Le possibili conseguenze di linee d’azione alternative


Regarding the opus of Sir Isaac Newton’s legacy, this enormous corpus of Newton’s documents, papers, manuscripts, including his correspondence, which have survived reveals to us a person with serious qualities of mind, physique, and personality extraordinarily favorable for the making of a great scientist: tremendous powers of concentration, ability to stand long periods of intense mental exertion (some of his contemporaries had affirmed that he was able to concentrate in a problem about 17 hours of mental exercise without a break), and objectivity uncomplicated by any frivolous interests.

When Sir Isaac Newton came to maturity, circumstances were auspiciously combined to make possible major changes in ways of thought and endeavor on human beings. The uniqueness of Newton’s achievement could be said to lie in his exploitation of these unusual circumstances. He alone among his gifted contemporaries fully recognized the implications of recent scientific discoveries. With these as a point of departure, he developed a unified mathematical interpretation of the cosmos, in the expounding of which he demonstrated method and direction for future elaboration. In shifting the emphasis from “quality” to “quantity,” from pursuit of answers to the question “Why?” to focus upon “What?” and “How?,” he effectively prepared the way for the age of technology and modern science.

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Games


A two-person game is any competition or conflict between two opponents, called players. A two-person zero-sum game is any two-person game where the amount won (or gained) by one player is lost by the other player.

In two-person zero-sum games we use a matrix, called a payoff matrix, to denote the gains and losses to player 1, called the row player. Gains appear as positive entries in the matrix and losses appear as negative entries in the matrix.

In a game with payoff matrix P, the row player can choose any of the rows of P and the column player can choose any of the columns of P. A player who makes the same choice each time the game is played is using a pure strategy. In terms of risk, the best pure strategy is the pure strategy that garantees the greatest gain to the player, not knowing the other player’s choise. The concept of best method to play is not based on the actual winnings but rather the notion that a player does the best he can under the circunstances.

The Hadwiger’s conjecture


Speaking about graphs, the Hadwiger number (G) of a graph G is the largest integer h such that the complete graph on h nodes K h is a minor of G. Equivalently, it is the largest integer in a way which any graph on at most (G) nodes is a minor of G. The Hadwiger’s conjecture states for any graph G, (G) number of G. It is well-known by mathematicians that for any connected undirected graph G, there exists a unique prime factorization with respect to Cartesian graph products. If the unique prime factorization of G is given as G1 G 2 ::: G k, where each G i is prime, then we say that the product dimension of G is k. Such a factorization can be computed efficiently.

Studying the Hadwiger’s conjecture for graphs in terms of their prime factorization. It shows the Hadwiger’s conjecture is true for a graph G if the product dimension of G is at least 2 log ( (G)) + 3. In fact, it is enough for

 

 

This approach also yields (almost) sharp lower bounds for the Hadwiger number of well-known graph products like d–dimensional hypercubes, Hamming graphs and the d–dimensional grids. In particular, we show that for a d–dimensional hypercube Hd, 2b

 

Hamiltonian circuits


The Hamiltonian cycle problem is a special case of the traveling salesman problem, obtained by setting the distance between two cities to a finite constant if they are adjacent and infinity otherwise.

In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path or a Hamiltonian cycle exists in a given graph (whether directed or undirected). Both problems are NP-complete. The problem of finding a Hamiltonian cycle or path is in FNP.

There is a simple relation between the two problems. The Hamiltonian path problem for graph G is equivalent to the Hamiltonian cycle problem in a graph H obtained from G by adding a new vertex and connecting it to all vertices of G.

The directed and undirected Hamiltonian cycle problems were two of Karp’s 21 NP-complete problems. Garey and Johnson showed shortly afterwards in 1974 that the directed Hamiltonian cycle problem remains NP-complete for planar graphs and the undirected Hamiltonian cycle problem remains NP-complete for cubic planar graphs.

A randomized algorithm for Hamiltonian path that is fast on most graphs is the following: Start from a random vertex, and continue if there is a neighbor not visited.

If there are no more unvisited neighbors, and the path formed isn’t Hamiltonian, pick a neighbor uniformly at random, and rotate using that neighbor as a pivot. (That is, add an edge to that neighbor, and remove one of the existing edges from that neighbor so as not to form a loop.) Then, continue the algorithm at the new end of the path.

Due to the complexity of the problem computers have to be used to solve what may seem to be minor tasks, for example to calculate the shortest route to tour the ten biggest cities in the UK over 3.5 million routes have to be analysed.

In July 2009, research published in the Journal of Biological Engineering showed that a bacterial computer can be used to solve a simple Hamiltonian path problem (using three locations).

A team of US scientists has engineered bacteria that can solve complex mathematical problems faster than anything made from silicon.

The research, published today in the Journal of Biological Engineering, proves that bacteria can be used to solve a puzzle known as the Hamiltonian Path Problem, a special case of the traveling salesman problem.

The researchers say that this proof-of-concept experiment demonstrates that bacterial computing is a new way to address NP-complete problems using the inherent advantages of genetic systems.

Kruskal’s algorithm


In general, a sequence of instructions that solve all instances of a well-defined problem is called an algorithm. The algorithm used to find a minimum cost spanning tree in a graph is called greedy. We greedily choose the edges of the graph to build our tree hat are of least weight, as long as they do not create a circuit.

The Kruskal Algorithm starts with a forest which consists of n trees. Each and everyone tree,consists only by one node and nothing else. In every step of the algorithm, two different trees of this forest are connected to a bigger tree.

Therefore, we keep having less and bigger trees in our forest until we end up in a tree which is the minimum genetic tree (m.g.t.).

In every step we choose the side with the least cost, which means that we are still under greedy policy. If the chosen side connects nodes which belong in the same tree the side is rejected, and not examined again because it could produce a circle which will destroy our tree.

Either this side or the next one in order of least cost will connect nodes of different trees, and this we insert connecting two small trees into a bigger one.

  1.  

    Set i=1 and let E0={}

  2. Select an edge ei of minimum value not in Ei-1 such that Ti=<Ei-1 cup {ei} >is acyclic and define Ei=Ei-1 cup {ei}. If no such edge exists, Let T=<Ei>and stop.
  3. Replace i by i+1. Return to Step 2.

The time required by Kruskal’s algorithm is O(|E|log|V|).

The Golden Mean


The Golden Mean (or Golden Section), represented by the Greek letter phi, is one of those mysterious natural numbers, like e or phi, that seem to arise out of the basic structure of our cosmos.

Unlike those abstract numbers, however, phi appears clearly and regularly in the realm of things that grow and unfold in steps, and that includes living things.

The decimal representation of phi is 1.6180339887499…

The golden mean, Phi, has been applied in diverse situations in art, architecture and music, and although some have claimed that it represents a basic aesthetic proportion.

Others have argued that it is only one of a large number of such ratios. We review its early history, especially its relationship to the Mount Meru of Pingala.

We examine the successive divisions of 3, 7, and 22 in Indian music and suggest derivation from variants of Mount Meru. We also speculate on the neurophysiological basis behind the sense that the golden mean is a pleasant proportion.

In pure mathematics, an increase in size can be any imaginable number, even one like e or phi. But in the world of nature.

Things always grow by adding some unit, even if the unit is as small as a molecule. So it’s not surprising that phi turns out to be an ideal rate of growth for things which grow by adding some quantity.

Some examples:

The Nautilus shell (Nautilus pompilius) grows larger on each spiral by phi.

The sunflower has 55 (see number list) clockwise spirals overlaid on either 34 or 89 (see number list) counterclockwise spirals, a phi proportion.

Leonhard Euler


Leonhard Euler was one of top mathematicians of the eighteenth century and the greatest mathematician to come out of Switzerland.

He made numerous contributions to almost every mathematics field and was the most prolific mathematics writer of all time. It was said that “Euler calculated without apparent effort, as men breathe….”

He was dubbed “Analysis Incarnate” by his peers for his incredible ability.

Leonhard Euler was born in Basel, Switzerland, on April 15, 1707. His father, a Calvinist pastor and former mathematician, planned the life of a clergyman for his son and originally Leonhard followed that path.

He graduated from the University of Basel in 1724 where he studied theology and Hebrew. During his time at the school, however, he was privately tutored in mathematics by Johann Bernoulli.

Johann was so impressed by his pupil’s ability that he convinced Euler’s father to allow Leonhard to become a mathematician.

Euler took up a position at the Academy of Sciences in St. Petersburg, Russia, in 1727 and became the professor of mathematics six years later.

During his stay, he was married and would over his lifetime have thirteen children, five of which would survive to adulthood. While in Russia, he lost sight in one eye after working day and night for three days to solve a problem.

The question, which was a public contest, took all the other mathematicians involved months to figure out. He also discovered that the Czar’s government was far from democratic as he was followed by secret police. He looked for a way out.

He found it in 1741, when he moved his family to Berlin to take over as director of mathematics at the Academy of Sciences under Frederick the Great.

While in Prussia, his home was destroyed by invading Russian armies, but he was held in such high esteem by both countries that he was compensated for more than he lost. He also frustrated Frederick’s mother to no end by refusing to engage in conversation.

When she finally asked him for a reason, he responded: “Madam, it is because I have just come from a country where every person who speaks in hanged.”

He also could not handle the intrigues and feuds that plagued the Academy. When the previous president died, Euler should have been the obvious successor except for the fact that Frederick disliked him.

The monarch asked D’Alembert, a French mathematician, to take the position. D’Alembert, who saw the injustice, refused on the basis that no one could be placed above Euler. However, it became clear it was time for Leonhard to find a new home.

Meanwhile, Russia had come under the rule of the more liberal Catherine the Great. In 1766, he returned to St. Petersburg and became the director of the Academy.

Soon afterwards, he went completely blind but continued his mathematical work by dictating to a secretary. His house burned down in 1771 and his life was saved only by the heroic efforts of a servant to carry him out of the flames.

He died of a stroke on September 7, 1783. Appropriately to this simple mathematician, his final words were simply “I die.”

Euler was especially famous from his writings. Simply put, he produced more scholarly work on mathematics than anyone.

It was said that he could produce an entire new mathematical paper in about thirty minutes and had huge piles of his works lying on his desk.

Even more impressive, Euler contemplated new problems not in quiet privacy but in the presence of his young children. It was not uncommon to find “Analysis Incarnate” ruminating over a new subject with a child on his lap.

Though Euler is best remembered for his contributions to mathematics, he was involved in some extent in almost all fields.

Especially close to his heart was philosophy. While in Berlin, he would constantly get involved in philosophical debates, especially with Voltaire. Unfortunately, Euler’s philosophical ability was limited and he often blundered to the amusement of all involved.

However, when he returned to Russia, he got his revenge. Catherine the Great had invited to her court the famous French philosopher Diderot, who to the chagrin of the czarina, attempted to convert her subjects to atheism.

She asked Euler to quiet him. One day in the court, the French philosopher, who had no mathematical knowledge, was informed that someone had a mathematical proof of the existence of God.

He asked to hear it. Euler then stepped forward and stated: “Sir, a+b/n=x, hence God exists; reply!” Diderot had no idea what Euler was talking about. However, he did understand the chorus of laughter that followed and soon after returned to France.

Euler’s contributions to every mathematical field that existed at the time. He standardized modern mathematics notation when he used symbols such as f(x), e, , i and in his textbooks.

He was the first person to represent trigonometric values as ratios and prove that e is an irrational number. His invention of the calculus of variations led to the general method to solve max and min value problems.

In physics, he developed the general equations for hydrodynamics and for motion. He was also one of the first people to recognize that infinite series had to be convergent to be used safely.

Possibly his most impressive work was his approximation of the three-body problem of the sun, earth and moon, which he solved while completely blind and performing all the computations in his head.

Among his other endeavors were proofs of Fermat’s final theorem for cubes and quads, the use of calculus in mechanics and the computation of logs for negative and imaginary numbers.

The origin of geometry


Speaking of the origin of geometry in the present world cycle, Aristotle tells us that the same ideas have repeatedly come to men at various periods of the universe (universe is here refere at the men’s universe or culture or world).

It is not, he goes on to say, in our time or in the time of those known to us that the sciences have first arisen, but they have appeared and again disappeared, and will continue to appear and to disappear, in various cycles, of which the number both past and future is countless.

But since we must speak of the origin of the arts and sciences with reference to the present world cycle, it was, we say, among the Egyptians that geometry is generally held to have been discovered. It owed its discovery to the practice of land measurement.

For the Egyptians had to perform such measurements because the overflow of the Nile would cause the boundary of each person’s land to disappear.

Yet, it should occasion no surprise that the discovery both of this science and of the other sciences proceeded from utility, since everything that is in the process of becoming advances from the imperfect to the perfect.

The progress, then, from sense perception to reason and from reason to understanding is a natural one.

And so, just as the accurate knowledge of numbers originated with the Phoenicians through their commerce and their business transactions, so geometry was discovered by the Egyptians for the reason we have indicated.

Manifold


In science though especially on math, but more specifically in topology and differential geometry, a Manifold is a tipical mathematical space which on a small enough scale resembles the Euclidean space of a certain dimension; it is called the Dimension of the Manifold.

Therefore a circle and a line are one-dimensional Manifolds, the surface of a ball and a plane are two-dimensional Manifolds, and so forth. Quite formally, every point of an any n-dimensional Manifold has a neighborhood homeomorphic to the n-dimensional space Rn.

Whereas Manifolds resemble Euclidean spaces near each point, or locally, the overall, or global; a structure of a manifold may be much more complicated.

For intance, any point on the usual two-dimensional surface of a sphere is surrounded by a circular region that can be flattened to a circular region of the plane, as in a geographical map. Yet, the sphere differs from the plane “in the large”: in the language of topology, they are not homeomorphic.

To illustrate this idea, consider the ancient belief that the Earth was flat as contrasted with the modern evidence that it is round. The discrepancy arises essentially from the fact that on the small scales that we see, the Earth does indeed look flat.

In general, any object that is nearly “flat” on small scales is a Manifold, and so Manifolds constitute a generalization of objects we could live on in which we would encounter the round/flat Earth problem, as first codified by Poincaré.

This concept of a Manifold is key to many parts of geometry and modern mathematical physics because they allow more complicated structures to be expressed and understood in terms of the relatively well-understood properties of simpler spaces.

So, a Manifold is typically provided with a differentiable structure that concedes one to do calculus, Riemannian metric that permits one to calculate distances and angles.

Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian Manifolds model space-time in general relativity.

The art of secrecy


Most of us conduct many of our daily personal, business and government transactions electronically. We do so many things online—from staying in touch with friends to buying and selling everything, including the kitchen sink—that getting comprehensive information about most people is as easy as logging, or recording, their online activities.

And for various reasons, ISPs are already logging our activities, such as which sites we have visited and when. They are not alone. Many entities we interact with online—stores, newspapers, dating sites,  and the like—keep close tabs on us as well. Thus, if we value privacy, we face the challenge of how to take advantage of everything the Internet has to offer without giving up our privacy.

An amazing discovery of modern cryptography is that virtually any task involving electronic communication can be carried out privately. Many people, including the editors of most dictionaries, mistakenly think that “cryptography” is synonymous with the study of encryption. But modern cryptography encompasses much more. It provides mathematical methods for protecting communication and computation against all kinds of malicious behavior—that is, tools for protecting our privacy and security.

Cryptography (or cryptology; from Greek κρυπτός, kryptos, “hidden, secret”; and γράφω, gráphō, “I write”, or -λογία, -logia, respectively) is the science that practices and studies how to hide information. Modern cryptography intersects the disciplines of mathematics, computer science, and engineering. Applications of cryptography include ATM cards, computer passwords, and electronic commerce.

The earliest forms of secret writing required little more than local pen and paper analogs, as most people could not read. More literacy, or opponent literacy, required actual cryptography. The main classical cipher types are transposition ciphers, which rearrange the order of letters in a message (e.g., ‘hello world’ becomes ‘ehlol owrdl’ in a trivially simple rearrangement scheme), and substitution ciphers, which systematically replace letters or groups of letters with other letters or groups of letters (e.g., ‘fly at once’ becomes ‘gmz bu podf’ by replacing each letter with the one following it in the English alphabet).

Simple versions of either offered little confidentiality from enterprising opponents, and still don’t. An early substitution cipher was the Caesar cipher, in which each letter in the plaintext was replaced by a letter some fixed number of positions further down the alphabet. It was named after Julius Caesar who is reported to have used it, with a shift of 3, to communicate with his generals during his military campaigns, just like EXCESS-3 code in boolean algebra.

Encryption attempts to ensure secrecy in communications, such as those of spies, military leaders, and diplomats. There is record of several early Hebrew ciphers as well. Cryptography is recommended in the Kama Sutra as a way for lovers to communicate without inconvenient discovery. Steganography (i.e., hiding even the existence of a message so as to keep it confidential) was also first developed in ancient times. An early example, from Herodotus, concealed a message – a tattoo on a slave’s shaved head – under the regrown hair. More modern examples of steganography include the use of invisible ink, microdots, and digital watermarks to conceal information.