Adding numerical value to node in TikZ
tikz relative coordinates
How to access a SQL Server database from other computer connected to the same workgroup? Proving the identity functions is continuous iff $\tau_1\supseteq\tau_2$ Exercise: Let $\tau_1$ and $\tau_2$ be two topological spaces on a  set $X$. Then $\tau_1$ is said to be a finer topology than $\tau_2$ if $\tau_1\supseteq\tau_2$. Prove the identity function $f:(X,\tau_1)\to (X,\tau_2)$ is continuous if and only if $\tau_1$ is a finer topology than $\tau_2$.  Attempted proof: Let $A\in\tau_1$. If $f$ is continuous then $\exists B\in\tau_2$ such $f(A)\subset B\implies A\subset B$ for any $A\in\tau_1$ such that all open sets of $\tau_1$ are open sets of $\tau_2$ hence $\tau_1\supseteq \tau_2$. If $\tau_1$ is a finer topology than $\tau_2$, for any $B\in\tau_2\implies B\in\tau_1$ which implies that $f^{-1}(B)=B$, then $B\in\tau_1$ which proves $f$ to be continuous. Questions: Is my proof right? If not. What is wrong? What are alternatives? Thanks in advance! Prove $\csc(x)=\sum_{k=-\infty}^{\infty}\frac{(-1)^k}{x+k\pi}$ $\textit{Why}$ the cases in 4 $\nmid n^2 - 3?$ (bases cases in modular induction) This question has already been answered . My question is, when we suppose the hypothesis is not true (proof by contradiction), namely that $4 \mid (n^2 - 3)$, we get $n^2 - 3 = 4k$ for some integer $k$ and then we have two cases. But where does the two cases come from? In addition, why does putting them into two cases work to prove the statment? My thought process would be to try to prove this statement by contradiction and somehow figure out a way to produce a contradiction similar to how $\sqrt{2}$ is proved by contradiction. Any help is great! Let me know if I need to clarify more! In $\mathbb{Z}\left [ x \right ]$, let $I = \left \{ f\left ( x \right ) \in \mathbb{Z}\left [ x \right ]: f\left ( 0 \right ) \text{ is an even integer} \right \}$. Is $I$ a Prime ideal of $\mathbb{Z}\left [ x \right ]$?  Let $f_{1}\left ( x \right ),f_{2}\left ( x \right ) \in \mathbb{Z}\left [ x \right ].$ Let $f_{1}\left ( x \right )f_{2}\left ( x \right ) \in I$. Clearly, I is a proper ideal of $\mathbb{Z}\left [ x \right ]$ since the elements in $\mathbb{Z}\left [ x \right ]$ evaluated at 0 can produce an odd integer but this cannot be true in I. Now, $f_{1}\left ( 0 \right )f_{2}\left ( 0 \right )$ is an even integer in I. Hence, either $f_{1}\left ( 0 \right )$ has an even constant integer or $f_{2}\left ( 0 \right )$ has an even constant integer. Thus, I is a prime ideal by definition. Is this correct? I would now like to show if I is a maximal ideal. Again, clearly, I is a proper ideal by the above argument. Let B be an ideal of $\mathbb{Z}\left [ x \right ]$ and I $\subseteq B\subseteq \mathbb{Z}\left [ x \right ]$. How should B be defined so that I would conclude either $B=I$ or $B=\mathbb{Z}[x]$? I'm looking for a book I read in Junior High (probably) about a world where everyone was basically the same race.  One genetically pure male and female teenager of each race still exists.  They are transporting a genetically pure boy to meet his genetically pure female counterpart and accidentally introduce him to a genetically pure female of a different race.  And, of course, chaos ensues.  Any thoughts? How do I prove that a graph is a topological manifold I'm having problems with that:  Prove that M is a topological manifold.  $f:U \to \mathbb R^k$, U $\subset \mathbb R^{n}$ open, f continuous  M = $\{\left(x,y\right)\in \mathbb R^{n+k} \mid x \in U, y=f(x) \}$  Can anyone help me please? Which "href" value should I use for JavaScript links, "#" or "javascript:void(0)"? The following are two methods of building a link that has the sole purpose of running JavaScript code. Which is better, in terms of functionality, page load speed, validation purposes, etc.?    function myJsFunc() {     alert("myJsFunc"); } &lt;a href="#" onclick="myJsFunc();"&gt;Run JavaScript Code&lt;/a&gt;     or    function myJsFunc() {     alert("myJsFunc"); }  &lt;a href="javascript:void(0)" onclick="myJsFunc();"&gt;Run JavaScript Code&lt;/a&gt; Lithium-6 isotope has an approximate magnetic momentum of $0.88\ \mu_N$ in its fundamental nuclear state. I'm trying to find its angular momentum and parity.  I found in a standard table: $I=1^+$ and while I get the parity part, I don't understand the rest. What I have done is filling the shells separately for protons and neutrons and finding that both have their outer particle in the $1 P_{3/2}$ state. Sum of these gives possible total angular momenta to be either $0$, $1$, $2$ or $3$.  So my question is: how do you choose between these to find the real value $I=1$? Thought Experiment - Poking a stick across a Black Hole's Event Horizon How is ISO implemented in digital cameras? If I change the ISO settings on my camera, obviously the gain of the system is increased, amplifying the signal from the sensor. What's not clear to me is where the amplification takes place. I see several possibilities:   In the sensor, by increasing voltage or some other mechanism Via an analog amplifier outside of the sensor Digitally, after the signal has been digitized, but before storing data in the RAW file As a parameter applied solely in creating an image from RAW   If #4 is true, then you could take a 4-stop overexposed RAW picture at ISO 1600, and then in post processing produce a JPEG at ISO 100 that would be the same as if the original photo had been shot at ISO 100.  If #1 or #2 is true, then a RAW file shot at ISO 1600 would actually contain more information about shadows, and an ISO 100 RAW would contain more information about highlights. REST API composite query: How do I use the result of a query subrequest in the referenceId of another subrequest? I am using Salesforce's REST API to (1) retrieve a list of object IDs according to a query, then (2) retrieve the records that correlate to those IDs.  I am familiar with the Salesforce composite documentation .    My request looks like this:  POST: /services/data/v48.0/composite  {     "allOrNone":false,     "compositeRequest": [ {         "method":"GET",         "url": "/services/data/v48.0/query?q=SELECT+Id+FROM+My_Entity+WHERE+(CreatedById+!=+'asdfghjkl')",         "referenceId": "myEntity"         },{         "method":"GET",         "url": "/services/data/v48.0/composite/sobjects/My_Entity?ids=@{myEntity.records[0].Id}",         "referenceId":"myObject"     }] }   My response:  {     "compositeResponse" : [ {         "body" : {             "totalSize" : 21,             "done" : true,             "records" : [ {                 "attributes" : {                     "type" : "My_Entity",                     "url" : "/services/data/v48.0/sobjects/My_Entity/a0D6g000xxxxxxx"                 },                 "Id" : "a0D6g000xxxxxxx"                 }, {*record2*}, {*etc*}             ]},             "httpHeaders" : { },             "httpStatusCode" : 200,             "referenceId" : "myEntity"         }, {         "body" : [ {         "errorCode" : "PROCESSING_HALTED",         "message" : "'myEntity.records' references an invalid Datatype. Only Strings and primitive data types are allowed to be referenced from previous operations."         } ],         "httpHeaders" : { },         "httpStatusCode" : 400,         "referenceId" : "myObject"     } ] }   The first subrequest (the query) executes without errors.  My problem is the second subrequest.  I understand from  that I can refer to the ID of a single item (e.g. myEntity.records[0].Id).  Question: How do I use all items from myEntity.records in the second subrequest? How should I use "eponym", "eponymous" and "namesake"? I was in London and visited the Tower Bridge. Afterwards, I went to a pub called The Tower Bridge. Just to be clear, the pub is named after the bridge.   Which of the following are correct?   The bridge is the eponym of the pub. The pub is the eponym of the bridge. From the pub, I could see the eponymous bridge. After I saw the bridge, I went to the eponymous pub. The bridge is the namesake of the pub. The pub is the namesake of the bridge. I've noticed that Visual Studio 2008 is placing square brackets around column names in sql. Do the brackets offer any advantage? When I hand code T-SQL I've never bothered with them.  Example:  Visual Studio:  SELECT [column1], [column2] etc...   My own way:  SELECT column1, column2 etc... Does it make sense to get a PhD if a person does not want to be in academia and prefers to stay in business? When do relativistic effects need to be explicitly included? pip: AttributeError: 'module' object has no attribute 'SSL_ST_INIT' Define the sequence $aₙ=20+n²$ and $dₙ=\gcd(aₙ , aₙ₊₁)$ Find with proof the values dₙ takes. Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum_{n=1}^\infty\frac{a_n}{n^s}$$ diverge? Is there ever a right time to upgrade to full frame camera body? Context  I've been shooting with an a6000 (APS_C mirrorless) and I plan to move on up to full frame, preferably mirrorless as well. I'm debating on when to upgrade, and probably don't plan to buy until next year.  I know I'll need to buy new full frame lens.  Question  Is there a "right" time to buy camera bodies, both mirrorless in my case as well as DSLRs? (I've seen the a7rII and like it but there might also be an a9 looming.) I feel that this is like smartphones in which the best time to buy is now as you might always be waiting until the next upgrade.   Related, are full frame bodies discounted during the holiday season or is that limited only to entry level bodies and older full frame bodies?  Related