formulasearchengine - User contributions [en]

Convergence of random variables

2014-12-27T21:05:11Z

128.189.217.142: /* Properties */

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Order theory

2014-12-01T20:42:20Z

128.189.187.168: /* Special types of orders */ Use ∧ and ∨ instead of ^ and v.

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Spectral abscissa

2014-03-18T18:23:19Z

128.189.203.21:

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Taxation in Norway

2014-02-18T21:52:34Z

128.189.202.198: /* Taxation of petroleum activities */

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t+you+intend+to+use+the+knife.+When+I+figured+out+how+the+thumb+locked+worked%2C+extracting+the+knife+from+the+sheath+was+a+breeze.+However%2C+I+find+acquiring+back+in+the+sheath+a+challenge+at+times.%0D%0A%0D%0AMeanwhile%2C+a+blade+length+of+not+significantly+less+than+two+and+a+half+inches+and+not+extra+than+four+inches+lengthy+is+ideal+if+you+prefer+putting+the+knife+inside+your+pocket.+In+addition%2C+this+length+is+sufficient+penetrate+the+essential+organs+of+a+individual.+Nonetheless%2C+longer+blades+appear+scarier+than+the+shorter+ones+specially+if+your+opponent+is+bigger+than+you.+Hand+Protection+The+finger+guard+and+the+pommel+are+each+covered+in+matte+gray+coating.+If+you+strategy+on+working+with+your+knife+for%C2%A0hammering%2C+then+you+can+expect+the+coating+of+the+pommel+to+come+off+rather+promptly%2C+therefore+exposing+the+raw+non-stainless+steel.+7.9+3.four+three.9+Sandvik+14C28N+%24%24+four.7+Spyderco+Military+G-10+Created+from+1095+Cro-Van+Steel+%28easy+to+sharpen%2C+holds+an+edge+well%29+THIS+IS+THE+ONLY+KNIFE+You+are+going+to+EVER+Want%0D%0A%0D%0AThe+slimmest+and+lightest+knife+in+our+gallery%2C+the+Snody+Instigator+could+be+the+ultimate+neck+knife.+The+complete-tang+skeletonized+knife+is+crafted+from+AUS-eight+stainless+steel+with+HRC+58-60+hardness.+The+modified+Wharncliff+blade+title+has+a+straight+edge+with+a+single+bevel%2C+and+a+%5Bhttp%3A%2F%2Fwww.Ehow.com%2Fsearch.html%3Fs%3Drounded%2Bspine+rounded+spine%5D+to+give+the+knife+a+talon-sharp+point.+This+knife+is+light+and+sleek+however+powerful.+The+knife+comes+with+a+snap-match+molded+Kydex+thermoplastic+sheath+with+tether+cord.+Knife+weight+is+lowered+and+grip+is+created+by+the+7+strategically+placed+holes+in+the+handle.+The+blade+length+is+3.28+inches%2C+the+overall+length+is+7.38+inches%2C+and+the+knife+weighs+just+2.3+ounces.%0D%0A%0D%0AAs+far+as+your+attacker+is+concerned+this+is+not+a+fight%2C+it+is+an+assassination.+He+is+not+going+to+want+to+stand+there+with+you+and+hack+it+out.+Regrettably%2C+this+is+precisely+the+fantasy+that+quite+a+few+so-named+knife+fighting+instructors+promote.+The+absolute+final+issue+you+want+to+do+is+to+try+to+%22fight.%22+Widespread+sense+tells+us+that+knife+fighting+is+risky.+And+yet%2C+like+a+dog+circling+a+bear%27s+den+-+exactly+where+a+smarter+element+of+it+knows+not+to+wake+that+sleeping+bear%2C+but+an+additional%2C+a+lot+more+instinctive+part+is+urging+it+on+-+quite+a+few+men+and+women+who+train+in+knife+fight+have+the+similar+torn+desires.+1+of+the+biggest+difficulties+goading+these+people+today+is+Do+they+have+what+it+takes%3F%22+In+a+knife+fight%2C+timing+is+right+up+there+close+to+the+major+of+importance.&create=Create performs nicely] in most functions including some heavy responsibility tasks and with very little blade play in either course. 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To properly install bi-fold doorways, it’s important to construct a square door body so there's an equal margin of space between the sting of each door panel and the aspect jambs. As I am writing this post, I've lodged between my lower proper jaw and sixth tooth from the center backwards, a DRY teabag which has healing properties because of the high quantity of 'tannins' in the black tea. Now any black tea variety will do Bentley, Lipton and even Twinings, but I've a Bentley-brand in there right [http://www.thebestpocketknifereviews.com/ www.thebestpocketknifereviews.com] now and I'm letting the tannins extract the pus and infectious material that has decided to reside within the pocket current in my decrease jaw (to not point out the other areas in my mouth wherein braces and extreme impacted tooth extraction, brought on dental caries to form. The Cryo’s deal with is titanium-carbo-nitride coated stainless-steel. It seems nice however I find it’s sub-par compared to the G-10 you [http://esmikorea.Ezconn.net/xe/?document_srl=83353 discover] on the Tenacious with regards to general grip. Now, for most functions you received’t find an issue but when the occasions comes when it’s raining and you’re in an emergency situation you might [http://www.thefreedictionary.com/pocket+knife usa made knives] be wishing you had G-10. Nonetheless, the jimping found on the top of the blade and backside of the deal with does help. In different decrease-priced choices from Kershaw we’ve been accustomed to seeing the black fiber bolstered nylon (FRN) so a minimum of the Cryo presents one thing a bit totally different in that department. By way of each fashion and size, nobody standard applies to all doors. Whereas hollow-core flat-surface doorways are generally used inside the house, exterior doors typically function panels and windows. Door widths vary from 28 to 36 inches, and the 2 commonest heights are 78 inches - often expressed as 6-foot-6 - and eighty inches, which is 6-foot-8. The latter top is the extra widespread measurement, and you will have to cut the door to fit right into a 6-foot-6 opening. To preserve the symmetry of the door, it is best to cut equal amounts from both ends.

Weak gravitational lensing

2014-02-12T09:18:42Z

128.189.187.104: /* Weak lensing by clusters of galaxies */ links

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Abietadiene synthase

2014-01-31T18:09:26Z

128.189.248.61: Added wiki links to conifer and the pine species

{{enzyme
| Name = aristolochene synthase
| EC_number = 4.2.3.9
| CAS_number =
| IUBMB_EC_number = 4/2/3/9
| GO_code = 0045483
| image =
| width =
| caption =
}}
In [[enzymology]], an '''aristolochene synthase''' ({{EC number|4.2.3.9}}) is an [[enzyme]] that [[catalysis|catalyzes]] the [[chemical reaction]]

:2-trans,6-trans-farnesyl diphosphate <math>\rightleftharpoons</math> aristolochene + diphosphate

Hence, this enzyme has one [[substrate (biochemistry)|substrate]], [[2-trans,6-trans-farnesyl diphosphate]], and two [[product (chemistry)|products]], [[aristolochene]] and [[diphosphate]].

This enzyme belongs to the family of [[lyase]]s, specifically those carbon-oxygen lyases acting on phosphates. The systematic name of this enzyme class is '''2-trans,6-trans-farnesyl-diphosphate diphosphate-lyase (cyclizing, aristolochene-forming)'''. Other names in common use include '''sesquiterpene cyclase''', '''trans,trans-farnesyl diphosphate aristolochene-lyase''', '''trans,trans-farnesyl-diphosphate diphosphate-lyase (cyclizing,''', and '''aristolochene-forming)'''. This enzyme participates in [[terpenoid biosynthesis]].

This protein may use the [[morpheein]] model of [[allosteric regulation]]. <ref name=pmid22182754>{{cite journal | author = T. Selwood and E. K. Jaffe. | title = Dynamic dissociating homo-oligomers and the control of protein function. | journal = Arch. Biochem. Biophys. | volume = 519| issue = 2| pages = 131–43| year = 2011 | pmid = 22182754 | url = http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Citation&list_uids=22182754 | doi=10.1016/j.abb.2011.11.020 | pmc=3298769}}</ref>

==Structural studies==

As of late 2007, two [[tertiary structure|structures]] have been solved for this class of enzymes, with [[Protein Data Bank|PDB]] accession codes {{PDB link|2E4O}} and {{PDB link|2OA6}}. They are both notable for the very high helix content of the structure.

==References==
{{reflist|1}}
* {{cite journal | author = Cane DE, Prabhakaran PC, Oliver JS and McIlwaine DB | date = 1990 | title = Aristolochene biosynthesis. Stereochemistry of the deprotonation steps in the enzymatic cyclization of farnesyl pyrophosphate | journal = J. Am. Chem. Soc. | volume = 112 | issue = 8 | pages = 3209–3210 | doi = 10.1021/ja00164a051 }}
* {{cite journal | author = Noguchi H and Rawlings BJ | date = 1989 | title = Aristolochene biosynthesis and enzymatic cyclization of farnesyl pyrophosphate | journal = J. Am. Chem. Soc. | volume = 111 | issue = 24 | pages = 8914–8916 | doi = 10.1021/ja00206a022 }}
* {{cite journal | author = Hohn TM, Plattner RD | date = 1989 | title = Purification and characterization of the sesquiterpene cyclase aristolochene synthase from Penicillium roqueforti | journal = Arch. Biochem. Biophys. | volume = 272 | pages = 137–43 | pmid = 2544140 | doi = 10.1016/0003-9861(89)90204-X | issue = 1 }}
* {{cite journal | author = Proctor RH, Hohn TM | date = 1993 | title = Aristolochene synthase. Isolation, characterization, and bacterial expression of a sesquiterpenoid biosynthetic gene (Ari1) from Penicillium roqueforti | journal = J. Biol. Chem. | volume = 268 | pages = 4543–8 | pmid = 8440737 | issue = 6 }}
* {{cite journal | author = Calvert MJ, Ashton PR, Allemann RK | date = 2002 | title = Germacrene A is a product of the aristolochene synthase-mediated conversion of farnesylpyrophosphate to aristolochene | journal = J. Am. Chem. Soc. | volume = 124 | pages = 11636–41 | pmid = 12296728 | doi = 10.1021/ja020762p | issue = 39 }}

{{4.2-enzyme-stub}}

[[Category:EC 4.2.3]]
[[Category:Enzymes of known structure]]

Time-to-digital converter

2013-12-12T00:07:52Z

128.189.91.231: /* Statistical counter */ Removed a superfluous comma.

In [[geometry]], a '''cross-polytope''',<ref>{{citation | last = [[E. L. Elte|Elte]] | first = E. L. | title = The Semiregular Polytopes of the Hyperspaces | publisher = University of Groningen | location = Groningen | year = 1912}} Chapter IV, five dimensional semiregular polytope [http://www.amazon.com/Semiregular-Polytopes-Hyperspaces-Emanuel-Lodewijk/dp/141817968X]</ref> '''orthoplex''',<ref>[[John Horton Conway|Conway]] calls it an n-'''orthoplex''' for ''[[orthant]] complex''.
</ref> '''hyperoctahedron''', or '''cocube''' is a [[regular polytope|regular]], convex [[polytope]] that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of (±1, 0, 0, …, 0). The cross-polytope is the [[convex hull]] of its vertices. Its facets are [[simplex]]es of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.

The ''n''-dimensional cross-polytope can also be defined as the closed [[unit ball]] (or, according to some authors, its boundary) in the [[L1-norm|ℓ1-norm]] on '''R'''''n'':
:<math>\{x\in\mathbb R^n : \|x\|_1 \le 1\}.</math>

In 1 dimension the cross-polytope is simply the [[line segment]] [−1, +1], in 2 dimensions it is a [[Square (geometry)|square]] (or diamond) with vertices {(±1, 0), (0, ±1)}. In 3 dimensions it is an [[octahedron]]—one of the five convex regular [[polyhedron|polyhedra]] known as the [[Platonic solid]]s. Higher-dimensional cross-polytopes are generalizations of these.
{| style="margin-left: auto; margin-right: auto"
|style="text-align: center;"|[[Image:2-orthoplex.svg|120px|A 2-dimensional cross-polytope]]
|width="50px"|
|style="text-align: center;"|[[Image:Octahedron.png|120px|A 3-dimensional cross-polytope]]
|width="50px"|
|style="text-align: center;"|[[Image:Schlegel wireframe 16-cell.png|120px|A 4-dimensional cross-polytope]]
|-
|style="text-align: center;"|2 dimensions [[square (geometry)|square]]
|
|style="text-align: center;"|3 dimensions [[octahedron]]
|
|style="text-align: center;"|4 dimensions [[16-cell]]
|}

The cross-polytope is the [[dual polytope]] of the [[hypercube]]. The 1-[[Skeleton (topology)|skeleton]] of a ''n''-dimensional cross-polytope is a [[Turán graph]] ''T''(2''n'',''n'').

== 4 dimensions ==

The 4-dimensional cross-polytope also goes by the name '''hexadecachoron''' or '''[[16-cell]]'''. It is one of six [[convex regular 4-polytope]]s. These [[polychoron|polychora]] were first described by the Swiss mathematician [[Ludwig Schläfli]] in the mid-19th century.

== Higher dimensions ==

The '''cross polytope''' family is one of three [[regular polytope]] families, labeled by [[Coxeter]] as ''βn'', the other two being the [[hypercube]] family, labeled as ''γn'', and the [[simplex|simplices]], labeled as ''αn''. A fourth family, the [[hypercubic honeycomb|infinite tessellations of hypercubes]], he labeled as ''δn''.

The ''n''-dimensional cross-polytope has 2''n'' vertices, and 2''n'' facets (''n''−1 dimensional components) all of which are ''n''−1 [[Simplex|simplices]]. The [[vertex figure]]s are all ''n'' − 1 cross-polytopes. The [[Schläfli symbol]] of the cross-polytope is {3,3,…,3,4}. The [[dihedral angle]] of the ''n''-dimensional cross-polytope is
:<math>\arccos\left(\frac{2-n}{n}\right)</math>.

The number of ''k''-dimensional components (vertices, edges, faces, …, facets) in an ''n''-dimensional cross-polytope is given by (see [[binomial coefficient]]):
:<math>2^{k+1}{n \choose {k+1}}</math>

The volume of the ''n''-dimensional cross-polytope is
:<math>\frac{2^n}{n!}.</math>

There are many possible [[orthographic projection]]s that can show the cross-polytopes as 2-dimensional graphs. [[Petrie polygon]] projections map the points into a regular ''2n''-gon or lower order regular polygons. A second projection takes the ''2(n-1)''-gon petrie polygon of the lower dimension, seen as a [[bipyramid]], projected down the axis, with 2 vertices mapped into the center.

{| class="wikitable"
|+
Cross-polytope elements
|-
! [[polytope|n]]
!βn k11
! Name(s) [[Graph (mathematics)|Graph]]
!Graph 2n-gon
!Graph 2(n-1)-gon
![[Schläfli symbol|Schläfli]]
![[Coxeter-Dynkin diagram|Coxeter-Dynkin diagrams]]
! [[Vertex (geometry)|Vertices]]
! [[Edge (geometry)|Edges]]
! [[Face (geometry)|Faces]]
! [[Cell (geometry)|Cells]]
! ''4''-faces
! ''5''-faces
! ''6''-faces
! ''7''-faces
! ''8''-faces
! ''9''-faces
|-
| [[1-polytope|1]]
!β1
|[[Edge (geometry)|Line segment]] 1-orthoplex
|[[Image:Cross graph 1.svg|50px]]
|
|{}
|{{CDD|node_1}} {{CDD|node_f1}}
|align=right| 2
|  
|  
|  
|  
|  
|  
|  
|  
|  
|-
| [[2-polytope|2]]
!β2 −111
|[[square (geometry)|square]] 2-orthoplex '''Bicross'''
|[[Image:Cross graph 2.png|50px]]
|[[Image:2-orthoplex B1.svg|50px]]
|{4} {}+{}
|{{CDD|node_1|4|node}} {{CDD|node_f1|2|node_f1}}
|align=right| 4
|align=right| 4
|  
|  
|  
|  
|  
|  
|  
|  
|-
| [[3-polytope|3]]
!β3 011
| [[octahedron]] 3-orthoplex '''Tricross'''
|[[Image:3-orthoplex.svg|50px]]
|[[Image:3-orthoplex B2.svg|50px]]
|{3,4} {30,1,1} {}+{}+{}
|{{CDD|node_1|3|node|4|node}} {{CDD|node_1|split1|nodes}} {{CDD|node_f1|2|node_f1|2|node_f1}}
|align=right| 6
|align=right| 12
|align=right| 8
|  
|  
|  
|  
|  
|  
|  
|-
| [[4-polytope|4]]
!β4 111
|[[16-cell]] 4-orthoplex '''Tetracross'''
|[[Image:4-orthoplex.svg|50px]]
|[[Image:4-orthoplex B3.svg|50px]]
|{3,3,4} {31,1,1} 4{}
|{{CDD|node_1|3|node|3|node|4|node}} {{CDD|node_1|3|node|split1|nodes}} {{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|align=right| 8
|align=right| 24
|align=right| 32
|align=right| 16
|  
|  
|  
|  
|  
|  
|-
| [[5-polytope|5]]
!β5 211
|[[5-orthoplex]] '''Pentacross'''
|[[Image:5-orthoplex.svg|50px]]
|[[Image:5-orthoplex B4.svg|50px]]
|{33,4} {32,1,1} 5{}
|{{CDD|node_1|3|node|3|node|3|node|4|node}} {{CDD|node_1|3|node|3|node|split1|nodes}} {{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|align=right| 10
|align=right| 40
|align=right| 80
|align=right| 80
|align=right| 32
|  
|  
|  
|  
|  
|-
| [[6-polytope|6]]
!β6 311
| [[6-orthoplex]] '''Hexacross'''
|[[Image:6-orthoplex.svg|50px]]
|[[Image:6-orthoplex B5.svg|50px]]
|{34,4} {33,1,1} 6{}
|{{CDD|node_1|3|node|3|node|3|node|3|node|4|node}} {{CDD|node_1|3|node|3|node|3|node|split1|nodes}} {{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|align=right| 12
|align=right| 60
|align=right| 160
|align=right| 240
|align=right| 192
|align=right| 64
|  
|  
|  
|  
|-
| [[7-polytope|7]]
!β7 411
|[[7-orthoplex]] '''Heptacross'''
|[[Image:7-orthoplex.svg|50px]]
|[[Image:7-orthoplex B6.svg|50px]]
|{35,4} {34,1,1} 7{}
|{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|4|node}} {{CDD|node_1|3|node|3|node|3|node|3|node|split1|nodes}} {{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|align=right| 14
|align=right| 84
|align=right| 280
|align=right| 560
|align=right| 672
|align=right| 448
|align=right| 128
|  
|  
|  
|-
| [[8-polytope|8]]
!β8 511
|[[8-orthoplex]] '''Octacross'''
|[[Image:8-orthoplex.svg|50px]]
|[[Image:8-orthoplex B7.svg|50px]]
|{36,4} {35,1,1} 8{}
|{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|4|node}} {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|split1|nodes}} {{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|align=right| 16
|align=right| 112
|align=right| 448
|align=right| 1120
|align=right| 1792
|align=right| 1792
|align=right| 1024
|align=right| 256
|  
|  
|-
| [[9-polytope|9]]
!β9 611
|[[9-orthoplex]] '''Enneacross'''
|[[Image:9-orthoplex.svg|50px]]
|[[Image:9-orthoplex B8.svg|50px]]
|{37,4} {36,1,1} 9{}
|{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}} {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|split1|nodes}} {{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|align=right| 18
|align=right| 144
|align=right| 672
|align=right| 2016
|align=right| 4032
|align=right| 5376
|align=right| 4608
|align=right| 2304
|align=right| 512
|  
|-
! [[10-polytope|10]]
! β10 711
|[[10-orthoplex]] '''Decacross'''
|[[Image:10-orthoplex.svg|50px]]
|[[Image:10-orthoplex B9.svg|50px]]
|{38,4} {37,1,1} 10{}
|{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}} {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|split1|nodes}} {{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}
|align=right|20
|align=right|180
|align=right|960
|align=right|3360
|align=right|8064
|align=right|13440
|align=right|15360
|align=right|11520
|align=right|5120
|align=right|1024
|-
| colspan=17 style="text-align:center" |...
|-
! '''n'''
! βn ''k''11
|''n''-orthoplex ''n''-cross
|
|
|{3''n'' − 2,4} {3''n'' − 3,1,1} n{}
|{{CDD|node_1|3|node|3|node}}...{{CDD|3|node|3|node|4|node}} {{CDD|node_1|3|node|3}}...{{CDD|node|3|node|split1|nodes}} {{CDD|node_f1|2|node_f1|2|node_f1|2|node_f1|2}}...{{CDD|2|node_f1}}
| colspan=11 style="text-align:center" | 2''n'' '''0-faces''', ... <math>2^{k+1}{n\choose k+1}</math> '''''k''-faces''' ..., 2''n'' '''(n-1)-faces'''
|}

The vertices of an axis-aligned cross polytope are all at equal distance from each other in the [[taxicab geometry|Manhattan distance]] ([[Lp space|L1 norm]]). [[Kusner's conjecture]] states that this set of 2''d'' points is the largest possible equidistant set for this distance.<ref>{{citation
| last = Guy | first = Richard K.
| issue = 3
| journal = American Mathematical Monthly
| pages = 196–200
| title = An olla-podrida of open problems, often oddly posed
| jstor = 2975549
| volume = 90
| year = 1983}}.</ref>

== See also ==

* [[List of regular polytopes]]
* [[Hyperoctahedral group]], the symmetry group of the cross-polytope

==Notes==
{{reflist}}

==References==
*{{cite book | first = H. S. M. | last = Coxeter | authorlink = H. S. M. Coxeter | year = 1973 | title = [[Regular Polytopes (book)|Regular Polytopes]] | edition = 3rd ed. | publisher = Dover Publications | location = New York | pages = 121–122 | isbn = 0-486-61480-8}} p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)

== External links ==
{{commons category|Cross-polytope graphs}}
* {{mathworld | urlname = CrossPolytope | title = Cross polytope}}
* [http://www.geocities.com/shapirojon34/tesseract/TesseractApplet.html Polytope Viewer]{{dead link|date=November 2010|bot=AnomieBOT}} (Click <polytopes...> to select cross polytope.)
*{{GlossaryForHyperspace | anchor=Cross | title=Cross polytope }}

{{Dimension topics}}
{{Polytopes}}

{{DEFAULTSORT:Cross-Polytope}}
[[Category:Polytopes]]
[[Category:Multi-dimensional geometry]]
[[Category:Four-dimensional geometry]]

Fraunhofer diffraction

2012-08-24T22:58:58Z

128.189.192.84: /* Far field */

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Grain growth

2012-08-14T07:06:54Z

128.189.137.87: /* Ideal grain growth */

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Skew-Hermitian matrix

2012-07-20T01:53:46Z

128.189.212.54: /* Example */ Having zeros on the diagonal might make people think that the diagonal must be zero

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Schrödinger equation

2005-03-21T07:49:25Z

128.189.194.140: Add a reference to Schrodingers original paper

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