The h Bar

2:03 AM

Hello Everyone
I am trying to study theoretically (Mathematically) the Effects of Flow Rate and Temperature on Thermoelectric Power Generation. I have experimental data but now I want to do mathematical analysis to compare it.
1. How can I find temperature difference for different flow rates? Is there any mathematical expression for this?
2. For different temperature I want to calculate the electricity produced.
Can someone please point me in the right direction?

uhoh

1

Q: Have cosmic ray muons or solar neutrinos been used to probe inside the Earth?

By looking at the attenuation of cosmic ray muons passing through pyramids as a function of their direction, it seems it is possible to "look inside" by mapping their 3D density distributions. Washington Post: Cosmic rays reveal mysterious void in Egypt’s Great Pyramid Nature: Cosmic-ray partic...

M.N.Raia

8:18 AM

Goodmorning/afternoon/night for everyone!

So what suppose to mean Wolfram's recent ideas?

phys.org/news/…

3

I mean, the guy explained, but I'm keep thinking on what he thinks he's going to achieve. For example, it's quite challenging to learn GR and QFT, then merge both the concepts together on semiclassical picture of gravity. Furthermore, publish and create n

Science are the next level of scientist's career

But

This is the tone for the majority of physics

How can an heuristic concept of Wolfram's could say that something about "a direction of right models?"

bolbteppa

(The constraint stuff is fine, it's supposed to be $\dot{q}_i = \{q_i,H_c + u_m \phi_m \} = \frac{\partial H_c}{\partial p_i} + \frac{\partial u_m}{\partial p_i} \phi_m + u_m \frac{\partial \phi_m}{\partial p_i} \approx \frac{\partial H_c}{\partial p_i} + u_m \frac{\partial \phi_m}{\partial p_i}$ where you can set $\phi_m \approx 0$ because it's after computing a P.B.)

M.N.Raia

Hey bolbteppa which book are that?

bolbteppa

See the video above

ACuriousMind

8:58 AM

@M.N.Raia My money is on "it can't'.

bolbteppa

Hawking: "However the split into three spatial dimensions and one time dimensions seems to be contrary to the whole spirit of relativity. Moreover it restricts the topology of spacetime to be the product of the real line with some three-dimensional manifold, whereas one would expect that quantum gravity would allow all possible topologies of spacetime including those which are not products"

ACuriousMind

Already his NKS wasn't really well-received by the majority of physicists, I see little reason to believe this will be any different.

bolbteppa

Apparently the way Dirac ignored and then re-introduced the surface terms in the EH action in that GR paper caused a big drama in the GR community in the late 50's and people like ADM and someone else re-derived these results without any controversy

Slereah

This isn't the first time Wolfram tried his hands at the One True Theory of Science

The last time he did so he was thoroughly mocked for his hubris and he then retreated, vowing to take revenge on science

bolbteppa

9:18 AM

'usually particles are treated as point particles, that's not what I do in my model, I wont explain how this doesn't violate relativity I'll just plow on to the next huge claim like a serious person'

I think this is Lisi $E_8$ level stuff

Slereah

From what I've skimmed, Wolfram's theory just looks like causal sets

ACuriousMind

Mar 18 at 16:22, by ACuriousMind

@NovaliumCompany You've also been told to speak English in this room often enough.

bolbteppa

Attaching particles to a hypergraph in Wolfram's case, whereas in Lisi's case it was to the $E_8$ root system, both stickerboard level thinking

Slereah

My new theory of everything is particles as moving on a chessboard

the graviton is the horse

Also it's a ten dimensional chessboard

bolbteppa

The 11th dimension is the person moving the chess pieces - MMMMMe theory

Slereah

9:26 AM

This is very much in line with Wolfram's old theory

Wolfram's big thing was graphs and cellular automata and whatnot

He is a man with a hammer in search of a nail

JingleBells

9:59 AM

Don't take life too seriously, you won't get out of it alive anways

I'm not a big fan of that philosophy but it applies here I guess

@ACuriousMind What's wrong with randomly typing stuff in different languages?

I'm not doing it all the time

PM 2Ring

I suspect we'll get a steady influx of questions about the new Wolfram stuff. I've only seen 2 so far, including this one from today: physics.stackexchange.com/q/544281/123208

Slereah

I used to know a guy who knew Wolfram

wolframscience.com/prizes/tm23/alex_smith_bio.html

That guy

From his testimony Wolfram does indeed have a huge ego

PM 2Ring

10:15 AM

I've read that he's not easy to work for, eg glassdoor.com/Reviews/…

PM 2Ring

10:26 AM

Not long after A New Kind of Science was published, I read a rant by a very disgruntled ex-employee. This guy claimed that he'd slaved away doing the "grunt" work of investigating lots of cellular automata, trying to find rules governing which CA rule sets led to interesting behaviour and which ones were boring. Allegedly, Wolfram took the credit for this work himself.

bolbteppa

After reading the overview to the new physics thing, the first thought I had was that the people who helped him were too afraid to question him about any of this, that review doesn't surprise me but there's two sides to everything

Slereah

@PM2Ring Basically what Alex Smith told me, yes

lettersofnote.com/2010/06/09/…

2

Of note

so infamous that even one of the most famous physicist calls you a prick

ACuriousMind

10:50 AM

@JingleBells Two reasons off the top of my head: 1. The language of SE is English, and users therefore expect that content here is in English. While chat is informal, users should still be able to read it. 2.The moderators cannot moderate content they cannot read/understand. How do I know you're "randomly typing stuff" and not hurling insults?

PM 2Ring

11:01 AM

FWIW, although CA show that complexity can arise from simple rules (given enough time & space), I don't think CA are very suitable for modeling real-world physics. It usually takes a lot of generations and space for a CA to exhibit sophisticated behaviour. But I certainly could be wrong: 't Hooft thinks CA approaches to fundamental physics are worth investigating.

Slereah

they're worth investigating

A bit early to claim they are the solution to everything

PM 2Ring

11:21 AM

I've spent a lot of time playing around with CA, mostly Conway's Life. IIRC, I read the Martin Gardner columns in Scientific American about Life in 1971 or maybe 1972. My first investigations were done on our bathroom tiles using counters made of plasticine (modeling clay). :)

JingleBells

@ACuriousMind thx, no negativity from me

PM 2Ring

Here are some of my patterns from a decade or so ago, on the conwaylife forum. Most Life programs can load the RLE format used on that site.

This thread has a larger pattern which "prints" a portrait of John Conway using lightweight spaceships. Conway himself discovered that spaceship, which is occasionally spontaneously generated by random "soups". There's also a pattern that prints a portrait of Marilyn Monroe in that thread.

Slereah

11:45 AM

Aaaaah

Slereah

1:20 PM

Current task is "Put the articles I opened in tabs in the bibliography of my articles"

So that I have less tabs opened

JingleBells

1:43 PM

I have invented time travel!

I will invent time travel!

Slereah

Hm

One of the paper opened is on holonomies

and its by... Urs Schreiber

Noooo

JingleBells

@Slereah How are you?

@Slereah How is your day going?

Charlie

2:09 PM

bolbteppa

It's looking like a Gauge theory is just a theory coming from a Lagrangian such that the accelerations for some of the variables are not specified by the Lagrangian or by playing with the EL equations (this playing, i.e. differentiating the EL equations w.r.t. time, is where the primary/secondary/first/second class stuff comes from in the Hamiltonian setting!). In such a case, if the Lagrangian has a symmetry involving this variable, it's going to involve a 'space-time' dependent function...

geocalc33

sup

Slereah

pretty much

underdetermined equations and all that

bolbteppa

There is an incredible constraint example in Sundermeyer's book, looking at one Lagrangian $L = \frac{1}{2} \dot{q}_1^2 + q_2 \dot{q}_1 + (1 - \alpha)q_1 \dot{q}_2 + \frac{\beta}{2}(q_1 - q_2)^2$ in four cases ($\alpha,\beta=0$, $\alpha=0,\beta \neq 0$, and $\alpha \neq 0$ which then naturally breaks into two cases), first (amazingly) completely in terms of Lagrangians only then re-doing it using Hamiltonians nicely motivating all Dirac's concepts.

In two of the cases by differentiating the $q_2$ EL equation you can uniquely specify the solutions $q_1,q_2$, but in two of the cases you can't, so $q_2$ is an undetermined function and the Lagrangian is invariant under a symmetry which involves an arbitrary function like in EM

Mad Spaces

can someone help e with this easy problem? i just dont know how to set up a correctly the parameters of the problem to integrate. I reliaze that the object moves it self on a straight . but how do i procceed?

JingleBells

2:24 PM

@Slereah Why are you ignoring me :(

geocalc33

When deriving the Einstein field equations...

what assumptions do I need for a fairly simple case

bolbteppa

@MadSpaces look up line integrals and some examples of computing the work between two points in 3D

ACuriousMind

@bolbteppa Yes, the Lagrangian definition of a gauge theory in the broad sense is that the general solutions to the equations of motion contain arbitrary functions of (space)time.

Mad Spaces

I did but i always get way too complicated articles and i just get a huge headache.
i know how to calculate the integral if give a vector R but my problem here is how to construct R not how to calculate the integral! i can integrate!

geocalc33

I made a lorentz transformation on desmos! wohoo

ACuriousMind

2:31 PM

Once again, I must recommend the book by Henneaux/Teitelboim for a very clear modern exposition of gauge theories in the broad sense in the first few chapters

bolbteppa

You don't need to construct $r$ to compute a work integral you need to compute a line integral involving $\mathbf{F} \cdot d \mathbf{r} = F_x dx + F_y dy + F_z dz$

ACuriousMind

("broad sense" I mean in opposition to the restrictive Yang-Mills motivated "a gauge theory is a theory with a gauge group and a gauge field and a covariant derivative")

Mad Spaces

But you need to construct r in order to derive r to get dr!

bolbteppa

I definitely missed this simple perspective on gauge theory from skimming that book before but yeah you have to know what's going on in that book eventually

@MadSpaces I think you're trying to say you need to know $\mathbf{r}(t)$ in order to compute $W = \int \mathbf{F}(\mathbf{r}(t)) \cdot \frac{d \mathbf{r}}{dt} dt$ and from the way the problem is stated (can barely read it :p) it's hard to figure out what $\mathbf{r}(t)$ should be, well you're only given two points and a vector field and asked to compute a line integral between those two points, is it conservative?

Mad Spaces

This is just the question there is nothing more writte ot then the question. you read it right.

bolbteppa

2:43 PM

Do you know what a conservative vector field is

Line integral

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane. The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with...

Mad Spaces

yes it that in a closed curve the work is zero

The point moves itself on a straight. Could this be helping to construct a straight as a equation for R(T)? i tried this but i didnt know how to continue

bolbteppa

What about a non-closed curve for a conservative vector field

Mad Spaces

I dont know. But i dont think this is relevant for the question is it.

bolbteppa

Look at the section of the wiki I linked to

Mad Spaces

I dont see how this should help me.

John Rennie

2:58 PM

@MadSpaces I would move between the points in two steps, first in the x direction then in the y direction. This makes the two integrals very easy to do.

We can get away with this because the field is conservative so it doesn't matter what path we take between the start and end point.

Mad Spaces

Hey John. Yes you are right. i thought of this. But i think maybe you guys are misunderstanding me. My problem is to write this mathematically. i had something in mind such as expressing the movement like $\vec{r}(t):= t* \vec{e_x}+2t* \vec{e_y}$ But i dont seem to make progress with the notation

JingleBells

@Slereah If you're not gonna chat me, at least tell me why. Did I do something wrong? Tell me my mistake, what have I done that you don't appreciate?

ACuriousMind

@JingleBells Other users are not required to explain themselves to you. If he doesn't respond, please don't ping him again.

JingleBells

:(

He was such a good person, really helped me... I don't know what happened

Aaron Stevens

0

Q: What are the physicists conclusion to chi, the alleged energy of life?

I guess this might be too much of a broad question for this forum, but there is something called 'chi' which is basically the energy of life. It tries to tell that every living thing has some sort of energy flowing through him or her. There are a lot of videos on YouTube, which there is n...

energy potential-energy

@ACuriousMind Trying to shut down biophysics I see? :P

ACuriousMind

3:06 PM

oh dear

John Rennie

@MadSpaces I think I see what you mean. You mean take the vector $\mathbf r = (1, 2)$ that joins the start and end points and integrate along this line?

JMac

Anyone else get like a nervous twitch any time they see someone call SE a "forum"?

Aaron Stevens

@JMac Yes. Or "Isn't this supposed to be a physics discussion site?"

ACuriousMind

Yes.

Mad Spaces

@JohnRennie yes

John Rennie

3:08 PM

Well the line is $y = 2x$, so everywhere on this line the force is $\mathbf F = (-2ax, -ax, 0)$.

And $d\mathbf r = (dx, 2dx)$.

So $\mathbf F \cdot \mathbf r = -2ax dx - 2axdx = -4ax dx$.

Tanuj

@JohnRennie hi

John Rennie

Now just integrate from $x = 0 to x = 1$

@Tanuj hi :-)

JMac

@AaronStevens Yeah that's a big one. I think what gets me about "forum" is that I don't even know whether to call someone out on it. I can't tell if they think of it as like a discussion board, or if they are just being imprecise when they say "forum" and basically just mean "website where people communicate through text".

Tanuj

@JohnRennie where can we talk

John Rennie

@Tanuj Do you want to chat in your room? I've lost the link so you'll need to invite me.

Tanuj

3:11 PM

That room seems to have disappeared

How do I invite you?

John Rennie

Found it! :-)

$Mathematics$ Room for Tanuj and John Rennie

Tanuj

Oh lol

Mad Spaces

@JohnRennie I have the exact same result but i put t instead of x

Aaron Stevens

@JMac I prefer to use a fiveum instead of a forum

Mad Spaces

@JohnRennie This. when i put t instead of x i didnt know how to put the integration limits like you did wait i will show you let me type it

@JohnRennie example express the vector such as $\vec{r}(t) := t*\vec{e_x}+2t*\vec{e_y}$ now subtitute that in your force field and you get the same result as you did but with $t$ and $dt$
if you derive $\vec{r}(t)$ you get the the original vector $(1,2,0)$ and put that now multiplied by the force field and multiply and integrate to t you get your result. My question. At which point do i put the integral limits for t like you did for x?

Oh.. Do i just assume it has to be from 0 to 1 since that satisfies the maximumv alues of the straight?

John Rennie

3:23 PM

@MadSpaces Yes, becasue $t=0$ at the start point and $t=1$ at the end point. So just integrate $\int_0^1 dt$

Mad Spaces

yup!! thanks john!!

bolbteppa

I can't believe this is what gauge theory is...

vzn

3:48 PM

@M.N.Raia am actually going to concur with "the management" in 1 respect: wolfram has "cried wolf" before and faces a major uphill battle. however, think he is generally on the right track, off on some specifics. like the open science/ source flavor of his endeavors. yes he has a universe-sized ego. one has to overlook his faults as with all humans. lets not forget (most physicists have!) newton was decades deep into alchemy. kepler believed in planetary alignments based on platonic solids...

How We I Got Here: The Backstory of the Wolfram Physics Project writings.stephenwolfram.com/2020/04/…

@JohnRennie funny, point taken, but (left unsaid, a possible elephant in the room) what is your scientific opinion on potential health effects of 5G? speaking of stupid humans, how many times in history have they inadvertently poisoned themselves, in increasingly sophisticated/ novel/ surprising ways?

vzn

4:03 PM

> Early in his career, he was a distinguished physicist, but later, left to found a computer company. ... They discussed delaying the release, but then decided that with much of the world locked down, there might be interest in a deeply cerebral project by isolated people who are growing bored. ← lol!

Physics Meta

2

Q: Is it unreasonable to expect feedback for writing an answer from the asker?

I find it quite frustrating if I write an answer and don't receive any feedback. On stack exchange the feedback is designed to work in the following manner: By accepting an answer. By providing comments. The recent case which caused me to write this post happened something like this: I wro...

feature-request users

Emilio Pisanty

1

Q: Gravity's effect on time

How can we say the time get faster or slower when we are measuring time using tangible objects (like caesium clock) . Can't we say that gravity is affecting the oscillation of caesium clock and hence it's showing error in time? Also does this mean that time is nothing if we don't measure it or ti...

general-relativity special-relativity spacetime

@JohnRennie is this a duplicate?

you know the relativity archives better than me.

John Rennie

@EmilioPisanty ugh that's a poorly defined question.

Physics Meta

4:27 PM

0

Q: Feature Request: Dark Mode

Would it be possible to implement a "Dark Mode" on the Physics Stack Exchange as is being trialed on the Stack Overflow? I personally would prefer that viewing mode. https://stackoverflow.blog/2020/03/30/introducing-dark-mode-for-stack-overflow/

feature-request

PM 2Ring

4:54 PM

@EmilioPisanty Maybe. The top answer here discusses how all clocks give the same time dilation, no matter what mechanism they use to measure time, but I don't think it's close enough to use as a dupe target.

Tanuj

I was recently reading about this effect called quantum locking or flux pinning

and it basically says that a type 2 semiconductor when placed above a strong magnetic field , gets fixed or pinned in the position in which it's initially kept in

I was thinking what are its applications really

I searched around and observed that it could be used in the future to make frictionless joints .

I don't really get what that is . Can someone explain please ?

Can anyone help please ?

vzn

6:05 PM

@Tanuj heres a nice intro from 2011 when it maybe went viral via israeli research video. afaict, this is the same principle behind superconducting qubits ie quantum computing and superconducting maglev trains etc., so yes there are diverse applications being pursued. Quantum levitating (locking) video goes viral phys.org/news/2011-10-quantum-levitating-video-viral.html

JingleBells

6:25 PM

Usher - Yeah (10D Bass Boosted)

►

PM 2Ring

6:36 PM

Does this really need to be on the HNQ? It's a check my work question!

3

Q: Why does my derivation of K.E seem correct but not agree with the formula of K.E?

We know that $$\vec{F}=m\vec{a}$$ $$Work = \vec{F}\cdot\vec{S}$$ where $\vec{S}$ is the displacement. So, we have, $$W = m\vec{a}\cdot \vec{S} = \frac{m(\vec{v}-\vec{u})}{t}\cdot \vec{S}$$ Assuming object started out at rest, $$W = \frac{m\vec{v}}{t}\cdot \vec{S}$$ Since $\vec{v}=\frac{\vec{S...

homework-and-exercises newtonian-mechanics energy kinematics work

Speaking of HNQ, the Maths mods are very active in kicking stuff off the HNQ:

3

in Tavern on the Meta on Meta Stack Exchange Chat, yesterday, by Catija

ACuriousMind

6:59 PM

@PM2Ring If I just kicked off stuff I don't like it would be far higher ;P But we generally only remove stuff when it's flagged with a good reason to do so. Note that that list does not include questions removed from HNQ by closure.

I think if we found that half of our HNQs needed to be removed we'd have to take a look at (i.e. start a meta discussion) whether that kind of questions shouldn't be off-topic to begin with.

skullpatrol

@PM2Ring there is one mod in particular who is doing most of it

Kicking stuff off they don't like is a thorny issue :-)

Mad Spaces

7:37 PM

why does a Maxwell wheel go up? what is the reason

Mad Spaces

7:52 PM

By one variant i read that the body rotates such that the ropes just start getting tied in the other direction and it goes up. but in the other variation apparently the wheel changes spin direction, how does this happen? in intutive way. Also why does the wheel get slower and lose kinetical linear energy to rotation? why does it not just gain kinetical linear energy more than it does for rotation?

geocalc33

8:18 PM

-1

Q: Deriving Einstein field equations

I posted this on math S.E. https://math.stackexchange.com/questions/3628370/deriving-einstein-field-equations. I think it will have more chance for an answer here. Basically I'm trying to derive the Einstein field equations and find the vacuum solution in a nonlinear space (nonlinear hyperbola...

general-relativity special-relativity vacuum

apparently this does not show research effort, is not helpful

incredible.

JMac

@geocalc33 What is the question? It seems like you're just showing a bunch of math without asking anything.

geocalc33

ill just delete it

bolbteppa

9:34 PM

@geocalc33 what is a nonlinear hyperbola

Mad Spaces

A body in rotational movement around an axis (for example a stone tied to a rope being rotated) does it have kinitcal energy? or is just rotational?

bolbteppa

Never heard of non-linear Minkowski space...

Mad Spaces

Do you consider the Tangetial speed of the object to be also its linear speed?

@bolbteppa could you help me out?

I am so confused i thought that an object in circular motion has only rotational energy, but why for a pendulum they use the linear kenetical energy instead of rotation around the axis ?

bolbteppa

Probably better if you post the full description of what's confusing you

Mad Spaces

We have a normal pendulum which is moving in its normal way. Then a person intruduces some kind of obstacle under the point of hanging at lenght x such that the pendulum hits it and makes a loop

Its called gallileos Pendulum or something

And i was asked to identify which kind of energy the pendulum has at the highest point of the loop

I thought, rotation and potentiell, right? But then i asked my self. If any object doing a circlular motion have only rotational energy or also kinitecal energy?
What about a normal pendulum? it makes a rotation around its axis. But we still use normal linear kinetical energy discreption for it, why?

bolbteppa

9:57 PM

Pendulum (mathematics)

A pendulum is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allow...

Do you mean this one

geocalc33

@bolbteppa I'm not confused it's just hard to explain it in one single post

bolbteppa

Look at the energy derivation, at the 'highest point' (in the small angle approximation) the energy is all potential energy, at the lowest point it's all kinetic energy, conservation of energy means you can equate them etc... etc...

@geocalc33 have you got a reference which refers to 'non-linear Minkowski space'

Then I think you asked about a conical pendulum which more obviously does circular motion

Conical pendulum

A conical pendulum consists of a weight (or bob) fixed on the end of a string or rod suspended from a pivot. Its construction is similar to an ordinary pendulum; however, instead of swinging back and forth, the bob of a conical pendulum moves at a constant speed in a circle with the string (or rod) tracing out a cone. The conical pendulum was first studied by the English scientist Robert Hooke around 1660 as a model for the orbital motion of planets. In 1673 Dutch scientist Christiaan Huygens calculated its period, using his new concept of centrifugal force in his book Horologium Oscillatorium...

geocalc33

10:24 PM

@bolbteppa no I don't have a reference

it doesn't appear in the literature so I had to call it something

it's equivalent to Minkowski so that's probably what's confusing lol

Charlie

Could someone answer a small but longstanding question I've had, do the wavefunctions of every quantum system live in a distinct Hilbert space or do some live in the same space, and if so, what determines whether two systems have state vectors that live in a common Hilbert space?

bolbteppa

A physical system is described by a Hilbert space of wave functions with a basis of wave functions so the space is complete, some other physical system will have a different Hilbert space with a different basis generating it

Charlie

10:40 PM

Am I right in saying that the only free parameter when we build the schrodinger equation for a system is the mass of the particle and the potential?

Sorry that this is quite basic stuff

bolbteppa

I guess so yeah

Charlie

Are you saying that each physical system corresponds to a distinct hilbert space? So if I build the schrodinger equation with two different potentials, the solutions cannot live in the same hilbert space?

bolbteppa

Yes and even for the same Schrodinger equation with different values for the mass you get different Hilbert spaces

Particle in a box

In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. In classical systems, for example, a particle trapped inside a large box can move at any speed within the box and it is no more likely to be found at one position than another. However, when the well becomes very narrow (on the scale of a few nanometers), quantum effects...

The basis eigenfunctions $\psi_n(x,t) = \sin(..) e^{-i \omega_n t}$ with $\omega_n \approx 1/m$ (explicit expressions in the wiki) will be different functions for different choices of $m$

Charlie

11:04 PM

If that is true then how can a "complete" hilbert space not contain all other wavefunctions from different systems? If the solutions to the TISE for the particle in the box system are complete, why can't I just take a linear combination and build the solutions to that of say, the harmonic oscillator system?

And if I can, how can they not live in the same inner-product space?

Room for Tanuj and John Rennie

Transcript for

$Mathematics$ Room for Tanuj and John Rennie