My question is not about the concept but about the use of these words.   Assuming this case:  int *a=new int[3]; int *b=a; // It is called shallow copy   My question: Is it right to call it a shallow copy? I can not see any shallow copy. int* is a pointer and it is all about storing some address. So, in the previous example every thing about it was totally copied because the actual data is not part of it. So, why it is called shallow not deep?  For more clarification, I see it like if someone do this:  std::string file_path="C:/file.txt"; std::string another_path=file path;   and called a shallow copy because the file was not really copied.  BTW, are deep/shallow words well-defined in the C++ standard?
What is the difference between a deep copy and a shallow copy?
Getting user provided data out of a database and into R I am animating a Rubik's cube, and as soon as I get to the second rotation the blocks that have previously been rotated start tripping out even though it should have been reset. I tried making the graph linear which fixed a lagging issue I had, but the blocks still start to spin instead of rotating around a single axis. PHP script not executing on Apache server Survival probability up to time $n$ in a branching process. Let $\{Z_n : n=0,1,2,\ldots\}$ be a Galton-Watson branching process with time-homogeneous offspring distribution $$\mathbb P(Z_{n,j} = 0) = 1-p = 1 - \mathbb P(Z_{n,j}=2), $$ where $0&lt;p&lt;1$. That is, $Z_0 = 1$ and $$Z_n = \sum_{j=1}^{Z_{n-1}}Z_{n,j}$$ for $n\geqslant 1$. Let  $$T=\inf\{n : Z_n=0\} $$ be the extinction time of the process. I want to find $$t_n := \mathbb P(T&gt;n), $$ i.e. the probability that the process survives up to time $n$. I found the following recurrence:  $$t_{n+1} = p(2t_n - t_n^2)$$ (with $t_0=1$) from this mathoverflow question:   I checked the recurrence for small values of $n$, and confirmed that $t:=\lim_{n\to\infty} t_n$ satisfies $$ t = p(2t - t^2),$$ both in the case where $t=0$ $\left(p\leqslant\frac12\right)$ and where $t=2-\frac1p$ $\left(p&gt;\frac12\right)$. So I'm fairly confident this recurrence is valid. However, I have no idea how to solve it.  For context, this problem comes from Adventures in Stochastic Processes by Sidney Resnick:   From @Did's comment, it appears to be intractable to find a closed form for $\mathbb P(T&gt;n)$. I find it curious that the question would be asked were it not, though. How can I get bold math symbols? To make Latin-letter variables bold I can use e.g. \mathbf{a}, but while putting Greek letters or symbols such as \nabla inside \mathbf doesn't cause any errors or warnings, it also doesn't do anything else.  What is the best way to make bold math symbols, in particular Greek letters and \nabla? Minion Pro, TL2016 and LuaLaTeX Is the function differentiable at $0$? Integer solutions to the equation $a^3+b^3+c^3=30$ Do full frame sensors gather more light than crop sensors? Lots of people say that full frame sensors receive more light than cropped sensors. I have never found a proof of this claim so I tried to do the computation by myself, and proved the contrary! Could you tell me if I am wrong ?  We want to compare the same frame with the same depth of field. I'm not concerned by the quantity of photons by photosite, which is unrelated to sensor size but to the density of pixels. I have neglected effects like vignetting or angle effect on the edge of the micro-lens array. Here is my simple reasoning :  If you want the same  α~2arctan(size/2f) with a full-frame sensor and a crop sensor with crop ratio c, you have to multiply the focal length by approximately c. Now, in order to maintain the , the f-number N have to be divided by c. If we measure the "amount of light" with the well defined  Ev provided by the same frame of the same scene (so the  is fixed), we have Ev ~ f/N.  Putting all together, Ev_crop = Ev_ff x c², so the cropped sensor receive more light than the full frame sensor !  For those who are interested in the price of two equivalent systems, one with a FF+50mm+135mm and the other with Crop+35mm+85mm, see this . Russian babel and chemformula In "Knots and Primes: An Introduction to Arithmetic Topology", the author uses the following proposition   Let $h: Y \to X$ be a covering. For any path $\gamma : [0,1] \to X$ and any $y \in h^{-1}(x) (x = \gamma(0))$, there exists a unique lift $\hat{\gamma} : [0,1] \to Y$ of $\gamma$ with $\hat{\gamma}(0) = y$. Furthermore, for any homotopy $\gamma_t (t \in [0,1])$ of $\gamma$ with $\gamma_t = \gamma(0)$ and $\gamma_t(1) = \gamma(1)$, there exists a unique lift of $\hat{\gamma_t}$ such that $\hat{\gamma_t}$ is the homotopy of $\hat{\gamma}$ with $\hat{\gamma_t}(0) = \hat{\gamma}(0)$ and $\hat{\gamma_t}(1) = \hat{\gamma}(1)$.  The author then follows with; "In the following, we assume that any covering space is connected. By the preceding proposition, the cardinality of the fiber $h^{-1}(x)$ is independent of $x \in X$."   I am not sure why this results is true. This is my attempt at explaining it to myself. Take two different $x_1, x_2$ and their fibers $h^{-1}(x_1), h^{-1}(x_2)$ such that $y_1 \in h^{-1}(x_1)$ and $y_2 \in h^{-1}(x_2)$. Take two paths $\gamma_1, \gamma_2$ such that $\gamma_1(0) = x_1$ and $\gamma_2(0) = x_2$. Then we get two lifts $\hat{\gamma_1}, \hat{\gamma_2}$ with $\hat{\gamma_1}(0) = y_1$ and $\hat{\gamma_2}(0) = y_2$. Now, since our covering space is connected we can continuously deform $\hat{\gamma_1}$ into $\hat{\gamma_2}$ and conclude that every elements in $h^{-1}(x_1)$ is also in $h^{-1}(x_2)$ and vice versa. I feel that this is wrong but cannot figure out the right way to see this. Any help would be appreciated. The unique style of the diagrams at  has an informative but nice hand-drawn touch. I guess they are actually drawn by hand but just recently on our partner site for Mathematica someone asked how to draw a similar diagram such as this one with Mathematica's plot functions ().    Now without Mathematica this should be possible as well but I have a hard time getting the details right. Can such a diagram be created in TeX with a similar design? My first attempt does not look particularly great but here it is (the font is from ):      \documentclass{standalone}     \usepackage{tikz,color}     \usepackage{pgfplots}     \usepackage{fontspec}     \setmainfont{Humor-Sans}      \begin{document}     \begin{tikzpicture}     \pgfplotsset{every axis/.append style={line width=1pt}}      \begin{axis}[%         axis x line=bottom,         axis y line=left,         xtick={1.2, 2.9, 4.5},         xticklabels={-3 days, -1 day, 5 minutes},         yticklabels={}     ]     % 1st plot     \addplot [cyan!80!white, samples=200, domain=0:6] {0.05*rnd+3+(sin(deg(x))^2)/sqrt(x)*exp(-(x-2))};     % white background for 2nd plot     \addplot [white, samples=200, domain=0:6,line width=4pt] {0.05*rnd+0.4*x+2+x^2*sin(deg(x))^2*exp(-x)};      % 2nd plot     \addplot [red!80!white, samples=200, domain=0:6] {0.05*rnd+0.4*x+2+x^2*sin(deg(x))^2*exp(-x)};      \end{axis}       \node at (2.7,4.6) {downhill};     \draw  plot[smooth, tension=.7,line width=2pt] coordinates {(1.2679,5.5206) (1.9454,5.493) (2.4017,4.9261)};      \end{tikzpicture}      \end{document} Can a mathematically sound prediction interval have a negative lower bound? I have used R to form a 95% prediction interval for the number of endemic species on an island. My lower bound is negative – is that mathematically sound?  In the linear model used in the prediction interval, the data used are:  Area Surface area of island, hectares DiscSC Distance from Santa Cruz, kilometres Elevation Elevation of higher point in metres and it is coded as such:   selected.model &lt;- lm(ES ~ Area + Elevation + DistSC + I(Elevation^2)                      + (Elevation:DistSC) + (A‌​rea:Elevation))    and stepwise regression was performed to find this "best" model   I'm not exactly sure how a prediction interval works. I just want to make sure it is OK. Obviously a negative number of species is incorrect, but I know it takes into account the uncertainty of the mean as well as data scatter. Well, this is an exercise problem from Herstein which sounds difficult:   How does one prove that if $|G|&gt;2$, then $G$ has non-trivial automorphism?   The only thing I know which connects a group with its automorphism is the theorem, $$G/Z(G) \cong \mathcal{I}(G)$$ where $\mathcal{I}(G)$ denotes the Inner- Automorphism group of $G$. So for a group with $Z(G)=(e)$, we can conclude that it has a non-trivial automorphism, but what about groups with center? How to use grep and cut together? Lets say I'm doing a database for fruits. Using a text file for it, inside contains, in this format,  (Fruit:Quantity:Cost) Apple:10:1 Orange:20:2 Pears:10:3 Banana:20:4  10 Apples at $1 each. 20 Oranges at $2 each. 10 Pears at $3 each. 20 Bananas at $4 each.   My simple script goes like this:  echo "Enter fruit to find" read fruit  echo "Enter quantity to find" read count  echo "Enter cost to find" read cost  grep $fruit FruitDB.txt     It will display the fruits in the database.  But how do i use grep and cut together to find just by the quantity or the cost. In the quantity count area, let's say i enter 20, it will display the Oranges and Banana. Or when i enter count and cost, it will display just those that meet the conditions. Uniform convergence of $\sum_{n=1}^{\infty}\frac{1}{n(1+(x-n)^2)}$ Is $$\sum_{n=1}^{\infty}\frac{1}{n(1+(x-n)^2)}$$ uniformly convergent on $(0,+\infty)$? Sadly, Weierstrass' criterion nor Cauchy's condition don't seem to work here and I just ran out of ideas. Copy material to another object? I saw a video where the person copied a material from one object to another (Cycles).  How can I accomplish this? Apply a string replacement on the citation as returned by \cite How to Redirect Pages with Query String? I want to Redirect as seen below:  https://example.com/p1/p2?sort=p.model&amp;order=ASC&amp;page=2   to  https://example.com/p3/p4?sort=p.model&amp;order=ASC&amp;page=2   Also, I have this code in my .htaccess file :  RedirectMatch 410 ^/p1/p2&amp;   I have used below code, but I just see 410 code for that URL :  RewriteRule ^/p1/p2?sort=p.model&amp;order=ASC&amp;page=2$ /p3/p4?sort=p.model&amp;order=ASC&amp;page=2 [QSD, R=301,L]   So, I have changed RedirectMatch 410 ^/p1/p2&amp; to Redirect 410 ^/p1/p2&amp; , but this time I have seen 404 Code for my Request URI. What should I do? Quantum mechanics textbooks that use path integrals I'm looking for a textbook in quantum mechanics that relies heavily on Green functions and the path integral formalism to supplement my QM books. I want to do some calculations using alternative methods (path integrals, green functions, Lagrangians etc).