Interior Definition and 149 Threads

For all the attention we‘ve paid to ring habitats, we haven’t talked that much about the interior design of the central trunk yet, around which the rings rotate. Just having one big hollow ship trunk, about 100 metre in diameter, would be a lot of wasted space. It would also be too easy for...

I've built an insulated chamber to protect a sensitive instrument at freeze temperatures in the winter. The instrument is mounted on a telescope, so the heat inside the chamber will slowly dissipate in the ambient. A digital PID thermostat is used to keep the temperature at a safeguard level...

This isn't homework, but I figured it's fine if I make it a HW problem and post here (if not, please let me know). Let ##z^*=0## be the vertex of the pyramid, and let ##z^*## run the altitude. It's easy to show the area of the base normal to the altitude is ##A = 4 \left.z^*\right.^2...

I'd love have a little discussion about the Interior Schwarzschild Solution. Here's a diagram I slapped together to illustrate the key points. (I assume everyone reading this familiar with embedding diagrams, and using an axis to 'project' a value, in this case the spatial z-axis is replaced by...

I am reading Stephen Willard: General Topology ... ... and am studying Chapter 2: Topological Spaces and am currently focused on Section 3: Fundamental Concepts ... ... I need help in order to prove Theorem 3.11 Part 1-a using the duality relations between closure and interior ... ..The...

I am reading Stephen Willard: General Topology ... ... and am currently focused on Chapter 2: Topological Spaces and am currently focused on Section 3: Fundamental Concepts ... ... I need help in order to fully understand a result or formula given by Willard concerning a link between...

It is possible to show via Gauss' law that the the net flux through a surface within the Faraday cage must be zero, however this is not a sufficient condition for the electric field to be zero. For the electric field to be zero in the interior of the cage, all points within the cage must be at...

What are the best methods for framing the locations where a new interior room meets the wall and ceilling of an existing interior room? I'll be adding a small closet to a room of a house. The rectangular closet will have 3 new walls and use one existing interior wall. The new walls will be...

PLEASE HELP1.What is the sum of the measures of the interior angles of a heptagon? A. 1260∘ B. 2520∘ C. 900∘ D. 1800∘ my answer is C 5.If the sum of the interior angle measures of a polygon is 3600∘, how many sides does the polygon have? A. 22 sides B. 20 sides C. 18 sides D. 10 sides MY...

I am working on a classical real analysis problem as follow: The answers from solution manual are respectively $ int (A) = \emptyset$ and $bd (A) = \{0\} \cup \{ \frac{1}{n} | n \in \mathbb N \}$. And here are my textbook's definition of interior point and border point: And then there is...

This paper by Christodolou and Rovelli discusses how to define the interior volume of a black hole: https://arxiv.org/abs/1411.2854 I'll try to condense their basic argument into a (heuristic) form that makes it easier to raise the issues I want to raise. The basic idea is to start with the...

Homework Statement Identify the boundary ##\partial c_{00}## in ##\ell^p##, for each ##p\in[1,\infty]## Homework Equations The interior of ##S## is ##\operatorname{int}(S) = \{a\in S \mid \exists \delta > 0 \text{ such that } B_\delta (a) \subseteq S\}##. ##\partial S = \bar{S}\setminus...

Hello! (Wave) Using Green's theorem, I want to compute the integral $$\oint_C ydx+xdy$$ where $C$ has the parametric representation $r(t)=2 \cos^3 t i+ 2 \sin^3 t j, (0 \leq t \leq 2 \pi)$. Using Green's theorem, we get that $\oint_C ydx+xdy=\iint_U (1-1)dxdy=0$. I am wondering if we could...

Homework Statement Define the interior A◦ and the closure A¯ of a subset of X. Show that x ∈ A◦ if and only if there exists ε > 0 such that B(x,ε) ⊂ A.The Attempt at a Solution [/B]

My question to you is this... Can the interior of the Sun be described as an ideal gas? From my knowledge, to describe a body of gas as an ideal gas, the separation between the particles must be much greater than the size of the actual particles. How could one justify whether the Sun fits this?

Hello PF, I am searching for a proof that I couldn't find on the internet. Theorem: E in X a metric space. p in E. p is an interior point of E if and only if p is not a limit point of (E complement)' Sorry for notations but I have no idea how to insert Latex here.

Homework Statement A hollow conducting sphere has inner radius R and outer radius 1.5R. A +40nC point charge is placed at the center. What is the total charge on the interior and exterior walls of the sphere? R=10cm Homework Equations Gauss's Law σ = Qenc/A The Attempt at a Solution I'm not...

Homework Statement How many sides does a polygon have if the measure of each interior angle is 8 times the measure of each exterior angle? Homework Equations Interior angle of a polygon: ((n-2) * 180)/n Exterior angle of a polygon: (360)/n The Attempt at a Solution ((n-2) * 180)/n = 8 *...

I have heard that whatever we know about stars experimentally is through only what we can see from its surface since the light from the interior is "hidden." However, when we look at the spectrum of a star, we do see absorption lines for heavy elements. I think the reason why that is is because...

Homework Statement A gas effuses into a vacuum though a small hole of area A. Show that if the particles effused into an evacuated sphere and the particles condensed where they collided that there would be a uniform coating. (7.6 of Blundell and Blundell) Homework Equations Angular...

There are $9$ points in the interior of a cube of side $1$. a) Show that at least two of them are less than $\frac{1}{2}\sqrt{3}$ apart. b) Can $\frac{1}{2}\sqrt{3}$ be replaced by a smaller number?

I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 6: Differentiation ... I need help in fully understanding the corollary to Theorem 6.2.1 ... Theorem 6.2.1 and its corollary ... ... read as follows: I am...

I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 6: Differentiation ... I need help in fully understanding the corollary to Theorem 6.2.1 ...Theorem 6.2.1 and its corollary ... ... read as follows: Can someone...

Hello! The interior product is defined as ##i_X:\Omega^r(M)\to \Omega^{r-1}(M)##, with X being a vector on the manifold and ##\Omega^r(M)## the vector space of r-form at a point p on the manifold. Now for ##\omega \in \Omega^r(M) ## we have ##i_X\omega(X_1, ... X_{r-1}) = \omega (X,X_1, ...

Schwarzschild black holes were discovered over 100 years ago, in 1915. Quite soon, in a couple years (by 1916...1918), charged Reissner-Nordström holes were discovered. Yet rotating black holes were only discovered by Kerr in 1963 - 48 years after Schwarzschild holes. Why? What was known about...

I'm looking influence of pressure on the general interior Schwarzschild metric (see for example the book by Weinberg, eq. 11.1.11 and 11.1.16. The radial component of the metric (usually called A(r)) depends only on the mass included up to radius r A(r) = \left(1-\frac{ 2G M(r)}{r}\right)^{-1}...

Can someone explain to me why there must be a real/meaningful space inside of a black hole? I have been autodidactically working on understanding the mathematical concepts that general relativity is based on, so I've never had anyone to ask questions to (until it occurred to me to find a forum...

Hey everyone! My name is Jing and I am participating in a one month summer program called SHAD where 800 high school students across Canada come together to try and solve some of the world's most multi-dimensional problems. One of the aspects of this program is the Entrepreneurial Project, where...

I've been playing around with Maxima and it's ctensor library for tensor manipulation. I decided to have a crack at deriving Schwarzschild's solution for the interior of a constant-density sphere. I've managed to derive a static, spherically symmetric solution, but am struggling a bit with the...

Hi. I'm working on a science fiction story about colonization of the solar system. Asteroid way stations are an important part of the story. I want to construct a realistic scenario about how we could convert the Eros asteroid into a shuttling habitat in it's current orbit (or something close...

Forgetting about the singularity for the moment, is the interior of a black hole a space-like region? That is, are any two events that occcur past the black hole horizon space-like separated? Thanks in advance.

Hello everyone, I was wondering if someone could assist me with the following problem: Let T be the topology on R generated by the topological basis B: B = {{0}, (a,b], [c,d)} a < b </ 0 0 </ c < d Compute the interior and closure of the set A: A = (−3, −2] ∪ (−1, 0) ∪ (0, 1) ∪ (2, 3) I...

Homework Statement $$ \bigtriangledown^2=0 for : 0<r<1 \\ BC : u(1,\Theta)= sin(\Theta), 0<\Theta<\pi \\ u(1,\Theta)= 0, pi<\Theta<2\pi \\ $$ Basically its an interior dirichlet problem for a circle. [/B]Homework EquationsThe Attempt at a Solution The answer is supposed to be $$U(r,\Theta) =...

If ##\bf{v}## is a vector and ##\alpha## is a ##p##-form, their interior product ##(p-1)##-form ##i_{\bf{v}}\alpha## is defined...

Does anybody know where to find schematics of the interior of modern processors for lithography? Not brand new, as far back as 32nm nodes? Or where to begin with this search? Thank

This could open a new frontier on seismic study of the Earth's interior. http://www.bbc.com/news/science-environment-37177575 From, http://science.sciencemag.org/content/353/6302/919 Seismic tomography is like an x-ray of Earth's interior, except that it uses earthquakes for the illumination...

Homework Statement Let ##(X,d)## be some metric space, and let ##A## be some subset of the metric space. The interior of the set ##A##, denoted as ##int A##, is defined to be ##\bigcup_{\alpha \in I} G_\alpha##, where ##G_\alpha \subseteq A## is open in ##X## for all ##\alpha \in I##. The...

Homework Statement Homework Equations sum interior angles (n-2)*180 angles of a quadrilateral: a+b+c=d = 360 [/B] The Attempt at a Solution What do you do with the exterior angle? 80+130+a+x=360

The sun we see has a measured surface temperature of 5800K. This is the temperature of the photosphere , a glowing layer of plasma radiating energy from the sun. But when there is a hole in this layer and we look deeper into the interior the temperature drops to 3800K. How can this possibly be...

Let's assume typical 10' X 20' greenhouse for growing veggies is empty and is all exposed concrete black floor with solid black (opaque) insulated knee high walls all around. The rest of it is glass sides and roof. Also, assume reflective radiant barrier was hung on those knee walls inside...

I am new to elastic theory. I have a question about elasticity. We assume we have a body with no internal forces. Surface forces are applied on the border. Can we leave the elastic domain (reach the yield surface) in an interior point without leaving the elastic domain on the boundary? If no...

Hi. I'm trying to self-study differential geometry and have come across interior products of vectors and differential forms. I will use brackets to show the interior product and I would just like to check I am understanding something correctly. Do I need to manipulate the differential form to...

Hi All, A simple question but one for which I cannot seem to get the intuition. 1. Homework Statement Find the interior point of {1/n : n ∈ ℕ}. Homework Equations N/A The Attempt at a Solution Let S = {1/n : n ∈ ℕ}, where S ⊆ℝ x is an interior point if ∃N(x ; ε), N(x ; ε) ⊆ S. My...

For shaping an interior PMSM rotor from top figure to as in bottom fig, I made an equation based curve, united with another curve , made a surface using coverline and united that shape with my rotor. When validated it showed band and rotor intersecting and I adjusted the band - DiaGap. Then...

Just attempted another exam question. Would you mind correcting me if I am wrong? The question is A triangle ABC has sides of length AB= 3.5 m, BC = 5.1 m and AC = 4.2m. a) Calculate the size of the angle B and the size of the angle C, in degrees correct to 1 decimal place, in each case...

A number of recent threads have discussed what happens when an observer falling into a massive black hole passes the event horizon. What I would like to know is this. For a massive BH of mass M, Schwartzchild Radius Rs, how long would it take for such an observer (who, presumably crosses the...

Homework Statement Deep sea fish experience a tremendous amount of atmospheric pressure. When you are at a depth of 2000 m in the sea, the pressure that you will experience is about 200 times atmospheric pressure (1atm=1.0×105N/m2)which you will experience on land. If you are lowering a...

Hey! :o Show that the measure of a rectangle is zero if and only if it has an empty interior. When a rectangle has an empty interior, does this mean that the length of the sides of the rectangle are equal to zero?? (Wondering)

Homework Statement . Let ##X## be a nonempty set and let ##x_0 \in X##. (a) ##\{U \in \mathcal P(X) : x_0 \in U\} \cup \{\emptyset\}## is a topology on ##X##. (b) ##\{U \in \mathcal P(X) : x_0 \not \in U\} \cup \{X\}## is a topology on ##X##. Describe the interior, the closure and the...

In Nakahara's book, the interior product is defined like this : i_{x} \omega = \frac{1}{r!} \sum\limits_{s=1}^r X^{\mu_{s}} \omega_{\mu_{1}...\mu_{s}...\mu_{r}}(-1)^{s-1}dx^{\mu_{1}} \wedge ...\wedge dx^{u_{s}} \wedge...\wedge dx^{\mu_{r}} Can someone give me please a concret example of...