Ring with maximal ideal not containing a specific expression
maximal ideal not containing some expression
Add new JDBC Driver to connect Mathematica to SQL database What do you call a person who is utterly independent and a loner? Valid references on origins of Machine Learning, Statistical Learning and Data Mining Proof for alternative definition of the derivative I read somewhere recently that you can define the derivative as follows:  $$f'(x) = \lim_{h, k \to 0^+} \frac{f(x +h) - f(x - k)}{h + k}$$   I have been trying to prove this for about 2 hours, and can't seem to get it done. How should I proceed?  Edit: Assume $f$ is differentiable for all $x \in (x - \delta, x + \delta)$ for some $\delta &gt; 0$. We are doing the research about applying remote sensed data on hydro logical model . After delineating, the river basin from DEM data using Argis9.3, We have to map the remote sensed data derived from products like PERSIANN. Our task now is to create the grid with the same resolution with the available remote sensed data( for example 0.25degree x 0.25 degree). and have a certain corner that matches with the obtained data.   Does anyone know how to deal with our problem? How does some of the black hole (BH) mass escape the event horizon (EH) of either BH, or the merged EH of two merged BHs? If $X$ and $Y$ are sets, their cartesian product $X\times Y$ is a set. In ZFC set theory, every element object is a set. We know or feel that if $X$ and $Y$ are sets, their cartesian product $X\times Y$ is a set.  My question is: Why if $X$ and $Y$ are sets, their cartesian product $X\times Y$ is a set? Permanently blocking a domain in iptables Suppose I want to block Facebook permanently. To do this, I have followed following processes:     #host -t a www.facebook.com   Sample output:     www.facebook.com has address 69.171.224.40   Find CIDR     #whois 69.171.224.40 | grep CIDR   Sample output:     CIDR:           69.171.224.0/19   To prevent outgoing access to www.facebook.com  Approach 1:     #iptables -I FORWARD -p tcp -d 69.171.224.0/19 -j DROP   Approach 2:     #iptables -I FORWARD -p tcp -d www.facebook.com -j DROP   Both Approaches work well. In approach 1, IP address may be changed so it will not block Facebook permanently. I don't know about approach 2 whether it will block Facebook permanently or not. If above approaches are not right way to block a domain permanently, how can I do it? Computing $\lim_{n\rightarrow \infty} \frac{1+\sqrt{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n}$. I have a problem with the calculation of the following limit. \begin{equation} \lim_{n\rightarrow \infty} \frac{1+\sqrt{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n} \end{equation} I do not know where to start! Thank you very much Switching motherboard without new install? A topological space $X$ is said to be irreducible if it cannot be written as a union $X = A∪B$, where $A$ and $B$ are proper closed subspaces of $X$  I want to try the following:  Show:  $(i)$ That every set with the indiscrete topology is irreducible, and every infinite set with the finite complement topology is irreducible.  $(ii)$ That an irreducible Hausdorff space contains at most one point.  Can you give me any suggestions on how to try this? Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer Let $a,b$ be positive integers.  When $$k = \frac{a^2 + b^2}{ab+1}$$ is an integer, it is a square.    Proof 1: (Ngô Bảo Châu): Rearrange to get $a^2-akb+b^2-k=0$, as a quadratic in $a$ this has two values: $a$ and $kb - a = (b^2-k)/a$. (The second root is determined in two different ways from the expansion $(x-r_1)(x-r_2) = x^2 - (r_1 + r_2)x + r_1 r_2$.)  Now suppose we have $a,b$ such that $k$ is an integer but not a square, by the investigation about roots we have that the second root is a nonzero integer since $k,b,a$ are integers and $k \not = b^2$, futhermore it is positive which is easily seen from its defining equation.  WLOG assume $a \ge b$ so that the second root is strictly smaller than $a$. This leads to a decent, replacing $a$ with the second root.    Proof 2 (Don Zagier): Apply reduction theory (specifically, Sätze 1 and 2 of Section 13 of my book on quadratic fields) to the quadratic form $x^2 + kxy + y^2$, which is the unique reduced quadratic form in its equivalence class.    Note that Proof 2 is pretty much the same as Proof 1 when written out in explicit detail, but I could not read Zagier's book because I cannot read German.  I would like to know more approaches to this and other alternative proofs of this result if possible! Thanks in advance.  I would also be interested in related problems (especially easier ones of a similar nature) and texts which cover the reduction theory in English. Require v8 only contrib module with v9 patch Given I have Drupal 9 composer installation with composer-patches plugin and given a contrib module with a stable v8 release, but no v9 release (not even dev branch) and given that contrib module has a working v9 patch in the issue queue  is there any method to install that module + patch in composer? Even if I manually add both, the package and patch, to my composer.json, I still can't require or update this module with composer due conflicting versions. I really want to avoid duplicating /contrib code into my project's /custom codebase. My current workaround is:  forking that module to my own, private git repo applying patch there creating a new composer.json in my private git, and changing the package vendor to my custom_private_vendor adding my private git as VCS repo in the D9 project's composer.json and then composer require custom_private_vendor/contrib_module  This fulfills my goal of not duplicating the contrib module in my project's custom codebase, but every time I do this I feel the urge to wash my dirty hands. Is there something more elegant like composer require drupal/contrib_module --apply-patch-first or can I somehow target drupal.org's git with a specific patch included? Speed of light in a gravitational field? How do I solve the speed of light in gravitational field?  Should I just add gravitational acceleration in speed of light?  $$c'=c_0+g(r)t~?$$ This doesn't work:  tar xf /tmp/foo.tar.gz foo/bar tar: foo/bar: Not found in archive   It's not obvious to me what would do this beyond extracting it in place and moving the files over. What are imaginary numbers? At school, I really struggled to understand the concept of imaginary numbers.  My teacher told us that an imaginary number is a number that has something to do with the square root of $-1$.  When I tried to calculate the square root of $-1$ on my calculator, it gave me an error.  To this day I still do not understand imaginary numbers.  It makes no sense to me at all.  Is there someone here who totally gets it and can explain it?  Why is the concept even useful? Create NSString by repeating another string a given number of times I know that the product of two Gaussians is a Gaussian, and I know that the convolution of two Gaussians is also a Gaussian. I guess I was just wondering if there's a proof out there to show that the convolution of two Gaussians is a Gaussian. This is similar to , except it's a bug report, not a feature request.  Currently, the nominee's comments on his/her own post are not highlights, unless you click "add / show more comments"  When the page is loaded it looks like:    AfterI click "add / show more comments", it becomes:    Could we have the blue highlighting on the post owner's comments there all the time? It becomes difficult to see the nominees responses to questions, especially on nominations where there aren't enough comments to enable the "add / show more comments" features, like Matt's. Dot Product Intuition I'm searching to develop the intuition (rather than memorization) in relating the two forms of a dot product (by an angle theta between the vectors and by the components of the vector ).      For example, suppose I have vector $\mathbf{a} = (a_1,a_2)$ and vector $\mathbf{b}=(b_1,b_2)$.  What's the physical or geometrical meaning that   $$a_1b_1 + a_2b_2 = |\mathbf{a}||\mathbf{b}|\cos(\theta)\;?$$  Why is multiplying $|\mathbf{b}|$ times $|\mathbf{a}|$ in direction of $\mathbf{b}$ the same as multiplying the first and second components of $\mathbf{a}$ and $\mathbf{b}$ and summing ?       I know this relationship comes out when we use the law of cosines to prove, but even then i cant get a intuition in this relationship.  This image clarifies my doubt:    Thanks