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1. ### B 4-colours theorem can be proved visually

Interesting graphs thank you. Do you mean a Planar graph G that contains no K5 can be colored with a minimum of more than 4 colors?
2. ### B 4-colours theorem can be proved visually

I think I just now understood what you meant by coloring. When I say 5 colors, I mean those particular 5 colors; say (red, green, blue, yellow, orange). Suppose we have a scanning device that walks across the map and is programmed such that it will beep when it finds a region of the map...
3. ### B 4-colours theorem can be proved visually

You can give me any map with any number of regions, but I think when 5 colors can not satisfy the minimum requirement mentioned above at any subgraph then this is equivalent to the 4 colors theorem. Or put it in this way; there is no subgraph that requires 5 colors across the entire graph. If...
4. ### B 4-colours theorem can be proved visually

I will try here to put it in a semiformal way; The map of regions n is isomorphic to a graph of vertices n where each region of distinct colour is represented by a vertex of a different colour. Adjcaney: two regions in the map are adjacent if there is an edge connecting the corresponding 2...
5. ### B 4-colours theorem can be proved visually

Instead, can you show me a part of any graph where the maximum number of distinct colors of the adjacent vertices is 5?
6. ### B 4-colours theorem can be proved visually

Do I need to prove that colouring a map requires local rule? What I am trying to say is if the 4-colours theorem is false, then the maximum number of required colours should be at least 5. But the picture shows that can not be the case, therefore the theorem must be true.
7. ### B 4-colours theorem can be proved visually

Because colouring a map requires only local rules, I believe. When this set of 4-nodes becomes part of a larger graph, one can treat each local set of 4-nodes separately without affecting other remote parts of the graph.
8. ### B 4-colours theorem can be proved visually

The 4-colour theorem states that the maximum number of colours required to paint a map is 4. The proof requires exhaustive computation with a help of a computer. But I thought that one can visually prove the theorem in the following way; If one replaces the map with a graph where each region...
9. ### I Would this experiment disprove that consciousness causes collapse?

I am not familiar with Quantum Decoherence, but I find it difficult to treat macroscopic states like; alive and deal on the same footing as quantum states that describes position, time, momentum and energy. Like in a recent paper published by Frauchiger and Renner, they described states like...
10. ### I Confused by nonlocal models and relativity

But still, no matter what frame you choose to ride with any time order of the events you get, the physical theory must explain the quantum correlation between the measurement outcomes. In other words, the problem of understanding the quantum entanglement is not because the time order of the...
11. ### Why the sum of cosines between "v" and any vector =1?

Exactly, ##1 = \sum_{k =1}^N (\frac{ \overrightarrow{v_k} \cdot \overrightarrow{q_j}}{ |\overrightarrow{v_k}||\overrightarrow{q_j}|})^2## I discovered this fact by coincidence but it turns out that it may have a nice link to the quantum mechanics. For example, if the cosine of the angle...
12. ### Challenge Math Challenge - April 2019

Is this function related to Riemann Zeta function in the critical strip?
13. ### Simple experiment with magnets

That was not my assumption anyway. I have also reached to the same conclusion; that is the pole is assigned to the larger face of the magnet too. This solves the first problem of the thread but does not solve the second problem mentioned in the thread no.1 and explained in no.5. If the same...
14. ### Simple experiment with magnets

Thank you for the reply. Here is the experiment in action. The first picture shows the broken pieces put together side by side. I stick a small yellow paper in one side and a blue paper on the opposite side (the color does not imply the polarity of the magnet). I assume that this should be the...
15. ### Simple experiment with magnets

Now, when I approach the broken magnet in your drawing to an intact magnet, a repulsion force happened as shown. How to explain this?
16. ### Simple experiment with magnets

I made a little experiment with magnets. I got two small bars of magnets. They obey the usual attraction-repulsion rules by approaching their faces together in various permutation. Accidentally, one of them is broken into two unequal pieces. When I managed to put the two broken pieces side by...
17. ### Online app which plots F(z) in the complex plane

I just need a 2D space, so that when I click any where in this plane (z), the app immediately plots the f(z) on the same plane. This should also draw f(z) as a continuous curve if the function is analytic.
18. ### Online app which plots F(z) in the complex plane

I am looking for an app that can instantaneously plot the function f(z) in the complex plane once z is given. It would be much favorable if this process is fast which allows one to visualize f(z) when the user is moving the mouse on the complex plane to the location of z. One possible...
19. ### 3D backprojection (potential project)

I like the 3D article in the last link. But it seems that it has a very limited capability and it is very small in size too (lens is used to watch it). My technique can be made on any scale.
20. ### 3D backprojection (potential project)

Sorry for the late in the reply. The light may be projected on a cloud or a crystal whose particles would shin when reflecting the light.
21. ### 3D backprojection (potential project)

I just thought about whether it is possible, or already exists, a method to construct a 3D optical scene using the method of backprojection. First, I will capture the scene (say a castle) from different views using HD camera. Then using one of the app to stitch the 2D images into a cloud of...
22. ### A solute added to a container with a small hole

But I do not see where to start from your post #9. You gave the variable in units and a boundary condition when t approaches infinity.
23. ### A solute added to a container with a small hole

Thank you, I think this should take me to my proposed solution under the assumption that the rate of the salt leaving the bowel is proportionated to the total salt in the bowel. (The proportion constant should be negative sign because the leaving salt reduces the total amount of salt in the...
24. ### A solute added to a container with a small hole

Correct, this is the one I am seeking to solve.
25. ### A solute added to a container with a small hole

Sorry that I was not clear in the description of the problem. I like the analogy of salt and water, so I will consider it here. Lets consider a tank has water and salt. The hole near the bottom leaks water and salt. To keep the volume of the fluid constant, we add from the top a volume of water...
26. ### A solute added to a container with a small hole

Yes, it leaks fluid and solute. I also assume that the added solute is instantaneously solved in the fluid with no sediments.
27. ### A solute added to a container with a small hole

Homework Statement Suppose there is solute ##s## in a bowel containing fluid. There is a tiny hole near the bottom which leaks a small fixed volume of solute ##\lambda## per unit time ##dt##. In addition, there is a small added solute to the fluid in a constant rate ##\alpha## so as the volume...
28. ### B Doubts about Light...

42 and 46 [Moderator's Note: Several posts were deleted after this post was made, so the post numbers above are no longer correct.]
29. ### B Doubts about Light...

I hope my consideration is answered before the thread is closed for moderation.
30. ### B Doubts about Light...

So, if my explanation of the aberration concentrated beam, 3 posts back, is right, what makes it is still valid for a hypothetically single source element, say one atom radiating light?