Gδ set: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>Melchoir
en>Tsirel
→Examples: wikilink instead of a strange source; see talk
 
Line 1: Line 1:
{{no footnotes|date=February 2013}}
It's a must to really feel just a little sorry for Tim Duncan. Labeled as boring all through his career, The Big Fundamental captured 4 rings however never the hearts of the basketball-loving public. And there's no secret why; the man lacks style. A 12-foot financial institution shot could also be an efficient tactic, however it's not going to persuade an informal fan to jump out of his or her seat to bellow a rowdy cheer. It is why a excessive-flying underachiever like Vince Carter was so highly touted while a multiple champion may elicit a collective shrug.
In [[mathematics]], in the study of [[dynamical system]]s, an '''orbit''' is a collection of points related by the [[evolution function]] of the dynamical system. The orbit is a subset of the [[Phase space (dynamical system)|phase space]] and the set of all orbits is a [[partition (set theory)|partition]] of the phase space, that is different orbits do not intersect in the phase space. Understanding the properties of orbits by using [[Topological dynamics|topological methods]] is one of the objectives of the modern theory of dynamical systems.


For [[discrete-time dynamical system]]s the orbits are [[sequence]]s, for [[real dynamical system]]s the orbits are [[curve]]s and for [[holomorphic function|holomorphic]] dynamical systems the orbits are [[Riemann surface]]s.
Style is the whole lot in basketball. NBA 2K14 embraces the inventive expression that surfaces only when buying and selling baskets with sweaty men, and in doing so, it's a powerful and exciting representation of the true sport. Final year's version of NBA 2K tinkered with proper analog stick control, however 2K14 goes full steam ahead with this initiative. Relying on what course you move the stick in and the way lengthy you hold it, you can mimic the movements of an actual NBA player.


== Definition ==
It is a worthwhile improvement that provides you greater control over the way you attack defenders, letting you infuse your own personality into the action. Easily performing crossovers, jab steps, sweeping hooks, and even flashier moves such as behind-the-again passes with ease allows you to orchestrate a show so entertaining even Miami Warmth followers would show up on time to watch. You'll be able to't blame Bucks fans for not showing up.
[[File:Simple Harmonic Motion Orbit.gif|right|thumb|300px|Diagram showing the periodic orbit of a mass-spring system in [[simple harmonic motion]]. (Here the velocity and position axes have been reversed from the standard convention in order to align the two diagrams)]]


Given a dynamical system (''T'', ''M'', Φ) with ''T'' a group, M a set and Φ the evolution function
There's an in depth tutorial to guide you thru the vagaries of your basketball repertoire, but you are higher off ignoring it completely.  Here's more information about [https://www.facebook.com/FreeNBA2K14LockerCodes NBA 2K14 Locker Code] look at the website. Oddly enough, the follow flooring is the worst place to learn to play. Fidgeting with the stick to tug off specific moves is frustrating, as a result of for those who're off by just a few levels, you do something utterly different. Basketball is all about context and reacting to what your opponent is doing, and the stiffness present within the tutorial smothers that freedom.


:<math>\Phi: U \to M</math> where <math>U \subset T \times M</math>
So simply head straight to the court. The one way you study is in the event you're pushing yourself towards the NBA elite. Defensive players swarm like rabid hawks, relentlessly attacking entry passes and hounding ball carriers. And the one way you are going to constantly rating is by being smart with both the ball and your positioning. Lazy passes are turned into quick-break baskets quicker than you may flop for a foul name, and the added defensive depth forces you to play good or lose miserably.


we define
  Master your again-to-the-basket sport, and you'll demand a double team. Pass to your open teammate for a few broad-open jumpers, nonetheless, and your defenders should concoct a brand new plan. There's an enticing chess match occurring as you and your opponent dimension each other up, implement methods, and then readjust on the fly.
 
:<math>I(x):=\{t \in T : (t,x) \in U \},</math>
 
then the set
 
:<math>\gamma_x:=\{\Phi(t,x) : t \in I(x)\} \subset M</math>
 
is called '''orbit''' through ''x''. An orbit which consists of a single point is called '''constant orbit'''. A non-constant orbit is called '''closed''' or '''periodic''' if there exists a ''t'' in ''T'' so that
:<math>\Phi(t, x) = x \,</math>
for every point ''x'' on the orbit.
 
=== Real dynamical system ===
 
Given a real dynamical system (''R'', ''M'', Φ), ''I''(''x'')) is an open interval in the [[real number]]s, that is <math>I(x) = (t_x^- , t_x^+)</math>. For any ''x'' in ''M''
:<math>\gamma_{x}^{+} := \{\Phi(t,x) : t \in (0,t_x^+)\}</math>
is called '''positive semi-orbit''' through ''x'' and
:<math>\gamma_{x}^{-} := \{\Phi(t,x) : t \in (t_x^-,0)\}</math>
is called '''negative semi-orbit''' through ''x''.
 
=== Discrete time dynamical system ===
For discrete time dynamical system  :
 
'''forward''' orbit of x is a set :
:<math> \gamma_{x}^{+} \  \overset{\underset{\mathrm{def}}{}}{=}  \    \{ \Phi(t,x) : t \ge 0 \} \,</math>
 
'''backward''' orbit of x is a set :
 
:<math>\gamma_{x}^{-} \ \overset{\underset{\mathrm{def}}{}}{=}  \    \{\Phi(-t,x) : t \ge 0 \} \,</math>
 
and '''orbit''' of x is a set :
 
:<math>\gamma_{x}  \  \overset{\underset{\mathrm{def}}{}}{=}  \  \gamma_{x}^{-} \cup \gamma_{x}^{+} \,</math>
 
where :
* <math>\Phi\,</math> is an evolution function <math>\Phi : X \to X \,</math> which is here an [[iterated function]],
* set <math>X\,</math> is '''dynamical space''',
*<math>t\,</math> is number of iteration, which is [[natural number]] and <math>t \in T \,</math>
*<math>x\, </math> is initial state of system and  <math>x \in X \,</math>
 
Usually different notation is used :
 
*<math>\Phi(t,x)\,</math> is noted as <math>\Phi^{t}(x)\,</math>
*<math>x_t = \Phi^{t}(x)\,</math> with <math>x_0 \,</math> is a <math>x \,</math> from above notation.
 
=== General dynamical system ===
For general dynamical system, especially in homogeneous dynamics, when one have a "nice" group <math>G</math> acting on a probability space <math>X</math> in a measure-preserving way, an orbit <math>G.x \subset X</math> will be called periodic (or equivalently, closed orbit) if the stabilizer <math>Stab_{G}(x)</math> is a lattice inside <math>G</math>.
 
In addition, a related term is bounded orbit, when the set <math>G.x</math> is pre-compact inside <math>X</math>.
 
The classification of orbits can lead to interesting questions with relations to other mathematical areas, for example the Oppenheim conjecture (proved by Margulis) and the Littlewood conjecture (partially proved by Lindenstrauss) are dealing with the question whether every bounded orbit of some natural action on the homogeneous space <math>SL_{2}(\mathbb{R})\backslash SL_{2}(\mathbb{Z})</math> is indeed periodic one, this observation is due to Raghunathan and in different language due to Cassels and Swinnerton-Dyer . Such questions are intimately related to deep measure-classification theorems.
 
=== Notes ===
 
It is often the case that the evolution function can be understood to compose the elements of a [[group (mathematics)|group]], in which case the [[orbit (group theory)|group-theoretic orbits]] of the [[group action]] are the same thing as the dynamical orbits.
 
== Examples ==
[[File:Critical orbit 3d.png|right|thumb|Critical orbit of discrete dynamical system based on [[complex quadratic polynomial]]. It  tends to weakly [[Attractor|attracting]] [[Fixed point (mathematics)|fixed point]] with multiplier=0.99993612384259]]
* The orbit of an [[equilibrium point]] is a constant orbit
 
== Stability of orbits ==
 
A basic classification of orbits is
* constant orbits or fixed points
* periodic orbits
* non-constant and non-periodic orbits
 
An orbit can fail to be closed in two ways.
It could be an '''asymptotically periodic''' orbit if it [[limit (mathematics)|converges]] to a periodic orbit.  Such orbits are not closed because they never truly repeat, but they become arbitrarily close to a repeating orbit.
An orbit can also be [[chaos theory|chaotic]].  These orbits come arbitrarily close to the initial point, but fail to ever converge to a periodic orbit.  They exhibit [[sensitive dependence on initial conditions]], meaning that small differences in the initial value will cause large differences in future points of the orbit.
 
There are other properties of orbits that allow for different classifications.  An orbit can be [[hyperbolic (dynamical systems)|hyperbolic]] if nearby points approach or diverge from the orbit exponentially fast.
 
==See also==
* [[Wandering set]]
* [[Phase space method]]
* [[Cobweb plot]] or Verhulst diagram
* [[Periodic points of complex quadratic mappings]] and multiplier of orbit
 
==References==
* {{cite book | author=Anatole Katok and Boris Hasselblatt | title= Introduction to the modern theory of dynamical systems | publisher= Cambridge | year= 1996 | isbn=0-521-57557-5}}
 
[[Category:Dynamical systems]]
[[Category:Group actions]]

Latest revision as of 10:37, 16 November 2014

It's a must to really feel just a little sorry for Tim Duncan. Labeled as boring all through his career, The Big Fundamental captured 4 rings however never the hearts of the basketball-loving public. And there's no secret why; the man lacks style. A 12-foot financial institution shot could also be an efficient tactic, however it's not going to persuade an informal fan to jump out of his or her seat to bellow a rowdy cheer. It is why a excessive-flying underachiever like Vince Carter was so highly touted while a multiple champion may elicit a collective shrug. 

Style is the whole lot in basketball. NBA 2K14 embraces the inventive expression that surfaces only when buying and selling baskets with sweaty men, and in doing so, it's a powerful and exciting representation of the true sport. Final year's version of NBA 2K tinkered with proper analog stick control, however 2K14 goes full steam ahead with this initiative. Relying on what course you move the stick in and the way lengthy you hold it, you can mimic the movements of an actual NBA player.

It is a worthwhile improvement that provides you greater control over the way you attack defenders, letting you infuse your own personality into the action. Easily performing crossovers, jab steps, sweeping hooks, and even flashier moves such as behind-the-again passes with ease allows you to orchestrate a show so entertaining even Miami Warmth followers would show up on time to watch. You'll be able to't blame Bucks fans for not showing up.

There's an in depth tutorial to guide you thru the vagaries of your basketball repertoire, but you are higher off ignoring it completely.  Here's more information about NBA 2K14 Locker Code look at the website. Oddly enough, the follow flooring is the worst place to learn to play. Fidgeting with the stick to tug off specific moves is frustrating, as a result of for those who're off by just a few levels, you do something utterly different. Basketball is all about context and reacting to what your opponent is doing, and the stiffness present within the tutorial smothers that freedom.

So simply head straight to the court. The one way you study is in the event you're pushing yourself towards the NBA elite. Defensive players swarm like rabid hawks, relentlessly attacking entry passes and hounding ball carriers. And the one way you are going to constantly rating is by being smart with both the ball and your positioning. Lazy passes are turned into quick-break baskets quicker than you may flop for a foul name, and the added defensive depth forces you to play good or lose miserably.

Master your again-to-the-basket sport, and you'll demand a double team. Pass to your open teammate for a few broad-open jumpers, nonetheless, and your defenders should concoct a brand new plan. There's an enticing chess match occurring as you and your opponent dimension each other up, implement methods, and then readjust on the fly.
Retrieved from "https://en.formulasearchengine.com/index.php?title=Gδ_set&oldid=227153"