Circulation problem: Difference between revisions

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In the study of metric spaces in mathematics, there are various notions of two metrics on the same underlying space being "the same", or equivalent.

In the following, $X$ will denote a non-empty set and $d_{1}$ and $d_{2}$ will denote two metrics on $X$ .

Topological equivalence

The two metrics $d_{1}$ and $d_{2}$ are said to be topologically equivalent if they generate the same topology on $X$ . The adjective "topological" is often dropped.^[1] There are multiple ways of expressing this condition:

a subset $A \subseteq X$ is $d_{1}$ -open if and only if it is $d_{2}$ -open;
the open balls "nest": for any point $x \in X$ and any radius $r > 0$ , there exist radii $r^{'}, r^{″} > 0$ such that

B_{r^{'}} (x; d_{1}) \subseteq B_{r} (x; d_{2})

and

B_{r^{″}} (x; d_{2}) \subseteq B_{r} (x; d_{1}) .

the identity function $I : X \to X$ is both $(d_{1}, d_{2})$ -continuous and $(d_{2}, d_{1})$ -continuous.

The following are sufficient but not necessary conditions for topological equivalence:

there exists a strictly increasing, continuous, and subadditive $f : ℝ_{+} \to ℝ$ such that $d_{2} = f \circ d_{1}$ .^[2]
for each $x \in X$ , there exist positive constants $α$ and $β$ such that, for every point $y \in X$ ,

α d_{1} (x, y) \leq d_{2} (x, y) \leq β d_{1} (x, y) .

Strong equivalence

Two metrics $d_{1}$ and $d_{2}$ are strongly equivalent if and only if there exist positive constants $α$ and $β$ such that, for every $x, y \in X$ ,

α d_{1} (x, y) \leq d_{2} (x, y) \leq β d_{1} (x, y) .

In contrast to the sufficient condition for topological equivalence listed above, strong equivalence requires that there is a single set of constants that holds for every pair of points in $X$ , rather than potentially different constants associated with each point of $X$ .

Strong equivalence of two metrics implies topological equivalence, but not vice versa. An intuitive reason why topological equivalence does not imply strong equivalence is that bounded sets under one metric are also bounded under a strongly equivalent metric, but not necessarily under a topologically equivalent metric.

All metrics induced by the p-norm, including the euclidean metric, the taxicab metric, and the Chebyshev distance, are strongly equivalent.^[3]

Even if two metrics are strongly equivalent, not all properties of the respective metric spaces are preserved. For instance, a function from the space to itself might be a contraction mapping under one metric, but not necessarily under a strongly equivalent one.^[4]

Properties preserved by equivalence

The continuity of a function is preserved if either the domain or range is remetrized by an equivalent metric, but uniform continuity is preserved only by strongly equivalent metrics.^[5]
The differentiability of a function is preserved if either the domain or range is remetrized by a strongly equivalent metric.^[6]

Notes

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References

Template:Refbegin

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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

Template:Refend

↑ Bishop and Goldberg, p. 10.
↑ Ok, p. 127, footnote 12.
↑ Ok, p. 138.
↑ Ok, p. 175.
↑ Ok, p. 209.
↑ Cartan, p. 27.

[1] Bishop and Goldberg, p. 10.

[2] Ok, p. 127, footnote 12.

[3] Ok, p. 138.

[4] Ok, p. 175.

[5] Ok, p. 209.

[6] Cartan, p. 27.

[1]

[2]

[3]

[4]

[5]

[6]

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+In the study of [[metric spaces]] in [[mathematics]], there are various notions of two [[metric (mathematics)|metrics]] on the same underlying space being "the same", or '''equivalent'''.
+In the following, <math>X</math> will denote a non-[[empty set]] and <math>d_{1}</math> and <math>d_{2}</math> will denote two metrics on <math>X</math>.
+==Topological equivalence==
+The two metrics <math>d_{1}</math> and <math>d_{2}</math> are said to be '''topologically equivalent''' if they generate the same [[topology]] on <math>X</math>. The adjective "topological" is often dropped.<ref>Bishop and Goldberg, p. 10.</ref> There are multiple ways of expressing this condition:
+* a subset <math>A \subseteq X</math> is <math>d_{1}</math>-[[open set|open]] [[if and only if]] it is <math>d_{2}</math>-open;
+* the [[open ball]]s "nest": for any point <math>x \in X</math> and any radius <math>r > 0</math>, there exist radii <math>r', r'' > 0</math> such that
+:<math>B_{r'} (x; d_{1}) \subseteq B_{r} (x; d_{2})</math> and <math>B_{r''} (x; d_{2}) \subseteq B_{r} (x; d_{1}).</math>
+* the [[identity function]] <math>I : X \to X</math> is both <math>(d_{1}, d_{2})</math>-[[continuous function|continuous]] and <math>(d_{2}, d_{1})</math>-continuous.
+The following are sufficient but not necessary conditions for topological equivalence:
+* there exists a strictly increasing, continuous, and [[subadditive]] <math>f:\mathbb{R}_{+} \to \mathbb{R}</math> such that <math>d_{2} = f \circ d_{1}</math>.<ref>Ok, p. 127, footnote 12.</ref>
+* for each <math>x \in X</math>, there exist positive constants <math>\alpha</math> and <math>\beta</math> such that, for every point <math>y \in X</math>,
+:<math>\alpha d_{1} (x, y) \leq d_{2} (x, y) \leq \beta d_{1} (x, y).</math>
+==Strong equivalence==
+Two metrics <math>d_{1}</math> and <math>d_{2}</math> are '''strongly equivalent''' if and only if there exist positive constants <math>\alpha</math> and <math>\beta</math> such that, for every <math>x,y\in X</math>,
+:<math>\alpha d_{1}(x,y) \leq d_{2}(x,y) \leq \beta d_{1} (x, y).</math>
+In contrast to the sufficient condition for topological equivalence listed above, strong equivalence requires that there is a single set of constants that holds for every pair of points in <math>X</math>, rather than potentially different constants associated with each point of <math>X</math>.
+Strong equivalence of two metrics implies topological equivalence, but not vice versa. An intuitive reason why topological equivalence does not imply strong equivalence is that [[Bounded set#Metric space|bounded sets]] under one metric are also bounded under a strongly equivalent metric, but not necessarily under a topologically equivalent metric.
+All metrics induced by the [[p-norm]], including the [[euclidean metric]], the [[taxicab metric]], and the [[Chebyshev distance]], are strongly equivalent.<ref>Ok, p. 138.</ref>
+Even if two metrics are strongly equivalent, not all properties of the respective metric spaces are preserved. For instance, a function from the space to itself might be a [[contraction mapping]] under one metric, but not necessarily under a strongly equivalent one.<ref>Ok, p. 175.</ref>
+==Properties preserved by equivalence==
+* The [[continuous function|continuity]] of a function is preserved if either the domain or range is remetrized by an equivalent metric, but [[uniform continuity]] is preserved only by strongly equivalent metrics.<ref>Ok, p. 209.</ref>
+* The [[differentiability]] of a function is preserved if either the domain or range is remetrized by a strongly equivalent metric.<ref>Cartan, p. 27.</ref>
+==Notes==
+{{reflist}}
+==References==
+{{refbegin}}
+* {{cite book
+  | authors = Richard L. Bishop, Samuel I. Goldberg
+  | title = Tensor analysis on manifolds
+  | year = 1980
+  | publisher = Dover Publications
+  | url = http://books.google.com/books?id=LAuN5-og4jwC
+  }}
+* {{cite book
+  | author = Efe Ok
+  | title = Real analysis with economics applications
+  | year = 2007
+  | publisher = Princeton University Press
+  | isbn = 0-691-11768-3
+  }}
+* {{cite book
+  | author = Henri Cartan
+  | title = Differential Calculus
+  | year = 1971
+  | publisher = Kershaw Publishing Company LTD
+  | isbn = 0-395-12033-0
+  }}
+{{refend}}
+[[Category:Metric geometry]]

Circulation problem: Difference between revisions

Revision as of 11:37, 17 December 2013

Contents

Topological equivalence

Strong equivalence

Properties preserved by equivalence

Notes

References

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Circulation problem: Difference between revisions

Revision as of 11:37, 17 December 2013

Topological equivalence

Strong equivalence

Properties preserved by equivalence

Notes

References

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