The syntax of "come to assume"
How is "I have come" + infinitive different from Present Perfect?
Oxford Comma Conventions According to the Wikipedia page for the , "Use of the comma is consistent with conventional practice" and "Use of the comma is inconsistent with conventional practice." Did the Oxford Comma come before its omission, or was the Oxford Comma traditionally omitted?  It makes logical sense that every item in a list would be separated in the same way: by a comma. If the Oxford Comma is conventionally correct, when and why did people begin to omit it? Single supply op amp to convert 0-5VDC to +/-2VDC (microcontroller to line level) I am generating a 0-5 volt sine wave using an arduino with a DAC. I would like to level shift this signal (DC bias?) by -2.5 volts and have a gain of .8, so that I end up with a line level sine wave.  Thus I am trying to design a single supply op amp with the following equation:  $$V_{\text{o}} = (V_{\text{i}}-2.5) \times 0.8$$  I found this white paper that is supposed to describe a way to do this but I'm having trouble following as I am new to op-amps.    Does anyone know of a way to do this with a single op amp or IC?  The frequency range of 20Hz to 20KHz must be preserved. Pattern in Isomorphisms between $\mathbb Z_m \times \mathbb Z_n$ and $\mathbb Z_{mn}$ In general, the group $\mathbb Z_{mn}$ under addition $\bmod mn$ is isomorphic to the direct product $\mathbb Z_m \times \mathbb Z_n$ if and only if $\gcd(m,n)=1.$ This is easy to see since $\mathbb Z_{mn}$ is generated by the element $(1,1)$, and if $\gcd(m,n)=1$, then this will only be zero if it is multiplied by itself $mn$ times (resulting in the cyclic group).   However I'm interested in constructing an isomorphism between the two groups as an explicit expression. I've been playing around a bit with some examples, one of which is the function $\phi : \mathbb Z_2 \times \mathbb Z_3 \to \mathbb Z_6$ where $\phi(a,b) = 3a + 4b \pmod 6$, which gives us the desired isomorphism. I'm interested here in the relationship between the integers $2$, $3$, and the coefficients $3, 4$ of $a$ and $b$. After playing around in Wolfram|Mathematica, I managed to obtain the following isomorphisms $\phi_{m,n} : \mathbb Z_m \times \mathbb Z_n \to \mathbb Z_{mn}$ for coprime $m,n$ in $\{1, 2, \dots, 10\}$.   First of all, the following results seem to hold in general.  $$\phi_{1, n}(a, b) = a + b \qquad \text{for all $n$}$$  $$\phi_{m, 1}(a, b) = a + b \qquad \text{for all $m$}$$  $$\phi_{2, n}(a, b) = na + (n + 1)b \qquad \text{for all $n$}$$ $$\phi_{m, n}(a, b) = K a + L b \iff \phi_{n,m}(a, b) = L a + K b \qquad \text{for all $m, n$}$$  Beyond this point, I could not spot a general pattern. The isomorphisms I obtained are:  $$\phi_{3, 4}(a, b) = 4a + 9b$$ $$\phi_{3, 5}(a, b) = 10a + 6b$$ $$\phi_{3, 7}(a, b) = 7a + 15b$$ $$\phi_{3, 8}(a, b) = 16a + 9b$$ $$\phi_{3, 10}(a, b) = 10a + 21b$$ $$\phi_{4, 5}(a, b) = 5a + 16b$$ $$\phi_{4, 7}(a, b) = 21a + 8b$$ $$\phi_{4, 9}(a, b) = 9a + 28b$$ $$\phi_{5, 6}(a, b) = 6a + 25b$$ $$\phi_{5, 7}(a, b) = 21a + 15b$$ $$\phi_{5, 8}(a, b) = 16a + 25b$$ $$\phi_{5, 9}(a, b) = 36a + 10b$$ $$\phi_{6, 7}(a, b) = 7a + 36b$$ $$\phi_{7, 8}(a, b) = 8a + 49b$$ $$\phi_{7, 9}(a, b) = 36a + 28b$$ $$\phi_{7, 10}(a, b) = 50a + 21b$$ $$\phi_{8, 9}(a, b) = 9a + 64b$$ $$\phi_{9, 10}(a, b) = 10a + 81b$$ If you want to try and generate a few yourself, here is the Mathematica function I implemented. I'm sure it can be optimised, but for the first 10 coprime integers it sufficed.  isomorphism[{m_, n_}] :=       Module[{x = Null},       For[i = 1, i &lt; m*n,           i++, (If[#[[2]] == Table[i, {i, 0, m*n - 1}],           x = ToString[#[[1]][[1]]] &lt;&gt; "a+" &lt;&gt; ToString[#[[1]][[2]]] &lt;&gt;          "b"; Break[]] &amp;) /@       Table[{{i, j},       Mod[i #[[1]] + j #[[2]], m*n] &amp; /@       Table[{Mod[i, m], Mod[i, n]}, {i, 0, m*n - 1}]}, {i, 1,      m*n}, {j, 1, m*n}][[i]]]; x];   I'd appreciate if anyone can help me find the pattern to determine $\phi_{m, n}$ in general. Summing Up A 2D Array Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$. I define a map $$\phi:D_{2n} \rightarrow \mathbb{Z}_2 \times D_n$$ by $R \mapsto (0,r^{\frac{n+1}{2}})$ and $M \mapsto(1,m)$. Then I am stuck at showing the map is bijective. By the way, do we need to show that $(0,r^{\frac{n+1}{2}})(1,m)(0,r^{\frac{n+1}{2}})^{-1}=(1,m)$ ? If we can prove this, then what can we conclude? How can I disable a tab inside a TabControl? Is there a way to disable a tab in a ? How do I send a cross-domain POST request via JavaScript? How do I send a cross-domain POST request via JavaScript?  Notes - it shouldn't refresh the page, and I need to grab and parse the response afterwards. Will reformatting my hard drive fix bad sectors on it? Smallest positive element of $ \{ax + by: x,y \in \mathbb{Z}\}$ is $\gcd(a,b)$ Let $S = \{ax + by: x,y \in \mathbb{Z}\}$  and let $e &gt; 0$ be the smallest element in $S$. Prove that $e \mid a$, and hence prove that $e = \gcd(a,b)$  I'm afraid I can't provide much of my initial working here because I'm at a loss of how to start.  I suppose that to show $e\mid a$ I would have to show that $eg = a$ for some $g \in \mathbb{Z}$. But I have no idea how I would do this. Prove that $(n-1)! \equiv -1 \pmod{n}$ iff $n$ is prime [Wilson's Theorem] How can I show that $(n-1)!\equiv-1 \pmod{n}$ if and only if $n$ is prime? Thanks. The question Is there any way to automatically scale and put an entire document on an A0 page? The thesis for which this is to be done is available as LaTeX code, as well as PDF output. Of course I know it is possible to do this manually, but I would prefer a LaTeX solution.  A small background Our university introduced a new page limit for master theses. This is not exactly appreciated by the teaching staff as well as students. To make a little joke, we decided to try to arrange an entire thesis (of approximately 200 pages) on a A0 poster, put it up in the hallway, and put a magnification glass next to it with a note saying: &quot;New page limit introduced: 1 page maximum for all upcoming theses!&quot;  The final result As some of you asked for this: below you can see a picture of the final result. (I &quot;professionally removed&quot; some of the sensitiv information.) We presented the poster during my final presentation and everyone liked it a lot! Now, it is on display next to my office and is still attracting some attention once in a while. How to make dried desert/cracked ground I am trying to create a cracked ground appearance for an animation that takes place in the desert. something like the image here:I would like to do this all in blender, and have blender generate the texture so I will not have to worry about unwanted ridges like those you would get in an image from the internet: I am wondering if there is a way to procedurally generate these cracks and use them with microdisplacements. Thank you! PreparedStatement IN clause alternatives? What are the best workarounds for using a SQL IN clause with instances of java.sql.PreparedStatement, which is not supported for multiple values due to SQL injection attack security issues: One ? placeholder represents one value, rather than a list of values.  Consider the following SQL statement:  SELECT my_column FROM my_table where search_column IN (?)   Using preparedStatement.setString( 1, "'A', 'B', 'C'" ); is essentially a non-working attempt at a workaround of the reasons for using ? in the first place.    What workarounds are available? Changing Default Apache version on MacOS Sierra A DHS sweep of the network showed that one of the researchers Macs is running a non-safe version of Apache (2.4.25) and they would like him to update to at least 2.4.27.  He is having difficulty getting MacOS to use the new version of Apache rather than the one installed by default with Sierra. His issues are:  Mac OS X 10.12.6 Sierra   I have installed httpd 2.4.27 . . .  /usr/local/apache2/bin/ but the default httpd running is located at /usr/sbin/ I cannot install a new httpd into /usr/sbin/  - no write persmissions for sudo I made a root user active - but "root" does not have write permissions for /usr/sbin/ either   I cannot find any source on the web that describes how to change the httpd path from the default system version (/usr/sbin/httpd) to a new USER installed version, like /usr/local/apache2/bin/httpd  What steps am I missing? Why would perfectly similar data have 0 mutual information? Why is my Spring @Autowired field null? How to merge all scene geometry into a single mesh? Where to place my local website starting with the 2.4.7 version of apache2? How would I go about detecting monopoles? A question needed for a "solid" sci-fi author: How to detect a strong magnetic monopole? (yes, I know no such thing is to be found on Earth).  Think of basic construction details, principles of operation and necessary components of a device capable of detecting/recognizing a macroscopic object emitting magnetic field of equivalent of order ~0.1-10 Tesla near its surface, but with only one pole, reliably distinguishing it from normal (2-pole) magnets, preferably at a distance.  Preferably a robust method, not involving extremely advanced technology. Detect the presence, possibly distance (or field strength) and direction.  I know of SQUIDs, but these concentrate on extreme sensitivity. I'm thinking of something less sensitive but more robust (like, no need for the monopole to fall through the loop) and still able to recognize a monopole against a magnet.  Also, how would such a macroscopic object behave practically? Such a "one-pole magnet" about the size and strength of a refrigerator magnets - how would it behave around ferromagnetics, normal magnets and so on? Could there be a bit of reputation leakage between the sites, please?