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Quantum channel: Difference between revisions

Latest revision as of 03:02, 11 September 2014

Contents

Nike Store Australia are absolutely adorable

Pandora Jewelry Australia . Replies Kane

Pandora Bracelet Australia this Monday

Air Max 1 do not take it personally.

Nike Air Jordan Australia etc.

Christian Louboutin Australia without me even asking

Cheap New Balance 574 Australia open from 07

Navigation menu

@@ Line 1: / Line 1: @@
-[[Image:Real projective line.svg|right|thumb|The real projective line can be thought of as a line whose endpoints meet at infinity.]]
+== Nike Store Australia  are absolutely adorable ==
-In [[real analysis]], the '''real projective line''' (also called the one-point [[compactification (mathematics)|compactification]] of the [[real line]], or the ''projectively extended real numbers''), is the set <math>\mathbb{R}\cup\{\infty\}</math>, also denoted by <math>\widehat{\mathbb{R}}</math> and by <math>\mathbb{R}P^1</math>.
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-The real projective line extends the [[field (mathematics)|field]] of [[real number]]s in the same way that the [[Riemann sphere]] extends the field of [[complex number]]s, by adding a single point called conventionally <math>\infty</math>.
+== Nike Air Jordan Australia  etc. ==
-Compare the [[extended real number line]] (also called the two-point [[compactification (mathematics)|compactification]] of the real line), which does distinguish between <math>+\infty</math> and <math>-\infty</math>.
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-==Order==
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-The order relation cannot be extended to <math>\widehat{\mathbb{R}}</math> in a meaningful way. Given a real number ''a'', there is no convincing reason to decide that <math>a > \infty</math> or that <math>a < \infty</math>.  Since <math>\infty</math> can't be compared with any of the other elements, there's no point in using this relation at all. As another example, both <math> 3 < 4</math> and <math>3 > 4</math> are true. However, order is used to make definitions in <math>\widehat{\mathbb{R}}</math> that are based on the properties of reals.
-==Geometry==
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-Fundamental to the idea that &infin; is a point ''no different from any other'' is the way the real projective line is a [[homogeneous space]], in fact [[homeomorphism|homeomorphic]] to a [[circle]]. For example the [[general linear group]] of 2×2 real [[invertible]] matrices has a [[transitive action]] on it. The [[group action]] may be expressed by [[Möbius transformation]]s, (also called linear fractional transformations), with the understanding that when the denominator of the linear fractional transformation is 0, the image is &infin;.
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-The detailed analysis of the action shows that for any three distinct points ''P'', ''Q'' and ''R'', there is a linear fractional transformation taking ''P'' to 0, ''Q'' to 1, and ''R''  to &infin; that is, the group of linear fractional transformations is triply [[transitive]] on the real projective line. This cannot be extended to 4-tuples of points, because the [[cross-ratio]] is invariant.
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-The terminology [[projective line]] is appropriate, because the points are in 1-1 correspondence with one-dimensional [[linear subspace]]s of '''R'''<sup>2</sup>.
+   <li>[http://task.jiacn.net/forum.php?mod=viewthread&tid=46929 http://task.jiacn.net/forum.php?mod=viewthread&tid=46929]</li>
-==Arithmetic operations==
+   <li>[http://bbs.liyufx.cn/forum.php?mod=viewthread&tid=82598 http://bbs.liyufx.cn/forum.php?mod=viewthread&tid=82598]</li>
-===Motivation for arithmetic operations===
-The arithmetic operations in this space are an extension of the same operations on reals. The motivation for the new definitions is the limits of functions of real numbers.
+   <li>[http://www.bbscan.com/news/html/?38699.html http://www.bbscan.com/news/html/?38699.html]</li>
-===Arithmetic operations which are defined===
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-:<math>
-\begin{align} \\
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-a + \infty = \infty + a & = \infty, & a \in \mathbb{R} \\
-a - \infty = \infty - a & = \infty, & a \in \mathbb{R} \\
+ </ul>
-a \cdot \infty = \infty \cdot a & = \infty, & a \in \mathbb{R}, a \neq 0 \\
-\infty \cdot \infty & = \infty \\
-\frac{a}{\infty} & = 0, & a \in \mathbb{R} \\
-\frac{\infty}{a} & = \infty, &  a \in \mathbb{R}, a \neq 0 \\
-\frac{a}{0} & = \infty, & a \in \mathbb{R}, a \neq 0
-\end{align}
-</math>
-===Arithmetic operations which are left undefined===
-The following cannot be motivated by considering limits of real functions, and any definition of them would require us to give up additional algebraic properties. Therefore, they are left undefined:
-:<math>
-\begin{align}
-& \infty + \infty \\
-& \infty - \infty \\
-& \infty \cdot 0 \\
-& 0 \cdot \infty \\
-& \frac{\infty}{\infty} \\
-& \frac{0}{0}
-\end{align}
-</math>
-==Algebraic properties==
-''The following equalities mean: Either both sides are undefined, or both sides are defined and equal.'' This is true for any <math>a, b, c \in \widehat{\mathbb{R}}</math>.
-:<math>
-\begin{align}
-(a + b) + c & = a + (b + c) \\
-a + b & = b + a \\
-(a \cdot b) \cdot c & = a \cdot (b \cdot c) \\
-a \cdot b & = b \cdot a \\
-a \cdot \infty & = \frac{a}{0} \\
-\end{align}
-</math>
-The following is true whenever the right-hand side is defined, for any <math>a, b, c \in \widehat{\mathbb{R}}</math>.
-:<math>
-\begin{align}
-a \cdot (b + c) & = a \cdot b + a \cdot c \\
-a & = (\frac{a}{b}) \cdot b & = \,\,& \frac{(a \cdot b)}{b} \\
-a & = (a + b) - b & = \,\,& (a - b) + b
-\end{align}
-</math>
-In general, all laws of arithmetic are valid as long as all the occurring expressions are defined.
-==Intervals and topology==
-The concept of an [[interval (mathematics)|interval]] can be extended to <math>\widehat{\mathbb{R}}</math>. However, since it is an unordered set, the interval has a slightly different meaning. The definitions for closed intervals are as follows (it is assumed that
-<math>a, b \in \mathbb{R}, a < b</math>):
-:<math>
-\begin{align}
-\left[a, a\right] & = \lbrace a \rbrace  \\
-\left[a, b\right] & = \lbrace x \vert x \in \mathbb{R}, a \leq x \leq b \rbrace  \\
-\left[a, \infty\right] & = \lbrace x \vert x \in \mathbb{R}, a \leq x \rbrace  \cup \lbrace \infty \rbrace  \\
-\left[b, a\right] & = \lbrace x \vert x \in \mathbb{R}, b \leq x \rbrace  \cup \lbrace \infty \rbrace \cup \lbrace x \vert x \in \mathbb{R}, x \leq a \rbrace  \\
-\left[\infty, a\right] & = \lbrace \infty \rbrace  \cup \lbrace x \vert x \in \mathbb{R}, x \leq a \rbrace  \\
-\left[\infty, \infty\right] & = \lbrace \infty \rbrace
-\end{align}
-</math>
-The corresponding open and half-open intervals are obtained by removing the endpoints.
-<math>\widehat{\mathbb{R}}</math> itself is also an interval, but cannot be represented with this bracket notation.
-The open intervals as [[base (topology)|base]] define a [[topological space|topology]] on <math>\widehat{\mathbb{R}}</math>. Sufficient for a base are the finite open intervals and the intervals <math>(b, a) = \{x | x \in \mathbb{R}, b < x\} \cup \{\infty\} \cup \{x | x \in \mathbb{R}, x < a\}</math>.
-As said, the topology is [[homeomorphism|homeomorphic]] to a [[circle]]. Thus it is [[metrizable]] corresponding (for a given homeomorphism) to the ordinary metric on this circle (either measured straight or along the circle). There is no metric which is an extension of the ordinary metric on '''R'''.
-==Interval arithmetic==
-[[Interval arithmetic]] is trickier in <math>\widehat{\mathbb{R}}</math> than in <math>\mathbb{R}</math>. However, the result of an arithmetic operation on intervals is always an interval. In particular, we have, for every <math>a, b \in \widehat{\mathbb{R}}</math>:
-:<math>x \in [a, b] \iff \frac{1}{x} \in \left [\frac{1}{b}, \frac{1}{a}\right ]</math>
-which is true even when the intervals involved include 0.
-==Calculus==
-The tools of [[calculus]] can be used to analyze functions of <math>\widehat{\mathbb{R}}</math>. The definitions are motivated by the topology of this space.
-===Neighbourhoods===
-Let <math>x \in \widehat{\mathbb{R}}, A \subseteq \widehat{\mathbb{R}}</math>.
-*A is a [[Neighbourhood (mathematics)|neighbourhood]] of ''x'', [[if and only if]] ''A'' contains an open interval ''B'' and <math>x \in B</math>.
-*A is a right-sided neighbourhood of x, if and only if there is <math>y \in \widehat{\mathbb{R}}, y > x</math> such that ''A'' contains <math>[x, y)</math>.
-*A is a left-sided neighbourhood of x, if and only if there is <math>y \in \widehat{\mathbb{R}}, y < x</math> such that ''A'' contains  <math>(y, x]</math>.
-*A is a (right-sided, left-sided) [[Neighbourhood_(mathematics)#Punctured_neighbourhood|punctured neighbourhood]] of ''x'', if and only if there is <math>B \subseteq \widehat{\mathbb{R}}</math> such that ''B'' is a (right-sided, left-sided) neighbourhood of x, and <math>A = B \setminus \{x\}</math>.
-===Limits===
-====Basic definitions of limits====
-Let <math>f : \widehat{\mathbb{R}} \to \widehat{\mathbb{R}}, p \in \widehat{\mathbb{R}}, L \in \widehat{\mathbb{R}}</math>.
-The [[limit of a function|limit]] of ''f(x)'' as ''x'' approaches ''p'' is ''L'', denoted
-:<math>\lim_{x \to p}{f(x)} = L</math>
-if and only if for every neighbourhood ''A'' of ''L'', there is a punctured neighbourhood ''B'' of ''p'', such that <math>x \in B</math> implies <math>f(x) \in A</math>.
-The [[one-sided limit]] of ''f(x)'' as ''x'' approaches ''p'' from the right (left) is ''L'', denoted
-:<math>\lim_{x \to p^{+}}{f(x)} = L</math> &nbsp;&nbsp; <math>\left (\lim_{x \to p^{-}}{f(x)} = L\right )</math>
-if and only if for every neighbourhood ''A'' of ''L'', there is a right-sided (left-sided) punctured neighbourhood ''B'' of ''p'', such that <math>x \in B</math> implies <math>f(x) \in A</math>.
-It can be shown that <math>\lim_{x \to p}{f(x)} = L</math> if and only if both <math>\lim_{x \to p^{+}}{f(x)} = L</math> and <math>\lim_{x \to p^{-}}{f(x)} = L</math>.
-====Comparison with limits in <math>\mathbb{R}</math>====
-The definitions given above can be compared with the usual definitions of limits of real functions. In the following statements, <math>p, L \in \mathbb{R}</math>, the first limit is as defined above, and the second limit is in the usual sense:
-*<math>\lim_{x \to p}{f(x)} = L</math> is equivalent to <math>\lim_{x \to p}{f(x)} = L</math>.
-*<math>\lim_{x \to \infty^{+}}{f(x)} = L</math> is equivalent to <math>\lim_{x \to -\infty}{f(x)} = L</math>.
-*<math>\lim_{x \to \infty^{-}}{f(x)} = L</math> is equivalent to <math>\lim_{x \to +\infty}{f(x)} = L</math>.
-*<math>\lim_{x \to p}{f(x)} = \infty</math> is equivalent to <math>\lim_{x \to p}{|f(x)|} = +\infty</math>.
-*<math>\lim_{x \to \infty^{+}}{f(x)} = \infty</math> is equivalent to <math>\lim_{x \to -\infty}{|f(x)|} = +\infty</math>.
-*<math>\lim_{x \to \infty^{-}}{f(x)} = \infty</math> is equivalent to <math>\lim_{x \to +\infty}{|f(x)|} = +\infty</math>.
-====Extended definition of limits====
-Let <math>A \subseteq \widehat{\mathbb{R}}</math>. Then ''p'' is a [[limit point]] of ''A'' if and only if every neighbourhood of ''p'' includes a point <math>y \in A</math> such that <math>y \neq x</math>.
-Let <math>f : \widehat{\mathbb{R}} \to \widehat{\mathbb{R}}, A \subseteq \widehat{\mathbb{R}}, L \in \widehat{\mathbb{R}}, p \in \widehat{\mathbb{R}}</math>, ''p'' a limit point of ''A''. The limit of ''f(x)'' as ''x'' approaches ''p'' through ''A'' is ''L'', if and only if for every neighbourhood ''B'' of ''L'', there is a punctured neighbourhood ''C'' of ''p'', such that <math>x \in A \cap C</math> implies <math>f(x) \in B</math>.
-This corresponds to the regular topological definition of continuity, applied to the [[subspace topology]] on <math>A\cup \lbrace p \rbrace</math>, and the restriction of ''f'' to <math>A\cup \lbrace p \rbrace</math>.
-===Continuity===
-Let
-: <math>f : \widehat{\mathbb{R}} \to \widehat{\mathbb{R}},\quad p \in \widehat{\mathbb{R}}.</math>
-''f'' is [[Continuous function|continuous]] at ''p'' if and only if ''f'' is defined at ''p'' and:
-:<math>\lim_{x \to p}{f(x)} = f(p).</math>
-Let
-: <math>f : \widehat{\mathbb{R}} \to \widehat{\mathbb{R}},\quad A \subseteq \widehat{\mathbb{R}}.</math>
-''f'' is continuous in ''A'' if and only if for every <math>p \in A</math>, ''f'' is defined at ''p'' and the limit of ''f''(''x'') as ''x'' approaches ''p'' through ''A'' is ''f''(''p'').
-An interesting feature is that every [[rational function]] ''P''(''x'')/''Q''(''x''), where ''P''(''x'') and ''Q''(''x'') have no common factor, is continuous in <math>\widehat{\mathbb{R}}</math>. Also, If tan is extended so that
-: <math>\tan\left(\frac{\pi}{2} + n\pi\right) = \infty\text{ for }n \in \mathbb{Z},</math>
-then tan is continuous in <math>\mathbb{R}</math>. However, many [[elementary function (differential algebra)|elementary function]]s, such as [[trigonometric functions|trigonometric]] and [[Exponential function|exponential]] functions, are discontinuous at <math>\infty</math>. For example, sin is continuous in <math>\mathbb{R}</math> but discontinuous  at <math>\infty</math>.
-Thus 1/''x'' is continuous on  <math>\widehat{\mathbb{R}}</math> but not on the [[affinely extended real number system]] <span style="text-decoration: overline">'''R'''</span>. Conversely, the function arctan can be extended continuously on <span style="text-decoration: overline">'''R'''</span>, but not on <math>\widehat{\mathbb{R}}</math>.
-==Hyperbolic involutions==
-An [[involution (mathematics)#Projective geometry|involution]] is hyperbolic when it has two [[fixed point (mathematics)|fixed point]]s. There are two elementary, arithmetic, hyperbolic involutions on the real projective line: [[additive inverse|negation]] and [[multiplicative inverse|reciprocation]]. Indeed, 0 and ∞ are fixed under negation, while 1 and &minus;1 are fixed under reciprocation.
-== See also ==
-* [[Real projective plane]]
-* [[Complex projective plane]]
-==External links==
-*[http://mathworld.wolfram.com/ProjectivelyExtendedRealNumbers.html Projectively Extended Real Numbers -- From Mathworld]
-[[Category:Real analysis]]
-[[Category:Topological spaces]]
-[[Category:Projective geometry]]
-[[Category:Infinity]]

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