Archive for February, 2010

This entry is going to be part of the Fourth Edition of the Carnival of the Physics, whose host is the blog RTFM.es.
What do not you know the Laws of Kepler? I cannot believe it. Anyway, we are going to begin by the simplest thing. The Laws of Kepler are those who govern the movements of the planets and they were discovered by the astronomer and mathematician German Johannes Kepler. But the most curious thing of all this is that the Kepler good one obtained them of the simple observation. In fact, it deduced them after studying meticulously the precise notes of his companion Tycho Brahe, who did it without the help of the telescope, invented with posteriority.
But let’s return to Kepler and his Three Laws (not, those of the Robotics are another Three laws that do not come to story). Kepler (although not in the same order in which today they are known and they are studied), enunción his famous three laws to explain the movement of the planets in his orbits about the Sun:

1. All the planets move about the Sun describing elliptical orbits, being the Sun placed in one of the foci.
2. I remove vector that joins the planet and the Sun it sweeps equal areas in equal times.
3. For any planet, the square of his orbital period (time that is late in giving a return about the Sun) is straight proportional to the bucket of the average distance with the Sun.

In this small article we are going to re-discover the second Kepler law, basing on the Law of Universal Gravitation of Newton:

The force that exercises an object met on mass m1 on other with mass m2 is straight proportional to product of mass, and inversely proportional to the square of the distance that separates them.

For ours intentions, we are going to fix like origin of our system of reference to the Sun, with mass M, and are going to suppose that we have a planet orbiting about him with mass m. And, also, we are going to adopt the system of polar coordinates. This way, if we fix the position of the planet (that we will suppose, as the sun, which is a point of polar coordinates (r, θ)), we are going to call ur to the unitary vector in the direction of the radiovector that joins the Sun with our planet and or θ to the unitary vector perpendicular to the previous one and in the direction in which it increases t.
Whole, which after all this foolish talk, we are going to calculate forces F that the Sun exercises on our planet. Of the Newton’s second law, we know that F =m to, where to es the acceleration of the planet. But if we want to write the acceleration in terms of the polar coordinates, it is necessary to do a few accounts (it avenges, costs, we are going to obviate them, that the stove is not for bollors), after which we will obtain that to = (r · θ "(t) +2r’ (t) · θ ‘(t)) or θ + (r" (t)-r · θ’ (t) 2) ur where t it represents, as almost always, the time.
So, if we decompose the force F in his central component Fr and tangentially F θ, we will obtain that F θ =m (r · θ "(t) +2r’ (t) · θ ‘(t)) and Fr =m (r" (t)-r · θ’ (t) 2)
But skylight, this, in fact, is valid for any type of force, that is to say, that this is the previous formulae there are only the Newton’s Second Law expressed in polar coordinates. Now we are going to introduce the fact that the force that we have is of gravitational type. In our case, only we are going to remain with an aspect of these forces, and the fact is that they are of central type, that is to say, that they do not have tangential component (remember the Law of Universal Gravitation).
Under this new prism, it turns out that the tangential component of our force must be, necessarily, void; which allows us to obtain a Distinguishing Equation r · θ "(t) +2r’ (t) · θ ‘(t) =0 If we multiply this equation for r, is obtained r2 · θ" (t) +2r · r’ (t) · θ ‘(t) =0 or what is the same, (r (t) 2 · θ’ (t)) ‘=0, so that the function between parentheses can be only a constant, that is to say, r (t) 2 · θ’ (t) =h for some constant h.
And now vámonos with the Second Law of Kepler. If To (t) the area is covered for r (t) from a fixed reference position, it is easy to verify (again there are only accounts with which I am not going to overwhelm you) ΔA = (r2 θ ‘(t)/2 · Δt=h/2 · Δt where the symbol Δ represents the increase of the function. And so, between two time moments t1 and t2, it is had that A (t2) – to (t2) =h/2 · (t2 - t1) that word saying is, exactly, what says the Second Law of Kepler:

I remove vector that joins the planet and the Sun it sweeps equal areas in equal times.

In another occasion, we all will make use these of calculations to verify that, as the gravitational force is inversely proportional to the square of the distance, the celestial orbits can be only conical.
I hope not to have bored you very much. Thanks for coming so far.

The mathematicians do not study objects, but the relations between them. Therefore, they are free to replace some of the objects by others, whenever the relations have not been modified. The content for them is irrelevant: they are interested in the form.

Jules Henry Poincaré, route MathDL.
I, in the fund, think that it is true. It does not matter to treat with functions, points, straight lines or sets. Everything is a question of the relations that exists between them. Do you agree?

The engaged thing debt is, and after having brought the compilation of all the articles published under the First Edition of the Carnival of Mathematics, today I bring to you the numbers that we could have extracted of him.
There have been published 70 new articles, between which 2 articles about press are included, blogs distintos.100 have been 47 the persons who have registered like participants in the web of the carnival until February 14 (today there are already 108). 18 of the articles that have been published have been promoted in the agregador meneame.net, 2 of which came to front. 28 articles have been include you in bitacoras.com, 5 of which have come to front.
During the week of from February 8 until February 14, in that the articles were published, the web of the carnival received 1429 visits of 897 only absolute users who visited in whole 5214 pages.
Regrettably, I do not have access to the times that each of the articles could have read of the Carnival of Mathematics, but to tenor of these numbers, can bet that the Mathematics have come very far during the last week.
As for the origin of the participant, the blogs are of very diverse nature and location: they are of Spain, Argentinian, perú, colombia… there are specifics of mathematics, of physics, of history, of touring by bicycle, personnels… and the authors have been professionals of the (teaching) mathematics, physicists, engineers, students, fans… But all of us something joins: the passion for the Mathematics.
And to finish magnificent news in form, how not, of number: 2. We already have in march II Edition of the Carnival of the Mathematics. It will take place next Monday, the 15th of March and will have the same format as this occasion. During the week from March 8 until March 14, the participants will have to publish his articles in his blogs, being necessary a link to the web of the carnival. The subject-matter, in this occasion, will be free again, although I remind to you that on March 14 the international day is celebrated of π. And the host of this occasion will be a blog who went continuing long ago: Juan de Mairena [v.2.71828]. I can only wish Juan Pablo a lot of luck and fortitudes, that the work, although very pleasant, it does not stop being important.
Anyway, that sure I that I am not going to be missing to II Edition of the Carnival of Mathematics: and you?

Today, in many places of the world, Monday of Carnival is celebrated, the most important day of the whole holiday. And in the mathematical world and, in particular, in the blogosfera of Hispanic speech, today it is the First Edition of the Carnival of Mathematics.
During the whole last week, earnings related somehow to the Mathematics have been published, in order to spread a little more the kind side of the same ones. More than 40 blogs they have taken part with more than 60 articles that right now I happen to review for subject-matter.
Some authors have decided to speak about Mathematics from the literature. This way, for example, from DesEquiLibros we meet a curious titled appetizer The Wise person of Palace in which they tell us a famous problem of share-outs of heredities. On the other hand, in Mathematics next to us they have preferred to tell us the History of Pi, the irrational pirate. Finally, our good friend @Zifra, in his Swap 3,14, has brought to us a compilation of his famous mathematical Minifictions in less than 140 characters.
Also we could have enjoyed all this week solving problems and puzzles. This way, from The Logical Mole they raise a Problem to us on Probabilities; in The Mathematicians they are not the serious people we meet a curious problem of areas and beards; from Blog Zona Press they tell us the pleasures of solving a puzzle the not only one, but two times. From interactive Mathematics and manipulativas, they offer us several problems: you add up in a triangle, add up in a square and a magic anti-square. We even have had participants very related to the cycling as there are a Bike for Barcelona with a problem of estimation and Plegaleando for Seville with his equations with words. Special mention I want to do for, probably, the youngest child of all of us who, from his blog Reprint Gauss II, tells us a problem on Five pirates, many coconuts and a monkey. And finally, and a little for the hair, it has brought in the puzzle on numerical series proposed from Life and Mathematics.
Other one of the things with which, during this week, they have delighted us is with photos and mathematical images. Of it they have taken charge in Mathematics next to us, where from they have presented a parabolic Clock, and the blog to us what I see in Saragossa, which diverse photos have taught us on Parallel bars, Stars, habitable Geometries, Spirals and geometric Fronts.
Also the Geometry has capacity in this carnival. From The Science for all they teach us to construct tables that never limp, while in Mathematical Notes they teach us to solve a problem on Farms and deposits. On the other hand, in Pandora’s Memories they have spoken to us on how using the Mathematics to adorn by means of homogeneous mosaics. In The Rafalillo world they give form to the credit cards, while in Physics in the Science fiction they have brought to us some squares and rectangles and more squares. From dynamic Geometry they bring to us Jansen’s Mechanism and in The Song of Malapata they teach us what a Diagram of Voronoi is and explain it to us to conscience. Finally, I want to emphasize another two university teachers of mathematics who have brought to us more advanced earnings. In particular, Juan de Mairena speaks to us about Three theorems on the water and his absence of form and Francis (th) And mule news he has made use of the carnival to speak to us about exotic Spheres and the unvariant of Arf-Kervaire from the point of view topológico and geometrically.
But: what would be of the mathematics, and of this carnival, without the numbers? Since of being right in presents several blogs have taken charge. In Betacontinua they have introduced us, with holder pardójico including, The Simple Complex numbers. In Numbers and Spreadsheet they speak about Frobenius and the MacNuggets, something not suitable for hungry, skylight. In Cinders in the trébede they teach us how the Muslims were multiplying, while in texnologia how there was the Chinese multiplication (with translation incluída). From Matgala, they present to us how to describe to the number Pi with sums and infinite products (with his original version in Catalan), in Gaussianos they speak to us about the numbers of Catalan, but not, it has not anything in common with the language, but with a called mathematician like that. On the other hand, from Mates and + they teach us what they have to do Homer Simpson y Fermat. And finally from Curiosities and thoughts they bring to us some curiosities of the numbers.
From Blog de Sangakoo they have done this week a monographic one on the infinite in which they have told us some paradoxes of the infinite; also they wonder if the real straight line is real to finish with the famous problem of the Bridges of Königsberg.
In some blogs they have offered us appointments and thoughts to reflect. In The adventure of the Mathematics a child asks his father why we like the mathematics; from DesEquiLIBROS they wonder for the alternative to the Mathematics absence and about the Swap 3,14 make to think a little our mathematical companions. In The Song of Malapata they remind to us that there is no place for ugly mathematics, while in Trip to Ítaca with Manoli they wonder if more logicians are the mathematicians. To stop reflecting, in science in the XXIst they believe that that almost sure that the sun will go out tomorrow, but well, that would be that demostralo mathematically.
There are those who have decided to do articles about opinion contrasfondo mathematician, as the case of the blog is Mathematical: 1,1,2,3,5,8,13… where it is said to us that in crisis times, to be matemátic is a future profession. Also like opinion, I have catalogued the article about The mathematical Small cave where the mathematics appear before us like amienemigas of the man.
From the blog Zurditorium, they speak to us about the mathematicians departing from the sets, while in the same blog also we have written about sets theory and, in particular, of how measuring sets of real numbers.
But not everything in mathematics is abstract. There are those who, departing from the relation that exists between the mathematics and the origami, offer to do mathematics to us with the hands. This way, in interactive Mathematics and manipulativas they teach us to construct flowers with geometric motives for San Valentín, in Mathematics next to us we will learn to do a good hat to ourselves for the carnival. Although if we want something rapid, the best thing is that you pass for Pi-Bugs so that we learn to construct instantaneous dodecahedrons.
Also there is the one who recommends to us some book. In particular, from Bibliotranstornados they make the most of the opportunity to show us a good compilation of ancient mathematical books. And our good friend Migui teaches us The big book of the random numbers, article that has come to front in Wiggle me.
But: what would be of this life without the music? Since in the carnival it was not missing either. And of that the gravity boys have entrusted themselves Zero who sing to us that 2+2=5, and the blog Looking Names, that speaks to us about the parallelisms and convergences of the mathematics and the Jazz.
The games and, since we are, the Games theory (that is something very different, but that in this entry they are going to go of the hand) has been made present in several blogs. From The Machine of Turing, they make use of a chapter of the series House to speak to us about Instant Karma: Dr House, Roy Randall and the deceit of the player, while in The Science for all they have chosen a passage of The Revenge of Don Mendo to tell us something on half after seven. Finally, in the Blog of Mathematics and TIC s they have spent the week to themselves playing the English recluse.
Next, I present to you a series of articles in which they teach us that, sometimes, the Mathematics we are them in the most unsuspected places. For example, from Kitchen and Mathematics they have managed to take the measures to him to a pancake, while in the Swap 3,14 of Zifra have found many mathematical postal stamps. In Wis Physics they have told us some mathematical milestones of the antiquity (article that also has been carried in Wiggle me), as it is the calculation of the distance of the Earth to the Moon. And there this time is where they have been in charge to The Unyielding Village to realize field work and to be able to tell us that The Moon is the biggest homage to the Mathematicians.
But let’s leave the Moon and let’s go to something more mundane. The carnival without humor would not be the same, and that itself (cost the redundancy) we have thought some. So from The Song of Malapata they do to us a compilation of vignettes of mathematical humor, while in

Today, like colophon to the week dedicated to the First Edition of the Carnival of Mathematics, I bring to you one of these tests that occasionally come to our mail mailboxes.
With this one, you are going to verify what important personage of the world is your model to be continued, Only you are going to take 30 segunditos of nothing, and can take a big surprise. Of course, do not do pitfall and do not see the answers in the end.

• He thinks a number of the 1 to 9
• Multiply it by 3
• Add 3
• Multiply it again by 3 (I hope that you should not have had to go for the calculator)
• You will obtain a result of 2 or 3 digits, add them between themselves (the times that it is necessary to have) until you remain with only one digit

DO I LIST?
Now there checks in the following list of personalities in accordance with the number that resulted to you from these operations and discovers the one who is your model to be continued:

1. Albert Einstein
2. Leonhard Euler
3. Carl Friedrich Gauss
4. John Von Neumann
5. Paul Erdös
6. Srnivisa Ramanujan
7. Martin Gardner
8. Henry Lebesgue
9. Tito Eliatron: magnificent person, friend of the whole world, his delight becomes irresistible for all the persons, the good people, faithfully, sincerely, affectionately, happily, communicatively. Without him next to you the life loses sense…. If you follow me imitating, some day you might be as I. Although I believe that that is unattainable, with me the mold broke.

This article is going to be part of the First initiative Edition of the Carnival of Mathematics that is celebrated during all this week and that will have his final colophon next Monday, the 15th, brother-in-law in the same blog will do to himself a compilation of all the released earnings.
In this entry we are going to speak about small sets of the Real Straight line R, but (almost) everything what we say can spread easily to the plane, to the space and, even, to the space n - dimensional.
Leaving apart to the finite sets, the smallest thing that we can be sets are the sets numerables, that is to say, for that a biyección exists with the Natural Numbers. Speaking in silver, a set is numerable if we can count his elements (the first element, the second, third…) and we will never stop. As Example (with capital letters) of set numerable, we have the natives N, but also there is more, as that of the points Z or that of the rational ones Q. All of them are numerables, then from this point of view, they all are small sets.
Nevertheless, the latter set possesses a characteristic that separates from others his two. The rational ones, on having seen them inside the real ones, fulfill a curious property that is called a Property Arquimediana:

Between any two different rational numbers, it is possible to find rational other different.

Even it is possible to say something more. Between two any different rational numbers, we can always find a rational number and irrational other.
Here we have the second way of measuring the magnitude of a real conjutno. A set is dense if any open interval intersect to the set, or saying of a simpler form, a set is dense (in the real ones), if given any real number (rational or irrational) we are capable of finding a rational number as close as let’s want. Let’s say that a dense set almost full the real numbers. Therefore, from this point of view, the set Q cannot be considered to be a child, but rather everything opposite: of big size.
Another way of measuring how small a set can be is closely related to the thickness concept. A set A of real numbers is not called densely or densely at all, if given any open interval, it is possible to find a subinterval that already does not contain points of A anywhere. Let’s say that it would be a property diametrically opposite to the thickness. Sometimes, they call them spread to these sets, since the idea is that they are much spread (cost the redundancy) by the real straight line.
A classic example of this type of sets is the Set of Singer. This one joint is obtained of the following form: we take the interval [0,1], divide it in 3 equal parts and remain with 2 parts of the ends, that is to say, [0,1/3] and [2/3,1]; Now we repeat the same procedure with the 2 you intercost that we have, later with the 4 that we would obtain and this way successively. In the step to the limit the Set of Singer is obtained. Well, from the previous point of view, the Set of Singer should be considered to be a child, but nevertheless, it is known that this conjutno has exactly the same cardinalidad as the real numbers, that is to say, that takes so many points as numbers relays there is. Therefore, from this another perspective, the Set of Singer should be considered to be big.
Even more, since the cardinalidad of the rational ones is ω (that of the natives), Q it should be considerdo smaller than the Set of Singer. Although under the crystal of the thickness, the rational ones are bigger than Singer.
In short, mathematically speaking, the extremely relative concepts of big or small sound. Here we have seen a pair of examples of how measuring sets sizes, but there are still several forms more as the length or measurement, and the Categories of Baire. But all this would give for several earnings more.

Today I do not bring to you any appointment. Simply I want that this post serves to remember all, that if you want to take part in the First Edition of the Carnival of Mathematics, you can publish your entry during this week and this way contribute to the publication of this fundamental part of the Sciences.
But not quite there are going to be reminders today. Making use that February is the month carnavalero excellently, I leave to you a humorous vignette, which comes much to story with this from the Carnival of Mathematics. The vignette comes from Mel’s Pleasantry and I came to her across a Swap 3,14.

So you know already, take part in the First Edition of the Carnival of Mathematics.

Today I bring to you one of these curious calendars that exists for Internet. One treats as the Calendar 12 theorems of mathematical women of 2010, of the magnificent web Theorem of the Day.
In words of the proper author of the web and of the calendar (that, apparently, is already the third edition), this year includes theorems that comprise the whole century, from the classification of 4-politopos arquimedianos of Alice Boole Stott in 1910 (third daughter of George Boole, inventor of the algebra of Boole), up to the works in dynamics of throwing of coins (coin-toss) released approximately 3 years ago for Susan Holmes (together with Persi Diaconis and Richard Montgomery).
Per months, these are the elected women:

• January: Vera Sós
• February: Hazel Perfect
• March: Annabelle McIver
• April: Jo Heath
• May: Alice Boole Stott (with links to works of Irene Polo Blanco)
• June: Jenny Baglivo
• July: Mireille Bousquet-Mélou
• I wither: Shafi Goldwasser
• September: Cheryl Praeger
• October: Fan Chung
• November: Susan Holmes
• December: Emmy Noether

In short a magnificent cast of mathematical women who demonstrate that the number there are no only men’s thing.
Route B log of the Library of Mathematics of the University of Barcelona.

An honor is for me, to bring again to this blog, the advance of the ranking of blogs on science that they realize from Wikio. In particular, because

If the people do not think that the mathematics are simple, it is only because they do not realize of it complicated that it is the life.

John von Neumann.
Simply, I do not have anything more that to add.