{{\ln x}}} \right) $$  Without using L'Hospital or Taylor approximations  Thanks in advance Proving $\frac{x}{y} +\frac{y}{z} + \frac{z}{x} \ge 3$ for positive $x,y,z$ Suppose $x,y,z$ are real positive numbers, prove that: $$\dfrac{x}{y} +\dfrac{y}{z} + \dfrac{z}{x} \ge 3$$ with equality when $x=y=z$.  Can someone help me find an easier solution? I started with assume only two are equal, without loss of generality, $x=y$. Then we get $$1 + \dfrac{x}{z} + \dfrac{z}{x}$$ $$1 + \dfrac{x^2 + z^2}{xz} = 1+ \dfrac{(x-z)^2 + 2xz}{xz} = 1 + 2 + \dfrac{(x-z)^2}{xz} \ge 3$$ To prove the general result I considered using a derivative with respect to $y$ to show that, starting with $x=y$ and increasing $y$ by a tiny amount, will show the equation's derivative as positive. I haven't gotten this to work yet. Any ideas? jQuery doesn't submit a form Difference between (in / for / ∅ / during / over) the (past / last) three years Microsoft has steadily improved its operating cash flows in the past three years. Microsoft has steadily improved its operating cash flows in the last three years. Microsoft has steadily improved its operating cash flows for the past three years. Microsoft has steadily improved its operating cash flows for the last three years. Microsoft has steadily improved its operating cash flows for the past three years. (&quot;for&quot; omitted) Microsoft has steadily improved its operating cash flows for the last three years. (&quot;for&quot; omitted) Microsoft has steadily improved its operating cash flows during the past three years. Microsoft has steadily improved its operating cash flows during the last three years. Microsoft has steadily improved its operating cash flows over the past three years. Microsoft has steadily improved its operating cash flows over the last three years.  Which of them are idiomatic and preferred? Are they interchangeable? sys.dm_db_index_physical_stats is extremly slow I have a database that is around 4.5TB, since we have parallel inserts (to reduce daily load time) on one table (partitioned by month) the clustered index on this table tends to be heavily fragmented. When I do a select from sys.dm_db_index_physical_stats (Limited) on this table it takes ages (> 4-5 hours). Is there a faster and better way to check the fragmentation levels on the partitions of this table, the current time this takes is completely inacceptable. Idiom or expression that would imply "someone is a habitual liar" I'm looking for an idiom or expression for describing people who tell lies out of habit (so you cannot trust them at all).   I know that these people are called , but I'm not looking for psychological terms.  Is there any idiom or expression for describing such people?   For example:     "Did X tell that to you?! Come on, don't believe him/her. Don't you know he/she ___ (=is a habitual liar)?" How to Compact geodatabases in multiple directories using ArcPy? I am trying to write a script to walk through all folders in my workspace and find and compact all geodatabases in my workspace.  I feel like I'm on the right track but I am stuck.  import acrpy, os from arcpy import env  arcpy.env.workspace "C:"  for (path, dirs, files) in os.walk(arcpy.env.workspace)     for file in files         if ".gdb" in file:         workspaces = arcpy.ListWorkspaces("*", "FileGDB")             for workspace in workspaces:             arcpy.Compact_management(workspace)   I am using ArcGIS 10.2.2 for Desktop. "Within" and "in" when referring to time Given that every class gets a minimum of 5 ability score improvements each at levels 4, 8, 12, 16, and 19 (Rogue and Fighter get more at other levels), do these ability score improvements occur by character level or by class level?  For example, is it possible to create a character who never once gains a single ability score improvement by reaching level 3 in a class and then switching to a different class repeatedly?  I had originally thought the 5 that every class gets would occur by character level, so for example a sorcerer1/bard3 would earn an ability score improvement for reaching character level 4. But there are others who have expressed the opinion that this is wrong. Why can't the Others go around the Wall at sea? As explained in a  earlier , the Wall does not extend all the way to the sea on the East, and the castle there is essentially undefended anyway. Thus, wildlings -- and Stannis' army -- were able to go around it undetected.  We know that the Others and the wights they raise can't cross through the Wall, and I think we can assume that it's built to stand up to any attacks they make trying to bring it down. But there does not appear to be any reason they can't just go around like anyone else.  Possibly related: in the TV Show, we actually see the attempted evacuation of Hardhome, and the Night King allows Jon Snow and his human army and civilians to flee back to their boats to safety, even after raising a massive army of wights. You could interpret this as merely a psychological ploy, but it may be a hint that the dead can't cross the water either.  Has there been any indication (books or show) that the Others/White Walkers and their army of wights are blocked by open water, or are somehow tied to the land of the Frozen North, or that there's any other obstacle keeping them from just sailing/swimming around the Wall? What is the name for the grammatical figure, where the present tense is substituted for a past event? I am preparing a book for publication which contains the following passage:     Stromberg in Religious Liberalism in Eighteenth-Century England   (1954) noted that 'the critical moment in the emergence of a positive   spirit of reform...consisted in the awareness of evil as being social   and remediable. It begins about 1750.'   My proof reader (my wife) has picked this out suggesting it should be the past tense. However if I say It began about 1750 it seems to convey too much certainty about the exact date. I just want to convey an impression of it being something which emerges in the mid-eighteenth century. I believe use of the present tense is the way to go.   But what is this artifice called, this use of the present to convey mild uncertainty? Can Raise Dead revive a character with zero maximum Hit Points? So, a character died from Draining Kiss of a succubus and his Max HP is 0. Can he be restored by Raise Dead, or does he need a higher level spell? Or a Greater Restoration cast on his lifeless body before Raise dead? I'm probably interested in RAW interpretation, since the character is in Adventurerer's league. New Beamer Version: Transparent Title Page Boxes Consider the following file:  \documentclass{beamer}  \setbeamercolor{title}{bg=red} \setbeamercolor{author}{bg=yellow} \setbeamercolor{date}{bg=} \setbeamercolor{institute}{fg=red,bg=green}  \title{TitelTitelTitel} \author{meiermueller} \institute{Institute} \date{lalala}  \begin{document}  \usebackgroundtemplate{\color{blue}\rule{20cm}{15cm}}    \begin{frame}    \titlepage  \end{frame} \end{document}   On one of my computers (using MiKTeX 2.9 and beamer 2015/01/05 3.36), the result is the following:    Note how, in particular, the date box has a transparent background  On my other computer, I installed a new MiKTeX version (2.9.6800), including beamer 2018/09/02 v3.52, and get this:    Here, the date field has a white background, and I cannot seem to get it to be transparent.   Apparently, the difference has some relation to the beamer version, or maybe some other change in the new MiKTeX version.  But what ist the reason, and how do I get transparent backgrounds? “While” loop without “do”; “if” statement without “then” I have a similar question as this :     How can I get rid of the do in a for loop and the then in an if   statement when typesetting algorithms using the algorithm and   algorithmicx packages?   but I want the same thing for algorithm2e package.  A minimal compilable example:  \documentclass[10pt]{article} \usepackage[linesnumbered,ruled]{algorithm2e} \SetKwRepeat{Do}{do}{while} \SetKw{KwGoTo}{go to}  \begin{document}  \begin{algorithm} \DontPrintSemicolon \KwIn{$A,B$} \KwOut{$C$} \While{$B &gt; 0$} {      $C = C+1$;      $B = B-1$;     \uIf{$C &gt; A$}        {            \textbf{break}\;        } } \caption{ABC} \end{algorithm}  \end{document} Cannot determine size of graphic I'm trying to include graphics in my Latex-file, which I compiled with latex+dvipdf on OS X. Latex however returns this error: "Cannot determine size of graphic"  My graphic is exported from PowerPoint, so I have tried both .pdf and .png. I get the same errors for both.  Sample code that doesn't work:  \begin{figure}[htb] \begin{center} \leavevmode \includegraphics[width=0.8\textwidth]{graph.png} \end{center} \end{figure}   If I have to manually set some measurements of the image, please tell me how I can find out those measurements. I need the image to take up about 80% of text width and be centered. Thanks. How to use measured step response to tune control system How does one use a measured step response to tune either a PID or convolution control scheme? Inspired by , I'm interested a more detailed explanation of how to implement a control system based on the measured step response.  I would not try to guess all the effects.  There are probably some non-obvious things going on, and you can't know all the parameters. I would measure the step response.  Find two pump settings that both result in the ball being within the measurable range within the tube.  Then have the controller suddenly switch from one setting to the other open loop.  Meanwhile measure what the ball does over time.  That is the step response.  You can take the derivative of that and get the impulse response.  From the impulse response you can predict the motion of the ball for any history of pump settings, assuming this is a linear system.  It is probably linear enough over the small range of settings to keep the ball within its normal range. You can use this as a simulation base to find the parameters for old fashioned PID control.  Or you can use the impulse response directly to do a convolution control.  You have to low pass filter the control input enough so that convolution kernel doesn't go negative unless your pump is actually reversible and can suck the ball back down.  How exactly does this work? ; I assume &quot;convolution control&quot; is the use of the pole-zero or transfer function, but don't see exactly how to get the parameters. RELATED:  * tikz: Distribute evenly and randomly circles I want to fill a part of a picture with more or less evenly and randomly distributed circles which should not overlap. It would be okay if circles are only partly inside the shape.  The density should be like what you see on this page  with values 80-100. (But I do not need so many circles, 30 should be enough, I could repeat the pattern).   I cannot use a grid and "wobble" the circles a bit: The circles are too large and it simply does not look randomly.   Currently I am thinking about a random attempt: Get some random point, measure its distance to existing circles, either paint or throw the point away. Try 100 times and hope that you get the right number of circles (and that it does not take too long). But perhaps someone has a better idea.   The example draws the circles simply manually:  \documentclass{article} \usepackage{tikz} \usetikzlibrary{backgrounds} \begin{document}   \begin{tikzpicture}[framed,gridded,radius=0.5cm]  \draw[red](0,0) rectangle (5,5);   \draw (1,1) circle;   \draw (2.3,1.1) circle ;   \draw (4.5,0.8) circle ;   \draw (5.1,1.8) circle ;   \draw (0.4,3.3) circle ;   \draw (2.1,2.8) circle ;   \draw (3.8,3.5) circle ;   \draw (4.8,4.2) circle ;   \draw (0.8,4.9) circle ;   \draw (2.1,4.1) circle ;   \draw (3.8,2.0) circle ;   \draw (3.5,0.6) circle ;   \draw (3.0,5.0) circle ;      \draw (4.1,5.1) circle ;             \draw (0.9,2.1) circle ;  \end{tikzpicture}  \end{document} Copying data over with rsync causes size discrepancies barchart with pgfplot - missing bar I am using PGFplot to make a barchart. There are 4 entries in my csv file and unfortunately I have only 3 bars in my figure.   my LaTeX code:  \begin{figure}     \centering     \begin{tikzpicture}          \begin{axis}[             x tick label style={/pgf/number format/1000 sep=},                 xlabel=Sequence,             ylabel=Heigth,                 ymin=0.7,             enlargelimits=0.05,             legend style={at={(0.5,-0.15)},                 anchor=north,legend columns=-1},             ybar interval=0.7,             width=\textwidth,             ]             \addplot table [col sep=comma,trim cells=true,y=height] {numericaltests/data/test.csv};              \addplot[red,sharp plot,update limits=false]                 coordinates {(-1,1.00) (34,1.00)}                 node[above] at (axis cs:15,1.00) {};             \legend{Ca2Fe2As2}         \end{axis}     \end{tikzpicture}     \caption{Sequence}     \label{fig:sequenceHeight} \end{figure}   my test.csv file:  sequence,height 1,2.0 2,1.5 3,1.8 4,2.4   Barchart: