Stochastic Integral martingale if no $dt$ term?
show that the solution is a local martingale iff it has zero drift
I am new to linux, but had a question. I am getting in the business of buying old computers, putting Linux on them, then donating them to students who don't have their own computer at home. I have gone through and installed Xubuntu on my system, installed/uninstalled all the software that I want/need, and now want to know if there is a way that I could make an ISO of my system, and put it onto ANOTHER computer, and still have all of my tweaks, and programs ready to go after the ISO has installed. That way I don't need to go through and spend an hour or two on every computer.   What programs would you recommend for creating an ISO like this?   Thanks, Tobin My professor proved that the kernel is a normal subgroup under a homomorphism f by saying. Let $h\in H$  by applying f to it  $f(ghg^{-1})=f(g)f(h)f(g)^{-1}=e $  My question is how is this a proof that the kernel is normal. What is the reason we apply f to it ? By applying f dont we just prove that $f(ghg^{-1})$ is normal ? How to change the list of file types which Safari will open immediately after download? I would like to change Safari Safe List of type files which open immediately after download. Any ideas how? Practical hyperparameter optimization: Random vs. grid search Would you please explain the difference between these two sentences?     Spain ceded (or gave) Puerto Rico to the United States. Since then, Puerto Rico has been a U.S. territory.      Spain ceded(or gave) Puerto Rico to the United States. Since then, Puerto Rico was a U.S. territory.   I am stuck between 'has been' and 'was' in the above sentences. If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$ Prove that if $G$ is abelian then the set $H$ of all elements of $G$ that are their own inverses is a subgroup of $G$.   Naturally in an abelian group, $ab = ba$ for $a, b \in G$, however I'm not sure how to show the set elements that are their own inverses is a subgroup of $G$ using arbitrary elements. Metrizability of Hausdorff continuous image of compact metric space Let $f$ be a continuous mapping of a compact metric space $(X, d)$ onto a Hausdorff space $(Y, \tau_1)$. Then $(Y, \tau_1)$ is compact and metrizable.   In one proof the following metric is constructed:  $d_1(y_1, y_2) = inf\{d(a, b) : a \in f^{-1}\{y_1\},\ b \in f^{-1}\{y_2\}\}$, for all $y_1$ and $y_2$ in $Y$.  I'm thinking about how to prove that triangle inequality holds for $d_1$, s.t. $d_1(x, z) \leq d_1(x, y) + d_1(y, z)$  And how about the case when triangle inequality does not hold?    Update: @daniel-schepler suggested to use metric $d_1(x, y) = inf\{d(x^*, x_1) + d(y_1, x_2) + \cdots + d(y_{n-1}, x_n) + d(y_n, y^*)\}$ where $f(x^*) = x$, $f(y^*) = y$, and $f(x_i) = f(y_i)$ for each $i$.  Now we need to prove 3 property of metric: positivity, symmetry and triangle inequality. This is my sketch:   Positivity. Chose two points $x,y \in Y$ s.t. $x \neq y$. Singleton sets $\{x\}$ and $\{y\}$ are disjoint closed sets by Hausdorfness of $Y$ and their preimages are also disjoint closed in $X$ by continuity of $f$. $X$ is compact metric space by hypothesis and therefore a normal space. In normal space every two disjoint closed sets have disjoint open neighborhoods. Let $f^{-1}(x) \subseteq U$ and $f^{-1}(y) \subseteq V$ s.t. $U,V$ are open and $U \cap V = \varnothing$. Therefore there exists an $\varepsilon &gt; 0$, s.t. $x^* \in B_\varepsilon(x^*) \subseteq U$ and $y^* \in B_\varepsilon(y^*) \subseteq U$ and distance between any two closed sets is strictly positive and $d_1$ is positive and zero iff $x = y$. Symmetry. Clearly $d_1$ is symmetric. Triangle inequality. For triangle inequality we can show that in the "worst" case $d_1(x, z) = d_1(x,y) + d_1(y, z)$. I've had the ability to review suggested edits for a while now, however I don't often do so because the list is usually full of questions about languages and technologies I'm not familiar with or not interested in, and can sometimes be hundreds of questions long.  I would like to help out more, but most of my rep has been gained by questions within a specific technology, so I only feel comfortable reviewing edits within that tag.   Can you allow us to filter suggested edits by tag?  Edit:   Actually, can you let us filter the entire Review section by tag, and not just Suggested Edits?  I would find it much more interesting to review questions in tags I'm familiar with and can  understand the question content of, and it would give me a bit of pride to help maintain "my" section of SO. Writing to NTFS drive in Linux Split $n$ into nontrivial factors via a nontrivial square-root of $1\!\pmod{\!n}$ Coming from an understanding of Fermat's primality test, I'm looking for a clear explanation of the Miller-Rabin primality test.  Specifically: I understand that for some reason, having non-trivial square roots of 1 mod p means that p is definitely composite; and I gather that you can find these non-trivial square roots by squaring x, but I don't really understand what these reasons are. Specific examples of non-trivial roots of a composite number would be helpful.  Cheers! May I summarize all answers on StackOverflow, I asked a . There are many good answers but I can not accept just one of it. I would prefer to create my own answer where I summarize all answers. Is that allowed? Calculating Mean for Multiple Rows in ArcGIS for Desktop? I am trying to calculate a mean across a multiple number of attribute rows and have this value added to a new field that I have entered in the attribute table. I have an attribute table which has a number of different ecoregion types associated with them.  For example:  If I select "Arid Canyonlands" there are 3 attribute rows that are selected, each with their own "mean". I am trying to add a new field to the attribute table which will take the "mean" of these three and populate that value in the new field created. I could manually do it through an editing session and using a calculator but just wanted to know if there is an easier way to actually do it through field calculator? PrintWriter and PrintStream never throw IOExceptions I am about to sell my old desktop PC and I am cautious about some of my sensitive information being available to the purchaser, even after reformatting the hard-drive, using data recovery software.  How can I securely wipe the hard drive so that the data on it cannot be recovered?  Although I specifically want help with my Windows PC, it wouldn't hurt if there were suggestions for Macs as well. Like most Python developers, I typically keep a console window open with the Python interpreter running to test commands, dir() stuff, help() stuff, etc.  Like any console, after a while the visible backlog of past commands and prints gets to be cluttered, and sometimes confusing when re-running the same command several times.  I'm wondering if, and how, to clear the Python interpreter console.  I've heard about doing a system call and either calling cls on Windows or clear on Linux, but I was hoping there was something I could command the interpreter itself to do.  Note: I'm running on Windows, so Ctrl+L doesn't work. bread that does not get moldy How can a bread have an expiration date of one year into the future?? The bread in questions is Deutsche Küche rye bread (from Aldi).  Yes, it is a dense bread, but it is not dry like a cracker, so there seems to be enough moisture for mold to grow.  There are no preservatives.  Edit (pictures added): Are there naturally occurring lasers? I wonder which of the two is correct:        I have a red and a blue car.   I have a red and a blue cars.      or        Everyone has a left and a right arms.   Everyone has a left and a right arm.      My (non-native) inclination would be for option 1, as it sounds better.  On the other hand, in the following examples, the plural sounds better to me.    Everyone has left and right arms. I have old and new cars. (when I only have only one old car and one new car) The left and the right arms are...   But are they correct? Green is the most common circuit-board color because it has become an industry standard.  What I'm interested in is How the traditional "PCB Green" become a standard in the first place?   Were there any interesting historical reasons for the initial choice, or was it just a product of what one particularly successful company was doing that became the de-facto standard? Tools to get a pictorial function call graph of code