Programming for a beginner (with GIS experience)
Learning Python programming with generic GIS goals in mind?
How to prove the midpoint of a chord is also the midpoint of the line segment defined by the points of intersection of other two chords with it? Bernhard Elsner, alias MathOMan, posted this exercise in plane Geometry,  on January $29$th, $2010.$  "Let $\mathcal{C}$ be a circle, $A,B$ two distinct points on $\mathcal{C}$ and $M$ be the midpoint of the chord $[AB]$. Take two other chords,$[PQ]$ and $[SR]$, that pass through $M$. Let $C$ (resp. $D$) be the intersection of $[AB]$ with $[PS]$ (resp. $[RQ]$). Prove that $M$ is the midpoint of the chord $[CD]$."    To prove it I've written the following (failed) argument, in the  of this post (translation of mine):    The figure is symmetric with respect to $M$: $\overline{AM}=\overline{MB}$, $\overline{PM}=\overline{MU}$, $\overline{RM}=\overline{MW}$, $\overline{QM}=\overline{MV}$, $\overline{QR}=\overline{VW}$, $\overline{SM}=\overline{MT}$. From $\dfrac{\overline{SC}}{\overline{DT}}=\dfrac{\overline{CM}}{\overline{MD}}=1$ follows that $\overline{CM}=\overline{MD}$.  Here is an extract of the author's reply (translation of mine):   "It is not clear that  $\overline{QM}=\overline{MV}$. Is the point $V$ on the line $(QM)$ defined by this equality or is $V$ defined as the intersection point of the lines $(QM)$ and $(WC)$? Why are both definitions to give the same point?  Let $C^{\prime }$ be the intersection of $(SP)$ and $(VW)$, and $D^{\prime }$ the intersection of $(TU)$ and $(QR)$. (...)  One still has to show that $C^{\prime }=C$ and $D^{\prime }=D$."   I have agreed with these objections.  Until now no proof has been posted. The author considers that the "proof is not quite simple".  Q. What is the theorem this exercise refers to? Or how does one prove it? Any good tutorials for learning how to post-process images? I don't really know where to start with post-processing my images. I normally just straighten, crop and sharpen. Do people know of any good resources on the internet for getting a grip with Photoshop and similar tools? We're running a couple SQL servers in Azure that are set up with an Always On availability group and Windows Failover Clustering. The servers are Windows 2019 and we're running SQL Server 2019. When we set up the cluster, it was set up to use a Distributed Network Name instead of a static cluster IP address. Thanks to this we shouldn't need an internal load balancer according to these notes: .  I'm struggling to understand exactly how this works though. Based on what I read, it seems like our connection strings will point to the DNS name of the cluster (let's call it AgCluster). If I look in DNS, there is an A record for AgCluster pointing to sql1 and another pointing to sql2. When I use AgCluster in my connection string it seems to always connect me to the primary server, even if I have ApplicationIntent=ReadOnly set. When I query @@SERVERNAME I always get the same server.  So with the Distributed Network Name setup, what should I use in my connection strings to make sure read/write queries go to the primary and read only go to a secondary? Any guides on setting this up in general would be helpful. Thanks! Prove that negative absolute temperatures are actually hotter than positive absolute temperatures Could someone provide me with a mathematical proof of why, a system with an absolute  (such that of a spin system) is  than any system with a positive temperature (in the sense that if a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system). Why is "Nature" usually spelt with an initial capital letter in scientific journals? I am mainly referring to life science here, in case this matters.  I am not talking about the obvious cases, like capitalisation in titles, at the beginning of a sentence, etc., but when it appears mid-sentence, where one would normally expect it to be lowercase.  I've heard some explanation that it gives the impression of Nature as a goddess (see comments on this post ) - but this is probably the last thing I would have expected in a scientific article... Is (or will be) there any sort of light linking in Cycles, or a method to get a light to affect only certain objects that doesn't involve compositing two or more different render layers? Sort of what 3Dstudio does with the Include/Exclude methods. Why doesn't a bike/bicycle fall if going with a high speed? Why does a bike/bicycle fall when its speed is very low or close to zero and is balanced when going with a high speed? Celebrating the same new year several times over? I have absolutely no experience with Linux, and I desperately need to get my computer back up and running again with Windows.  How do I remove Ubuntu and reinstall Windows?    Editor's note: many of the answers are about removing Ubuntu from dual-boot but keeping Windows (which is a bit complicated), while other answers are about removing Ubuntu from single-boot (which is easy: basically just format the disk while installing Windows). The question as written is ambiguous between dual-boot or single-boot. I am trying to represent trade flows on a world map where the size and direction of arrows indicate the exports/imports of a given country.  This type of map is commonly found in magazines and newspapers. However, I find it extremely difficult to make.   Is there any function in ArcGIS for Desktop or any free software that would help me in making it? With all the various races in the federation, how do the humans manage to dominate it? I've been at this for almost two hours and no one seems to have the same problem as I and I doubt this will be solved but I'll try anyway.  this is the code  /tellraw @p ["Would you like to teleport?",{"text":"Yes","color":"green","bold":"true","clickEvent":{"action":"run_command","value":"/tp @p ~ ~4 ~ "},"hoverEvent":{"action":"show_text","value":{"text":"Confirm"}},{"text":"No","color":"red","bold":"true","clickEvent":{"action":"run_command","value":"/tellraw @p{"text":"Come back when you want to teleport"}"}}}]   this part is fine  /tellraw @p ["Would you like to teleport?",{"text":"Yes","color":"green","bold":"true","clickEvent":{"action":"run_command","value":"/tp @p ~ ~4 ~ "},"hoverEvent":{"action":"show_text","value":{"text":"Confirm"}}   it's just when I add this  ,{"text":"No","color":"red","bold":"true","clickEvent":{"action":"run_command","value":"/tellraw @p{"text":"Come back when you want to teleport"}"}}}]   Minecraft gives me the error:     Invalid JSON at line 1 column 203   I've tried to space it out and figure out what column 203 is but I can't, I don't know what I did. Anyone got any ideas? If  $\mathbb R^*$ and $\mathbb C^*$ be respectively multiplicative groups of non-zero real and complex number then whether $\operatorname{Hom}(\mathbb Z_6,R^*\oplus C^*)$  is  isomorphic to $\mathbb Z_{12}$? or $\mathbb Z_6$? or $\mathbb Z_6\oplus \mathbb Z_2$? It seems I put a   Bounty assigned by outside party, two lengthy, reference-citing answers, one "-1" (awarded the bounty), one "-2", two others scored "0" and "-2" respectively, the answers suggesting one or the other is correct, 73 comments and no consensus so far - and me, as the asker, lost without a clue what to think of the answers anymore -  this is no longer "Learners' Level" question, so I thought I'd bring it here and hear what the experts have to say.    Original question:  Plurality of verb depending on plurality of list elements  An edit was suggested to my sentence.     There was were an orange, some grapes, two apples and a small pile of cherries on the plate.   In my native language plurality of the verb always follows plurality of the first element on the list. There were an orange,... sounds awkward to me, no matter what follows up. My simple solution was reordering:     There were some grapes, an orange, two apples and a small pile of cherries on the plate.   But that's not the first time I faced this situation and I'd like to know what the rules of grammar say about that - was my editor overzealous or am I trying to copy rules of my language that don't apply in English?    Someone linked a  for negatives, where the situation is more clear-cut ("There is no..."). Same applies to connection with "or" apparently, per accepted answer there. Still, nothing for "and".  It seems there is a consensus that in that if the verb goes after the list, it will be plural.     an orange, some grapes, two apples and a small pile of cherries were on the plate.   There doesn't seem to be clear consensus for:     On the plate { was | were } an orange, some grapes, two apples and a small pile of cherries   Adding more to the confusion is the abbreviated form: "There's"     One last note. In North America, at least, there is a widespread use of using a singular form like "there is" and "there was", without regard for the subject item or items, and this "there is" is often shortened to "there's":           There's three apples on the table!      Could you please clarify this mess? Error notification on plone 4 Elementary formula for permutations? This question is updated to request additional answers and details in light of information now available in the Dungeon Master's Guide. In the released DM basic rules, Lost Mine of Phandelver, Hoard of the Dragon Queen, and the DMG teaser there are wands, staves, and rings that you can use to cast spells.  Do abilities that trigger from casting a spell activate when casting a spell from an item?  Can the character modify the spell using class abilities when casting a spell from an item?   Examples include:  An abjurer Wizard's arcane ward recovering hit points from a Dispel Magic cast from a Ring of Spell Storing  A sorcerer using Twinned Spell on a ray of enfeeblement from a Staff of Power  An evocation wizard using Sculpt Spell on a fireball from a Wand of Fireballs.   Now that the DMG is out we have some clearer rules on casting spells from items (DMG p. 141):  Some magic items allow you to cast a spell from the item.  The spell is cast at the lowest possible spell level, doesn't expend any spell slots, and requires no components, unless the item's description says otherwise  The spell uses its normal casting time, range and duration, and the user of the item must concentrate if the spell requires concentration. A magic item, such as certain staffs, may require you to use your own spellcasting ability when you cast a spell from the item.  Does whether or not you can apply class abilities to the casting of the spell depend whether the item uses the user's own spellcasting ability (such as the Staff of Fire) versus an item that comes with a fixed spellcasting ability built-in (such as the Wand of Fireballs)? Does it matter if it's something that triggers more or less automatically like Arcane Ward, versus something you have to control, like Careful Spell? I'm working on a 2nd order passive low pass filter, consisting of two passive low pass filters chained together.         &ndash; Schematic created using   Let \$ H(s) = H_1(s)H_2(s) \$ where \$ H_1(s) \$ and \$ H_2(s) \$ are the transfer functions for each separate filter stage.  Then \$ |H(s)| = |H_1(s)||H_2(s)| \$  Knowing the magnitude of a passive low pass filter,   $$|H(s)| = \dfrac1{\sqrt{ (\omega R_1C_1)^2 + 1} } \times \dfrac1{\sqrt{ (\omega R_2C_2)^2 + 1} } = \dfrac1{\sqrt{((\omega R_1C_1)^2 + 1)((\omega R_2C_2)^2 + 1)}} $$  Then trying to find the cutoff frequency:  $$ \left(\dfrac1{\sqrt{2}}\right)^2  = \dfrac1{\sqrt{((\omega R_1C_1)^2 + 1)((\omega R_2C_2)^2 + 1)}} $$ $$ 2  = \sqrt{((\omega R_1C_1)^2 + 1)((\omega R_2C_2)^2 + 1)} $$ $$ 4  = ((\omega R_1C_1)^2 + 1)((\omega R_2C_2)^2 + 1) $$ $$ 4  = (\omega R_1C_1)^2(\omega R_2C_2)^2 + (\omega R_1C_1)^2 + (\omega R_2C_2)^2 + 1 $$  And I'm stuck. Research on the web tells me \$ \omega_c = \dfrac1{\sqrt{R_1C_1R_2C_2}} \$, but I can't find why? Can anybody show my the derivation to find this? Is there a list of the names used for Sauron in Lord of the Rings? Use of inequality $1 - \cos (x) \leq x^2 /2$ In  answer the value of $1 - \cos(x)$ has to be evaluated in order to find its upper limit, if it exists.  In particular, $x = 2 \pi / n$. The answer is related to the length of a side of a regular $n$-gon inscribed into a unit-radius circumference; because the perimeter of the $n$-gon is always less than $2 \pi$, the single side must always be less than $2 \pi / n$.  The inequality  $$1 - \cos (x) \leq \displaystyle \frac{x^2}{2}$$ (1)  is used and the proof is completed with   $$2(1 - \cos(x)) \leq (2 \pi / n)^2$$  $$\sqrt{2(1 - \cos(x))} \leq 2 \pi / n$$  But it is well known that the cosine is a function $f(x) \in [-1;1]$, so $1 - \cos (x) \in [0,2]$. By using this information, we would obtain  $$1 - \cos (x) \leq 2$$ (2)  The proof would provide  $$2(1 - \cos(x)) \leq 4$$  $$\sqrt{2(1 - \cos(x))} \leq 2$$  which is a completely different result.   Why in that case it is preferable to use (1) instead of (2)? How to choose when it is convenient to use (1) and when to use (2) in a proof?