PERSONAL SUBJECT ASSESSMENT

First of all, before focus on the personal subject assessment and finally closing the course 2013-2014, I would like to say that it's been a placer for me to share with us this scientific blog for spreading purposes.

In this subject coursed on 2014, which I know it's very importan for the future as an engineer and very interesting world because everything change and gather together at the same time no matter the point of view you choose. A bit confused but I found it interesting.

Personally, I couldn't spent all time I wished to focused on this subject and try to perform all my potential on this particular subject which, I repeat again, blows my mind. Well, to be honest, at the beginning I should say that i get a bit stock because my basic knowledge about algebra was completely null, so that, I didn't understand anything and thought that this subject was far away from funny or interesting. But as long as I was studying it, it was getting more "funny" because at least what I studied I think I understand "completely" and I say completely between comas because I know that I still ain't understand this world completely, I just understand little pieces of each section.

I know also that there's too much information left to study, and even the one who only a few privilege people can understand, but I will take it as a personal challenge.

Also, I want to mention that if i pass midterm exam it's because the facilites that our teacher gave us, uploading every single lecture and with a lot of comments for those who couldn't assist to class. And also, the fact of all the tutor hour he conceded to me that I know i didn't deserve them.

Not to mention, the care showed by the teacher among the students, solving individual question, wasting time for put the concepts into our brains in such a very formal and funny way which makes Algebra softer than it is.

But anyways, next year I will take it with a bit more confident, responsibility and trying to deal for remain my life a bit more stabilized and try to study not only what we do in class, trying to search a bit more (as I did in high school) in order to get a completely understand of what i'm doing, that actually that's by best gift given ever, that I try to understand very good what I'm studying relating things with simple things and common things and i try to get used to them quickly, so once i got it everything gather and consequently I can give my best.

So that, let's be a bit hopeful and next year I will spent more time on it for sure.

Thanks for paying attention.

PROOF, SETS AND BASIC CONCEPTS

Here I attach you a video of mine making a basic explanation about basic algebra.
I hope you enjoy and also i hope it's going to be useful for you.
Best Regards,

Enjoy:)

INTERVIEW

Thanks to Mr Luis Balbuena Castilian, I'm able to attach you here this amazing interview about such a great professor.

1) In order to teach a subject is necessary to know as much as possible. You dedicated most of your time teaching maths, what is your educational background? Do you consider it enough?

First I studied the teaching career (three years) and even as such exercised in unit schools although only a few months. Also I completed the Bachelor of Mathematics (five years) at the University of Santiago de Compostela. At thit time you could only do this degree in four Spanish Universities. Just when the last courses were leftover some Universities began offering me many other opportunities, including "La Laguna", so finally I accepted as a "founding member"

Of course I think that is more than enough my scientific training to teach mathematics in secondary education which is what always attracted me. So I also set about preparing the opposition for a place in a school and I showed up in 1974 as I passed the good fortune of the first Professor. Since then I have been at that level continuously.

2) By the dates indicated, did you life at Franco's times. What would you highlight at your time as PNN in college?

Indeed, the PNN was the figure most abundant professor at the University of that time (late "Franquismo") and early the transition). It means "No tenured Professor" because we had not fixed but each course, if we were good, we were renewed the workplace. This group claimed a lot, among other things, the stability of that broad set of teachers they had, in general, a great responsibility to provide really strong subjects and required a lot of time to prepare lessons.

I was representative of the PNN, Faculty of Science and doing a couple of courses and I was able to follow the subject from the first line. I remember a semi-clandestine meeting was in Granada where we were greeted by Mayor Zaragoza was the Rector. Many of the leaders who assumed important responsibilities in the early stages of Democracy, came from this group. PNN The term was later extended to other levels and became so popular that the word PNN appears in the 2001 edition of the dictionary of the Real Academia de la Lengua.

3) You have attempt several times to "Giner de los Ríos" Award organized by the Ministry of Education of Spain and financed by the BBVA and I understand that at all times has earned one of the prizes. I guess that it will be one of those jobs that you have developed. Could you comment us some of those jobs?

Yes, actually until now we could do some works like that. And notice that i'm using the plural tense because I have developed them collaborating with colleagues and with pupils as well. In particular I would like to note that in almost all collaborated professor Dolores Coba, Department partner in the "Viera y Clavijo" she helped too much in every single project.

In this institute we make a Maths Week for many years ago. Initially it meant a little extra effort, but with the reiteration i turned out into a stream ideas about mathematics world. A year we submit a report about our activity in order to be awarded at "Giner de los Ríos" because we felt it was an worthy idea. We knew that this ideas maybe is not that worthy for trying to get the awards because of there's a lot of competition, and a lot of talent, but at least we tried out. And finally they award out project. Really, I swear makes you feel very good in a deeply way.

AN SPECIAL GROUP !

Hi everyone again !! Today we are going to discuss about an special group ---> The Abelian Group. 

Let's say that an abelian group (G,+)  is a set G paired with a commutative binary operation +, where G has a special identity element called 0r which acts as an identity for +. The typical example of an abelian group is, the integers Z with the addition as a condition, with zero playing the role of the null element.

The easiest way to think of a ring is as an abelian group with more structure. This structure comes in the form of a multiplication operation which is “compatible” with the addition coming from the group structure.

A ring (R,+, · ) is a set R which forms an abelian group under + (with neutral element --> 0), and has an additional operation with an element 1 as a neutral element . Furthermore,  distributes over + in the sense that for all 

 and 

The most important thiing to note is that multiplication is not comutative both in general rings and for most rings in practize. If multiplication is comutative, then the ring is called commutative. Some easy examples of commutative rings include rings of numbers like Z,Q,R, which are just the abelian groups that we already know.

INTERSECTION OF LINES

Today we are going to talk about the interception of lines, where is the basic stuff in order to understand de basic geometric part from Algebra.

Let's get started !

When two lines intersect they form two pairs of opposite angles, A + C and B + D. Another words for opposite angles are vertical angles. 

Vertical angles are always congruent, which means that they are equal. Adjacent angles are angles that come out of the same vertex. Adjacent angles share a common ray and do not overlap. 

The size of the angle xzy in the picture above is the sum of the angles A and B. Two angles are said to be complementary when the sum of the two angles is 90°. 

PD: Specially I liked to this post about this because i found interesting that humans beings makes science from intersection between two lines, where we analyze points, interception points angles between lines...
I found it interesting because if i were scientist maybe i wouldn't give it as much importance as it deserves for --------> algebraic or geometric purposes.

RIGHT TRIANGLES

In geometry, is called every triangle right triangle with a right angle, ie an angle of 90 grados.1 The ratios of the lengths of the sides of a right triangle is an approach of plane trigonometry. In particular, a right triangle is defined by the Pythagorean theorem.

A right triangle tour, having as one of its legs axis as its hypotenuse generator generates a cone of radius equal to the non-axial leg and height equal to the axial leg.

If two triangles similar rectangles generate two cones, under the conditions of the preceding sentence, then their volumes are proportional to the cubes of any two corresponding sides. Also, the areas are proportional to the squares of any two corresponding sides.

FROM TRIANGLE TO SQUARE !

The problem of haberdashery (Haberdasher's puzzle) is to cut a square into four distinct parts that can be rearranged to form an equilateral triangle. This problem was proposed and solved by Henry Dudeney in the early twentieth century. Furthermore, placing four hinges between the surfaces forming the original shape, the transformation is quite dynamic.

Based on this problem, a  Company decided to designed a house that can perform this transformation offering different configurations depending on the season. You may see it in the video attached below.

INTRODUCTION TO ALGEBRA

Hi ! everyone again this blog also is going to be part from Algebra I from 1rst year of Aerospace engineering. The main goal is trying out to show that Algebra is closer to us than ever thought.

 This blog is also special for me just because was the first time i created something like this.
I hope you also enjoy as much as me reading all the posts and videos.

Interview to Mr.Professor Michio Kaku.

This is my own interview to one of the best researchers nowadays, the professor Michio Kaku, from the  University of New York.

I hope you enjoy it.

Solving a kinematic problem.

This is my video for the proof of how to solve a kinematic problem.
I hope you enjoy it.

The Unexpected Hanging Paradox.

The unexpected Hanging Paradox is a paradox about a person's expectations about the timing of a future event. This paradox is a situation of mathematic rules applied in wrong situations in real life; Let me explain you:

A judge condemns a prisoner on death row to be hanged next week by the warden, but before leaving, he makes sure the prisoner understands that it is going to be unexpected, so they will not tell him the exact date.

After thinking about how to escape from there, he remember what the warden said: You will be hanged in an unexpected date for you; therefore, the prisoner open his eyes and starts jumping, screaming and celebrating it: He knows he is not going to die.

After reading this, you might think either he is crazy or he has a way to escape, but non of them are right. Let's think about it: The prisoner knows he is going to die be in an unexpected date next week. Then, in case he is not hanged on thursday, then he cannot be hanged on friday, knowing that it would not be unexpected (only 1 day left of the week, obviously he will die on friday because there are no more days in the week, but it would not be unexpected, so it cannot be). Once we have eliminated Friday, same happens with thursday. If on wednesday he was not hanged, he will be on thursday, knowing that there are no more days left. The prisoners applies this hypothesis for the rest of the days of the week, and this makes him reach the conclusion that he will not die next week for sure. Notwithstanding, he is hanged on wednesday at 6 am.

Supposedly, even though the warden's statement to the prisoner was paradoxical, it ended up being true anyway. However, if the prisoner is no better at making inferences than he is in the problem, the warden's statement is true and not paradoxical; the prisoner was executed at noon within the week, and was surprised. This just shows that you can mess with the minds of people who can't make inferences properly. Nothing new there. But in case we follow logic and apply strictly the warden´s statement, and the fact that our prisoner in this case is is able to do all this reasoning, then his statement would be false and paradoxical.

Therefore, in conclusion, depending on the prisoner, I would or would not be able to "escape from the death".

This paradox give us another way of seeing math, in this case probability and way we have to predict future events: Mathematical-rules or statements applied in wrong situations give us results not possible in real life.

Photoelectric effect + Einstein's Biography.

Planck's hypothesis of 1900 about the main features of light was very badly received by the scientific community because it was breaking all the precepts of classical physics. It was not accepted until, in 1905, Einstein used it to explain an unexplained  phenomenon "the  photoelectric effect"  described by Hertz in 1887.

This mentioned "photoelectric effect" goes about electron emission of a metal when it's irradiated with electromagnetic radiation. However, not all electromagnetic radiations works, I mean, it has to be an specific type one, for example, it has to display a determinated frequency, and a minimum energy called " threshold frequency ". If the energy of the way does not reach this amount of energy, no electrons will be emitted.

The Albert Einstein's equation is the next one:

Where E=hxv ( energy of a quark), and E=hxv0 (where v0 it's the threshold frequency). So,
E0 + Ke(Kinetic Energy of the electron) = Ef( Final Energy).

Also this formula was so outstanding by the fact of new concept of quarks, as I mention before.
The concept of quarks was firstly introduced by Planck. But time later, Einstein tended to called fotons, those particles which forms the light itself.

Here I attach you a link in order to improve your knowledge about Einstein's history, one of the most important scientific in the world:

  http://www.youtube.com/watch?v=Xc6f_qF81BU

 

Polyhedron by Euler's formula.

I'm going to introduce you into a field full of beautiful mathematical formulas and probably one of the most useful due to its broad applications in chemistry, engineering, maths... As I said we are going to enter within one of the most beautiful mathematical formulas, Euler's formula. Space dedicated to issues of geometric objects that are more or less daily, the polyhedra, including the cube, triangular pyramid or tetrahedron, dodecahedron (whose faces are pentagons), the truncated icosahedron (the soccer ball), etc ... 

The study of polyhedra is of vital importance not only for the geometric study of them, for mathematical research, but also for its applications in diverse fields such as chemistry, mineralogy, biology, engineering, architecture, design or even art....

If  "C" represents all the faces of a simple polyhedron and "A" the polyhedron's number of edges/ridges and "V" represent the number of corners, then :

C+V-A = 2

The most outstanding fact is that doesn't matter how many cuts can displays a polyhedron, cause this cut is going to form another polyhedron and the relationship between them is going to be the same between its faces, corners, ridges/edges...( C+V-A=2).

Let's make an example of it.

We can see that this cube displays 6 faces (C=6), and also it displays 8 edges (V=8), and finally, we can see that the number of edges/ridges is equal to 12 (A=12).

So;

     C+V-A=2 -----------> 6+8-12 =14-12 = 2

And now, we can see here another a cut polyhedron, so as I mention the relationship has to be same as shown in the picture.

Prime numbers.

A prime number is an integer bigger than zero which can not be expressed as the division of two positive integer numbers with the exception of number 1 and itself.

Indeed, prime numbers can be defined as: "Numbers which can not be expressed as the product of two positive integer numbers smaller than the him."

Notice that number 1 is not included into the prime numbers. 

Let's make an example:

   a)Number 7 only can be divided by 1 and by 7(itself). And It only can be expressed as the multiplication of 7 x 1. In conclusion, as number  7 can no be expressed by either the division of two number different from 1 and itself we can say that 7 is a prime number.

    b) Number 15 can be divided by 1,3,5 and 15, because we can multiply 3x5, divide by 3, divide by 5 ...etc. In conclusion, as number 15 can be expressed by more than 2 number it is not an integer.

The prime term does not mean being relative of someone. It came from the Latin "primus", and it means "the first" (protos in Greek). The fundamental theorem of arithmetic states that every integer is expressed uniquely as a product of prime numbers. That is why they are considered "the first", because from the most primitive ones we are able to get all other integers. (Ex: The number 15 is obtained by multiplying the primes 3 and 5).

Finally, the firsts 25 primes numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 y 97, which all are smaller than 100.

   

The power of the integral

Regarding the power of the integral we can see differents aplications, for example calculate distances, areas, even volumes that surround us.

That's why our houses are part of simple shapes like squares. Even the columns/pillars are like a finite cylinder.

Now we are able to calculate it with a simple doble or maybe triple integral if necessary.
Also, in order to calculate the volume of some functions where "ROU" remains the same regarding the z axis, we also can create two planes( z=4. =0) in order to calculate the volume under the curve.

The possibility to be able to calculate the surface or area of a non flat shape is amazing to me. Because we can solve it trough a doble integral and with a simple procedure, thanks to the revisor coeficient.

Moreover, if you look further, even the lamp's shape can be inferred like a finit cone. That's wonderful.

In conclusion, when we talk about Calculus we are talking about a useful tool to explain our world.