Wikipedia:Reference desk/Archives/Mathematics/2019 August 15

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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)

@Jasper Deng: 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)

@Jasper Deng: 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)