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# August 15

## "Strong metric"

What do we call a metric on a topological vector space that satisfies the additional property that the only case of equality for the triangle equality is collinear points? Or on a graph, the only case for equality for the triangle inequality is where every path between two points must pass through the third point. The Euclidean metric satisfies this: the distance between points a and b is always strictly less than the distance between a and c plus the distance between b and c, unless the points are collinear. The taxicab metric obviously does not satisfy this property.--Jasper Deng (talk) 07:22, 15 August 2019 (UTC)

Is Strictly convex space what you are looking for? —Kusma (t·c) 08:17, 15 August 2019 (UTC)
@Kusma: This doesn't look straightforward to show even if it is, though I am thinking so: a handwavy argument I've given myself is that, being strictly convex means that the shortest of all paths (a line) is contained in our space, which in a normed vector space means being collinear, but don't know how to formalize it.--Jasper Deng (talk) 23:37, 15 August 2019 (UTC)
In the "Properties" section of that article, the third property follows from the second (which is just the definition of strict convexity): Take ${\frac {x}{\|x\|}}$ for x, ${\frac {y}{\|y\|}}$ for y and $\alpha ={\frac {\|x\|}{\|x\|+\|y\|}}$ . With that property, $\|a-c\|=\|a-b\|+\|b-c\|$ implies $b-c=\lambda (a-b)$ , so c is on the straight line through a and b. —Kusma (t·c) 06:18, 16 August 2019 (UTC)
Ahhh, yeah I knew I was missing something elementary. How do we generalize this to a graph though? It's a different setting but the basic idea is the same. --Jasper Deng (talk) 06:46, 16 August 2019 (UTC)
You probably need something like uniqueness for the shortest connections, which may mean you need a tree. But that is just guessing. —Kusma (t·c) 07:43, 16 August 2019 (UTC)
There certainly are non-tree graphs with unique shortest connections, for example polygons with an odd number of vertices. —Kusma (t·c) 11:42, 16 August 2019 (UTC)
It's not necessarily unique shortest connections; basically, a direct connection must always be shorter than any indirect one. Not sure how this plays out for pairs of nodes without direct connections though.--Jasper Deng (talk) 16:14, 16 August 2019 (UTC)

───────────────────────── I am not sure what your definitions of "direct connection" and "indirect connection" are on a general graph. Is your graph weighted? —Kusma (t·c) 09:11, 17 August 2019 (UTC)

A direct connection is just one edge; an indirect connection passes through another node. This is for weighted graphs.--Jasper Deng (talk) 16:09, 17 August 2019 (UTC)

## Schrödinger equation in math

The article Schrödinger equation is entirely about the Schrödinger equation in quantum mechanics. But I believe it is also important in PDE's as a math topic, not particularly related to physics. Are there some non-physics areas where it arises? Is it still about complex Hilbert spaces? What I'm actually wondering is whether something like complex probability amplitudes and the Born rule arise naturally in math, not counting math inspired directly by quantum mechanics. Areas like classical fluid mechanics are ok though. Thanks. 173.228.123.207 (talk) 22:14, 15 August 2019 (UTC)

The nonlinear Schrödinger equation is a generalization of the Schrödinger equation that involves a complex field. It is studied in physics and has bee used to model wave phenomena in optics and water waves. The fields are not quantum probability amplitudes, however. --{{u|Mark viking}} {Talk} 22:28, 15 August 2019 (UTC)
Diffusion can be described by a Schrödinger equation in imaginary time, see here. Count Iblis (talk) 23:40, 15 August 2019 (UTC)
Thanks. The NLSE article is interesting. I'm trying to make sense of the path integral article section. 173.228.123.207 (talk) 03:11, 16 August 2019 (UTC)

## Stuck on a vector problem.

I have gotten a vector assignment that is unlike any I've seen before and I dont even know how to begin to tackle it, Im hoping someone can give me some hints.

you have info on 3 coordinates: A(2,3) B(6,4) C(6,6). Vector(AB) = vector(v), vector(BC)=(0 2)

For all possible numbers t include [0;1]

Point D is defined by vector(AD)=t*vector(v)

Task: Solve t so the area of triangle ADC and DBC are the same.

PS: not sure how to write math things on wikipedia so I hope its understandable.

91.101.26.175 (talk) 23:14, 15 August 2019 (UTC)

Why not just take $t={\frac {1}{2}}$ ? Geometrically, Cavalieri's principle guarantees that this will get the areas to be equal. If you want to solve it by brute force, use the cross product area formulae for the smaller triangles and equate them: ${\frac {1}{2}}||\mathbf {v} t\times {\vec {AC}}||={\frac {1}{2}}||\mathbf {v} (1-t)\times {\vec {BC}}||$ . Since $\mathbf {v} ={\vec {AB}}+{\vec {BC}}$ , we can relate the cross products on both sides and get the same result. If we substitute that in for $\mathbf {v}$ and use the fact that the cross product of a vector with itself is zero, we obtain $||t{\vec {BC}}\times {\vec {AC}}||=||(1-t){\vec {AC}}\times {\vec {BC}}||$ and thus $t=1-t$ or $t={\frac {1}{2}}$ (note this manipulation is valid only with the restriction on t you placed as it assumes both $t,1-t$ aren't negative).--Jasper Deng (talk) 23:34, 15 August 2019 (UTC)

I did guess t=0,5 and I've been trying to reverse engineer my way back to the result without success. I'm familiar with Cavallieri's Principle and the wiki entry isnt the most user friendly. Can you advice a place where its written a bit more comprehensively? Also not sure why you state v = AB+BC let me try to write it a bit better:

${\vec {v}}={\vec {AB}}$ ${\vec {BC}}={\begin{bmatrix}0\\2\\\end{bmatrix}}.$ t include all numbers in [0;1]
${\vec {AD}}=t*{\vec {v}}$ Ive tried to do cross product calculations but I keep ending up at t=t-1 rather than t=1-t. Ive reviewed it dusins of times and i cant see i missed anything.

91.101.26.175 (talk) 00:46, 16 August 2019 (UTC)

My argument still holds, sorry for misreading your notation though. Replace AB with AC, and BC with CB in the cross product equations above. I don't think you're familiar with Cavalieri's principle since it is plain obvious why it applies here: if I divide every cross-section of the triangle in half then I end up dividing the whole thing in half, and it is an elementary geometric proof to show that. The analogy with the usual coins example of the principle is, lean the stack to some side, cut every coin in half in one slicing motion. The volume will be halved. Your computations with the cross product ignore the fact that you need to take the absolute value of it first.--Jasper Deng (talk) 00:56, 16 August 2019 (UTC)

Sorry I meant to write im UNfamiliar with it, can you recommend a place where its a bit more comprehensive and easier to read?

Im unsure how you got to those calculations, where did the (t-1) come from and how is it possible you only added it to one side? Yes dividing everything in half will... Well divide things in half but why wouldnt a 9:11 ratio for example give the triangles equal area since they dont have the same starting line why are we treating them as identical? 91.101.26.175 (talk) 01:02, 16 August 2019 (UTC)

Let's be a bit more explicit: ${\frac {1}{2}}||\mathbf {v} t\times {\vec {AC}}||={\frac {1}{2}}||\mathbf {v} (1-t)\times {\vec {CB}}||$ , by substitution and cancellation we have $||t{\vec {CB}}\times {\vec {AC}}||=||(1-t){\vec {AC}}\times {\vec {CB}}||$ , then since t and 1 - t aren't negative, we can pull both of them out of the absolute values, $t||{\vec {CB}}\times {\vec {AC}}||=(1-t)||{\vec {AC}}\times {\vec {CB}}||$ , and then since $||\mathbf {a} \times \mathbf {b} ||=||-\mathbf {b} \times \mathbf {a} ||=||\mathbf {b} \times \mathbf {a} ||$ (this is the part you've overlooked), we have $t||{\vec {AC}}\times {\vec {CB}}||=(1-t)||{\vec {AC}}\times {\vec {CB}}||$ . Canceling the $||{\vec {AC}}\times {\vec {CB}}||$ (allowable since the area isn't zero) yields $t=1-t$ . To better understand Cavalieri's principle, note that the area of a triangle depends only on the length of the base (vt here) and the height, and nothing else.--Jasper Deng (talk) 01:08, 16 August 2019 (UTC)

${\frac {1}{2}}||\mathbf {v} t\times {\vec {AC}}||$ so this part makes sense, vt is equal to AD and we need AD and AC to do the cross product. But this part doesnt make sense to me ${\frac {1}{2}}||\mathbf {v} (1-t)\times {\vec {CB}}||$ since one is CB im assuming the other is suppose to be CD, but by my calculations CD should be CD=D-C=${\begin{bmatrix}4t-4\\t-3\\\end{bmatrix}}.$ how did you get CD to be CD=v(1-t), and then you suddenly substitute v for CB and v for AC.

91.101.26.175 (talk) 01:30, 16 August 2019 (UTC)

Clearly you didn't bother to fully read my first reply, which explains why I was able to relate the cross products. You forgot that $\mathbf {v} t=({\vec {AC}}+{\vec {CB}})t$ and therefore its cross product with ${\vec {CB}}$ is $({\vec {AC}}+{\vec {CB}})t\times {\vec {CB}}=({\vec {AC}}t+{\vec {CB}}t)\times {\vec {CB}}={\vec {AC}}t\times {\vec {CB}}+{\vec {CB}}t\times {\vec {CB}}=t({\vec {AC}}\times {\vec {CB}})+t({\vec {CB}}\times {\vec {CB}})=t({\vec {AC}}\times {\vec {CB}})+t\mathbf {0} =t({\vec {AC}}\times {\vec {CB}})$ . The cross product of any vector with itself is zero. And as for how I got the ${\vec {DB}}=(1-t)\mathbf {v}$ part, this is simple vector arithmetic so I'll leave that as an exercise for you. I'm not using ${\vec {CD}}$ but rather, ${\vec {DB}}$ .--Jasper Deng (talk) 04:39, 16 August 2019 (UTC)

# August 17

## Math question at WP:CANCER

Obviously not going to get an answer to the math question, and the spending question is off topic. --Guy Macon (talk) 20:54, 18 August 2019 (UTC)

At User:Guy Macon/Wikipedia has Cancer I asked the following math question:

"The good news is that the donations have also continued to grow, but anyone who thinks that this will continue forever simply does not understand basic economics. I really ought to calculate how soon the present rates of increase will bring us to the point where donations to the WMF are larger than the amount of money in the world. Anyone willing to run those numbers is invited to discuss them on the talk page for this essay."

I can do the calculation with ease, but deciding on reasonable assumptions for the future growth of total world wealth (or maybe I should specify GDP?) and deciding on what curve the donations are following is a lot harder, and I am a bit wary that I may have an unconscious bias.

Does anybody want to give it a shot? --Guy Macon (talk) 17:15, 17 August 2019 (UTC)

Sigmoid function is the applicable article. However, note that, in your case, 1.0 on the vertical axis is not 100% of the money in the world, but the maximum donation rate Wikipedia will get, say per year. Thus it should start leveling off as it approaches this upper limit. SinisterLefty (talk) 17:25, 17 August 2019 (UTC)
A sigmoid function isn't a specific function, but a whole class whose graphs have a similar basic shape. Particular sigmoid functions can have dramatically different decay rates. And yeah, it depends what you're measuring donations in, dollars at the time of donation, 2019 dollars, etc. So sure, you can continue to grow indefinitely, especially if you don't account for inflation; to really say something meaningful you'd have to more carefully specify how donations are measured and if you really mean "grow", or maybe "grow at the current rate". This might be a better question for WT:ECON. –Deacon Vorbis (carbon • videos) 18:14, 17 August 2019 (UTC)
What I was looking for is an argument like this:
Reductio ad absurdum argument:
IF you assume that the rate of donations will continue to increase at the current rate forever (so no sigmoid function -- even though we all know it will either level off or start going down, this is a "what if" argument aimed at those who insist that it will continue to increase the rate of growth forever)...
And IF you assume that the [total wealth of the world / global GDP -- take your pick] will continue growing linearly at the current rate...
THEN in the year XXXX wikipedia will have more money than there is money in the world.
Note that I am saying that in this imaginary scenario the world economy will grow at a linear rate and Wikipedia donations will continue to grow at an increasing (exponential? look at the curve for yourself and decide, but it certainly is not following a straight line) rate.
THEREFOR the initial assumption that donations will never level off must be wrong,
And THEREFOR sooner or later the WMF will have to stop the ever-increasing spending.
This is the same basic argument as "your cancer cannot keep growing at that rate. If it did, in 20 years it would weigh 20 times more than your current weight" or "you cannot put a thousand dollars in a bank account with 5% compounding interest, use a time machine to go into the far future, and end up being rich enough to buy everything on earth" or "at the current rate of membership increase, in twenty years there will be more Southern Baptists than there are people". --Guy Macon (talk)
This is all true, but not exploiting the growth potential for as long as it lasts is a bad business decision. So, the point you want to make with this calculation may not be valid. Count Iblis (talk) 01:02, 18 August 2019 (UTC)
Well, it is an op-ed, so I don't expect everybody to agree. I am all for exploiting the growth potential in donations for as long as it lasts. I strongly advocate exploiting the growth potential of the endowment and growing our savings for as long as it lasts. I am even for exploiting the growth potential in`the actual work the WMF accomplishes for as long as it lasts and as long as they can identify actual new work to do. What I am not for is huge growth in spending and in numbers of employees while essentially doing the same amount of actual work. We were running an encyclopedia just fine with X number of dollars, and now we are pretty much running the same encyclopedia (OK, maybe 10X bigger) but spending has increased by several hundredfold and the growth is accelerating. That doesn't seem like a good business decision to me. --Guy Macon (talk) 01:58, 18 August 2019 (UTC)
Mathematics is silent on a business decision of this sort. Whether such a decision is the "right" one or not will always be political and while math will be done to support one side or another for every budgetary decision anywhere, there's no strictly mathematical reasons anyway. Your attempted "argument" above lacks rigor and you should not attempt to present it as a proof by contradiction. Be careful what you mean by "rate". You may be meaning the logarithmic derivative rather than the usual derivative. For example, an annual growth rate of 5 percent means that the logarithmic derivative with respect to the number of years is $\ln(1.05)$ .--Jasper Deng (talk) 07:47, 18 August 2019 (UTC)
One thing is clear from the data, if only the WMF had not exchanged the Bitcoins they got for dollars when Bitcoin was worth thousands of times less than it is today, they would be financially in much stronger position. Count Iblis (talk) 09:13, 18 August 2019 (UTC)
I only read the first 5 paragraphs of your essay. The problem with your analysis doesn’t pertain to mathematics or economics (fwiw, I have 3 bachelor’s degrees in math, finance, and economics); it has to do with the WMF’s business model. You’re comparing exactly 1 content project that the WMF offered in 2005 to that 1 product now while contrasting it with all of its business expenses back then vs now. The problem with doing that is that the number of meta:Wikimedia projects has grown and the number of languages for which it provides content has also increased since that time; the costs it incurs is relative to the size and use of all those projects combined, not Wikipedia alone. I really doubt the number of projects will continue to increase at the same rate as it has the past. Seppi333 (Insert ) 13:42, 18 August 2019 (UTC)
The modern Wikipedia hosts 11–12 times as many pages as it did in 2005, but the WMF is spending 33 times as much on hosting, has about 300 times as many employees, and is spending 1,250 times as much overall. I am completely open to replacing that "hosts 11–12 times as many pages" metric with any reasonable alternative, such as number of wikis of all kinds, number of page views, number of edits per year, or whatever else you prefer. I think we can all agree that Wikipedia is doing more, but I need a definition of "more" that I can put a number on. Got any suggestions?
That being said, while we can all agree that Wikipedia is doing more, are you willing to claim that Wikipedia is doing 1,250 times more?
Let's look at this another way. Instead of going back to 2005, let's go back ten years to 2007-2008 when we spent $5,032,981 USD to do everything. In 2017-2018 we spent$104,505,783 USD.
In 2008 Wikipedia had over 5 million registered editors, 250 language editions, and 7.5 million articles. wikipedia.org was the 10th-busiest website in the world. The number of regularly active editors on the English-language Wikipedia peaked in 2007 and has since been declining.
Are you willing to make the claim that we are doing twenty times more than we were doing 10 years ago and thus need to spend twenty times as much money? I was here in 2015. I did not notice any evidence of pressing needs that were not funded because we were spending 5% of what we are spending now. --Guy Macon (talk) 18:34, 18 August 2019 (UTC)
Why not just ask the WMF what its expenditures in excess of $1,000,000 are? You won’t have to ask hypothetical questions if you know what it’s spending money on. Seppi333 (Insert ) 18:54, 18 August 2019 (UTC) I have been asking the WMF to give details of expenditures for years. They have never given me any response other than a link to the existing non-detailed financial reports. To give you a feel for the kind of things they refuse to reveal, here is a question that I have been asking since 2015. (I started asking this particular question after hearing complaints that asking the WMF to explain how they spend our donations was too much work, so I asked about one tiny detail just to see if they will ever reveal any details.) Extended content Some here have, quite reasonably, asked "where does the money I donate to the Wikipedia Foundation go?" Well, about two and a half million a year goes to buy computer equipment and office furniture. That's roughly twelve thousand dollars per employee. The report says "The estimated useful life of furniture is five years, while the estimated useful lives of computer equipment and software are three years." so multiply that twelve thousand by three or more -- and we all know that at least some employees will be able to keep using a PC or a desk longer than that. I would really like to see an itemized list of exactly what computer equipment and office furniture was purchased with the$2,690,659 spent in 2012 and the \$2,475,158 spent in 2013. Verifying that those purchases were reasonable and fiscally prudent would go a long way towards giving me confidence that the rest of the money was also spent wisely.
If I can't get an itemized list of where the money was spent, could I at the very least get a breakdown as to how much was spent on computer equipment and how much was spent on office furniture? It wouldn't be an actual answer to my question, but it would at least allow me to either ask a question about computer equipment or ask a question about office furniture instead of repeatedly asking the same question about computer equipment and office furniture.
A little bit of financial transparency would go a long way here. -- Guy Macon
Also see: User talk:Guy Macon#WMF Financial Transparency.
I see no point is asking another financial question when they have not answered any of the previous ones.
Speaking of unanswered questions, I ask you again, are you willing to make the claim that we are doing twenty times more than we were doing 10 years ago and thus need to spend twenty times as much money? --Guy Macon (talk) 19:45, 18 August 2019 (UTC)
I strongly consider continued discussion of the WMF's particular case to be off-topic here. It's not our (the reference desk's) job or prerogative to do their math for them.--Jasper Deng (talk) 20:22, 18 August 2019 (UTC)

I don't think there was a reasonable math question there, so hatting/closing the thread was the right thing for the math refdesk. To some extent there is a reasonable economics/politics question about how growth rates of firms max out, that would belong on the humanities desk. The WMF isn't that good a subject for such a question but one but one could reasonably ask the maximum possible future size of Amazon or Microsoft, or under what conditions the historical Dutch East India Company's growth might have continued unchecked. 67.164.113.165 (talk) 05:49, 19 August 2019 (UTC)