Suppose I consider the affine cipher defined by $E(x) = ax + b \bmod n$  on the standard alphabet of 26 letters. Which choices of $a$ and $b$ lead to proper encryption functions?  Using the information provided from this :  They say that $a=5$ and $b=8$ works, but $a=4$ and $b=5$ does not work.  My understanding is that in cases where it does not work then different letters prior to applying this cipher results in the same ciphertext having applied the cipher.  However, I consider this explanation to be quite woolly. How would I know which choices would work prior to applying the cipher?
In the , the key $k = (a, b)$ should have the following property, i.e. GCD of $a$ and the modulus should be 1:     $\gcd (a, m) = 1$, where $m$ is the set of possible moduli   My question is, why it is required that $a$ should have the above property? What is the significance of this? And why is $b$ excluded from this condition?
Why is the installer not showing any partitions? Flagging "look at my answer pls" comments I've started to flag a few of these once I had seen one too many of them, but I'm not sure whether I'm doing the right thing.  This is an example of what I'm talking about (this comment is on the question, but sometimes on their own answers too):    Are they something that should be considered acceptable, and if not, are they bad enough to be "flaggable"? Printing all instances of a class A property of every real polynomial. Question: For every polynomial $P(x)$ of degree at n we have $$\sum_{i=0}^{n+1}(-1)^i\binom{n+1}{i}P(i)=0$$  Well I was inducting on degree of the polynomial, for $n=0$ its true and I got a bit confused in the inductive step, actually I cant figure the inductive step.   Is this property (if true) a famous one?  Thank you for your help. I know that a continuous function which is a BV may not be absolutely continuous.  Is there an example of such a function?   I was looking for a BV whose derivative is not Lebesgue integrable but I couldn't find one. Is there the easiest way to toggle (show/hide) navigational grids in TikZ? In PSTricks, we can toggle the grids easily by showgrid=[false/true]. How to do it in TikZ? Doing as follows is tedious.    \documentclass{minimal} \usepackage{tikz}  \begin{document}   \tikzpicture  \draw[gray,very thin] (-4,-2) grid (4,2); \foreach \x in {-4,...,4}   \draw[red] (\x,-1.9) -- (\x,-2.1) node [below] {\x};   \foreach \y in {-2,...,2}   \draw[red] (-3.9,\y) -- (-4.1,\y) node [right] {\y};  \draw (0,0) circle (2);   \endtikzpicture    \end{document}   Is there the easiest way to toggle (show/hide) navigational grids in TikZ?  Edit  The navigation needs not only the grids but also the labeling numbers. Clipping a GEE Image with Python API I am looking for a way to clip image collection by a given polygon geometry with Python API, The GEE JavaScript equivalent is simply ImageCollection.clip(geom) () With the python API I get the error: 'ImageCollection' object has no attribute 'clip'.  I'm adding my code after using .mean() as a reducer My code: set the boundary polygon my_boundary = ee.Geometry.Polygon([ [[-5, 40],[65, 40],[65, 60],[-5, 60],[-5, 60]]] ,proj=None) #set the ImageCollection and Image img_Collection = ee.ImageCollection('COPERNICUS/S5P/NRTI/L3_NO2').filterDate('2020-03-01', '2020-03-27').select('NO2_column_number_density') my_Image=img_Collection.mean().clip(my_boundary) #Create Ipyleaflet map my_map = ipyleaflet.Map(zoom=7) my_map.add_layer(layer=my_Image) my_map error: AttributeError: 'Image' object has no attribute 'model_id' I have basic programming skills (I program simple vba python and lisp scripts and had gone through a java course a long time ago), but it seems to me (judging by ESRI publications) that there is no escaping the need to learn .NET for future relevance in the GIS job market (even as an analyst). I've gone through a basic c# course and can program (regrettably) quite poor programs, but I have no idea how to link the c# knowledge to actually programming for ArcGIS in order to progress.  I know the differences aren't big, but does ArcGIS use .NET or C#?  I know the best way to further my experience in .NET is to program more and more but are there any good .NET programming books or courses I can learn techniques from?  In your opinion, what is the best way to learn to program in .NET for ArcGIS? Are there any useful books or websites?   I seek your comments regarding ArcGIS 10 (and will be saving them for later) but also for ArcGIS 9.3.1. I have a file with a number of lines in a file filename.  I want to count how many lines start with character 'a', with 'b' and so on in one go.  What command i should execute.? The quotient norm on $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$. I try to show that the norm on the quotient space $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$, where $x = (x_n)_{n \in \mathbb{N}} \in \ell^{\infty} (\mathbb{N})$. My attempt is the following:     By definition, the quotient norm $\| \cdot \|_{\ell^{\infty} / c_0}$ is defined as   $$ \| x + c_0 (\mathbb{N}) \|_{\ell^{\infty} / c_0} = d( x + c_0 (\mathbb{N}), 0 + c_0 (\mathbb{N}) ), $$   which can be re-written as    $$ \| x + c_0 (\mathbb{N}) \|_{\ell^{\infty} / c_0} = \inf \{ \| x + y \|_{\infty} \; | \; y \in c_0 (\mathbb{N}) \}. $$   Since $0 = (0,0,...) \in c_0 (\mathbb{N})$, it follows that   $$ \| x + c_0 (\mathbb{N}) \|_{\ell^{\infty} / c_0} \leq \| x \|_{\infty} = \sup_{n \in \mathbb{N}} |x_n|. $$   I thought about using Bolzano-Weiserstrass' Theorem at a certain point, but I am not sure how to obtain the wished equality from here.    Any help is appreciated. Thanks in advance. Number of automorphisms of a direct product of two cyclic $p$-groups How do I set the bootloader so that in my dualboot with Windows 7/Ubuntu 11.04, Windows start up as standard instead of Ubuntu? Let $x$ and $y$ be positive integers such that $xy \mid x^2+y^2+1$. Let $x$ and $y$ be positive integers such that $xy \mid  x^2+y^2+1$. Show that $$ \frac{x^2+y^2+1}{xy}= 3 \;.$$ I have been solving this for a week and I do not know how to prove the statement. I saw this in a book and I am greatly challenged. Can anyone give me a hint on how to attack the problem? thanks Dragging layer into new file changes its size Let's say I have a file that's 4'x12' and represents a space on an actual wall. I want to open certain files (of different sizes) and drag each file into the large file in order to arrange the images correctly on the computer before physically hanging the framed finals.  When I drag a new image into the larger file, the new image's size and the size of its copy don't match. That is, Say I drag an 8x10 into the large file, well on the large file (trying to control size with guides and rulers) the new image is some other size, but not the right one.   What am I doing wrong? How do I make sure that when I click and drag the new image that it will be properly represented? Array of zero length I am working on refactoring some old code and have found few structs containing zero length arrays (below). Warnings depressed by pragma, of course, but I've failed to create by "new" structures containing such structures (error 2233). Array 'byData' used as pointer, but why not to use pointer instead? or array of length 1? And of course, no comments were added to make me enjoy the process...  Any causes to use such thing? Any advice in refactoring those?  struct someData {    int nData;    BYTE byData[0]; }   NB It's C++, Windows XP, VS 2003 Proof of the Hockey-Stick Identity: $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$ After reading , the most popular answer use the identity $$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.$$  What's the name of this identity? Is it the identity of the  modified.  How can we prove it? I tried by induction, but without success. Can we also prove it algebraically?  Thanks for your help.    EDIT 01 : This identity is known as the  because, on Pascal's triangle, when the addends represented in the summation and the sum itself are highlighted, a hockey-stick shape is revealed. I am trying to use a  as input and then creating a query with it - so I need a .    I was surprised to find that it couldn't do the conversion implicitly or explicitly - but I don't even know how I would do this, as the Java API is still fairly new to me. Determining Min - Max within zip code (statistics by area)? I'm new to ArcGIS Desktop. I have created a layer representing zip codes and overlayed it with a layer containing some data points. I want to find the min and max value of the data points in each zip code. Currently, I am manually checking the values if all the data points in each zip code but this is becoming very time-consuming and cumbersome.   Is there a faster way to do this? This is a question in Ahlfors in the section on the argument principle: How many roots of the equation $f(z)=z^4+8z^3+3z^2+8z+3=0$ lie in the right half plane?  He gives a hint that we should "sketch the image of the imaginary axis and apply the argument principle to a large half disk."  Since $f$ is an entire function, I think I understand that the argument principle tells us that for any closed curve $\gamma$ in $\mathbb{C}$, the winding number of $f(\gamma)$ around 0 is equal to the number of zeros of $f$ contained inside $\gamma$.  How would you go about actually applying the hint though? I am having trouble figuring out what the image of a large half disk under $f$ would look like. How do I create a script file for terminal commands? In Windows I can write a file containing commands for cmd (usually .cmd or .bat files).  When I click on those files it will open cmd.exe and run them commands the file contains.  How would I do this in Ubuntu?         I'm sure this is a duplicate, but I can't find my answer. Its similar to these questions, but they don't answer the question: