Save byte array to file
Can a Byte[] Array be written to a file in C#?
Is it alright to use lowercase "i" or should you always use "I" (uppercase)? Why is method overloading and overriding needed in java? How should references in non-Latin script language be treated in English paper? Let's say I want to cite a Russian text in my own paper which is written in English. Should I:   reference it as is, in the original Cyrillic alphabet, although the readers won't be able even to pronounce it? transliterate the authors' names, while keeping the rest as is? transliterate the names and translate the title of the work, journal name, etc. into English? One-dimensional [Noetherian] UFD is a PID What is the difference between "thee" and "thou"? Is spell level a character concept or a player concept? Is there anything in RAW that supports spell level as a character concept, or alternatively that supports it as solely a player concept?  On the one hand, I see nothing that explicitly states that a character knows that there are spell levels. On the other hand, I don't see how a spellcaster could manage spell slots without knowing about spell levels. After Google Play Service update to version 13 I got an error Maya is sister of dhara who is doctor. Maya is sister of dhara, who is doctor.   In both statements who is used for which person??? Good books on complex numbers I searched on this site but I found posts asking for books on complex analysis which I guess is not what I want.   I want a book on problems concerning with roots of unity, argand plane, exponential form, inequalities etc. Some examples : , , .  I don't think these types of problems are covered in a analysis book but I don't know. I think I want a book on algebra of complex numbers.   Are these topics covered in analysis book ? If not, please suggests some books on this topic. I guess my question is bit vague, sorry for that. :).      As suggested in the comments I would like to have books similar to . My question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook.  Page 57. Exercise 12. Show that the following are polynomials by converting them to the form $\sum_{k=0}^{m}a_kx^k$ for a suitable $m$. In each case $n$ is a positive integer. $a)$ $(1+x)^{2n}.$ $b)$ $\frac{1-x^{n+1}}{1-x}, x\not=1.$ $c)$ $\prod_{k=0}^{n}(1+x^{2^k}).$  The attempt at a solution: a) part of the problem is pretty easy I guess, it is example of binomial theorem, so the answer would be $(1+x)^{2n}=\sum_{k=0}^{2n}(^{2n}_{k})x^k.$ Answer to part b) would be the following: $$\frac{1-x^{n+1}}{1-x}=\frac{(1-x)(1+x+x^2+\cdots+x^n)}{1-x}=1+x+x^2+\cdots+x^n=‌​\sum_{k=0}^{n}x^k.$$thanks to @DiegoMath's hint. As for part c), we have $$\prod_{k=0}^{n}(1+x^{2^k})=(1+x)(1+x^2)(1+x^4)(1+x^8)\cdots(1+x^{2^n})$$ and I have trouble &quot;converting&quot; this to sum which would be of a form of polynomial. How to prove that $E[Y]=E[E[E[Y|X_1, X_2]]]$ ?  PS. I don't see how $E[E(Y|X_{1},X_{2})|X_{1}]=Y[Y|X_{1}]$ and $E[Y]=E[E(Y|X_{1})]$ can be used here. But it feels close. Please help, I'm stuck PPS. Thanks @Deep North. Let's consider the case when X1 is independent of X2 Prove that if $ f\mathrm{(}z\mathrm{)} $ is analytic at the infinity  ,then :  $ {\lim}_{{z}\mathrm{\rightarrow}\mathrm{\infty}}{f}\prime{\mathrm{(}}{z}{\mathrm{)}}\mathrm{{=}}{0} $ We are working on a service that is quite new and unknown to people. It's is oriented on small business. The target audience is divided into 3 specific different groups. As for now, we have one landing page, which communicates to 3 groups.   We have very low conversion mainly because people don't understand how they can use the service and benefit from it.  So, we decided to change the logic:  There will be one entrance page, where user will have to choose his group (1 of 3) and then he will be forwarded to the page of the group he chose before, where all the explanations how to use the service are provided.  So instead of one page we will have four pages. Will it improve the communication with users and are there any suggestions how to deal with 3 different target audiences? I am getting the following error in a Drupal 8 site.     The following module is missing from the file system   It looks as if a custom module was installed and deleted without uninstalling it.   How can I fix this error? How to fix slow wireless on machines with Intel wireless cards? I have looked through all the previous questions but turning power management off for the Wi-Fi card didn't work. This problem seems to be with the Lenovo Thinkpads. I switched to Ubuntu 11.10 from Windows 7 3 days ago so I'm new to the operating system. Adding/updating field to polygon feature class that lists bordering (neighbor) polygons? I have a dataset of zipcodes.  I want to add a new field "NEIGHBORS" that includes bordering polygons in a comma delimited manner such as attached image: Change bounding box of math glyphs in LuaTeX I'm trying to use the Garamond-Math font in LuaTeX (despite the documentation saying I should use it with XeTeX only :-). It's working more or less OK with LuaTeX, except for some unfortunate glyph combinations, mostly at the end of inline math, such as “where $T$ is the period”.  I used \fbox to draw the bounding box of the glyph and noticed that the bounding boxes are different in LuaTeX:        and in XeTeX:        In XeTeX the glyph fits almost entirely in the bounding box, but in LuaTeX it overshoots far too much.  I'm not a font guy, so I have no idea why this happens. I thought the bounding box was a property of the font, so in my imagination the result should be the same. Thus the first question is: what actually happens that the bounding boxes are different?  The second question, in the title, is: can I change the bounding box of specific glyphs (using LuaTeX) so that the output looks more like XeTeX's?  I tried manually adding some spacing after the glyphs, but this looks hideous and will go down the drain if I happen to change the font or engine, so I'd really like not to do that. I also tried using the italic correction \/ but it seems to have no effect in math.  Here's the code to produce the pictures above:  \documentclass{standalone} \usepackage{unicode-math} \setmathfont{Garamond-Math.otf} \begin{document} \fboxrule=1sp \fboxsep=0pt In Lua\TeX{} this \fbox{$T$} looks horrible, but the \fbox{$X$} looks OK. \Huge\fbox{$T$}\quad. \end{document}   I'm using TL 2018:  This is LuaTeX, Version 1.07.0 (TeX Live 2018)  (format=lualatex 2019.3.29)  13 APR 2019 12:43  restricted system commands enabled. **test.tex (./test.tex LaTeX2e &lt;2018-12-01&gt; Lua module: luaotfload-main 2019-02-14 2.96 luaotfload entry point Pointwise convergence of $h_n(x) = x^{1+\frac{1}{2n-1}}$ Okay, maybe I'm missing something obvious but consider $h_n(x) = x^{ 1+ \frac{1}{2n-1}}$ on the set $[-1,1]$. Now consider a fixed $x \in [-1,1]$, we have:  $$\lim_{n\rightarrow \infty} h_n(x) = x \lim_{n \rightarrow \infty} x^\frac{1}{2n-1} = x$$  But my book says it's $|x|$? Why? How do I programmatically obtain the version in JBoss AS 5.1? Prove $\int\limits_{0}^{\infty} \mathrm{exp}(-ax^{2}-\frac{b}{x^{2}}) \mathrm{d} x = \frac{1}{2}\sqrt{\frac{\pi}{a}}\mathrm{e}^{-2\sqrt{ab}}$ This is Example 8.4.1 from Chapter 8 (The Normal Integral) of  by Boros and Moll. The authors outlined a solution method, which I will provide in full here. My question: is there another way to obtain this result?  Here is the solution of Boros and Moll with their notation.  Let \begin{equation} \mathrm{L}(a,b) := \int\limits_{0}^{\infty} \mathrm{exp}(-ax^{2}-\frac{b}{x^{2}}) \mathrm{d} x \label{eq:iic8-1} \tag{1} \end{equation} Making the substitution $t=x\sqrt{a}$ yields \begin{equation} \int\limits_{0}^{\infty} \mathrm{exp}(-ax^{2}-\frac{b}{x^{2}}) \mathrm{d} x = \frac{1}{\sqrt{a}} \int\limits_{0}^{\infty}  \mathrm{exp}(-t^{2}-\frac{ab}{t^{2}}) \mathrm{d} t \label{eq:iic8-2} \tag{2} \end{equation} Letting $ab=c$ we call the integral in equation \eqref{eq:iic8-2} $f(c)$, \begin{equation} f(c) = \int\limits_{0}^{\infty} \mathrm{exp}(-t^{2}-\frac{c}{t^{2}}) \mathrm{d} t \label{eq:iic8-3} \tag{3} \end{equation} so that  \begin{equation} \mathrm{L}(a,b) = \frac{f(ab)}{\sqrt{a}} \label{eq:iic8-4} \tag{4} \end{equation} In equation \eqref{eq:iic8-3} we let $y=\sqrt{c}/t$ \begin{equation} f(c) = \sqrt{c} \int\limits_{0}^{\infty} \mathrm{exp}(-y^{2}-\frac{c}{y^{2}}) y^{-2} \mathrm{d} y \label{eq:iic8-5} \tag{5} \end{equation}  Combining equations \eqref{eq:iic8-3} and \eqref{eq:iic8-5}, we have \begin{equation} f(c) = \frac{1}{2} \int\limits_{0}^{\infty} \mathrm{exp}(-t^{2}-\frac{c}{t^{2}}) \left(1+\frac{\sqrt{c}}{t^{2}} \right) \mathrm{d} t \label{eq:iic8-6} \tag{6} \end{equation} Now we let $s = t - \sqrt{c}/t$ \begin{equation} f(c) = \frac{\mathrm{e}^{- 2\sqrt{c}}}{2} \int\limits_{-\infty}^{\infty} \mathrm{exp}(-s^{2}) \mathrm{d} s = \frac{\sqrt{\pi}}{2} \mathrm{e}^{- 2\sqrt{c}} \lim_{z \to \infty} \mathrm{erf}(z) = \frac{\sqrt{\pi}}{2} \mathrm{e}^{- 2\sqrt{c}} \label{eq:iic8-7} \tag{7} \end{equation} Combining equations \eqref{eq:iic8-1}, \eqref{eq:iic8-4}, and \eqref{eq:iic8-7} yields our result.