The embarrassing way in which Spain were heavily defeated by Netherlands in the World Cup can only be interpreted as a reminder that those who fail to evolve with the times are usually left behind. While the current world champions and Barcelona are different organisations, the Blaugrana board should take note of such an alarming result and continue to take difficult, yet crucial decisions ahead of next season.
303: 名無しさん：14/06/19 06:29 ID:Atc
Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation. Such ideas go back to Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place"). The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics.
304: 名無しさん：14/06/19 06:30 ID:Atc
Euclide originariamente formul？ il problema geometricamente, per trovare una "misura" comune per la lunghezza di due segmenti, e il suo algoritmo procedeva sottraendo ripetutamente il pi？ corto dal pi？ lungo. Questo procedimento ？ equivalente alla implementazione seguente, che ？ molto meno efficiente del metodo indicato sopra:
305: 名無しさん：14/06/19 06:30 ID:Atc
Questi algoritmi possono essere usati, oltre che con i numeri interi, in ogni contesto in cui ？ possibile eseguire la divisione col resto. Ad esempio, l'algoritmo funziona per i polinomi ad una indeterminata su un campo K, o polinomi omogenei a due indeterminate su un campo, o gli interi gaussiani. Un oggetto algebrico in cui ？ possibile eseguire la divisione col resto ？ chiamato anello euclideo.
306: 名無しさん：14/06/19 06:31 ID:Atc
L'algoritmo di Euclide ？ un algoritmo per trovare il massimo comun divisore (indicato di seguito con MCD) tra due numeri interi. ？ uno degli algoritmi pi？ antichi conosciuti, essendo presente negli Elementi di Euclide[1] intorno al 300 a.C.; tuttavia, probabilmente l'algoritmo non ？ stato scoperto da Euclide, ma potrebbe essere stato conosciuto anche 200 anni prima. Certamente era conosciuto da Eudosso di Cnido intorno al 375 a.C.; Aristotele (intorno al 330 a.C.) ne ha fatto cenno ne I topici, 158b, 29-35. L'algoritmo non richiede la fattorizzazione dei due interi.
307: 名無しさん：14/06/19 06:31 ID:Atc
Quando si analizza il tempo di calcolo dell'algoritmo di Euclide, si trova che i valori di input che richiedono il maggior numero di divisioni sono due successivi numeri di Fibonacci, e il caso peggiore richiede O(n) divisioni, dove n ？ il numero di cifre nell'input. Occorre anche notare che le divisioni non sono operazioni atomiche (se i numeri sono pi？ grandi della dimensione naturale delle operazioni aritmetiche del computer), visto che la dimensione degli operandi pu？ essere di n cifre. Allora il tempo di calcolo reale ？ quindi O(n？).
308: 名無しさん：14/06/19 06:31 ID:Atc
L'algoritmo di Euclide ？ ampiamente usato nella pratica, specialmente per numeri piccoli, grazie alla sua semplicit？. Un algoritmo alternativo, l'algoritmo del MCD binario, utilizza la rappresentazione binaria dei computer per evitare le divisioni e quindi aumentare l'efficienza, sebbene anch'esso sia dell'ordine di O(n？): infatti su molte macchine reali permette di diminuire le costanti nascoste nella notazione "O grande".
309: 名無しさん：14/06/19 06:32 ID:Atc
I quozienti che compaiono quando l'algoritmo euclideo viene applicato ai valori di input a e b sono proprio i numeri che compaiono nella rappresentazione in frazione continua della frazione a/b. Si prenda l'esempio di a = 1071 e b = 1029 usato prima. Questi sono i calcoli con i quozienti in evidenza:
310: 名無しさん：14/06/19 06:32 ID:Atc
Questo metodo pu？ anche essere usato per valori di a e b reali; se a/b ？ irrazionale allora l'algoritmo euclideo non ha termine, ma la sequenza di quozienti che si calcola costituisce sempre la rappresentazione (ora infinita) di a/b in frazione continua.
311: 名無しさん：14/06/19 06:33 ID:Atc
In questo algoritmo ？ stato usato per la rappresentazione numerica il tipo "int16" ma pu？ essere cambiata piacimento con qualsiasi altro tipo di variabile numerica secondo i bisogni del programma.
Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation. Such ideas go back to Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place"). The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics.
319: 名無しさん：14/07/14 00:21 ID:x/k
Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series.
Henri Poincar？ published Analysis Situs in 1895,[4] introducing the concepts of homotopy and homology, which are now considered part of algebraic topology.
320: 名無しさん：14/07/14 00:21 ID:x/k
Topology as a branch of mathematics can be formally defined as "the study of qualitative properties of certain objects (called topological spaces) that are invariant under a certain kind of transformation (called a continuous map), especially those properties that are invariant under a certain kind of equivalence (called homeomorphism)." To put it more simply, topology is the study of continuity and connectivity.
321: 名無しさん：14/07/14 00:22 ID:x/k
One of the first papers in topology was the demonstration, by Leonhard Euler, that it was impossible to find a route through the town of K？nigsberg (now Kaliningrad) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges, nor on their distance from one another, but only on connectivity properties: which bridges are connected to which islands or riverbanks. The Seven Bridges of K？nigsberg is a famous problem in introductory mathematics, and led to the branch of mathematics known as graph theory.
322: 名無しさん：14/07/14 00:22 ID:x/k
Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle. A precise definition of homeomorphic, involving a continuous function with a continuous inverse, is necessarily more technical.
323: 名無しさん：14/07/14 00:23 ID:x/k
An introductory exercise is to classify the uppercase letters of the English alphabet according to homeomorphism and homotopy equivalence. The result depends partially on the font used. The figures use the sans-serif Myriad font. Homotopy equivalence is a rougher relationship than homeomorphism; a homotopy equivalence class can contain several homeomorphism classes. The simple case of homotopy equivalence described above can be used here to show two letters are homotopy equivalent. For example, O fits inside P and the tail of the P can be squished to the "hole" part.
324: 名無しさん：14/07/14 00:23 ID:x/k
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3 (Note that since we're using topology the concept of circle isn't bound only to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
325: 名無しさん：14/07/14 00:24 ID:x/k
The original motivation for the founders of knot theory was to create a table of knots and links, which are knots of several components entangled with each other. Over six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century.
To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other three-dimensional spaces and objects other than circles can be used; see knot (mathematics). Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space.
326: 名無しさん：14/07/14 00:24 ID:x/k
A knot invariant is a "quantity" that is the same for equivalent knots (Adams 2004)(Lickorish 1997)(Rolfsen 1976). For example, if the invariant is computed from a knot diagram, it should give the same value for two knot diagrams representing equivalent knots. An invariant may take the same value on two different knots, so by itself may be incapable of distinguishing all knots. An elementary invariant is tricolorability.
"Classical" knot invariants include the knot group, which is the fundamental group of the knot complement, and the Alexander polynomial, which can be computed from the Alexander invariant, a module constructed from the infinite cyclic cover of the knot complement (Lickorish 1997)(Rolfsen 1976). In the late 20th century, invariants such as "quantum" knot polynomials, Vassiliev invariants and hyperbolic invariants were discovered. These aforementioned invariants are only the tip of the iceberg of modern knot theory.
327: 名無しさん：14/07/14 00:25 ID:x/k
Actually, there are two trefoil knots, called the right and left-handed trefoils, which are mirror images of each other (take a diagram of the trefoil given above and change each crossing to the other way to get the mirror image). These are not equivalent to each other, meaning that they are not amphicheiral. This was shown by Max Dehn, before the invention of knot polynomials, using group theoretical methods (Dehn 1914). But the Alexander？onway polynomial of each kind of trefoil will be the same, as can be seen by going through the computation above with the mirror image. The Jones polynomial can in fact distinguish between the left and right-handed trefoil knots (Lickorish 1997).
328: 名無しさん：14/07/14 00:25 ID:x/k
Geometry lets us visualize what the inside of a knot or link complement looks like by imagining light rays as traveling along the geodesics of the geometry. An example is provided by the picture of the complement of the Borromean rings. The inhabitant of this link complement is viewing the space from near the red component. The balls in the picture are views of horoball neighborhoods of the link. By thickening the link in a standard way, the horoball neighborhoods of the link components are obtained. Even though the boundary of a neighborhood is a torus, when viewed from inside the link complement, it looks like a sphere. Each link component shows up as infinitely many spheres (of one color) as there are infinitely many light rays from the observer to the link component. The fundamental parallelogram (which is indicated in the picture), tiles both vertically and horizontally and shows how to extend the pattern of spheres infinitely.
329: 名無しさん：14/07/14 00:26 ID:x/k
There are a number of introductions to knot theory. A classical introduction for graduate students or advanced undergraduates is Rolfsen (1976), given in the references. Other good texts from the references are Adams (2001) and Lickorish (1997). Adams is informal and accessible for the most part to high schoolers. Lickorish is a rigorous introduction for graduate students, covering a nice mix of classical and modern topics.
330: 名無しさん：14/07/14 00:27 ID:x/k
Knopentheorie is een deelgebied van de topologie. De topologie bestudeert eigenschappen van lichamen die niet veranderen bij continue vervorming. Knopentheorie onderzoekt welke knopen in elkaar kunnen worden vervormd. Daarbij is een knoop een wiskundige idealisering van een stuk touw waarvan de eindjes zijn samengebonden.
331: 名無しさん：14/07/14 00:28 ID:x/k
Een knoop is een equivalentieklasse van inbeddingen (continue injecties) van de topologische cirkel in de driedimensionale Euclidische ruimte. Twee inbeddingen zijn equivalent als er een continue vervorming van de Euclidische ruimte bestaat die de ene inbedding in de andere vervormt (en die tijdens de vervorming een inbedding blijft, dus de lus snijdt zichzelf niet):
332: 名無しさん：14/07/14 00:28 ID:x/k
Alle inbeddingen van de cirkel hebben beeldverzamelingen die per definitie topologisch equivalent zijn, want ze zijn allemaal homeomorf met de cirkel zelf. Het interessante topologische object is het knoopcomplement van de beeldverzameling. Krachtens een stelling van Gordon en Luecke zijn twee knopen die topologisch equivalente complementen hebben, gelijk of elkaars spiegelbeeld. De studie van knopen kan dus worden herleid tot de studie van driedimensionale vari？teiten.
333: 名無しさん：14/07/14 00:29 ID:x/k
In twee van de voorbeelden hierboven worden knopen grafisch voorgesteld door het beeld van de inbedding te projecteren op een vlak, waarbij het vlak zodanig wordt gekozen dat het beeld zichzelf slechts een eindig aantal keren snijdt, in duidelijk gescheiden kruispunten. Bij elk kruispunt wordt een conventioneel teken aangebracht om aan te geven welke van de twee delen van het touw "boven" het andere ligt, bijvoorbeeld door het "onderste" deel onderbroken te tekenen. Dit heet een knoopdiagram.
334: 名無しさん：14/07/14 00:29 ID:x/k
Als twee inbeddingen van de cirkel in de ruimte door hetzelfde diagram worden voorgesteld, behoren ze tot dezelfde equivalentieklasse (dezelfde knoop). Ook als de diagrammen in elkaar kunnen worden vervormd zonder het aantal of de ori？ntatie van de snijpunten te wijzigen, stellen ze dezelfde knoop voor.
Het omgekeerde is echter niet waar. Eenzelfde knoop kan worden voorgesteld door knoopdiagrammen met verschillende aantallen snijpunten. Bijvoorbeeld: een tekening van een cirkel zonder snijpunten stelt de triviale knoop voor, maar een tekening van het cijfer acht doet dat ook.
335: 名無しさん：14/07/14 00:30 ID:x/k
1. een eenvoudige lus verwijderen;
336: 名無しさん：14/07/14 00:30 ID:x/k
2. bij twee segmenten met nabijliggende onderlinge snijpunten met gelijke ori？ntatie, de snijpunten verwijderen (de segmenten parallel hertekenen);
337: 名無しさん：14/07/14 00:31 ID:x/k
3. bij drie segmenten met nabijliggende onderlinge snijpunten, waarbij ？？n van de drie segmenten de andere twee met gelijke ori？ntatie snijdt, het derde snijpunt aan de andere kant van het ene segment hertekenen.