Graph Theory: Pure or Applied Mathematics?

Graph Theory worries about the study of graphs. That connects nicely to Topology and Geometry and therefore to what we traditionally call Pure Mathematics. 

For those who have doubts:

Watch:

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Mobius Band

Band brings a woman, a bagel, and, possibly (not my area), a huge inaccuracy: two sides and one surface... 


A band, by definition, has two faces: if we flatten it up, then we have upper and lower or over and under, whatever we want to call them, but it is still two sides. 


I liked the idea of the bagel being cut like that...


It is a short video.


I think it has a huge inaccuracy. 


The knife is not lifted? 


Do we lift the knife when spreading cream cheese on a bagel that is opened traditional way?


Why would we invest so much time to open a bagel if we can simply do it in a normal way?


Only female... 1 minute and 47 seconds it says... record time, shortest ever, or something like that.


Apparently, we say face in Mathematics, not side. 


In this way, we could stick to the normal definition of side. 



This defines side as a surface (American Heritage Dictionary). 


According to the same source, the Collins English Dictionary says that side is face. 


In this case, she might be right sometimes. 


We say that we are measuring surfaces when we calculate area, so that we must think of how to calculate the area of the band. 


In this case, it would be only one surface, but then all other figures would have only one surface, since we flatten them all before calculating area. 


Surprisingly, Wolfram, which used to be my preferred source for Mathematics in 2001, does accept her definition of side, and then says that the band has one side. 


See: Wolfram



"The Möbius strip, also called the twisted cylinder (Henle 1994, p. 110), is a one-sided nonorientable surface obtained by cutting a closed band into a single strip, giving one of the two ends thus produced a half twist, and then reattaching the two ends (right figure; Gray 1997, pp. 322-323)."




A few inaccuracies inhabit Wolfram's pages... 


Side cannot be the top or the bottom face, therefore any face but top and bottom, so that the band could also have no sides if we go with what people say: not top or bottom




The best argument I have to state that the Mobius Band has two sides or two faces is that we could use the same reasoning, of the walk, for any common solid, so say a prism.


 We start with the finger on one face and we go all over the figure coming back to the same place. 


That cannot mean that all those faces became one. 


It cannot... 


Face is about the aspect: As the dictionary says, it is what we see in front of us. 


In this case, we see two faces or two sides in the untouched band: The inner and the outer. 


We would have to have at least these two sides or faces to form a three dimensional figure, since only two dimensions could show one-sided things, the flat shape in the Cartesian Plane. 


Prism brings a nice collection of prisms. Say we choose this one:

Now, do the same: Run your finger, index finger, over the lateral surface of this prism (and notice that top and bottom is relative: put it standing and what was side according to one of the sources (not top or bottom) will become a non-side). 


And now? Do we have only three sides? 


You could run your finger over four of the surfaces and come back to the same place... 


That is not a good argument... 


I found no definition relating finger walk to sides or faces, so that not even the non-mathematical sources support this view. 


We must have a mathematical agreement on what a side or face is, however. 


We cannot define face based on what we had before connecting elements, before forming the shape, since otherwise all shapes would be the initial flat surface, and therefore would have one face. 


Notwithstanding, any three-dimensional shape, perhaps taking away the line, would have to have at least two faces, for otherwise it would fit the Cartesian Plane and it would be in 2D instead. 


The definition should be visual, and therefore based on the source that said it is what we see in front of us: if we look at the band from the front, we see one face. If we look at the band from the back, we see another, to the back of the face we just mentioned. 


We put a number on what we see and we will have two. 


For instance, take the picture that follows (it came from band):

Write 1 in all you see when starring at this. 


Now put it upside down and write 2. 


Any other angle will return the same numbers, so that we would definitely have 2 in the end.


Perhaps we define it as being the largest number we may have when changing angles of sight of a 3D-shape. 


In this way, our selected prism would be positioned in a way to show one face at a time to us, so that we get six. 


Where we have an edge, we have the encounter of two faces. We can then just count the edges of the band. 


We definitely have two: The top of the band and the bottom.


We also have the same faces meeting there, so two. 


Got the idea from here (edge):

See:

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Circumferences

YouTube brings a SAT question. 


We have a circle of radius that is 1/3 of the radius of another circle. 


They ask how many times the smaller circle goes around the bigger circle. 


The answer should be 3: 2 Pi r/3 would be the length of the circumference of the smaller circle. 


With this, we need to multiply it by 3 to get 2 Pi r, which is the circumference of the bigger circle. 


That means that the length of the smaller circumference will mean 3 turns over the bigger one for it to go back to the initial point. 

Please write to drmarciapinheiro@gmail.com if you want to converse about any of the contents of my blogs here.

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Pizza and Mathematics

Pizza brings an interesting question and an even more interesting correction of the student's answer: the student seems to have used good logic. 

I thought in the same way, to be sincere. 

It is confusing. 

Marty is told to have eaten 4/6 of his pizza. 

Luis is told to have eaten 5/6 of his pizza. 

Marty ate more pizza than Luis. 

How is that possible? 

The student answered: Marty's pizza was bigger. 

That sounds really logical: You just have a larger radius for this pizza, and therefore his 4/6 ends up being more value in pizza than Luis' 5/6. 


If you do not specify to the level you are thinking, the student has to win on this one. 


If the intentions were saying that that was unreasonable, as the presenter states, the teacher would have to have written pizzas of the same size. 


It says it is about being reasonable. 


When you ask us why, reasonable is assuming that whatever you described is a fact, has already happened, not that you are lying or inventing. 


Reasonable has to be where the average thinker goes with their thinking when reading. 


Maybe those who know Mathematics would think like the boy did... 

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Division: Exact?

MM brings an interesting YouTube video about the number zero and why dealing with it is really hard. 


The most interesting thing that I found here is the alternative way of talking about division. 


The guy makes use of subtraction to explain it. 


If your numerator is larger than your denominator, all works relatively OK, is it not? 5/4, for instance, can be explained in this way: 5-4=1. Therefore 5 can be divided by 4. With 1/4, 1-4 is negative, so that we cannot do it. 


We then have one and one fourth as a result. 4/5 could be explained in this way: 4-5 gives you negative, so that we cannot do it. 


We get 4/5 or 0 and something. 


What if you have negative in the upper or lower part of the fraction? 

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Prime Numbers: Competition

Chasing the largest primes is an incredible adventure... 


Watch Lucas 


The program he mentioned, the underdog one, is something similar to what SETI used to do: An acquaintance of mine  frequently helped them calculate things. 


SETI used people's computers - private people's computers - to study the signals somehow. 


They multiplied their power of calculation by much each time someone volunteered and offered them their computer.

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Master Yaser and Dr. Pinheiro: S1-Convexity

Master Yaser Maleki                               Master Science (Mathematics) Tarbiat Modares University Tehrān  E-mail yasermaleki71@gmail.com Web: Research Gate

Dr. Marcia Pinheiro Lecturer at IICSE University Certified Translator and Interpreter Portuguese & English NAATI  40296          Member: PROz, RGMIA, Ancient Philosophy PhD in Philosophy and Mathematics Master in Philosophy Certified TESOL/TEFL professional Licentiate in Mathematics PO Box 12396 A’Beckett St Melbourne, VIC, AU, 8006 Tel 0416915138 E-mail drmarciapinheiro@gmail.com Web Research Gate

Master Yaser, have you heard of Hudzik, Maligranda, and their S₁-Convexity before? Have you heard of convex functions? 

Yes, I know convex functions but I haven't heard anything about S₁-Convexity. 

Usually people like the concept of convex function because it is a lot graphical, is it not? In the Universe of the Real Numbers, a convex function is a function with a graph that is built in such a way that regardless of which couple of points we pick on its line (it will be a line for our eyes, right?), the curve representing the function will always be either over or under that line. 

Yes it is graphical and this property helps us understand convex functions better than other types of function. Also people who do not have a mathematical knowledge like convex function more than other in fact they understand their eyes. But if we want to talk about Mathematics, in particular Pure Mathematics, we know that it is not necessary for a function to have a graphical property, but yes it is interesting.

As we can see in the graph above, whatever that is part of the curve (blue) between one intersection of the straight line (red) with the curve and another is under the straight line. It does not matter where we draw this straight line that contains two points of our original curve; it is always going to be the same: It is all under the line, at most over it, that is, never above it. We can then say that, in analytical terms, that, for any two elements of the domain of the function we pick, so say x and y, it is always true that the image of ax + by is always less than or equal to a times the image of x plus b times the image of y if a+b=1. 

In our first internationally known paper on S-convexity, which got published by the WSEAS group, due to their conference in Cancun, which we did not actually attend, we wrote: 

That was in 2004, Master Yaser. You can already notice some difference between this picture, from the WSEAS paper, and what I wrote before. Can you?

No, I can't see any differences between them: Everything you say is shown in this definition. I just zoom on the (f:X--->R)². 2 is for footnote or it is part of the definition? 

I think you are talking about a footnote, Master Yaser. You are right, it is all very subtle, but I have been working on this for a while because I think it all matters quite a lot. Please observe the coefficients: In one definition, you see a and b, therefore two constants. In another definition, you see lambda only. You are also immediately told, in the second definition, the one presented at the WSEAS, that it is lambda and 1-lambda, and therefore the sum of the coefficients leads to 1, so that you don't need to write that down, like not only we have reduced the constants to one (we had two constants, now we have one), in terms of coefficients, but we also deleted the extra piece of information: That the coefficients together give us 1. That saves us and makes the definition look more elegant, which can only be part of the objectives of Science when we talk about refining mathematical definitions: more objectivity, simplest presentation as possible, more immediate application, etc. 

The first difference between convex and S-convex functions is their domain. The second difference is the way of choosing a and b, and the third is the conditions on a and b. For instance, a+b=1. The difference between S₁-convex and S₂-convex is in the condition on a+b, perhaps that a+b=1 for one of them.  I can imagine some things, as you can see. Are they true? In the convex function, we see that if we choose two elements in the domain, x and y, then the image of the graph for all points between x and y is under the line (red line) across f(x) and f(y), but I imagine that this S-convex concept does not lead to a line!! I think it is a curve and its  Curvature depends on S. 

                                                                    

You are very right, Master Yaser. It is precisely that, the main points are precisely those. I saw things in the same way you see them in that 2001, since it is all very much obvious. One more detail catches our eyes: Because S is between 0 and 1, and both a and b would have to be between 0 and 1, and you will notice that I actually produced a proof for that in my paper with WSEAS, the first one on the topic that got published by a major vehicle, and the proof follows this paragraph, a to the s would have to be greater than a. That was the key for my understanding of the shape of S-convexity: It is actually a lift on the limiting line for Convexity, and that is why both Hudzik and Maligranda thought that they had a proper extension of the concept. See the proof regarding the coefficients first:

Before we talk about the rest, I think I would like to know if you agree with what I stated before after seeing the proof we presented at the WSEAS (a and b would both be between 0 and 1, and, therefore, we can rewrite the definition of S-convexity given by Hudzik and Maligranda in the way you see in the last picture of the paper presented at the WSEAS in that 2004). Do you agree with that, please, Master Yaser?

I understand the new definition of   S₂-convex  and agree with this, but I don't understand S₁-convexity. In fact I don't understand why you use (1-lambda ^ s)^ (1/s)!!!!

Yes, exactly. Perhaps the first intuition is that it is important to keep the coefficients unaltered because we want to keep the percentages we take in the mix unaltered in terms of base. You will notice however that easy counter-examples exist to prove to us that S₁-Convexity is not a proper extension of Convexity. Perhaps the main question to be asked was always what does extension mean? When we extend something in Mathematics, that means that we have included what we had before in what we have after the extension and we have added a little bit, as a minimum thing, is it not? If we lose something that was part of the something we claim to now be extending, then we must not be extending: We must be creating another class instead. 

See the counter-example to the claim that S₁ extended convexity, which is right below this line. I want to know if you agree with all that is said here. Perhaps you could give your take on extension as well. 

  

I agree with your take on extension in Mathematics. We can also notice that if we find a new class of something, then we also extend the Mathematics involved. However maybe we don't extend an old definition. Now something to make my mind busy: Is S-convex function a subclass of convex function? In other words, is every S-convex function a convex function? 

Master Yaser, the reason for your confusion is probably the fact that you agreed that we have a genuine counter-example to the claim that S₁-convexity extends convexity. I wonder if you have validated my every step in the proof above. Please confirm. 

Yes. Every part of the above extract proves that it is all true and I can't add any comments to that. 

Great, Master Yaser! I think I was eager to get more people saying yes to my results in a meaningful manner, people who are not journal editors. Thanks for that. That means we both agreed that S₁-convexity cannot extend Convexity because, for instance, the group of functions we have just mentioned is part of the class convex real functions but is not part of the class S₁-convex functions. As said before, if a class extends another, we should have at least the group we claim to be extending inside of it. As for convexity, please observe that whenever s=1 you would be recovering this notion both in the definition of S₁- and in the definition of S₂-convexity.

                            References

Pinheiro, M. R. (2004). Exploring the Concept of S-convexity – Proceedings of the 6th WSEAS Int. Conf. on Mathematics and Computers in Physics (MCP '04).

Pinheiro, M. R. (2015). Second Note on the Definition of S1-Convexity. Advances in Pure Mathematics, 5, 127–130.

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