|
|
24.05.2007., 00:21
|
#81
|
nezainteresiran
Registracija: Apr 2006.
Postova: 623
|
sta onda ne dobijemo singularnost?
|
|
|
24.05.2007., 00:47
|
#82
|
dungeon-Keeper
Registracija: Apr 2007.
Postova: 548
|
singularnost je sta?
__________________
Wtf? :-O
|
|
|
24.05.2007., 00:54
|
#83
|
nezainteresiran
Registracija: Apr 2006.
Postova: 623
|
Quote:
Dungeon Keeper kaže:
singularnost je sta?
|
evo sto wikipedia kaze link
|
|
|
24.05.2007., 01:23
|
#84
|
dungeon-Keeper
Registracija: Apr 2007.
Postova: 548
|
Quote:
cosmonaut kaže:
evo sto wikipedia kaze link
|
For other mathematical uses, see Mathematical singularity. For non-mathematical uses, see Singularity.
In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor. Probably there will appear a number of double points, at which the string crosses itself in an approximate 'X' shape. These are the simplest kinds of singularity. Perhaps the string will also touch itself, coming into contact with itself without crossing, like an underlined 'U'. This is another kind of singularity. Unlike the double point, it is not stable, in the sense that a small push will lift the bottom of the 'U' away from the 'underline'.
Contents [hide]
1 How singularities may arise
2 Algebraic curve singularities
3 The general position of singularities in algebraic geometry
4 The smooth theory, and catastrophes
5 Arnold's view
6 Duality
7 Other possible meanings
8 See also
[edit] How singularities may arise
In singularity theory the general phenomenon of points and sets of singularities is studied, as part of the concept that manifolds (spaces without singularities) may acquire special, singular points by a number of routes. Projection is one way, very obvious in visual terms when three-dimensional objects are projected into two dimensions (for example in one of our eyes); in looking at classical statuary the folds of drapery are amongst the most obvious features. Singularities of this kind include caustics, very familiar as the light patterns at the bottom of a swimming pool.
Other ways in which singularities occur is by degeneration of manifold structure. That implies the breakdown of parametrization of points; it is prominent in general relativity, where a gravitational singularity, at which the gravitational field is strong enough to change the very structure of space-time, is identified with a black hole. In a less dramatic fashion, the presence of symmetry can be good cause to consider orbifolds, which are manifolds that have acquired 'corners' in a process of folding up resembling the creasing of a table napkin.
[edit] Algebraic curve singularities
Historically, singularities were first noticed in the study of algebraic curves. The double point at (0,0) of the curve
y2 = x3 − x2
and the cusp there of
y2 = x3
are qualitatively different, as is seen just by sketching. Isaac Newton carried out a detailed study of all cubic curves, the general family to which these examples belong. It was noticed in the formulation of Bézout's theorem that such singular points must be counted with multiplicity (2 for a double point, 3 for a cusp), in accounting for intersections of curves.
It was then a short step to define the general notion of a singular point of an algebraic variety; that is, to allow higher dimensions.
[edit] The general position of singularities in algebraic geometry
Such singularities in algebraic geometry are the easiest in principle to study, since they are defined by polynomial equations and therefore in terms of a coordinate system. One can say that the extrinsic meaning of a singular point isn't in question; it is just that in intrinsic terms the coordinates in the ambient space don't straightforwardly translate the geometry of the algebraic variety at the point. Intensive studies of such singularities led in the end to Heisuke Hironaka's fundamental theorem on resolution of singularities (in birational geometry in characteristic 0). This means that the simple process of 'lifting' a piece of string off itself, by the 'obvious' use of the cross-over at a double point, is not essentially misleading: all the singularities of algebraic geometry can be recovered as some sort of very general collapse (through multiple processes). This result is often implicitly used to extend affine geometry to projective geometry: it is entirely typical for an affine variety to acquire singular points on the hyperplane at infinity, when its closure in projective space is taken. Resolution says that such singularities can be handled rather as a (complicated) sort of compactification, ending up with a compact manifold (for the strong topology, rather than the Zariski topology, that is).
[edit] The smooth theory, and catastrophes
At about the same time as Hironaka's work, the catastrophe theory of René Thom was receiving a great deal of attention. This is another branch of singularity theory, based on earlier work of Hassler Whitney on critical points. Roughly speaking, a critical point of a smooth function is where the level set develops a singular point in the geometric sense. This theory deals with differentiable functions in general, rather than just polynomials. To compensate, only the stable phenomena are considered. One can argue that in nature, anything destroyed by tiny changes is not going to be observed; the visible is the stable. Whitney had shown that in low numbers of variables the stable structure of critical points is very restricted, in local terms. Thom built on this, and his own earlier work, to create a catastrophe theory supposed to account for discontinuous change in nature.
[edit] Arnold's view
While Thom was an eminent mathematician, the subsequent fashionable nature of elementary catastrophe theory caused a reaction, in particular on the part of Vladimir Arnold. He may have been largely responsible for applying the term singularity theory to the area including the input from algebraic geometry, as well as that flowing from the work of Whitney, Thom and other authors. He wrote in terms making clear his distaste for the too-publicised emphasis on a small part of the territory. The foundational work on smooth singularities is formulated as the construction of equivalence relations on singular points, and germs. Technically this involves group actions of Lie groups on spaces of jets; in less abstract terms Taylor series are examined up to change of variable, pinning down singularities with enough derivatives. Applications, according to Arnold, are to be seen in symplectic geometry, as the geometric form of classical mechanics.
[edit] Duality
An important reason why singularities cause problems in mathematics is that, with a failure of manifold structure, the invocation of Poincaré duality is also disallowed. A major advance was the introduction of intersection cohomology, which arose initially from attempts to restore duality by use of strata. Numerous connections and applications stemmed from the original idea, for example the concept of perverse sheaf in homological algebra.
[edit] Other possible meanings
The theory mentioned above does not directly relate to the concept of mathematical singularity as a value at which a function isn't defined. For that, see for example isolated singularity, essential singularity, removable singularity. The monodromy theory of differential equations, in the complex domain, around singularities, does however come into relation with the geometric theory. Roughly speaking, monodromy studies the way a covering map can degenerate, while singularity theory studies the way a manifold can degenerate; and these fields are linked.
Ovo? Hehehe, necu ni poceti citati. ajde mi ukratko reci!
__________________
Wtf? :-O
|
|
|
24.05.2007., 01:26
|
#86
|
nezainteresiran
Registracija: Apr 2006.
Postova: 623
|
Quote:
Endimion17 kaže:
|
|
|
|
24.05.2007., 01:29
|
#87
|
dungeon-Keeper
Registracija: Apr 2007.
Postova: 548
|
dobro, i sta je to na kraju?
__________________
Wtf? :-O
|
|
|
24.05.2007., 01:34
|
#88
|
Registrirani korisnik
Registracija: Jun 2006.
Postova: 26,953
|
Quote:
Dungeon Keeper kaže:
dobro, i sta je to na kraju?
|
pa čitaj! :balkanski smiley s brčinama i raširenim rukama:
|
|
|
24.05.2007., 04:31
|
#89
|
Registrirani korisnik
Registracija: May 2006.
Postova: 667
|
Quote:
Dungeon Keeper kaže:
Zanimljivo pitanje, da. Mozda ide u drugu dimenziju, puno mi toga ne znamo. Mozda imamo krivu podlogu znaju da bi shvatili, mozda smo krenuli u krivom smjeru shvacanja....
|
Jebena je ta svemirska "hevolucija"!!!
|
|
|
24.05.2007., 10:06
|
#90
|
Registrirani korisnik
Registracija: Apr 2007.
Lokacija: Osijek
Postova: 188
|
Quote:
stojecival kaže:
meni je najzanimljivje pitanje crne rupe sto se to zapravo dogodi kada se materija urusi u najmanju mogucu tocku sto se dogodi kad se materija urusi do planckove duljine koja je nedjeljiva
|
Pa takva masa ima beskonačnu težinu. bilo kakva materija se stisne na max. rekobiniraju se protoni i neutroni i tako tvore supergusti i superteški materijal.
|
|
|
24.05.2007., 10:57
|
#91
|
Registrirani korisnik
Registracija: Jun 2006.
Postova: 26,953
|
zdrobe se i kvarkovi, sve se potpuno zdrobi. prestaju vrijediti zakoni fizike. hawking zna
|
|
|
24.05.2007., 13:09
|
#92
|
long live rock'n'roll
Registracija: Mar 2007.
Lokacija: 'rvacka metropola
Postova: 64
|
e da....
bojim se da na to pitanje niko nema točan odgovor...
a ni netočan... :kes:
__________________
Volim te. Ne mrzim te. Ne želim razgovarati s tobom.
|
|
|
25.05.2007., 12:46
|
#93
|
Registrirani korisnik
Registracija: Apr 2007.
Lokacija: Osijek
Postova: 188
|
Quote:
Endimion17 kaže:
zdrobe se i kvarkovi, sve se potpuno zdrobi. prestaju vrijediti zakoni fizike. hawking zna
|
Ne, zakoni i dalje vrijede samo što su dovedeni do samih granica! zakoni se ne mogu mjenjati mogu se rastezati donekle ali su u svojoj prirodi fiksni. današnja znanost i zakone koje je otkrila su previše površni da bi bili primjenjivi u cijelom svemiru... još smo premaladi....
|
|
|
25.05.2007., 13:21
|
#94
|
Registrirani korisnik
Registracija: Jun 2006.
Postova: 26,953
|
Quote:
ElessarOne kaže:
Ne, zakoni i dalje vrijede samo što su dovedeni do samih granica! zakoni se ne mogu mjenjati mogu se rastezati donekle ali su u svojoj prirodi fiksni. današnja znanost i zakone koje je otkrila su previše površni da bi bili primjenjivi u cijelom svemiru... još smo premaladi....
|
vrijeme će pokazati.
|
|
|
25.05.2007., 14:20
|
#95
|
dungeon-Keeper
Registracija: Apr 2007.
Postova: 548
|
Nismo premladi nego je sve krenulo u krivom smjeru od pocetka...
__________________
Wtf? :-O
|
|
|
27.05.2007., 12:26
|
#96
|
Registrirani korisnik
Registracija: May 2007.
Lokacija: Samobor
Postova: 16
|
hm zasto postojim? nemam blage,no ovo dal svemir ima svoju svijest(lako moguce) jel nesto tako slozeno i tako nerazumljivo moze imat svasta
|
|
|
28.05.2007., 14:53
|
#97
|
Registrirani korisnik
Registracija: Apr 2007.
Lokacija: Osijek
Postova: 188
|
Quote:
Endimion17 kaže:
vrijeme će pokazati.
|
Naravno ali zakoni ne mogu malo vrijediti pa ne vrijediti u istom prostoru... možda mogu ali to dodatno komplicira stvari bespotrebno...
|
|
|
28.05.2007., 15:09
|
#98
|
Registrirani korisnik
Registracija: Jun 2006.
Postova: 26,953
|
Quote:
ElessarOne kaže:
Naravno ali zakoni ne mogu malo vrijediti pa ne vrijediti u istom prostoru... možda mogu ali to dodatno komplicira stvari bespotrebno...
|
singularitet baš i nije prostor
|
|
|
28.05.2007., 15:21
|
#99
|
U samoizolaciji
Registracija: Jan 2007.
Lokacija: iza tebe
Postova: 607
|
Quote:
Dungeon Keeper kaže:
For other mathematical uses, see Mathematical singularity. For non-mathematical uses, see Singularity.
In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor. Probably there will appear a number of double points, at which the string crosses itself in an approximate 'X' shape. These are the simplest kinds of singularity. Perhaps the string will also touch itself, coming into contact with itself without crossing, like an underlined 'U'. This is another kind of singularity. Unlike the double point, it is not stable, in the sense that a small push will lift the bottom of the 'U' away from the 'underline'.
Contents [hide]
1 How singularities may arise
2 Algebraic curve singularities
3 The general position of singularities in algebraic geometry
4 The smooth theory, and catastrophes
5 Arnold's view
6 Duality
7 Other possible meanings
8 See also
[edit] How singularities may arise
In singularity theory the general phenomenon of points and sets of singularities is studied, as part of the concept that manifolds (spaces without singularities) may acquire special, singular points by a number of routes. Projection is one way, very obvious in visual terms when three-dimensional objects are projected into two dimensions (for example in one of our eyes); in looking at classical statuary the folds of drapery are amongst the most obvious features. Singularities of this kind include caustics, very familiar as the light patterns at the bottom of a swimming pool.
Other ways in which singularities occur is by degeneration of manifold structure. That implies the breakdown of parametrization of points; it is prominent in general relativity, where a gravitational singularity, at which the gravitational field is strong enough to change the very structure of space-time, is identified with a black hole. In a less dramatic fashion, the presence of symmetry can be good cause to consider orbifolds, which are manifolds that have acquired 'corners' in a process of folding up resembling the creasing of a table napkin.
[edit] Algebraic curve singularities
Historically, singularities were first noticed in the study of algebraic curves. The double point at (0,0) of the curve
y2 = x3 − x2
and the cusp there of
y2 = x3
are qualitatively different, as is seen just by sketching. Isaac Newton carried out a detailed study of all cubic curves, the general family to which these examples belong. It was noticed in the formulation of Bézout's theorem that such singular points must be counted with multiplicity (2 for a double point, 3 for a cusp), in accounting for intersections of curves.
It was then a short step to define the general notion of a singular point of an algebraic variety; that is, to allow higher dimensions.
[edit] The general position of singularities in algebraic geometry
Such singularities in algebraic geometry are the easiest in principle to study, since they are defined by polynomial equations and therefore in terms of a coordinate system. One can say that the extrinsic meaning of a singular point isn't in question; it is just that in intrinsic terms the coordinates in the ambient space don't straightforwardly translate the geometry of the algebraic variety at the point. Intensive studies of such singularities led in the end to Heisuke Hironaka's fundamental theorem on resolution of singularities (in birational geometry in characteristic 0). This means that the simple process of 'lifting' a piece of string off itself, by the 'obvious' use of the cross-over at a double point, is not essentially misleading: all the singularities of algebraic geometry can be recovered as some sort of very general collapse (through multiple processes). This result is often implicitly used to extend affine geometry to projective geometry: it is entirely typical for an affine variety to acquire singular points on the hyperplane at infinity, when its closure in projective space is taken. Resolution says that such singularities can be handled rather as a (complicated) sort of compactification, ending up with a compact manifold (for the strong topology, rather than the Zariski topology, that is).
[edit] The smooth theory, and catastrophes
At about the same time as Hironaka's work, the catastrophe theory of René Thom was receiving a great deal of attention. This is another branch of singularity theory, based on earlier work of Hassler Whitney on critical points. Roughly speaking, a critical point of a smooth function is where the level set develops a singular point in the geometric sense. This theory deals with differentiable functions in general, rather than just polynomials. To compensate, only the stable phenomena are considered. One can argue that in nature, anything destroyed by tiny changes is not going to be observed; the visible is the stable. Whitney had shown that in low numbers of variables the stable structure of critical points is very restricted, in local terms. Thom built on this, and his own earlier work, to create a catastrophe theory supposed to account for discontinuous change in nature.
[edit] Arnold's view
While Thom was an eminent mathematician, the subsequent fashionable nature of elementary catastrophe theory caused a reaction, in particular on the part of Vladimir Arnold. He may have been largely responsible for applying the term singularity theory to the area including the input from algebraic geometry, as well as that flowing from the work of Whitney, Thom and other authors. He wrote in terms making clear his distaste for the too-publicised emphasis on a small part of the territory. The foundational work on smooth singularities is formulated as the construction of equivalence relations on singular points, and germs. Technically this involves group actions of Lie groups on spaces of jets; in less abstract terms Taylor series are examined up to change of variable, pinning down singularities with enough derivatives. Applications, according to Arnold, are to be seen in symplectic geometry, as the geometric form of classical mechanics.
[edit] Duality
An important reason why singularities cause problems in mathematics is that, with a failure of manifold structure, the invocation of Poincaré duality is also disallowed. A major advance was the introduction of intersection cohomology, which arose initially from attempts to restore duality by use of strata. Numerous connections and applications stemmed from the original idea, for example the concept of perverse sheaf in homological algebra.
[edit] Other possible meanings
The theory mentioned above does not directly relate to the concept of mathematical singularity as a value at which a function isn't defined. For that, see for example isolated singularity, essential singularity, removable singularity. The monodromy theory of differential equations, in the complex domain, around singularities, does however come into relation with the geometric theory. Roughly speaking, monodromy studies the way a covering map can degenerate, while singularity theory studies the way a manifold can degenerate; and these fields are linked.
Ovo? Hehehe, necu ni poceti citati. ajde mi ukratko reci!
|
totalno si mašio...singularnost u matematici nema veze sa singularnosti crne rupe
singularnost je po hawkingu točka koja je samo središte crne rupe, u kojoj je beskonačna masa sabijena ispod planckovog promjera, stoga ima ogromnu gravitaciju te povlači u sebe čak i svjetlost
pretpostavlja se da u toj točci ne vrijede nikakvi zakoni fizike, vrijeme ne postoji, a također se pretpostavlja da ta kroz tu točku se dolazi u neku drugu dimenziju vrijeme il šta ti ja znam šta...
|
|
|
28.05.2007., 17:17
|
#100
|
dungeon-Keeper
Registracija: Apr 2007.
Postova: 548
|
Quote:
fiki21 kaže:
totalno si mašio...singularnost u matematici nema veze sa singularnosti crne rupe
singularnost je po hawkingu točka koja je samo središte crne rupe, u kojoj je beskonačna masa sabijena ispod planckovog promjera, stoga ima ogromnu gravitaciju te povlači u sebe čak i svjetlost
pretpostavlja se da u toj točci ne vrijede nikakvi zakoni fizike, vrijeme ne postoji, a također se pretpostavlja da ta kroz tu točku se dolazi u neku drugu dimenziju vrijeme il šta ti ja znam šta...
|
To je cosmonaut link dao, nisam ni procitao to, ni ne znam sta je, tako da nema veze samnom
__________________
Wtf? :-O
|
|
|
|
|
Sva vremena su GMT +2. Trenutno vrijeme je: 16:52.
|
|
|
|