Assume that the set $I \subseteq \mathbb{Z}$ satisfies the following properties:  (a) There exists $n \in I$ such that $n \not = 0$  (b) If $m, n \in I$, then $m + n \in I$  (c) If $m \in  I$ and $a \in \mathbb{Z}$, then $am \in I$  Prove that there exists $n_0 \in \mathbb{Z}$ such that $I = \{kn_0 | k \in \mathbb{Z}\}$  Not too sure if I am correct but I am thinking if $n_0$ is actually $gcd(m, n)$, then $I$ would be the set of all integer combinations of $m$ and $n$. So essentially this question is to relate the set of all integer linear combinations to the set of all integer multiples of the $gcd(m, n)$.
I have the following problem:  Let $a, b \in\mathbb{Z}$. Show that $\,\{ ax + by\ :\ x, y \in \mathbb{Z}\} = \{ n \gcd(a,b)\ :\ n\in \mathbb{Z} \}$   I understand that the Bezout's lemma says that $gcd(a,b) = ax +by$, so Im not really how you would go about proving the above, it doesn't really make sense to me. Any help is appreciated1
$f:\mathbb R \to \mathbb R$ be a continuous such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , is $f(x)=x , \forall x \in \mathbb R$? Let $f:\mathbb R \to \mathbb R$ be a continuous function such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , then is it true that $f(x)=x , \forall x \in \mathbb R$ ? I can see that $f$ is bijective , hence strictly monotone . But I cannot progress further . Please help. Thanks in advance What intuitively is "bias"? I'm struggling to grasp the concept of bias in the context of linear regression analysis.    What is the mathematical definition of bias? What exactly is biased and why/how? Illustrative example? Probability of false positive key matching two plaintext/ciphertext pairs Given a keyspace of $ 2^{80} $ and plaintext space of $2^{64}$. And two plaintext and ciphertext pairs $(x_1, y_1)$ , $(x_2, y_2)$. Now we have $2^{80}/2^{64} = 2^{16}$ keys that encrypt $x_1$ to $y_1$ and another $2^{16}$ keys that encrypt $x_2$ to $y_2$, with only one key that is supposed to be the target key (correct key). What is the probability that once brute-force identifies a first key ($k_1$) this same key happen by mistake to also encrypt $x_2$ to $y_2$, that is this this key happen to be a False-positive (that is, this key will likely not encrypt $x_3$ correctly). What is the equation used and how is it derived? I am trying to prove that any conjugate of $a$ has the same order as $a$.  Let $G$ be a group and let $a \in G$.  An element $b \in G$ is called a conjugate of $a$ if $b=xax^{-1}$.  My professor gave us a hint.  Prove $a^k=e$ iff $b^k=e$ for all $k \in \mathbb{Z}$ implies $o(a) = o(b)$.  My first move is to $b^k = (xax^{-1})^k = (x^{-1})^ka^kx^k = (x^{-1})^kex^k = (x^{-1})^kx^k=x^{-k+k}=x^0$  I am not sure that I am doing this correctly?  I am trying to get $b^k=e$.  However when I raise $(xax^{-1})^k$, I am not sure if I what I have done is correct?  I am kind of confused here. Access Denied using TakeOwn.exe I have got this file that I can't delete. It happened after a system crash, so the CHKDSK kicked in upon next reboot. After that, I can't delete the file.  THis is on Windows Home Server, and the file is one of those hidden Thumbs.db, and my WHS reports a "File conflict" on the file, the reason: Access Denied"  What I have tried so far, running as an Administrator: Delete: Access Denied TakeOwn.exe : Access Denied Attrib.exe -s -h : Access Denied cacls.exe : Access Denied   (even whe trying to display the current owner) Rename: File not found (its hidden) Re-boot in to safe mode and tried the above: Access Denied Running the above commands as Local System: Access Denied   Tizer Unlocker: Fail (Doesn't find anything locking the file) File Assasin: Crashes ("FindRemoteFileHandles returned NULL value.")  I have used the CHKDSK /f again, rebooted since some suggestions is that the file has been corrupted, but that didn't change anything. Any suggestions ?  Update: In a total separate scenario, I experienced a similar situation; Access Denied when trying to take ownership of a file. It turned out that the file was owned by TrustedInstaler, which is not possible to override just by being an Administrator. In this case, one can use psexec /ids cmd to elevate yourself to System and then TakeOwn.exe /a Explanation of Spikes in training loss vs. iterations with Adam Optimizer I am training a neural network using i) SGD and ii) Adam Optimizer. When using normal SGD, I get a smooth training loss vs. iteration curve as seen below (the red one). However, when I used the Adam Optimizer, the training loss curve has some spikes. What's the explanation of these spikes?   Model Details:  14 input nodes -> 2 hidden layers (100 -> 40 units) -> 4 output units  I am using default parameters for Adam beta_1 = 0.9, beta_2 = 0.999, epsilon = 1e-8 and a batch_size = 32.  i) With SGD  ii) With Adam When to use the double not (!!) operator in JavaScript How to quickly move one layer to a different group in large Photoshop files, without dragging? leaving oil on a frying pan for use later I live alone, so I cook for myself. And often times I cook steaks/burgers on a pan, and later in the same day would return to use the same pan/oil without washing/using new oil.  Is this safe, or should one wash and use a new pan before cooking every steak/burger patties? I've actually done this for more than a day once (cooked burgers in the morning, steaks later that afternoon, and then burgers for the next morning). Converting vector layer to raster using QGIS? In qgis in the raster menu there is an item "convert vector to raster".  This behaves strangely - I do not understand how it is supposed to work.  I can select a vector file, an attribute and an output raster. But, for some reason the output file is only accepted if it exists and even if I select an existing raster nothing seems to happen.  Also I do not understand how the program knows what kind of raster to create, what cell-size, projection etc.  How can I convert vector to raster? Options to move the school rucksack during the trip I'm working with a date in this format: yyyy-mm-dd.  How can I increment this date by one day? Why $\dim U+\dim U^\perp=\dim V$? Let $V$ be a vector space. Given $U\le V$, define $U^\perp=\{f\in V^\ast\mid f(u)=0\,,\;\forall u\in U\}$ and given $W\le V^\ast$, define $W^\perp=\{u\in V\mid f(u)=0\;,\;\forall f\in W\}$. This defines a Galois connection between subspaces of $V$ and subspaces of $V^\ast$. If $V$ is finite dimensional, why is $\dim U+\dim U^\perp=\dim V$? Unfortunately, I am not very comfortable with linear algebra. Any hint will be appreciated. In the , if I select any site in the site switcher it will be set as Stack Overflow or some random site in the drop down list's selected site name.  For the Stack Overflow, Super User, Server Fault and Meta Stack Exchange, the site switcher is set to the selected site name, but the remaining do not retain the selected site name. For English Language and Usage, it changes to Physical Fitness for some reason. All the others I've tried  switch to Stack Overflow.  It can be replicated in Chrome and Firefox.  GIF for the same: I have customer data from 2 brands. The data structure are the same, but I expected the customer behaviour to be different in different brand.  So I could train 2 models, 1 for each brand, or I could train one model on both brands, but flag which brand the customers belongs to within the data.  So my question is, would training on both brands with flags be too much more work for the Machine Learning (I'm using neural network), and therefore be harder to achieve good accuracy than to have dedicated models for each brand? Calculate $I=\int_0^1\frac{\ln(1+x)}{1+x^2}\,dx$ How can I calculate the integral $I=\int_0^1\frac{\ln(1+x)}{1+x^2}\,dx$ by substituting $t=\frac{1-x}{1+x}$. Limit of ${a_n}^{1/n}$ is equal to $\lim_{n\to\infty} a_{n+1}/a_n$ Which is the simplest way to update a Label from another Thread?   I have a Form running on thread1, and from that I'm starting another thread (thread2).  While thread2 is processing some files I would like to update a Label on the Form with the current status of thread2's work.   How could I do that? Number of points on a line in a finite projective plane I've been reading some proofs regarding finite projective planes of order n, and often they start out by assuming that each line contains n+1 points. Is this a fact that follows from the axioms for finite projective planes? Or does this relate to the order of the finite field being projected? (for example in the Fano plane, does the fact that each line contains 3 points follow from the fact that there are 3 elements {0,1,2} in Z_2?) I haven't been able to find a clear explanation of this (clearly I do not yet have a solid understanding of how such planes are constructed). If someone could enlighten me that would be excellent. I have been working with a string[] array in C# that gets returned from a function call. I could possibly cast to a Generic collection, but I was wondering if there was a better way to do it, possibly by using a temp array.  What is the best way to remove duplicates from a C# array?