Realized variance: Difference between revisions

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{{distinguish|Rotation (mathematics)}}
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In [[mathematics]], a '''rotation map''' is a function that represents an undirected edge-[[labeled graph]], where each vertex enumerates its outgoing neighbors. Rotation maps were first introduced by Reingold, Vadhan and Wigderson (“Entropy waves, the zig-zag graph product, and new constant-degree expanders”, 2002) in order to conveniently define the [[zig-zag product]] and prove its properties.
Given a vertex <math>v</math> and an edge label <math>i</math>, the rotation map returns the <math>i</math>'th neighbor of <math>v</math> and the edge label that would lead back to <math>v</math>.
 
==Definition==
For a ''D''-regular graph ''G'', the rotation map <math>\mathrm{Rot}_G : [N] \times [D] \rightarrow [N] \times [D]</math> is defined as follows: <math>\mathrm{Rot}_G (v,i)=(w,j)</math> if the ''i''th edge leaving ''v'' leads to ''w'', and the ''j''th edge leaving ''w'' leads to&nbsp;''v''.
 
==Basic properties==
From the definition we see that <math>\mathrm{Rot}_G</math> is a permutation, and moreover <math>\mathrm{Rot}_G \circ \mathrm{Rot}_G</math> is the identity map (<math>\mathrm{Rot}_G</math> is an [[involution (mathematics)|involution]]).
 
==Special cases and properties ==
* A rotation map is consistently labeled if all of the edges leaving each vertex are labeled in such a way that at each vertex, the labels of the incoming edges are all distinct. Every regular graph has some consistent labeling.
* A rotation map is <math>\pi</math>-consistent if <math>\forall v \ \mathrm{Rot}_G(v,i)=(v[i],\pi (i))</math>. From the definition, a <math>\pi</math>-consistent rotation map is consistently labeled.
 
==See also==
*[[Zig-zag product]]
*[[Rotation system]]
 
==References==
{{Refbegin}}
* {{Citation
| first1=O. | last1=Reingold
| first2=S. | last2=Vadhan
| first3=A. | last3=Widgerson
| title=Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors
| journal=41st Annual Symposium on Foundations of Computer Science
| year=2000
| doi=10.1109/SFCS.2000.892006
| pages=3–13
}}
 
* {{Citation
| first=O
| last=Reingold
| title=Undirected connectivity in log-space
| journal=[[Journal of the ACM]]
| year=2008
| volume=55
| issue=4
| pages=Article 17, 24 pages
| doi=10.1145/1391289.1391291
}}
 
* {{Citation
| first1=O. | last1=Reingold
| first2=L. | last2=Trevisan
| first3=S. | last3=Vadhan
| title=Pseudorandom walks on regular digraphs and the RL vs. L problem
| journal=Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
| year=2006
| doi=10.1145/1132516.1132583
| pages=457
}}
 
{{Refend}}
 
[[Category:Extensions and generalizations of graphs]]
[[Category:Graph operations]]

Revision as of 04:35, 21 February 2014

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Use your knives on the correct cutting surfaces and stick to the knife manufacturer's care and upkeep recommendations. I suggest obtaining your knives sharpened by certified pros anytime achievable. The new cooks simply don't fully grasp the value of the proper knives. Peeling garlic requires a lot of time, that is unless you use a garlic peeler. Use it to get your dishes completed quicker, regardless of whether it is a marinade or lemonade.

It works just as nicely with a serrated edge as it does a straight blade and its design and style tends to make it incredibly basic to use. The non-slip pad adds to its stability when in use and it hones time and again without the need of losing its personal edge. The Presto Pro is the most effective of the electric knife sharpeners and is a surprisingly excellent high quality kitchen tool. Most sharpeners function with knives alone and, on, occasion scissors.

Wusthof Classic Chef's Knife Made for chopping or slicing on a cutting board, the chef's knife has a rigid blade to absorb impact with a slight curve that..,Wusthof Classic 12-Inch Cook's Knife four.9 out of 5.. This Classic 14-inch Heavy Cooks knife is the ultimate function horse.. As a basic-goal chef's knife,..,12 Inch Chef Knife - 366 outcomes like Wusthof Grand Prix II 4 1/two-inch Challenging Cheese Knife, Wusthof Classic 12-inch Heavy Chef's Knife, Wusthof Culinar 2 Knife Set 8..,Wusthof Classic 12 inch Heavy Chef's Knife NEW $499.95 Time Remaining: 15d 22h 47m Invest in It Now for only: $499.95. Wusthof 9746 Classic 5-Piece Chef Knife Set: $494.92,Save on the Wusthof Classic 12 inch Chef's Knife WU-4582-32.

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The Schmidts' debut line of knives, out now at Williams-Sonoma and other kitchenware shops about the nation, are light enough for novice cooks and sharp and balanced adequate to impress the experts in the F&W Test Kitchen. Most chef knives have a pointy "heel" next to the handle in contrast, heels of Schmidt knives are curved (a patented design and style function) so that you cannot get nicked. Knives that don't have a scallop or serrated edge have to have frequent sharpening to preserve their keenness.

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