Fundamental mathematics
The theory of numbers, the Queen of Mathematics, as Gauss called it, is perhaps the oldest mathematical discipline and that has exerted on men of all times a deeper fascination. It is full of unresolved problems, cuvos terms are perfectly intelligible to everyone and who, however, have defied the sagacity of the most potent mathematicians for centuries. Here are a few examples: Is it true that every positive pair number is a sum of two prime numbers? This is the conjecture of Goldbach, formulated at the beginning of the eighteenth century. Is it true that if n is greater than or equal to 3 there are not three positive integers x, y, z, such as x ^ n + y ^ n = z ^ n? This is Fermat's theorem, which has not yet come to prove its truth or its falsehood. , there are infinite pairs of twin prime numbers, ie, infinite pairs of consecutive odd numbers that are cousins, such as 17 and 19, 71 and 73 ... ? Limit Calculators.
A characteristic phenomenon of modern number theory is the versatility of its methods. of many theorems in whose hypothesis and conclusions do not enter but the concepts of number and its arithmetic operations are not known more than demonstrations based on Nuy theorems deep of the analytical theory of complex variable, or of the theory of the probability, and even of the mathematical logic. The first demonstrations of prime number distribution theorem that states that the number of prime numbers between 2 and N when n is large is approximately.
A characteristic phenomenon of modern number theory is the versatility of its methods. of many theorems in whose hypothesis and conclusions do not enter but the concepts of number and its arithmetic operations are not known more than demonstrations based on Nuy theorems deep of the analytical theory of complex variable, or of the theory of the probability, and even of the mathematical logic. The first demonstrations of prime number distribution theorem that states that the number of prime numbers between 2 and N when n is large is approximately.
They were based on the theory of functions of complex variable and the theory of probability.
The current number theory has been enriched by the emergence of new automatic computing techniques. The computers can be used in it experimentally, for example to verify that a conjecture is false, and also as an aid in the methods of demonstration where the reasoning are branched beyond the possibilities of the human mind without TA help.
The current number theory has been enriched by the emergence of new automatic computing techniques. The computers can be used in it experimentally, for example to verify that a conjecture is false, and also as an aid in the methods of demonstration where the reasoning are branched beyond the possibilities of the human mind without TA help.
Combinatorial analysis
The fundamental question of Combinatori analysis is the following: How many different ways can you perform a particular task? For example: A secretary has ten different letters written for ten different people. He has also written in ten envelopes the signs of these people. How many different ways can you make mistakes by putting the cards in the envelopes? Such problems are often easy to enunciate and difficult to solve, requiring, in general, nothing more or less than what Gauss called intelligently to tell. A collapsible map of Madrid that has three folds vertically and four horizontally, how many different ways can you fold? The general problem, which asks for a formula that gives the number of different ways to bend a known map the number of folds is still to be resolved.
One of the problems opened for more than a century and fourth is that of the four colors. It's halfway between the Combinatory and the topology. It is a question of coloring a map with the least possible number of colors in such a way that two adjacent countries do not have the same color. It is very easy to see that three colors are not generally enough and you can show that five are enough to properly color any map. Can it be done with four? The problem, solved affirmatively in 1976 by Appel and Haken, with the essential help of the computer, has opened, as we will see later, deep questions about the very nature of the mathematical demonstration.
It is surprising the number of Science v fields of technology in which combinatorial problems arise and their results are applied. The genetics, the biochemistry, the statistical mechanics, the design of circuits, the theory of programming languages ... take advantage of the results obtained in this area of the mathematics. The traveler's is one of those problems where the applicability of the theory is well patent. It is a question of establishing the most economical itinerary that a commercial traveler can follow in his visit by a number of cities, taking into account the expense that the displacement from one to another implies.
One of the problems opened for more than a century and fourth is that of the four colors. It's halfway between the Combinatory and the topology. It is a question of coloring a map with the least possible number of colors in such a way that two adjacent countries do not have the same color. It is very easy to see that three colors are not generally enough and you can show that five are enough to properly color any map. Can it be done with four? The problem, solved affirmatively in 1976 by Appel and Haken, with the essential help of the computer, has opened, as we will see later, deep questions about the very nature of the mathematical demonstration.
It is surprising the number of Science v fields of technology in which combinatorial problems arise and their results are applied. The genetics, the biochemistry, the statistical mechanics, the design of circuits, the theory of programming languages ... take advantage of the results obtained in this area of the mathematics. The traveler's is one of those problems where the applicability of the theory is well patent. It is a question of establishing the most economical itinerary that a commercial traveler can follow in his visit by a number of cities, taking into account the expense that the displacement from one to another implies.