To denote non-empty subsets, I repeatedly find myself writing $A\subset S, A\neq \emptyset$. Is there any established shorthand for this, you know, like $A\subset S$ can be seen as a shorthand for $A\subseteq S, A\neq S$?
Define $\mathcal N$  $\mathcal N (A):=\mathcal P(A)\setminus\{\varnothing\}$  Does $\mathcal N$ has a special name and standard notation?
How to edit command line in full screen editor in ZSH? What are the increased possibilities (if any) of being able to crack a private key given the following:   The associated bitcoin (ECDSA Secp256k1-based) public key is known. The private key has been used to generate a public key from a different ECDSA curve and that public key is also known. It is known that the two public keys are derived from the same private key. It is known which ECDSA curves were used to generate the public keys. It is known which ECDSA curve was used to generate which public key.   Does it make any more difference if there is data that is known to have been signed by the private key and the signature(s) are known and the raw data is known?  EDIT: I should have pointed out that the conditions are ANDed, I.e. All conditions are known. Sorry, I should have been clearer the first time. Hazard Function - Survival Analysis I just started taking survival analysis class and I'm stumped on this question.  Let $T_1,...,T_n$ random independent continuous variables, with hazard function of $h_1(t),...,h_n(t)$. $T=min(T_1,...,T_n)$. And we need to show that the hazard function of T is $\sum_j{h_j(t)} $  Any help or direction are welcome :) I googled it but I couldn't find the answer... maybe I didn't seek it in the proper way. My question is if &quot;to&quot; must be repeated in sentences such as the following ones:   In the past, women used to do the housework and to take care of children.  The aim of this meeting is to control their progress and to discuss the new working plan.    If I had to bet, I would omit the second to... But I am not sure about it. ARMA lag selection for ARMA-GARCH models Is it possible to remove a terminal command? For example, if there was a command call lookup (not an alias), how would I go about removing it? Rotating symbols per attribute value in QGIS? I am trying to apply a rule to rotate symbols based on the value of an attribute.  I tried to copy , which is essentially my question, however it does not work for me.   I have a layer with two attributes, the slope and aspect.  I want to colour the symbols by the slope (this part works) and rotate by the aspect.  The steps I take are:   Layer Properties -> Style -> Graduated -> Symbol -> Change Select Simple Marker (arrow) and press the button to the right of angle -> field type (double) -> Aspect-sel    Applying these commands changes the symbol to an arrow, and I can easily apply the colour gradient, but the symbols do not rotate. The attribute Aspect-sel varies from 2.87 to 334.23.    ETA: I was using Version 2.8.3 - on suggestion I updated to 2.14.10, which solved the problem. I am using 64bit 2.10.1-Pisa on Windows 10.  When I set up my composer to print, everything looks fine until I try to print or export. The composer goes completely blank and outputs a corrupted PDF using the Export to PDF function. The PDF file won't even open.  I had the same result exporting to PNG.  When I try to use the PDF document printer instead of exporting, QGIS stops responding indefinitely. List of tables, db schema, dump etc using the Python sqlite3 API What is a plain English explanation of "Big O" notation? I'd prefer as little formal definition as possible and simple mathematics. WIFI connection not working I ran these commands:  sudo rmmod iwlwif sudo modprobe iwlwifi 11n_disable=1 sudo su echo "options iwlwifi 11n_disable=1" &gt;&gt; /etc/modprobe.d/iwlwifi.conf   Now, my WiFi connection is lost and there is no WiFi option in the network manager. How can I restore it? Simple approximation of Inverse Incomplete gamma function Limit of $L^p$ norm when $p\to0$ Let ($\Omega$, $\cal{F}$, $\mu$) be a probability space and $f\in L^1(\Omega)$. Prove that  $$\displaystyle\lim_{p\to 0} \left[ \int_{\Omega}|f|^pd\mu \right]^{\frac{1}{p}}=\exp \left[ \int_{\Omega}\log|f| d\mu  \right],$$  where $\exp[-\infty]=0$. To simplify the problem, we may assume $\log|f|\in L^1(\Omega).$ I'm trying to make a conditional formatting rule in Google Spreadsheet that would highlight a row green based on text in multiple cells.   If cells L2, M2, and N2 = APPROVED; and cell O2 = X; then row 2 = highlighted green. Firstly, I apologise that this question is specific to two polynomials. I understand that this post will not help a lot of people, and for that I am sorry. I will make it up to you all by anwsering lots more questions in the future! However, as my lecturers enjoy time off on the weekend, I am unable to ask this to anyone else who would know what the heck I'm talking about. Sorry if my notation is not widely used, I can elaborate more if required. Please excuse my poor Latex too.  Let:  $B = Y^2 - X^3 + X$, and   $F = (X^2 + Y^2)^3 - 4X^2Y^2$.  I wish to find the intersection number of $B$ and $F$ at $P = (0,0)$  i.e $I(P,F \cap B). $  However, I want to use the following proposition to do this:  If $P$ is simple on $B$ then $I(P,F \cap B) = ord_P^B(F) $.  I will omit proving that $P$ is simple on $B$ as I hope it is clear to see it is.  I know that $I(P,F \cap B) &gt; M_p(F) * M_p(B) = 1 * 4 $, where $M_P(G)$ is the multiplicity of $P$ on $G$.  Now, consider the Image of $X$ and $Y$:  $K[X,Y] \longrightarrow \frac{K[X,Y]}{Y^2 - X^3 + X} \longrightarrow O_P(Y^2 - X^3 + X)$  $X \longrightarrow x$  $Y \longrightarrow y$  Where lower case denotes the image.  Then $ord_P^{Y^2 - X^3 + X}((X^2 + Y^2)^3 - 4X^2Y^2)$  Can be found by substituting $y^2 = x^3 - x$ into $(x^2 + y^2)^3 - 4x^2y^2$  Which gives:  $ord_P^{Y^2 - X^3 + X}(x^9+3x^8 -5x^6 -4x^5 +3x^4 +3x^3)=3$  i.e $I(P,F \cap B) = 3$, a contradiction to  $I(P,F \cap B) &gt; 4$ as above.  The correct answer I know is 6, so where have I gone wrong? Does there exist a function such that $\lim_{x \to a} f(x) = L$ for all $a \in \mathbb R$ but $f(x)$ is never $L$? Does there exist a function $f:\mathbb R \to \mathbb R$ such that $\lim_{x \to a} f(x) = L$ for   all $a \in \mathbb R$ but $f(x) \neq L$ for all $x$?   I found such a function in $\mathbb Q \to \mathbb Q$, where $f(\frac{p}{q})= \text{first p+q digits of }\pi$ satisfies the above condition. However, I have not been able to extend it to reals. So, is such a function possible, and if it is, is there any explicit example preferably related to the above function in rationals? What's wrong with this demonstration?:  $$A \iff 1 = 1^1$$ $$A \implies 1 = 1^\frac{2}{2}$$ $$A \implies 1 = (1^2)^\frac{1}{2}$$ $$A \implies 1 = ((-1)^2)^\frac{1}{2}$$ $$A \implies 1 = (-1)^\frac{2}{2}$$ $$A \implies 1 = (-1)^1 = -1$$  So finaly we get  $$1 = -1$$ Showing that every field is an integral domain. Should my first trip be to the country which issued my Schengen Visa? In case, if I get a multiple-entry Schengen visa in one of the embassies in Ukraine, Kyiv - should my first trip be to the country which issued the visa?  Any there any requirements of this sort? Does this condition vary depending on embassy which issued the visa?  What are the possible consequences on not visiting that country first? Does it somehow depend on country where the embassy is (i.e. non-Schengen and not EU members)? what is the relationship between the dynamical casimir effect and virtual particles?